Algebraic Topology
Edwin H. Spanier
Algebraic Topology
Springer-Verlag Publishers
New York Berlin Heidelberg London Paris
Tokyo Hong Kong Barcelona Budapest
Edwin H. Spanier
University of California
Department of Mathematics
Berkeley, CA 94720
Library of Congress Cataloging in PubUcation Data
Spanier, Edwin Henry, 1921Algebraic topology.
Includes index.
1. Algebraic topology.
QA612.S6
514'.2
I. Title.
81-18415
ISBN 978-0-387-94426-5
This book was originally published by McGraw-Hill, 1966.
©
1966 by Springer-Verlag New York, Inc.
All rights reserved. No part of this book may be translated or reproduced in any form without
written permission from Springer- Verlag, 175 Fifth Avenue, New York, New York 10010,
U.S.A.
The use of general descriptive names, trade names, trademarks, etc. in this publication, even
if the former are not especially identified, is not to be taken as a sign that such names, as
understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely
by anyone.
First Corrected Springer Edition.
9 8 765 432 I
ISBN 978-0-387-94426-5
DOI 10.1007/978-1-4684-9322-1
ISBN 978-1-4684-9322-1 (eBook)
Algebraic Topology
PREFACE TO THE SECOND
SPRINGER PRINTING
IN THE MORE THAN TWENTY YEARS SINCE THE FIRST APPEARANCE OF
Algebraic Topology the book has met with favorable response both in its use
as a text and as a reference. It was the first comprehensive treatment of the
fundamentals of the subject. Its continuing acceptance attests to the fact that
its content and organization are still as timely as when it first appeared. Accordingly it has not been revised.
Many of the proofs and concepts first presented in the book have become
standard and are routinely incorporated in newer books on the subject. Despite
this, Algebraic Topology remains the best complete source for the material
which every young algebraic topologist should know. Springer-Vcrlag is to be
commended for its willingness to keep the book in print for future topologists.
For the current printing all of the misprints known to me have been corrected and the bibliography has been updated.
Berkeley, California
December 1989
vi
Edwin H. Spanier
PREFACE
THIS BOOK IS AN EXPOSITION OF THE FUNDAMENTAL IDEAS OF ALGEBRAIC
topology. It is intended to be used both as a text and as a reference. Particular
emphasis has been placed on naturality, and the book might well have been
titled Functorial Topology. The reader is not assumed to have prior knowledge
of algebraic topology, but he is assumed to know something of general topology
and algebra and to be mathematically sophisticated. Specific prerequisite
material is briefly summarized in the Introduction.
Since Algebraic Topology is a text, the exposition in the earlier chapters
is a good deal slower than in the later chapters. The reader is expected to
develop facility for the subject as he progresses, and accordingly, the further
he is in the book, the more he is called upon to fill in details of proofs.
Because it is also intended as a reference, some attempt has been made to
include basic concepts whether they are used in the book or not. As a result,
there is more material than is usually given in courses on the subject.
The material is organized into three main parts, each part being made up
of three chapters. Each chapter is broken into several sections which treat
vii
viii
PREFACE
individual topics with some degree of thoroughness and are the basic organizational units of the text. In the first three chapters the underlying theme is
the fundamental group. This is defined in Chapter One, applied in Chapter
Two in the study of covering spaces, and described by means of generators
and relations in Chapter Three, where polyhedra are introduced. The concept
of functor and its applicability to topology are stressed here to motivate
interest in the other functors of algebraic topology.
Chapters Four, Five, and Six are devoted to homology theory. Chapter
Four contains the first definitions of homology, Chapter Five contains further
algebraic concepts such as cohomology, cup products, and cohomology operations, and Chapter Six contains a study of topological manifolds. With each
new concept introduced applications are presented to illustrate its utility.
The last three chapters study homotopy theory. Basic facts about homotopy groups are considered in Chapter Seven, applications to obstruction
theory are presented in Chapter Eight, and some computations of homotopy
groups of spheres are given in Chapter Nine. Main emphaSiS is on the application to geometry of the algebraic tools introduced earlier.
There is probably more material than can be covered in a year course.
The core of a first course in algebraic topology is Chapter Four. This contains
elementary facts about homology theory and some of its most important
applications. A satisfactory one-semester first course for graduate students
can be based on the first four chapters, either omitting or treating briefly
Secs. 5 and 6 of Chapter One, Secs. 7 and 8 of Chapter Two, Sec. 8 of
Chapter Three, and Sec. 8 of Chapter Four. A second one-semester course
can be based on Chapters Five, Six, Seven, and Eight or on Chapters Five,
Seven, Eight, and Nine. For students with knowledge of homology theory and
related algebraic concepts a course in homotopy theory based on the last
three chapters is quite feasible.
Each chapter is followed by a collection of exercises. These are grouped
into sets, each set being devoted to a single topic or a few related topics.
With few exceptions, none of the exercises is referred to in the body of the
text or in the sequel. There are various types of exercises. Some are examples
of the general theory developed in the preceding chapter, some treat special
cases of general topics discussed later, and some are devoted to topicS
not discussed in the text at all. There are routine exercises as well as more
difficult ones, the latter frequently with hints of how to attack them. Occasionally a topic related to material in the text is developed in a set of exercises
devoted to it.
Examples in the text are usually presented with little or no indication of
why they have the stated properties. This is true both of examples illustrating
new concepts and of counterexamples. The verification that an example has
the desired properties is left to the reader as an exercise.
The symbol - is used to denote the end of a proof. It is also used at the
end of a statement whose proof has been given before the statement or which
follows easily from previous results. Bibliographical references are by footnotes
PREFACE
ix
in the text. Items in each section and in each exercise set are numbered con-
secutively in a single list. References to items in a different section are by
triples indicating, respectively, the chapter, the section or exercise set, and the
number of the item in the section. Thus 3.2.2 is item 2 in Sec. 2 of Chapter
Three (and 3.2 of the Introduction is item 2 in Sec ..3 of the Introduction).
The idea of writing this book originated with the existence of lecture
notes based on two courses I gave at the University of Chicago in 1955. It is
a pleasure to acknowledge here my indebtedness to the authors of those notes,
Guido Weiss for notes of the first course, and Edward Halpern for notes of
the second course. In the years since then, the subject has changed substantially and my plans for the book changed along with it, so that the present
volume differs in many ways from the original notes.
The final manuscript and galley proofs were read by Per Holm. He made
a number of useful suggestions which led to improvements in the text. For
his comments and for his friendly encouragement at dark moments, I am
sincerely grateful to him. The final manuscript was typed by Mrs. Ann
Harrington and Mrs. Ollie Cullers, to both of whom I express my thanks for
their patience and cooperation.
I thank the Air Force Office of Scientific Research for a grant enabling
me to devote all my time during the academic year 1962-63 to work on this
book. I also thank the National Science Foundation for supporting, over a
period of years, my research activities some of which are discussed here.
Edwin H. Spanier
LIST OF SYMBOLS
VAj
Tor A, p(A)
Tr <p
'lTy, 'lTY, h#> f#
[X,A; Y,B]x', [flx
'lTn(X)
hr
'IT(X,xo)
f[w]
G(XI X)
Pn(C), Pn(Q)
s, s, Kq, K(G([\), Kl * Kz
IKld, lsi, IKI
(s)
st v
sd K
E(K,vo)
Z(C), B(C), H(C), T*
C(K), /1q
/1(X)
c,R
/1(K)
0*
A*B
z X z'
C*, R*
Ext (A,B)
h
uxv
uvv
ff\C
Hn( {Aj }, X'; G)
2
8
9
19
24
43
45
50
73
85
91
109
III
112
114
123
136
157
160
161
168
170
181
220
231
237
241
242
249
251
254
261
Sqi
c* Ic'
8(X), Yu, H* (A,B)
Yu
Hqc
C*, H*
H~,
C * (~G)l')
C~, H~
t
H*(X;f)
Wi
c\c*, Yu
Wi
C(X,A), Cr
aTf3
'lTn(X,A)
a
0'
<p
'lT~, <p'
/1(X,A,xo)n
Hq(n)
<p", bn
(X,A)k
Tu
!f;
c(f)
d(jO,fl)
/1(O,u), S/1(O,u)
EI,t, dr
E~,t, dr
e
270
287
289
292
299
308
311
320
325
327
349
351
354
365
370
372
377
378
388
390
391
393
394
401
408
427
433
434
450
466
493
505
CONTENTS
INTRODUCTION
I
I
Set theory
2
General topology
3
Group theory
1
4
6
4
Modules
5
Euclidean spaces
7
9
HOMOTOPY AND THE FUNDAMENTAL GROUP
I
Categories
14
2
Functors
3
Homotopy
4
Retraction and deformation
5
6
7
8
H spaces
Suspension
18
22
27
33
39
45
The fundamental groupoid
The fundamental group
Exercises
12
50
56
xi
xii
2
CONTENTS
COVERING SPACES AND FIBRATIONS
I
3
Covering proiections
62
2
The homotopy lifting property
3
Relations with the fundamental group
4
The lifting problem
65
70
74
:.
The classification of covering proiections
6
Covering transformations
7
8
Fiber bundles
79
85
89
Fibrations
96
Exercises
103
POLYHEDRA
106
I
Simplicial complexes
2
3
Linearity in simplicial complexes
Subdivision
121
4
Simplicial approximation
108
:.
Contiguity classes
6
The edge-path groupoid
7
8
Graphs
114
126
129
134
139
143
Examples and applications
Exercises
4
60
149
HOMOLOGY
I
154
Chain complexes
156
162
2
Chain homotopy
3
The homology of simplicial complexes
4
Singular homology
:.
Exactness
6
7
8
Some applications of homology
173
179
Mayer- Vietoris sequences
186
193
Axiomatic characterization of homology
Exercises
167
205
199
xiii
CONTENTS
it
PRODUCTS
210
1 Homology with coefficients
212
2 The universal-coefficient theorem for homology
3 The Kunneth formula
227
4 Cohomology
236
it The universal-coefficient theorem for cohomology
6
Cup and cap products
6
255
8
The cohomology algebra
9
The Steenrod squaring operations
263
269
276
GENERAL COHOMOLOGY THEORY AND DUALITY
The slant product
2
3
4
Duality in topological manifolds
..
292
299
The fundamental class of a manifold
306
The Alexander cohomology theory
The homotopy axiom for the Alexander theory
311
315
Tautness and continuity
'1 Presheaves
323
329
8
Fine presheaves
9
Applications of the cohomology of presheaves
10 Characteristic classes
Exercises
338
346
356
HOMOTOPY THEORY
362
1
2
Exact sequences of sets of homotopy classes
3
4
Change of base points
379
The Hurewicz homomorphism
:5
The Hurewicz isomorphism theorem
..
CW complexes
Higher homotopy groups
371
400
'1 Homotopy functors
406
8 Weak homotopy type
412
Exercises
284
286
1
it
7
241
248
'1 Homology of fiber bundles
Exercises
219
418
387
393
364
xiv
8
CONTENTS
OUSTRUCTIO~
I
Eilenberg-MacLane spaces
2
3
Moore-Postnikov factorizations
Principal fibrations
4
Obstruction theory
5
The suspension map
Exercises
9
,.22
THEORY
424
432
437
445
452
460
SPECTRAL SEqUENCES AND HOMOTOPY
GROUPS OF SPHERES
464
I
Spectral sequences
2
3
Applications of the homology spectral sequence
466
The spectral sequence of a fibration
473
4
Multiplicative properties of spectral sequences
5
Applications of the cohomology spectral sequence
6
Serre classes of abelian groups
7
Homotopy groups of spheres
Exercises
I~DEX
481
490
498
504
512
518
521
ALGEBRAIC TOPOLOGY
INTRODUCTION
THE READER OF THIS BOOK IS ASSUMED TO HAVE A GRASP OF THE ELEMENTARY
concepts of set theory, general topology, and algebra. Following are brief
summaries of some concepts and results in these areas which are used in this
book. Those listed explicitly are done so either because they may not be
exactly standard or because they are of particular importance in the subsequent text.
I
SET THEORY'
The terms "set," "family," and "collection" are synonyms, and the term
"class" is reserved for an aggregate which is not assumed to be a set (for
example, the class of all sets). If X is a set and P(x) is a statement which is
either true or false for each element x E X, then
1 As a general reference see P. R. Halmos, Nafve Set Theory, D. Van Nostrand Company, Inc.,
Princeton, N.J., 1960.
1
2
INTRODUCTION
{x E X I P(x)}
denotes the subset of X for which P(x) is true.
If ] = {i} is a set and {Aj} is a family of sets indexed by], their union is
denoted by U Aj (or by U jEJ Aj), their intersection is denoted by n Aj (or by
njEJAj), their cartesian product is denoted by X Aj (or by XjEJAj), and
their set sum (sometimes called their disioint union) is denoted by V Aj (or
by VjEJAj) and is defined by V Aj = U (; X A j). In case] = {l,2, . . . ,n},
we also use the notation Al U Az U ... U An, Al n Az n ... nAn,
Al X Az X ... X An, and Al v Az v ... v An, respectively, for the union,
intersection, cartesian product, and set sum.
A function (or map) f from A to B is denoted by f: A ~ B. The set of all
functions from A to B is denoted by BA. If A' C A, there is an inclusion map
i: A' ~ A, and we use the notation i: A' C A to indicate that A' is a subset of
A and i is the inclusion map. The inclusion map from a set A to itself is called
the identity map of A and is denoted by lAo If J' C ], there is an inclusion
map
An equivalence relation in a set A is a relation - between elements of A
which is reflexive (that is, a - a for all a E A), symmetric (that is, a - a'
implies a' - a for a, a' E A), and transitive (that is, a - a' and a' - a"
imply a - a" for a, a', a" E A). The equivalence class of a E A with
respect to - is the subset {a' E A I a - a'}. The set of all equivalence classes
of elements of A with respect to - is denoted by AI - and is called a
quotient set of A. There is a proiection map A ~ AI - which sends a E A to
its equivalence class. If J' is a nonempty subset of ], there is also a proiection map
pJ': X Aj
JEJ
~
X Aj
iEJ'
(which is a projection map in the sense above).
Given functions f: A ~ Band g: B ~ C, their composite g a f (also denoted by gf) is the function from A to C defined by (g a f)(a) = g(f(a)) for
a E A. If A' C A and f: A ~ B, the restriction of f to A' is the function
fl A': A' ~ B defined by (fl A')(a') = f(a') for a' E A' (thus fl A' = fa i,
where i: A' C A), and the function f is called an extension of f I A' to A.
An iniection (or iniective function) is a function f: A ~ B such that
f(al) = f(az) implies al = az for aI, az EA. A suryection (or sur;ective
function) is a function f: A ~ B such that b E B implies that there is a E A
with f(a) = b. A biiection (also called a biiective function or a one-to-one
correspondence) is a function which is both injective and surjective.
A partial order in a set A is a relation :S between elements of A which is
reflexive and transitive (note that it is not assumed that a :S a' and a' :S a
imply a = a'). A total order (or simple order) in A is a partial order in A such
that for a, a' E A either a :S a' or a' :S a and which is antisymmetric
(that is, a :S a' and a' :S a imply a = a'). A partially ordered set is a set with
a partial order, and a totally ordered set is a set with a total order.
SEC.
1
3
SET THEORY
I
ZORN'S LEMMA
A partially ordered set in which every simply ordered
subset has an upper bound contains maximal elements.
A directed set A is a set with a partial-order relation ~ such that for
a, {3 E A there is yEA with a ~ y and {3 ~ y. A direct system of sets
{A",f"P} consists of a collection of sets {A"} indexed by a directed set
A = {a} and a collection of functions f"P: A" ~ AP for every pair a ~ {3 such
that
= lA.: A" C A" for all a E A
= fpY f"P: A" ~ AY for a ~ {3 ~
(a) f,,"
(b) f"Y
0
y in A
The direct limit of the direct system, denoted by lim~ {A"}, is the set
of equivalence classes of V A a with respect to the equivalence relation
a" ~ aP if there is y with a ~ y and {3 ~ y such that f"yaa = fp YaP. For each
a there is a map i,,: A" ~ lim~ {Aa}, and if a ~ {3, then i" = ip fa P.
0
2
Given a direct system of sets {Aa,f"p} and given a set B and for every
a E A a function g,,: A a ~ B such that g" = gp f"P if a ~ {3, there is
a unique map g: lim~ {A"} ~ B such that g 0 ia = ga for all a E A.
0
3
With the same notation as in theorem 2, the map g is a bijection if and
only if both the following hold:
=
(a) B
U ga(Aa)
(b) g,,(a") = gp(a P) if and only if there is y with a
that f"Y(a")
fpY(a P)
=
~
y and {3
~
y such
Let {Aj} be a collection of sets indexed by J = U}. Let A be the
collection of finite subsets of J and define a ~ {3 for a, {3 E A if a C {3.
Then A is a directed set and there is a direct system {A a} defined by
A" = VjE" A j, and if a ~ {3, then fa P: Aa ~ AP is the injection map.
Let g,,: A" ~ ViE J Aj be the injection map.
4
With the above notation, there is a bijection g: lim~ {A"} ~ V j E J Aj
such that go ia
g" (that is, any set sum is the direct limit of its finite
partial set sums).
=
An inverse system of sets {Aa,fol} consists of a collection of sets {Aa} indexed by a directed set A = {a} and a collection of functions f"P: Ap ~ A"
for a ~ {3 such that
(a) f,,"
(b) faY
= lA.: A" C A" for a E A
= fa P fpY: Ay ~ A" for a
0
~
{3
~ y
in A
The inverse limit of the inverse system, denoted by lim_ {A,,}, is the subset of
X A" consisting of all points (a,,) such that if a ~ {3, then aa = f"Pap. For
each a there is a map pa: lim_ {A,,} ~ A", and if a ~ {3, then pa = f"P pp.
0
£i Given an inverse system of sets {Aa,fa P} and given a set B and for every
a E A a function g,,: B ~ A" such that g"
f"P 0 gp if a ~ {3, there is
a unique function g: B ~ lim_ {Aa} such that ga
p" 0 gfor all a E A.
=
=
4
INTRODUCTION
6
With the same notation as in theorem 5, the map g is a biiection if and
only if both the following hold:
(a) g,,(b) = g",(b') for all a E A implies b = b'
(b) Given (a,,) E X A" such that a" = f,,(3af3 if a ::; [3, there is b E B
such that g,,(b) = a" for all a E A
Let {Ai} be a collection of sets indexed by J = {i}. Let A be the collection
of finite nonempty subsets of J, and define IX ::; [3 for a, [3 E A if a C [3.
Then A is a directed set and there is an inverse system {A,,} defined by
A" = XiE" Ai, and if a ::; [3, f"f3: Af3 ~ Aa is the projection map. For each
IX E A let g,,: XiEJ Ai ~ A" be the projection map.
7
With the above notation, there is a biiection g: XiEJAi ~ lim~ {A,,}
such that g" = PiX g (that is, any cartesian product is the inverse limit of its
finite partial cartesian products).
0
2
GENERAL TOPOLOGY'
A topological space, also called a space, is not assumed to satisfy any separation
axioms unless explicitly stated. Paracompact, normal, and regular spaces will
always be assumed to be Hausdorff spaces. A continuous map from one
topological space to another will also be called simply a map.
Given a set X and an indexed collection of topological spaces {Xi} i E J and
functions jj: X ~ Xj, the topology induced on X by the functions {h} is the
smallest or coarsest topology such that each h is continuous.
I
The topology induced on X by functions {jj: X ~ Xi} is characterized
by the property that if Y is a topological space, a function g: Y ~ X is
continuous if and only if fi g: Y ~ Xi is continuous for each i E ].
0
A subspace of a topological space X is a subset A of X topologized by
the topology induced by the inclusion map A C X. A discrete subset of a
topological space X is a subset such that every subset of it is closed in X. The
topological product of an indexed collection of topological spaces {Xi }iEJ is
the cartesian product X Xj, given the topology induced by the projection
maps Pi: X Xi ~ Xi for i E J. If {X"}"EA is an inverse system of topological
spaces (that is, X" is a topological space for IX E A and f"f3: Xf3 ~ X" is continuous for a ::; [3) their inverse limit lim~ {X,,} is given the topology induced
by the functions PiX: lim~ {X,,} ~ X" for a EA.
Given a set X and an indexed collection of topological spaces {Xi}iEJ
and functions gi: Xi ~ X, the topology coinduced on X by the functions {gi}
is the largest or finest topology such that each ~ is continuous.
As general references see J. L. Kelley, General Topology, D. Van Nostrand Company, Inc.,
Princeton, N.J., 1955, and S. T. Hu, Elements of General Topology, Holden-Day, Inc.,
San Francisco, 1964.
1
SEC.
2
5
GENERAL TOPOLOGY
2
The topology coinduced on X by functions {&: Xi ~ X} is characterized
by the property that if Y is any topological space, a function f: X ~ Y is continuous if and only iff gi: Xi ~ Y is continuous for each i E J.
0
A quotient space of a topological space X is a quotient set X' of X topologized by the topology coinduced by the projection map X ~ X'. If A c X,
then XI A will denote the quotient space of X obtained by identifying all of A
to a single point. The topological sum of an indexed collection of topological
spaces {Xi}iEJ is the set sum V Xi, given the topology coinduced by the
injection maps ir Xi ~ V Xi for i E J. If {X"}"EA is a direct system of topological spaces (that is, X" is a topological space for (X E A and f,,/3: X" ~ X/3 is
continuous for (X S /3) their direct limit lim ~ {X''} is given the topology
coinduced by the functions i,,: X" ~ lim~ {X''} for (X E A.
Let a = {A} be a collection of subsets of a topological space X. X is said
to have a topology coherent with a if the topology on X is coinduced from the
subspaces {A} by the inclusion maps A C X. (In the literature this topology
is often called the weak topology with respect to a.)
3 A necessary and sufficient condition that X have a topology coherent
with a is that a subset B of X be closed (or open) in X if and only if B n A
is closed (or open) in the subspace A for every A E if.
4
If a is an arbitrary open covering or a locally finite closed covering of X,
then X has a topology coherent with a.
5
Let X be a set and let {Ai} be an indexed collection of topological spaces
Ai n Ai' is a closed (or
each contained in X and such that for each i and
open) subset of Ai and of Ai' and the topology induced on Ai n Ai' from Ai
equals the topology induced on Ai n Ai' from Ai" Then the topology coinduced on X by the collection of inclusion maps {Ai C X} is characterized by
the properties that Ai is a closed (or open) subspace of X for each i and X has
a topology coherent with the collection {Ai}'
t,
The topology on X in theorem 5 will be called the topology coherent
with {Ai}' A compactly generated space is a Hausdorff space having a topology
coherent with the collection of its compact subsets (this is the same as what
is sometimes referred to as a Hausdorff k-space).
6
A Hausdorff space which is either locally compact or satisfies the first
axiom of countability is compactly generated.
7
If X is compactly generated and Y is a locally compact Hausdorff space,
X X Y is compactly generated.
If X and Y are topological spaces and A C X and BeY, then (A;B)
denotes the set of continuous functions f: X ~ Y such that f(A) C B.
yx denotes the space of continuous functions from X to Y, given the compactopen topology (which is the topology generated by the subbase {( K; U) },
where K is a compact subset of X and U is an open subset of Y). If A C X
6
INTRODUCTION
and BeY, we use (Y,B)(X,A) to denote the subspace of yx of continuous
functions f: X ~ Y such that f(A) C B. Let E: yx X X ~ Y be the evaluation map defined by E(f,x) = f(x). Given a function g: Z ~ yx, the composite
Z XX
gXI)
yx
X XL Y
is a function from Z X X to Y.
8
THEOREM OF EXPONENTIAL CORRESPONDENCE
If X is a locally compact
Hausdorff space and Y and Z are topological spaces, a map g: Z ~ yx is continuous if and only if E 0 (g Xl): Z X X ~ Y is continuous.
9
EXPONENTIAL LAW
If X is a locally compact Hausdorff space, Z is
a Hausdorff space, and Y is a topological space, the function 1/;: (YX)z ~ yzxx
defined by 1/;(g) = Eo (g X 1) is a homeomorphism.
10 If X is a compact Hausdorff space and· Y is metrized by a metric d, then
yx is metrized by the metric d' defined by
d'(f,g) = sup {d(f(x),g(x)) I x E X}
3
GROUP THEOR'·!
A homomorphism is called a monomorphism, epimorphism, isomorphism,
respectively, if it is injective, surjective, bijective. If {G j }iEJ is an indexed collection of groups, their direct product is the group structure on the cartesian
product X Gj defined by (g;)(gj) = (g;gj). If {G,,} is an inverse system of
groups (that is, G" is a group for each IX andf"f3: Gf3 ~ G" is a homomorphism
for IX ::; (3), their inverse limit lim~ {G,,} (which is a set) is a subgroup of X G".
Let A be a subset of a group G. G is said to be freely generated by A and A
is said to be a free generating set or free basis for G if, given any function
f: A ~ H, where H is a group, there exists a unique homomorphism cp: G ~ H
which is an extension of f. A group is said to be free if it is freely generated
by some subset. For any set A a free group generated by A is a group F(A)
containing A as a free generating set. Such groups F(A) exist, and any two are
canonically isomorphic.
I
Any group is isomorphic to a quotient group of a free group.
A presentation of a group G consists of a set A of generators, a set
B C F(A) of relations, and a function f: A ~ G such that the extension of f
to a homomorphism cp: F(A) ~ G is an epimorphism whose kernel is the norAs a general reference for elementary group theory see G. Birkhoff and S. MacLane, A
Survey of Modem Algebra, The Macmillan Company, New York, 1953. For a discussion of free
groups see R. H. Crowell and R. H. Fox, Introduction to Knot Theory, Ginn and Company,
Boston, 1963.
1
SEC.
4
7
MODULES
mal subgroup of F(A) generated by B. If A and B are both finite sets, the presentation is said to be finite and G is said to be finitely presented.
4
MODULES'
We are mainly interested in R modules where R is a principal ideal domain.
However, we shall begin with some properties of R moJules where R is
a commutative ring with a unit which acts as the identity on every module.
If q;: A ~ B is a homomorphism of R modules, then we have R modules
= {a E A I q;(a) = O} c A
im q; = {b E Bib = cp( a) for some a E A}
coker cp = B/im cp
ker q;
C B
I
NOETHER ISOMORPHISM THEOREM
Let A and B be submodules of a
module C and let A + B be the submodule of C generated by A U B. The
inclusion map A C A + B sends A n B into B and induces an isomorphism
of A/(A n B) with (A + B)/B.
If {Aj};EJ is an indexed collection of R modules, their direct product
X Aj is an R module and their direct sum <f)Aj is an R module (<f)Aj is the
submodule of X Aj consisting of those elements having only a finite number
of nonzero coordinates). The inverse limit lim~ {A,,} of an inverse system of
R modules (and homomorphisms f"f3: Af3 ~ A" for a S {3) is an R module, and
the direct limit of a direct system of R modules (and homomorphisms) is an
R module.
2 Any R module is isomorphic to the direct limit of its finitely generated
submodules directed by inclusion.
If A and Bare R modules, their tensor product A ® B (also written
A ® B) is an R module. For a E A and b E B, there is a corresponding element
R
a ® b E A ® B. A ® B is generated by the elements {a
with the relations (for a, a' E A, b, b' E B, and r, 1" E R)
(ra + r'a') ® b
a ® (rb + r'b')
® b I a E A, b E B}
= r(a ® b) + r'(a' ® b)
= r(a ® b) + r'(a ® b')
In case A or B is also an R' module, then so is A ® B.
R
3
For any R module A the homomorphisms a ~ a ® 1 and a
define isomorphisms of A with A ® Rand R ® A.
~
1® a
• As general references see H. Cartan and S. Eilenberg, Homological Algebra, Princeton
University Press, Princeton, N.J., 1956 and S. MacLane, Homology, Springer-Verlag OHG,
Berlin, 1963.
8
INTRODUCTION
4
For R modules A and B there is an isomorphism of A Q9 B with B Q9 A
taking a ® b to b Q9 a.
it
If A and Bare R modules and Band Care R' modules, there is an isomorphism of (A ~ B) ~ C with A ~ (B ~ C) (both being regarded as Rand
R' modules) taking (a Q9 b) Q9 c to a ® (b ® c).
If A and Bare R modules, their module of homomorphisms Hom (A,B)
[also written HomR (A,B)] is an R module whose elements are R homomorphisms A _ B. In case A or B is also an R' module, then so is HomR (A,B).
If A and Bare R madules and Band Care R' modules, there is an isomorphism of HomR' (A ® B, C) with HomR (A, HomR' (B,C)) (both being
6
R
regarded as Rand R' modules) taking an R' homomorphism cp: A ~ B to the R homomorphism cp': A _ HomR' (B,C) such that cp'(a)(b)
C
= cp(a Q9 b).
A subset 5 of an R module A is said to be a basis for A (and A is said to
be freely generated by 5) if any function f: 5 _ B, where B is an R module,
admits a unique extension to a homomorphism cp: A _ B. If a module has a
basis, it is said to be a free module. For any set 5 the free module generated
by 5, denoted by FR(5), is the module of all finitely nonzero functions from
5 to R (with pointwise addition and scalar multiplication) and with s E 5
identified with its characteristic function. FR(5) contains 5 as a basis, and any
module containing 5 as a basis is canonically isomorphic to FR(5).
7
Any R module is isomorphic to a quotient of a free R module.
8
If A' is a submodule of A, with A/A' free, then A is isomorphic to the
direct sum A' E8 (A/ A').
We now assume that R is a principal ideal domain (that is, it is an
integral domain in which every ideal is principal). If A is an R module, its
torsion submodule Tor A is defined by
= 0 for some nonzero r E R}
A is said to be torsion free or without torsion if Tor A = O.
Tor A
9
= {a
E A I ra
Over a principal ideal domain, a submodule of a free module is free.
10 Over a principal ideal domain, a finitely generated module is free if and
only if it is torsion free.
I I Over a principal ideal domain, A/Tor A is torsion free.
If A is a finitely generated module over a principal ideal domain R, its
rank p(A) is defined to be the number of elements in a basis of the quotient
module A/Tor A.
12 If A' is a submodule of a finitely generated module A (over a principal
ideal domain), then
p(A)
= p(A') + p(A/A')
SEC.
5
9
EUCLIDEAN SPACES
Let cp: A -7 A be an endomorphism of a finitely generated module (over
a principal ideal domain R). The trace of cp, Tr cp, is the element of R which
is the trace of the endomorphism cp' induced by cp on the free module
A/Tor A [that is, if A/Tor A has a basis aI, . . . , an, then cp'(ai) = ~ rijaj
and Tr cp = ~ riiJ.
13 Let cp be an endomorphism of a finitely generated module A and let A'
be a submodule of A such that cp(A') C A'. Then cp I A' is an endomorphism
of A' and there is induced an endomorphism cp" of A/A'. Their traces satisfy
the relation
Tr cp = Tr (cp I A')
+ Tr cp"
A module with a single generator is said to be cyclic. Over a principal
ideal domain R such a module A is characterized, up to isomorphism, by the
element rA E R which generates the ideal of elements annihilating every
element of A (rA is unique up to multiplication by invertible elements of R).
14 STRUCTURE THEOREM FOR FINITELY GENERATED MODULES Over a principal
ideal domain every finitely generated module is the direct sum of a free
module and cyclic modules AI, . . . , Aq whose corresponding elements
rl, . . . , rq E R have the property that ri divides ri+l for 1 s i s q - 1. The
elements rl, . . . , rq are unique up to multiplication by invertible elements
of R and, together with the rank of the module, characterize the module up
to isomorphism.
5
EI;CUDEAN SPACES
We use the following fixed notations:
o = empty set
Z
= ring of integers
Zm = ring of integers modulo m
= field of real numbers
R
C
= field of complex numbers
Q = division ring of quaternions
Rn = euclidean n-space, with Ilxll = V~ Xi 2 and (x,y) = ~ XiYi
o = origin of Rn
I
= closed unit interval
i
= {O,I}
C I
In = n-cube = {x E Rn I 0 :<:::: Xi :<:::: 1 for 1 :<:::: i :<:::: n}
in
= {X E In I for some i, Xi = 0 or Xi = I}
En = n-ball = {X E Rn I I xii :<:::: I}
Sn-l = (n - I)-sphere = {x E Rn Illxll = I}
pn = proiective n-space = quotient space of Sn with x and -x identified
for all X E Sn
10
INTRODUCTION
If X and yare points of a real vector space, the closed line segment joining them, denoted by [x,y], is the set of points of the form tx + (1 - t)y for
o :::; t :::; 1 (thus I = [0,1]). If x -=1= y, the line determined by them is the set
{tx + (1 - t)y I t E R}. A subset C of a real vector space is said to be
an affine variety if whenever x, y E C, with x -=1= y, then the line determined
by x and y is also in C. A subset C is said to be convex if x, y E C imply
[x,y] c c. A convex bodyl in Rn is a convex subset of Rn containing a nonempty open subset of Rn (thus In and En are convex bodies in Rn).
I
If C is a convex body in Rn and C' is a convex body in Rm, then C X C'
is a convex body in Rn X Rm
Rn+m.
=
2
Any two compact convex bodies in Rn are homeomorphic.
A subset S of a real vector space is said to be affinely independent
if, given a finite number of distinct elements xo, Xl. . . . , Xm E Sand
to, tl, . . . , tm E R such that ~ ti = 0 and ~ tiXi = 0, then ti = 0 for
o :::; i :::; m (this is equivalent to the condition that
Xl -
XO, X2 - XO, . . . ,Xm - Xo
be linearly independent).
3
There exist affinely independent subsets of Rn containing n + 1 points,
but no subset of Rn containing more than n + 1 points is affinely independent.
4
Given points xo, Xl. . . . ,Xm ERn, the convex set generated by them is
the set of all points of the form ~ tiXi. with 0 :::; ti :::; 1 and ~ ti = 1. The set
{XO,Xl' . . . ,xm} is affinely independent if and only if every point x in the
convex set generated by this set has a unique representation in the form
~ tiXi, with 0 :::; ti :::; 1 for 0 :::; i :::; m and ~ ti
1.
x
=
=
SOME BOOKS ON ALGEBBAIC TOPOLOGY
Alexandrafl', P. and H. Hopf: Topologie, Springer-Verlag, 1935.
Bott, R. and L. W. Tu: Differential Fonns in Algebraic Topology, Springer-Verlag, 1982
Bourgin, D.G.: Modern Algebraic Topology, Macmillan, 1963.
Bredon, G.E.: Sheaf Theory, McGraw-Hill, 1967.
Cairns, 5.5.: Introductory Topology, Ronald Press, 1962.
Dold, A.: Lectures on Algebraic Topology, Springer-Verlag, 1980.
Eilenberg, S. and N.E. Steenrod: Foundations of Algebraic Topology, Princeton University Press, 1952.
Godement, R.: Topologie algebrique et theorie des faisceaux, Hermann and Cie, 1958.
1 For general properties of convex sets see F. A. Valentine, Convex Sets, McGraw-Hill Book
Company, New York, 1964.
SOME BOOKS
11
Gray, B.: Homotopy Theory, An Introduction to Algebraic Topology, Academic Press,
197,5.
Greenberg, M.J. and J.R. Harper: Algebraic Topology, A First Course Benjamin/Cummings, 1981.
Hilton, P.J. and S. Wylie: Homology Theory, Cambridge University Press, 1960.
Hocking, J.G. and G.S. Young: Topology, Addison-Wesley, 1961.
Hu, S.T.: Homotopy Theory, Academic Press, 1959.
Lefschetz, S.: Algebraic Topology, American Math Society, 1942.
Lefschetz, S.: Introduction to Topology, Princeton Univ. Press, 1949.
Massey, W. S.: Algebraic Topology, An Introduction, Harcourt, Brace and World, 1967.
Massey, W.S.: Homology and Cohomology Theory, An Approach Based on AlexanderSpanier Cochains, Dekker, 1978.
Massey, W.S.: Singular Homology Theory, Springer-Verlag, 1980.
Maunder, C.R.F.: Algebraic Topology, Cambridge Univ. Press, 1980.
Munkres, J.R.: Elements of Algehraic Topology, Addison-Wesley, 1984.
Pontryagin, L. S.: Foundations of Combinational Topology, Graylock Press, 1952.
Schubert H.: Topologie, Teubner Verlagsgesellschaft, 1964.
Seifert, H. and W. Threlfall: Lehrbuch der Topologie, Teubner Verlagsgesellschaft,
1934.
Steenrod, N. E.: The Topology of Fiber Bundles, Princeton University Press, 19.51.
Switzer, R.M.: Algebraic Topology-Homotopy and Homology, Springer-Verlag, 1975.
Vick, J. W.: Homology Theory, An Introduction to Algebraic Topology, Academic Press, 1973.
Wallace, A.H.: An Introduction to Algebraic Topology, Pergamon Press, 1957.
Wallace, A. H.: Algebraic Topology, Homology and Cohomology, Benjamin, 1970.
Wilder, R.L.: Topology of Manifolds, American Math Society, 1949.
CHAPTER ONE
HOMOTOPY AND THE
FUNDAMENTAL GROUP
TOPOLOGY IS THE STUDY OF TOPOLOGICAL SPACES AND CONTINUOUS FUNCTIONS
between them. A standard problem is the classification of such spaces and
functions up to homeomorphism. A weaker equivalence relation, based on
continuous deformation, leads to another classification problem. This latter
classification problem is of fundamental importance in algebraic topology,
since it is the one where the tools available seem to be most successful.
As a working definition for our purposes, algebraic topology may be
regarded as the study of topological spaces and continuous functions by means
of algebraic objects such as groups, rings, homomorphisms. The link from
topology to algebra is by means of mappings, called functors. For this reason,
Sees. 1.1 and 1.2 are devoted to the basic concepts of category and functor.
In Sees. 1.3 and 1.4 the concept of continuous deformation, known technically as homotopy, is introduced. We then define the homotopy category
and certain functors on this category, all of which are important for the subject. Sections 1.5 and 1.6 are devoted to a study of conditions under which
these functors on the homotopy category take values in the category of groups.
As examples, the homotopy group functors are briefly mentioned.
13
14
HOMOTOPY AND THE FUNDAMENTAL GROUP
CHAP.
1
The first functor considered in detail is the fundamental group functor,
introduced and discussed in Sees. 1.7 and 1.8. This is an intuitively appealing
example of the kind of functor considered in algebraic topology. Some applications of this functor are presented in the exercises at the end of the chapter.
In Chapter Two this functor is used in a systematic study and classification of
covering spaces.
I
CATEGORIES
An algebraic representation of topology is a mapping from topology to algebra.
Such a representation converts a topological problem into an algebraic one to
the end that, with sufficiently many representations, the topological problem
will be solvable if (and only if) all the corresponding algebraic problems are
solvable.
The definition of a representation, formally called a functor, is given in
the next section. This section i~ devoted to the concept of category, because
functors are functions, with certain naturality properties, from one or several
categories to Illlother.
A category may be thought of intuitively as consisting of sets, possibly
with additional structure, and functions, possibly preserving additional structure. More precisely, a category consists of
e
(a) A class of obfects
(b) For every ordered pair of objects X and Y, a set hom (X, Y) of
morphisms with domain X and range Y; if f E hom (X, Y), we write
f: X --:) Y or X -4 Y
(c) For every ordered triple of objects X, Y, and Z, a function associating
to a pair of morphisms f: X --:) Y and g: Y --:) Z their composite
These satisfy the following two axioms:
Associativity. If f: X --:) Y, g: Y --:) Z, and h: Z --:) W, then
h(gf)
= (hg)f:
X --:) W
Identity. For every object Y there is a morphism Iv: Y --:) Y such that if
f: X --:) Y, then Ivf f, and if h: Y --:) Z, then hl y
h.
=
=
If the class of objects is a set, the category is said to be small. For most
of our purposes we could restrict our attention to small categories, but it
would be inconvenient to have to specify a set of objects before obtaining a
category. For example, we should like to consider categories whose objects
are sets or groups, and we prefer to consider the class of all sets or groups,
rather than some suitable set of sets or groups in each instance.
From the two axioms it follows that ly is unique (see lemma 1 below),
SEC.
I
15
CATEGORIES
and it is called the identity morphism of Y. Given morphisms f: X -7 Y and
g: Y -7 X such that gf = lx, g is called a left inverse of f and f is called a
right inverse of g. A two-sided inverse (or simply an inverse) of f is a
morphism which is both a left inverse of f and a right inverse of f. A morphism
f: X -7 Y is called an equivalence, denoted by f: X :::::: Y, if there is a morphism
g: Y -7 X which is a two-sided inverse of f. If g': Y -7 X is a left inverse of f
and gil: Y -7 X is a right inverse of f, then
g' = g'ly = g'(fg") = (g'f)g" = lxg" = gil
showing that g' = gil. Therefore we have the following lemma.
I
LEMMA
If f: X -7 Y has a left inverse and a right inverse, they
are equal, and f is an equivalence. •
In particular, it follows that an equivalence f: X :::::: Y has a unique inverse, denoted by f- 1 : Y -7 X, and f- 1 is an equivalence. If there is an
equivalence f: X :::::: Y, X and Yare said to be equivalent, denoted by X :::::: Y.
Because the composite of equivalences is easily seen to be an equivalence,
the relation X :::::: Y is an equivalence relation in any set of objects of e.
We list some examples of categories.
2
The category of sets and functions [that is, the class of objects is the class
of all sets, and for sets X and Y, hom (X, Y) equals the set of functions from
X to Y]
3
The category of topological spaces and continuous maps
4
The category of groups and homomorphisms
:;
The category of R modules and homomorphisms
6
The category of normed rings (over R) and continuous homomorphisms
7
The category of sets and injections (or surjections or bijections)
8
The category of pointed sets (a pointed set is a nonempty set with a distinguished element) and functions preserving distinguished elements
9
The category of pointed topological spaces (a pointed topological space
is a nonempty topological space with a base point) and continuous maps
preserving base points
I 0 The category of finite sets and functions
I I Given a partial order ::;: in X, there is a category whose objects are the
elements of X and such that hom (x,x') is either the singleton consisting of the
ordered pair (x,x') or empty, according to whether x ::;: x' or x :$ x'
12 The category of groups and conjugacy classes of homomorphisms (that is,
a morphism G -7 G' is an equivalence class of homomorphisms from G to G',
two homomorphisms being equivalent if they differ by an inner automorphism
of G')
16
HOMOTOPY AND THE FUNDAMENTAL GROUP
A subcategory
e
c
CHAP.
1
8 is a category such that
(a) The objects of 8' are also objects of 8
(b) For objects X' and Y' of e, hOIIle' (X', Y') C hOIIle (X', Y')
(c) Iff': X' ~ Y' and g': Y' ~ Z' are morphisms of e, their composite
in 8' equals their composite in 8
e
8' is called a full subcategory of 8 if is a subcategory of 8 and for objects X' and Y' in e, home' (X',Y')
home (X',Y'). The category in example 7
above is a subcategory of the one in example 2, and the category in example
10 is a full subcategory of the one in example 2. The categories in examples
3, 4, 5, 6, and 8 are not subcategories of the category of sets, because each
object of one of these categories consists of a set, together with an additional
structure on it (hence, different objects in these categories may have the same
underlying sets). In examples 11 and 12, the morphisms in the respective
categories are not functions, and so neither of these categories is a subcategory of the category of sets.
A diagram of morphisms such as the square
=
X~ Y
g~
~h
X' ~ Y'
is said to be commutative if any two composites of morphisms in the diagram
beginning at the same place and ending at the same place are equal. This
square is commutative if and only if hf = f'g.
Following are descriptions of some categories which are associated to a
given category. Given a category 8, there is an associated category called
the category of morphisms of 8. Its objects are morphisms X L Y, and
its morphisms with domain X -4 Y and range X' -4 Y' are pairs of morphisms
g: X ~ X' and h: Y ~ Y' such that the square
X~ Y
g~
~h
X' ~ Y'
is commutative. In a similar way, diagrams of morphisms in 8 more general
than X -4 Yare the objects of a suitable category associated to 8.
Let 8 be a category whose objects are sets with additional structures
(such as distinguished elements or topologies) and whose morphisms are
functions preserving the additional structures. For example, 8 might be any
of the categories in examples 2 through 10. There is a category associated to
8, called the category of pairs of e, whose objects are injective morphisms
i: A ~ X (because each morphism in such a category is a function, it is meaningful to consider those which are injective) and whose morphisms are commutative squares
SEC.
1
CATEGORIES
17
Thus the category of pairs of 2is a full subcategory of the category of morphisms
of e The notation (X,A) will denote the pair consisting of X and i: A c X,
and the notation f: (X,A) --0> (Y,B) will mean that f: X --0> Y is a morphism of
8 such that f(i(A)) C ;(B). The category of pairs of 8, therefore, has as objects
the pairs (X,A) and has as morphisms the morphisms f: (X,A) --0> (Y,B).
If 8 1 and ~ are categories, their product 8 1 X ~ is the category whose
objects are ordered pairs (Yr, Y 2 ) of objects Y1 in 21 and Y2 in ~ and whose
morphisms (X 1 ,X 2 ) --0> (Y1 , Y2 ) are ordered pairs of morphisms (h,h), where
h: Xl --0> Y1 in 21 and h: X 2 --0> Y2 in ~. Similarly, there is a product of an
arbitrary indexed family of categories.
Given a category 2, there is an opposite category 2* whose objects y*
are in one-to-one correspondence with the objects Y of 2and whose morphisms
f*: Y * --0> X * are in one-to-one correspondence with the morphisms f: X --0> Y
[withf*g* defined to equal (gf)* for X L Y J4 Z in 2]. We identify (2*)*
with 2, so that (X *) * = X and (f*) * = f.
We next show how to interpret sums and products, as well as direct and
inverse limits in arbitrary categories. An object X in a category 2 is said to be
an initial ob;ect if for each object Yin 2 the set hOIll (X, Y) contains exactly
one element. Dually, an object Z of 2 is said to be a terminal ob;ect if for each
Y of 8 the set hom (Y,Z) contains exactly one element. Note that any two
initial objects of 2 are equivalent and any two terminal objects of 2 are
equivalent. In examples 2 and 3 the empty set is an initial object and any
one-point set is a terminal object. In example 4 the trivial group is both an
initial and a terminal object. In example 7 the category of sets and bijections
has neither an initial object nor a terminal object.
Let {Yj }iEJ be an indexed collection of objects of a category e Let ~{Yj}
be the category whose objects are indexed collections of morphisms {fj}iEJ of
8 having the same range and whose morphisms with domain {k Yj --0> Z}
and range {ff: Yj --0> Z/} are morphisms g: Z --0> Z' of 2 such that gfj = jj' for
every; E J. An initial object of S){ Yj} is called a sum of the collection {Yj}. A
given collection mayor may not have a sum in e The set sum is a sum in
the category of sets, the topological sum is a sum in the category of topological spaces, the free product is a sum in the category of groups, \and the direct
sum is a sum in the category of R modules. In the category of finite sets, in
general only finite collections have a sum. Similarly, in the category of finitely
generated R modules, in general only finite collections have a sum.
Dually, given an indexed collection of objects {YdiEJ in 2, let 0l{Yj} be
the category whose objects are indexed collections of morphisms {&}jEJ of 2
having the same domain and whose morphisms with domain {gj: X --0> Yj }
and range {g): X' --0> Yj} are morphisms f: X --0> X' of 2 such that gjf = & for
every; E J. A terminal object of ':P{ Yj} is called a product of the collection
{ Yj}. The cartesian product of sets is a product in the category of sets, the
topological product is a pro-cIuct in the category of topological spaces, and the
direct product is a product in the category of groups, or R modules. In the
category of finite sets (or finitely generated R modules), in general only finite
collections have a product.
18
HOMOTOPY AND THE FUNDAMENTAL GROUP
CHAP.
1
e
A direct system {ya,fa ll } in a category
consists of a collection of
objects {Ya} indexed by a directed set A = {a} and a collection of morphisms
{fall: Ya ~ YIl} in e for a ::;: f3 in A such that
=
(a) fa a lya for a E A
(b) faY = fllYfa ll : Y" ~ YY for a ::;:
f3 ::;: y in
A
There is then a category dir {ya,f"ll} whose objects are indexed collections of
morphisms {ga: ya ~ Z}"E A such that g"
gllfa ll if a ::;: f3 in A and whose
morphisms with domain {ga: ya ~ Z} and range {g~: Y" ~ Z'} are
morphisms h: Z ~ Z' such that hg" = g~ for a E A. An initial object
of dir {Y",f"ll} is called a direct limit of the direct system {ya,fa ll }. The direct
limits of sets, topological spaces, groups, and R modules are examples of
direct limits in their respective categories.
Dually, an inverse system {Y",fa ll } in consists of a collection of objects
{Ya } indexed by a directed set A = {a} and a collection of morphisms
{fall: Yll ~ Y,,} in e for a ::;: f3 in A such that
=
e
(a) fa"
(b) f"Y
= lYa for a E A
= fallfllY: Yy ~ Y" for a ::;: f3 ::;: y in A
There is then a category inv {Ya,fa ll } whose objects are indexed collections of
morphisms {ga: X ~ Ya}"EA such that go = f"llgll if a ::;: f3 in A and whose
morphisms with domain {g,,: X ~ Y,,} and range {g~: X' ~ Y,,} are morphisms
h: X -+ X' of ~ such that g~h = g" for a E A. A terminal object of inv {Y",/all\
is called an inverse limit of the inverse system {Ya,fa ll }. The inverse limits of
sets, topological spaces, groups, and R modules are examples of inverse limits
in their respective categories.
By similar considerations it is possible to define a direct or inverse limit
for an arbitrary indexed collection of objects in a category and an indexed
collection of morphisms in between these objects. We omit the details.
e
e
2
FUN(;TORS
Our main interest in categories is in the maps from one category to another.
Those maps which have the natural properties of preserving identities and
composites are called functors. This section is devoted to the definition
of functors of one or more variables, some examples and applications, and
the definition of natural transformations between functors.
Let e and OJ) be categories. A covariant functor (or contravariant functor)
T from e to OJ) consists of an object function which assigns to every object X
of e an object T(X) of oj) and a morphism function which assigns to every morphism f: X ~ Y of a morphism T(f): T(X) ~ T(Y) [or T(f): T(Y) ~ T(X)]
of OJ) such that
e
(a) T(lx)
(b) T(gf)
= IT(Xl
= T(g)T(f)
[or T(gf)
= T(f)T(g)]
SEC.
2
19
FUNCTORS
We list some examples of functors.
There is a covariant functor from the category of topological spaces and
I
continuous maps to the category of sets and functions which assigns to every
topological space its underlying set. This functor is called a forgetful functor
because it "forgets" some of the structure of a topological space.
There is a covariant functor from the category of sets and functions to
2
the category of R modules and homomorphisms which assigns to every set
the free R module generated by it.
a Given a fixed R module Mo, there is a covariant functor (or contravariant
functor) from the category of R modules and homomorphisms to itself which
assigns to an R module M the R module HomR(Mo,M) [or HomR(M,Mo)].
4
For any category C; and object Y of C; there is a covariant functor 'lTy (or
contravariant functor 'lTY) from C; to the category of sets and functions which
assigns to an object Z (or X) of C; the set 'lTy(Z) = hom (Y,Z) [or 'lTY(X)
hom (X, Y)] and to a morphism h: Z ---,) Z' [or f: X ---,) X'] the function
=
h#: hom (Y,Z) ---,) hom (Y,Z')
defined by h#(g)
= hog for g:
[or f#: hom (X',Y) ---,) hom (X,Y)]
Y ---,) Z [or f# (g')
= g'
0
f for g': X' ---,) Y]
5
There is a contravariant functor C from the category of compact Hausdorff
spaces and continuous maps to the category of normed rings over R and continuous homomorphisms which assigns to X its normed ring of continuous
real-valued functions.
6
There is a covariant functor Ho from the category of topological spaces
and continuous maps to the category of abelian groups and homomorphisms
such that Ho(X) is the free abelian group generated by the set of components
of X, and if f: X ---,) Y, then Ho(f): Ho(X) ---,) Ho(Y) is the homomorphism such
that if C is a component of X and C' is the component of Y containing f(c),
then Ho(f)C = C'.
7 A direct system (or inverse system) in a category C; is a covariant functor
(or contravariant functor) from the category of a directed set (defined as in
example 1.1.11) to c:.
8
For any category C; there is a contravariant functor to its opposite category C;* which assigns to an object X of C; the object X* of C;* and to
a morphism f: X ---,) Y of C; the morphism f*: y* ---,) X*.
Note that any contravariant functor on C; corresponds to a covariant
functor on C;*, and vice versa. Therefore any functor can be regarded as covariant on a suitable category. Despite this, we shall find it convenient to consider contravariant as well as covariant functors on 8, rather than consider
only covariant functors on two categories.
Any functor from the category of topological spaces and continuous
maps to an algebraic category (such as the category of abelian groups and
20
HOMOTOPY AND THE FUNDAMENTAL GROUP
CHAP.
1
homomorphisms) is a representation of the topological category by an algebraic one. Algebraic topology is the study of such functors; we show that
simple remarks about functors can be used to obtain necessary conditions for
the solvability of topological problems.
Let T be a functor from a category
T maps equivalences in e to equivalences in 6j).
9
THEOREM
e to a category
oj).
Then
PROOF
Assume that T is a covariant functor (the argument is similar if Tis
contravariant). Let f: X -7 Y be an equivalence in e. Then f-lf = Ix.
Therefore
In¥) = T(Ix) = T(f-l)T(f)
Similarly, T(f)T(f-l) = IT(y). Therefore T(f-l) is a two-sided inverse of T(f),
and T(f) is an equivalence in 6)). •
In particular, if T is an algebraic functor on the category of topological
spaces and continuous maps, a necessary condition that X be homeomorphic
to Y is that T(X) be equivalent to T( Y). Thus the functor Ho of example 6
shows that the real line R and the real plane R2 are not homeomorphic
[if they were homeomorphic, then R - 0 would be homeomorphic to R2 - P
for some p E R2, but Ho(R - 0) is a free abelian group on two generators,
while Ho(R2 - p) is a free abelian group on one generator]. This is a trivial
example. However, the homology functors Hq defined in Chapter 4 generalize
Ho and can be used in much the same way to prove that Rn and Rm are not
homeomorphic if n =1= m.
In applications of algebraic functors to topological problems the algebra
will frequently play an essential role. For example, let To(X) be the functor
obtained by composing the functor Ho with the forgetful functor, which
assigns to every abelian group its underlying set. The functor To contains less
information than the functor Ho and does not give as strong a necessary condition for homeomorphism [for example, To(R - 0) and To(R2 - p) are both
countably infinite sets and are equivalent in the category of sets and functions]. For this reason it is important to provide functors with as much algebraic structure as possible. Later we shall consider functors which depend on
a chosen topological space. These functors take values in the category of sets
and functions, but some of them, depending on properties of the particular
spaces which define them, are functors to the category of groups and homomorphisms. The added algebraic strurture in such cases will prove useful.
To show how functors can be applied to another problem, let A be
a subspace of a topological space X and let f: A -7 Y be continuous. The extension problem is to determine whether f has a continuous extension to Xthat is, whether the dotted arrow in the triangle
A
C
X
r\ ./
Y
corresponds to a continuous map making the diagram commutative.
SEC.
2
21
FUNCTORS
10 THEOREM Let T be a covariant functor (or contravariant functor) from
the category of topological spaces and continuous maps to a category e. A
necessary condition that a map f: A ~ Y be extendable to X (where i: A eX)
is that there exist a morphism cp: T(X) ~ T(Y) [or cp: T(Y) ~ T(X)] such that
cp T(i)
T(f) [or T(f)
T(i) cp].
=
0
=
0
Assume that f': X ~ Y is an extension of f. Then f'i = f. Therefore
TU') T(i)
T(f) [or T(f)
T(i) T(f')], and T(f') can be taken as the
morphism cpo •
PROOF
0
=
=
0
The above result can be applied to prove that the identity map of i cannot be extended to a continuous map I ~ i. We use the functor Ho and obtain the necessary condition that there must exist a homomorphism
cp: Ho(1) ~ Ho(i) such that cp Ho(i) = Ho(lj) (where i: i C 1). Because Ho(t) is a
free abelian group on two generators and Ho(I) is a free abelian group on one
generator, there is no such homomorphism cpo Again, this is a trivial example,
but it illustrates the method, and the general homology functors Hq defined
later can be used in the same way to show that there is no continuous map
En+! ~ Sn that is the identity map on Sn.
Thus we see that a functor yields necessary conditions for the solvability
of topological problems. There are situations in which these necessary conditions are also sufficient. For example, the functor C of example 5 gives a
necessary and sufficient condition for homeomorphism-that is, two compact
Hausdorff spaces X and Yare homeomorphic if and only if C(X) and C(Y)
are isomorphic. l This is not a particularly useful result, however, because it
seems to be no easier to determine whether or not two normed rings are isomorphic than it is to determine whether or not two compact Hausdorff
spaces are homeomorphic. We seek functors to categories that are somewhat
simpler than the category of topological spaces, so that the algebraic problems
that arise in these categories can be effectively solved. One big problem of
algebraic topology is to find, and compute, sufficiently many such functors
that the solvability of a particular topological problem is equivalent to the
solvability of the corresponding (and simpler) algebraic problems.
We shall also have occasion to compare functors with each other. This is
done by means of a suitable definition of a map between functors. Let Tl and
T2 be functors of the same variance (either both covariant or both contravariant)
from a category to a category Gj). A natural transformation cp from Tl to
£2 is a function from the objects of to morphisms of 6] such that for every
morphism f: X ~ Y of
the appropriate one of the follOWing diagrams
is commutative:
Tl(X) (T,(f) T1(y)
Tl(X) T,(f» Tl(y)
0
e
e
e
<p(X)l
T2(X)
l<p(Y)
TM»
T2(y)
<p(X)l
T2 (X)
l<p(Y)
(TM)
T 2 (y)
T1, T2 contravariant
1 See
Theorem D on page 330 of G. F. Simmons, Introduction to Tapology and Modem Analysis, McGraw-Hili Book Company, New York, 1963.
22
HOMOTOPY AND THE FUNDAMENTAL GROUP
CHAP.
I
If cp is a natural transformation from T1 to T2 such that cp(X) is an equivalence in Gj) for each object X in
then cp is called a natural equivalence.
As an example of a natural transformation, let Y1 and Y2 be objects of a
category and let g: Y1 ~ Y2 be a morphism in e. There is a natural transformation g# from the covariant functor 7TY2 to the covariant functor 7TYI and
a natural transformation g# from the contravariant functor 7T Y1 to the contravariant functor 7T Y2 • If g is an equivalence in 8, both these natural transformations are natural equivalences.
It is also of interest to consider functors of several variables. Thus, if 8 1 ,
and l'J) are categories, a covariant functor from 8 1 X
to oIl is called a
functor of two arguments covariant in each. A covariant functor from
8 1 X 8~ to oj), regarded as a function from ordered pairs (X 1 ,X2 ), where Xl is
an object of 8 1 and Xz is an object of 8 2 , is called a functor of two arguments
covariant in the first and contravariant in the second. In a similar fashion,
functors of more arguments with mixed variance are defined.
If 8 is any category, there is a functor of two arguments in 8 to the category of sets and functions which is contravariant in the first argument and
covariant in the second. This functor assigns to an ordered pair of objects X
and Y of 8 the set hom (X, Y) and to an ordered pair of morphisms f: X' ~ X
and g: Y ~ Y' in 8 the function f#~ = ~f#: hom (X, Y) ~ hom (X', yl).
e,
e
ez,
3
ez
HOMOTOPY
The problem of classifying topological spaces and continuous maps up to
topological equivalence does not seem to be amenable to attack directly by
computable algebraic functors, as described in Sec. 1.2. Many of the computable functors, because they are computable, are invariant under continuous
deformation. Therefore they cannot distinguish between spaces (or maps) that
can be continuously deformed from one to the other; the most that can be
hoped for from such functors is that they characterize the space (or map) up
to continuous deformation.
The intuitive concept of a continuous deformation will be made precise
in this section in the concept of homotopy. This leads to the homotopy category which is fundamental for algebraic topology. Its objects are topological
spaces and its morphisms are equivalence classes of continuous maps (two
maps being equivalent if one can be continuously deformed into the other).
For technical reasons we consider not just the homotopy category of topological spaces, but rather the larger homotopy category of pairs.
A topological pair (X,A) consists of a topological space X and a subspace
A C X. If A is empty, denoted by 0, we shall not distinguish between the
pair (X, 0) and the space X. A subpair (X',A') C (X,A) consists of a pair with
X' C X and A' C A. A map f: (X,A) ~ (Y,B) between pairs is a continuous
function f from X to Y such that f(A) C B, and as in Sec. 1.1, there is
SEC.
3
23
HOMOTOPY
a category of topological pairs and maps between them which contains as full
subcategories the category of topological spaces and continuous maps, as well
as the category of pointed topological spaces and continuous maps.
Given a pair (X,A), we let (X,A) X [ denote the pair (X X [, A X I). Let
X' C X and suppose that fo, II: (X,A) ~ (Y,B) agree on X' (that is, fo I X' =
iI I X'). Then fo is homotopic to iI relative to X', denoted by fo ~ f1 reI X', if
there exists a map
F: (X,A) X [ ~ (Y,B)
such that F(x,O) = fo(x) and F(x,l) = iI(x) for x E X and F(x,t) = fo(x) for
x E X' and t E 1. Such a map F is called a homotopy relative to X' from fo to
iI and is denoted by F: fo ~ iI reI X'. If X' = 0, we omit the phrase "relative to 0." Clearly, fo ~ iI reI X' implies fo ~ it reI X" for any X" C X'. A
map from X to Y is said to be null homotopic, or inessential, if it is homotopic
to some constant map.
For t E [define ht: (X,A) ~ (X,A) X [by ht(x) = (x,t). If F:fo ~iI reIX',
then Fho = fo, Fh1 = iI, and Fht I X' = fo I X' for all t E 1. Therefore the collection {Fhdt€I is a continuous one-parameter family of maps from (X,A) to
(Y,B), agreeing on X', which connects fo = Fho to it = Fh11. Hence fo ~ iI
reI X' corresponds to the intuitive idea of continuously deforming fo into iI
by maps all of which agree on X'. Note that if fo ~ iI rel X' there will usually
be many maps F which are homotopies relative to X' from fo to f1 (see
example 3 below).
I
EXAMPLE
Let X = Y = Rn and define fo(x) = x and f1(X) = 0 for
x E Rn (that is, fo = 1Rn and iI is the constant map of Rn to its origin).
If F: Rn X [ ~ Rn is defined by
F(x,t) = (1 - t)x
then F: fo ~ iI reI O.
2
EXAMPLE
Let X = Y
F: [ X [ ~ [ is defined by
= [ and define fo(t) = t and f1(t) = 0 for t E 1. If
F(t,t') = (1 - t')t
then F: fo ~ fl reI O.
3
EXAMPLE
Let X = Y = E2 = {z E C Iz = re iO , 0 ::::: r ::::: I} and let
A = B = 51 = {z E C Iz = eiO }. Define fo: (E2,5 1 ) ~ (E2,5 1 ) to be the
identity ma~ andiI: (E2,5 1 ) ~ (E2,5 1 ) to be the reflection in the origin [that
is, iI(re iO ) = re i(O+7r»). Define a homotopy F: fo ~ iI reI 0 by F(reiO,t) =
rei(o+t7r). Another homotopy F': fo ~ f1 reI 0 is defined by F' (reiO,t) = re i (O-t7r).
1 A one-parameter family ft: (X,A) -> (Y,B) for tEl is continuous if ft(x) is jOintly continuous
in t and x, in which case the function (x,t) -> ft(x) is a homotopy from fa to [J. The corresponding function t -> ft from I to (Y,B)tx.A) is always continuous [where (Y,B)(X,A) = {g: (X,A)->
(Y,B)} topologized by the compact-open topology]. Conversely, in case X is a locally compact
Hausdorff space, it follows from theorem 2.8 in the Introduction that for any continuous map
cp: 1-> (Y,B)(X,A) the one-parameter family cp(t) is continuous and defines a homotopy from cp(O)
to cp(1).
24
HOMOTOPY AND THE FUNDAMENTAL CROUP
CHAP.
1
4
EXAMPLE
Let X be an arbitrary space and let Y be a convex subset of
Rn. Let fo, f1: X --') Y be maps which agree on some subspace X' C X. Then
fo c:::::: f1 rei X', because the map F: X X 1--') Y defined by
F(x,t)
= tft(x) + (1 -
t)fo(x)
is a homotopy relative to X' from fo to ft.
Example 4 is a generalization of examples 1 and 2. In example 3 the space E2
is convex, but the homotopy between fo and f1 cannot be taken to be a particular case of the homotopy in example 4, because it must keep S1 mapped into
itself at all stages, and S1 is not convex.
To define the homotopy category we need the following easy results.
5
THEOREM
Homotopy relative to X' is an equivalence relation in the set
of maps from (X,A) to (Y,B).
Reflexivity. For f: (X,A) --') (Y,B) define F: f c:::::: frel X by F(x,t) = f(x).
Symmetry. Given F: fo c:::::: f1 rei X', define F: ft c:::::: fo rel X' by F/(X,t) =
F(x, 1 - t).
Transitivity. Given F: fo ~ f1 rei X' and G: ft ~ h rei X', define
H: fo ~ h rei X' by
PROOF
H(x,t)
= { F(x,2t)
G(x, 2t - 1)
o ::; t ::; lh
lh ::; t ::; 1
Note that H is continuous because its restriction to each of the closed sets
X X [O,lh] and X X [lh,l] is continuous. •
It follows that the set of maps from (X,A) to (Y,B) is partitioned into disjoint equivalence classes by the relation of homotopy relative to X'. These
equivalence classes are called homotopy classes relative to X'. We use
[X,A; Y,B]X' to denote this set of homotopy classes. Givenf: (X,A) --') (Y,B), we
use [f]x' to denote the element of [X,A; Y,B]X' determined by f. Homotopy
classes relative to the empty set will be denoted by omitting the subscript X'.
6
THEOREM
Composites of homotopic maps are homotopic.
Let fo, ft: (X,A) --') (Y,B) be homotopic relative to X' and let go, g1:
(Y,B) --') (Z,C) be homotopic relative to yl, where f1(X') C Y'. To show that
gofo, gd1: (X,A) --') (Z,C) are homotopic relative to X', let F: fo c:::::: ft rel X'
and G: go c:::::: g1 rei Y'. Then the composite
PROOF
(X,A) X I ~ (Y,B) ~ (Z,C)
is a homotopy relative to X' from gofo to goft, and the composite
(X,A) X I ~ (Y,B) X I ~ (Z,C)
is a homotopy relative to ft-1(Y' ) from goft to gd1' Since X' C f1- 1(Y'), we
have shown that gofo c:::::: goft rei X' and gOf1 c:::::: gdl rei X'. The result now
follows from theorem 5. •
SEC.
3
25
HOMOTOPY
The last result shows that there is a homotopy category of pairs whose
objects are topological pairs and whose morphisms are homotopy classes
(relative to 0). This category contains as full subcategories the homotopy
category of topological spaces (also shortened to homotopy category) and the
homotopy category of pointed topological spaces. There is a covariant functor
from the category of pairs and maps to the homotopy category of pairs whose
object function is the identity map and whose mapping function sends a map
f to its homotopy class [fl. As pointed out at the beginning of the section,
most of the algebraic functors we consider will be defined from the appropriate homotopy category. A diagram of topological pairs and maps is said to
be homotopy commutative if it can be made a commutative diagram in the
homotopy category (that is, when each map is replaced by its homotopy
class).
As in example 1.2.4, for any pair (P,Q) there is a covariant functor '/T(P,Q)
(or a contravariant functor '/T(P,Q») from the homotopy category of pairs to the
category of sets and functions defined by '/T(P,Q) (X,A) = [P,Q; X,A] (or
'/T(P,Q) (X,A) = [X,A; P,Q]), and if f: (X,A) ~ (Y,B), then '/T(P,Q) ([fl) = f#
(or '/T(P,Q) ([fl) = f#), where f#[g] = [fg] for g: (P,Q) ~ (X,A) (or f#[h] = [hf]
for h: (Y,B) ~ (P,Q)). If 0': (P,Q) ~ (P',Q'), there is a natural transformation
a# from '/T(P',Q') to '/T(P,Q) and a natural transformation a# from '/T(P,Q) to '/T(P',Q').
A map f: (X,A) ~ (Y,B) is called a homotopy equivalence if [fl is an
equivalence in the homotopy category of pairs. A map g: (Y,B) ~ (X,A)
is called a homotopy inverse of f if [g] = [fl- 1 in the homotopy category.
Pairs (X,A) and (Y,B) are said to have the same homotopy type if they are
equivalent in the homotopy category.
The simplest nonempty space is a one-point space. We characterize the
homotopy type of such a space as follows. A topological space X is said to be
contractible if the identity map of X is homotopic to some constant map of X
to itself. A homotopy from Ix to the constant map of X to Xo E X is called a
contraction of X to Xo. Examples 1 and 2 show that Rn and I are contractible,
and example 4 shows that any convex subset of Rn is contractible. The following lemma may be regarded as a generalization of the result of example 4.
7
LEMMA
Any two maps of an arbitrary space to a contractible space are
homotopic.
PROOF
Let Y be a contractible space and suppose I y ~ c, where c is a constant map of Y to itself. Let fo, fl: X ~ Y be arbitrary. By theorem 6,
fo
lyfo ~ cfo, and similarly, fl ~ c/1. Since cfo
c/1, it follows from
theorem 5 that fo ~ /1. •
=
=
8
COROLLARY
If Y is contractible, any two constant maps of Y to itself
are homotopic, and the identity map is homotopic to any constant map of Y
to itself. •
It is interesting to observe that lemma 7 cannot be strengthened to the
case of relative homotopy. That is, if fo and /1 are maps of ~ into a contract-
26
HOMOTOPY AND THE FUNDAMENTAL CROUP
CHAP.
1
ible space Y which agree on X' C X, it need not be true that fo = it reI X'
(although example 4 shows this to be true for convex subsets of Rn). The following example illustrates this and will be referred to again later.
9
EXAMPLE
The comb space Y illustrated in the diagram
(0,1)
(l/n,l)
(0,0)
(1/n,O)
(%,1)
(1,1)
(1,0)
Comb space
is defined by
Y
= {(x,y)
E R21 0::;; y ::;; 1, x
= 0,
lin
or
y
= 0, 0::;;
x ::;; I}
Let F: Y X I ---7 Y be defined by F((x,y), t) = (x, (1 - t)y). Then F is a
homotopy from ly to the projection of Y to the x axis. Since the latter map is
homotopic to a constant map, Y is contractible. Let c: Y ---7 Y be the constant
map of Y to the point (0,1). By corollary 8, ly = c, but even though these
two maps agree on (0,1), there is no homotopy relative to (0,1) between them.
The following theorem shows that contractible spaces are homotopically
as simple as possible.
10 THEOREM A space is contractible if and only if it has the same homotopy
type as a one-point space.
PROOF
Assume that X is contractible and let F: X X I ---7 X be a contraction
of X to a point Xo E X. Let P be the one-point space consisting of Xo and let
f: X ---7 P and i: P C X. Then fi = Ip and F: Ix ~ if. Therefore [i] = [f)-I,
and f is a homotopy equivalence from X to P.
Conversely, if X has the same homotopy type as a one-point space P, let
f: X ---7 P be a homotopy equivalence with homotopy inverse g: P ---7 X. Then
Ix
gf. Because gf is a constant map, X is contractible. -
=
SEC.
4
27
RETRACTION AND DEFORMATION
1 1 COROLLARY Two contractible spaces have the same homotopy type, and
any continuous map between contractible spaces is a homotopy equivalence.
PROOF
The first part follows from theorem 10 and the transitivity of the
relation of having the same homotopy type. The second part follows from the
first part and lemma 7 (and from the obvious fact that any map homotopic to
a homotopy equivalence is itself a homotopy equivalence). •
The next result establishes an important relation between homotopy and
the extend ability of maps.
12 THEOREM Let po be any point of Sn and let f: Sn
are equivalent:
(a) f is null homotopic
(b) f can be continuously extended over En+l
(c) f is null homotopic relative to po
---0>
Y. The following
(a) ==;. (b). Let F: f ~ c, where c is the constant map of Sn to yo E Y.
Define an extension f' of f over En+l by
PROOF
f'(x) =
{~(X/IIXII, 2 -
211xll)
o s: Ilxll
1J2
s:
I xii
s: Y2
s: 1
Since F(x,l) = yo for all x E Sn, the map f' is well-defined. f' is continuous
because its restriction to each of the closed sets {x E En+l lOS: Ilxll
Y2}
and {x E En+! 1112 ::::; I xii ::::; I} is continuous. Since F(x,O) = f(x) for x E Sn,
f' I Sn = f and f' is a continuous extension of f to En+l.
(b) ==;. (c). If f has the continuous extension f': En+l ---0> Y, define
F: Sn X I ---0> Y by
s:
F(x,t) = f'((1 - t)x
+ tpo)
Then F(x,O) = f'(x) = f(x) and F(x,l) = f'(po) for x E Sn. Since F(po,t) = f'(po)
for tEl, F is a homotopy relative to po from f to the constant map to f'(po).
(c) ==;. (a). This is obvious. •
Combining theorem 12 with lemma 7, we obtain the following result.
13 COROLLARY Any continuous map from Sn to a contractible space has a
continuous extension over En+l. •
4
RETRACTION AND
D":FOR~IATION
This section is concerned mainly with inclusion maps. We consider whether
such a map has a left inverse, a right inverse, or a two-sided inverse in either
the category of topological spaces and continuous maps or the homotopy
category. 1
1 Many of the results in this section can be found in R. H. Fox, On homotopy type and deformation retracts, Annals of Mathematics, vol. 44, pp. 40-50, 1943 (see also H. Samelson,
Remark on a paper by R. H. Fox, Annals of Mathematics, vol. 45, pp. 448-449, 1944).
28
HOMOTOPY AND THE FUNDAMENTAL GROUP
CHAP.
1
A subspace A of X is called a retract of X if the inclusion map i: A C X
has a left inverse in the category of topological spaces and continuous maps.
Hence A is a retract of X if and only if there is a continuous map r: X --'> A
such that ri = lA [that is, r(x) = x for x E A]. Such a map r is called a retraction of X to A.
A subspace A of X is called a weak retract of X if the inclusion map
i: A C X has a left homotopy inverse (that is, a left inverse in the homotopy
category). Thus A is a weak retract of X if and only if there is a continuous
map r: X --'> A such that ri c::::: lAo Such a map r is called a weak retraction of
X to A.
Anyone-point subspace is a retract of any larger space containing it. A
discrete space with more than one point is never a weak retract of a connected
space containing it. If A is a retract of X, it is a weak retract of X. The converse is not true, as is shown by the following example.
EXAMPLE
Let X be the closed unit squar~ 12 in R2 and let A C X be
I
the comb space of example 1.3.9. Then A and X are both contractible, and
by corollary 1.3.11, the inclusion map A C X is a homotopy equivalence.
Therefore A is a weak retract of X. However, it can be shown that A is not a
retract of X.
Despite the fact that, in general, a weak retract need not be a retract,
these concepts do coincide when A is a suitable subspace of X. This occurs
frequently enough to warrant special consideration and will prove of use later.
Let (X,A) be a pair and Y be a space. (X,A) is said to have the homotopy extension property with respect to Y if, given maps g: X --'> Y and G: A X 1--,> Y
such that g(x) = G(x,O) for x E A, there is a map F: X X I --'> Y such that
F(x,O) = g(x) for x E X and FI A X I = G. If g is regarded as a map of X X 0
to Y, the existence of F is equivalent to the existence of a map represented by
the dotted arrow which makes the following diagram commutative:
AXO
C
n
Y
XXO
y
r
'"
C
A X I
n
,
X X I
If (X,A) has the homotopy extension property with respect to Y and fa,
A --'> Yare homotopic, then if fa has an extension to X, so does il; for if
g: X --'> Y is an extension of fa and G: A X I --'> Y is a homotopy from fa to
il, the homotopy extension property implies the existence of a map
F: X X I --'> Y which is an extension of G, therefore F(x,l) is an extension
of il. It follows that whether or not a map A --'> Y can be extended over X is
a property of the homotopy class of that map. Therefore the homotopy
extension property implies that the extension problem for maps A --'> Y is a
problem in the homotopy category.
il:
SEC.
4
29
RETRACTION AND DEFORMATION
Of particular importance is the case when (X,A) has the homotopy
extension property with respect to any space. More generally, a map f: X' ----',> X is
called a cofibration if, given maps g: X ----',> Yand G: X' X I ----',> Y (where Y is
arbitrary) such that g(f(x')) = G(x',O) for x' E X', there is a map F: X X 1----',> Y
such that F(x,O) = g(x) for x E X and F(f(X'), t) = G(X',t) for x' E X' and tEl.
If g is regarded as a map of X X to Y, the existence of F is equivalent to
the existence of a map represented by the dotted arrow which makes the following diagram commutative:
X' X I
X' X
C
°
°
f x
10]
~
Y
~
XXo
c XXI
Thus an inclusion map i: A C X is a co fibration if and only if (X,A) has the
homotopy extension property with respect to any space.
2
THEOREM
If (X,A) has the homotopy extension property with respect to
A, then A is a weak retract of X if and only if A is a retract of X.
PROOF
We show that any weak retraction r: X ----',> A is, in fact, homotopic to
a retraction. Let i: A C X; then ri ':-::: 1A . Let G: A X I ----',> A be a homotopy
from ri to 1A ; then G(x,O) = r(x) for x E A. Because (X,A) has the homotopy
extension property with respect to A, there is a map F: X X I ----',> A which
extends G such that F(x,O) = r(x) for x E X. If r': X ----',> A is defined by
r'(x) = F(x,l), then r' is a retraction of X to A, and F is a homotopy from
r to r'. •
We can just as well consider inclusion maps with right homotopy
inverses as those with left homotopy inverses. This leads to the following
definitions. Given X' C X, a deformation D of X' in X is a homotopy
D: X' X I ----',> X
such that D(x',O) = x' for x' E X'. If, moreover, D(X' X 1) is contained in a
subspace A of X, D is said to be a deformation of x' into A and X' is said to
be deformable in X into A. A space X is said to be deformable into a subspace
A if it is deformable in itself into A. Thus a space X is contractible if and only
if it is deformable into one of its points.
3
LEMMA
A space X is deformable into a subspace A if and only if the
inclusion map i: A C X has a right homotopy inverse.
PROOF
If i has a right homotopy inverse f: X ----',> A, then if ~ Ix. Let
F: X X I ----',> X be a homotopy from Ix to if; then F(x,O) = x, so F is a deformation of X, and F(X X 1) = if(X) C A, so X is deformable into A.
Conversely, if X is deformable into A, let D: X X I ----',> X be a deformation such that D(X X 1) C A. Let f: X ----',> A be defined by the equation
if(x)
= D(x,l)
x EX
30
HOMOTOPY AND THE FUNDAMENTAL GROUP
Then D: Ix
~
if, showing that f is a right homotopy inverse of i.
CHAP.
1
•
Note that an inclusion map i: A C X never has a right inverse in the
category of topological spaces and continuous maps except in the trivial case
A
X.
We now consider inclusion maps which are homotopy equivalences. A
subspace A C X is called a weak deformation retract of X if the inclusion
map i: A C X is a homotopy equivalence. From lemma 1.1.1 and lemma 3
above we obtain the following result.
=
4
LEMMA A is a weak deformation retract of X if and only if A is a weak
retract of X and X is deformable into A. •
As was the case with the concept of weak retract, there are more useful
concepts than that of weak deformation retract. The subspace A is a strong
deformation retract of X if there is a retraction r of X to A such that if
i: A C X, then Ix ~ ir reI A. If F: Ix ~ ir reI A, F is called a strong deformation retraction of X to A.
There is an intermediate concept useful in comparing the weak and
strong forms already defined. A subspace A is called a deformation retract of
X if there is a retraction r of X to A such that if i: A C X, then Ix ~ ir. If
F: Ix ~ ir, F is called a deformation retraction of X to A. A homotopy
F: X X I ~ X is a deformation retraction if and only if F(x,O) = x for
x E X, F(X X 1) C A, and F(x,l) = x for x E A. It is a strong deformation
retraction if and only if it also satisfies the condition F(x,t) = x for x E A and
tEl.
:;
EXAMPLE
It follows from example 1.3.4 that anyone-point subset of a
convex subset of Rn is a strong deformation retract of the convex set.
6
EXAMPLE
Sn is a strong deformation retract of Rn+l map F: (Rn+1 - 0) X I ~ Rn+l - 0 defined by
F(x,t)
= (1
- t)x
tx
+W
o.
In fact the
x E Rn+1 - 0, tEl
is a strong deformation retraction of Rn+1 - 0 to Sn.
It is clear that a strong deformation retract is a deformation retract, and a
deformation retract is a weak deformation retract. The following examples
show that neither of these implications is reversible.
7 EXAMPLE As in example 1 above, let X be the closed unit square and A
be the comb space. As pointed out in example 1, the inclusion map A C X is
a homotopy equivalence, but A is not a retract of X. Therefore A is a weak
deformation retract of X which is not a deformation retract of X.
8
EXAMPLE
Let X be the comb space and A be the one-point subspace of
X consisting of the point (0,1). Because X is contractible, there is a homotopy
F from Ix to the constant map of X to A. Such a map F is a deformation re-
SEC.
4
31
RETRACTION AND DEFORMATION
traction of X to A. However, as was remarked in example 1.3.9, there is no
homotopy relative to A from Ix to the constant map to A; therefore A is a
deformation retract of X which is not a strong deformation retract of X.
In the presence of suitable homotopy extension properties the three concepts of deformation retract coincide, and we shall now prove this.
9
LEMMA
tract of X.
If X is deformable into a retract A, then A is a deformation re-
Let r: X ~ A be a retraction and let i: A C X. Then r is a left
homotopy inverse of i. Because X is deformable into A, it follows from
lemma 3 that i has a right homotopy inverse. By lemma 1.1.1, r is also a right
homotopy inverse of i. Since Ix c::::o ir, A is a deformation retract of X. •
PROOF
Combining lemma 9 with theorem 2 yields the following corollary.
10 COROLLARY If (X,A) has the homotopy extension property with respect
to A, then A is a weak deformation retract of X if and only if A is a deformation retract of X. •
I I THEOREM If (X X I, (X X 0) U (A X I) U (X X 1)) has the homotopy
extension property with respect to X and A is closed in X, then A is a deformation retract of X if and only if A is a strong deformation retract of X.
PROOF
If A is a deformation retract of X, let F: X X I ~ X be a homotopy
from Ix to ir, where r: X ~ A is a retraction and i: A C X. A homotopy
G: [(X X 0) U (A X 1) U (X X 1)] X I
~
X
is defined by the equations
G((x,O), t') = x
G((x,t), t') = F(x, (1 - t')t)
G((x,I), t') = F(r(x), 1 - t')
x E X, t' E I
x E A; t, t' E I
x E X, t' E I
G is well-defined, because for x E A
G((x,O), t')
= x = F(x,O)
by the first two equations and
G((x,I), t')
= F(x, 1 -
t')
= F(r(x), 1 -
t')
by the last two equations. G is continuous because its restriction to each of
the closed sets (X X 0) X I, (A X I) X I, and (X X 1) X I is continuous. For
(x,t) E (X X 0) U (A X 1) U (X XI), G((x,t), 0) = F(x,t) [because F(x,O) = x,
and since ris a retraction, F(r(x), 1) = ir(r(x)) = r(x) = F(x,l)]. Therefore G restricted to [(X X 0) U (A X 1) U (X X 1)] X 0 can be extended to (X X I) X O.
From the homotopy extension property in the hypothesis, G restricted to
[(X X 0) U (A X I) U (X X 1)] X 1 can be extended to (X X I) X 1. Let
G': (X X 1) X 1 ~ X be such an extension, and define H: X X I ~ X
32
HOMOTOPY AND THE FUNDAMENTAL GROUP
by H(x,t)
CHAP.
1
= G'((x,t), 1). Then we have the equations
H(x,O)
H(x,I)
H(x,t)
= G'((x,O), 1) = G((x,O), 1) = x
= G((x,I), 1) = F(r(x),O) = r(x)
= G((x,t), 1) = F(x,O) = x
xEX
xEX
x E A, tEl
Therefore H is a homotopy relative to A from Ix to ir, so A is a strong deformation retract of X. •
The next result asserts that any map is equivalent in the homotopy
category to an inclusion map that is a co fibration. Let f: X ~ Y and let Zr
denote the quotient space obtained from the topological sum of X X I and Y
by identifying (x,I) E X X I with f(x)'E Y. Zr is called the mapping cylinder
of f and is depicted in the diagram
D - - - -,
X
I
,,
,
."
,
'"
---'---------L-
Y
,
,, t
II-
Zr
Y
Mapping cylinder
We use [x,t] to denote the point of Zr corresponding to (x,t) E X X I under
the identification map and [y] to denote the point of Zr corresponding
to y E Y (thus [x,I] = [f(x)] for x E X). There is an imbedding i: X ~ Zr
with i(x) = [x,O] and an imbedding i: Y ~ Zr with i(Y) = [y]. X and Yare
regarded as subspaces of Zr by means of these imbeddings. A retraction
r: Zr ~ Y is defined by r[x,t]
[f(x)] for x E X and tEl and r[y]
[y] for
y E Y.
=
I2
THEOREM
=
Given a map f: X
~
X
Y, there is a commutative diagram
i
--?
Zr
t\ Ir
Y
such that (a) I z, ::::: ir rei Y (b) i is a cofibration
PROOF
By definition, ri = f, and the triangle is commutative.
(a) A homotopy F: Zr X I ~ Zr is defined by
F([x,t], t') = [x, (1 - t')t
F([y],t') = [y]
Then F: I z , ::::: ir rei Y.
+ t']
x E X; t, t' E I
y E Y, t' E I
(F is continuous because '4 x I has the topology coinduced by the maps X x
I x 1-+ Zf X I sending (x, t, t/) to ([x, t], t/) and Y x 1-+ Zf X I sending (y, t/) to
([y], t/).)
SEc.5
H
33
SPACES
(b) Let g: Z, -') Wand G: X X I -') W be such that g([x,O])
for x E X. If H: Z, X I -') W is defined by the equations
H([y],t') = g[y]
,
H([x,t], t)
=
= G(x,O)
yEY,t'EI
(g[x, (2t - t')/(2 - t')]
G(x, (t' _ 2t)/(1 - t))
0 ~ t' ~ 2t ~ 2, x E X
0 ~ 2t ~ f ~ 1, x E X
then H([x,t], 0) = g[x,t] and H([y],O) = g[y], and
HI X X 1=
G.
•
It follows that the map i: X C Z, is a cofibration equivalent in the
homotopy category to the map f: X -') Y. The mapping cylinder can be used
to prove the following amusing result.
13 THEOREM Two spaces X and Y have the same homotopy type if and
only if they can be imbedded as weak deformation retracts of the same
space Z.
PROOF
If X and Y can be imbedded as weak deformation retracts of the
same space Z, then X and Y each have the same homotopy type as Z. Therefore X and Y have the same homotopy type.
Conversely, if f: X -') Y is a homotopy equivahmce, it follows from
theorem 12 that if Z, is the mapping cylinder of f, then the composite
X --4 z, -4 Y is a homotopy equivalence. Because r is a homotopy equivalence, this implies that i is a homotopy equivalence. By theorem 12a, i: Y -') Z,
is a homotopy equivalence. Therefore X and Yare imbedded as weak deformation retracts in Z,. •
All the foregoing concepts can also be considered for pairs. For example,
a pair (X',A') C (X,A) is a strong deformation retract if there is a map
F: (X,A) X I -') (X,A) such that F(x,O)
x for x EX, F(X X 1) ex',
F(A X 1) C A', and F(x',t) = x' for x' E X' and tEl. The mapping cylinder
of a map f: (X,A) -') (Y,B), where A is closed in X, is the pair (Z,,,Z'2)' where
Z" is the mapping cylinder of the map II: X -') Y defined by f and Z'2 is the
mapping cylinder of the map 12: A -') B defined by f. A map f: (X',A') -')
(X,A) is a cofibration if, given maps g: (X,A) -') (Y,B) and G: (X' ,A') X 1-')
(Y,B) [where (Y,B) is arbitrary] such that G(x',O)
gf(x') for x' E X', there
exists a map F: (X,A) X I -') (Y,B) such that F(x,O)
g(x) for x E X and
G(x',t) = F(f(x'), t) for x' E X' and t E 1. All the results remain valid when
suitably formulated for pairs.
=
=
:;
H
=
SPACES
In some cases it is possible to introduce a natural group structure in the set
of homotopy classes of maps from one space (or pair) to another. In this
section we consider spaces P such that [X;P] admits a group structure for all
X. It is not surprising that there is a close relation between natural group
structures on [X;P] for all X and "grouplike" structures on P.
34
HOMOTOPY AND THE FUNDAMENTAL CROUP
CHAP.
1
We shall work in the homotopy category of pointed topological spaces,
although much of what we do is also valid in the homotopy category of
topological spaces. If X and Yare pointed topological spaces, [X; Y] will denote the set of base-paint-preserving homotopy classes of continuous maps
X ~ Y (with all homotopies understood to be relative to the base point).
Thus [X; Y] is the set of morphisms from X to Y in the homotopy category of
pointed topological spaces.
One method of obtaining a group structure on [X;P] is to start with a
group structure on P. Thus, let P be a topological group with identity element
as base point. There is a law of composition in the set of all base-pointpreserving continuous maps from X to P defined by pointwise multiplication
of functions. That is, if gl, g2: X ~ P, then glg2: X ~ P is defined by
glg2(X) = gl(X)g2(X), where the right-hand side is the group product in P. With
this law of composition, the set of base-paint-preserving continuous maps
from X to P is a group (which is abelian if P is abelian). The law of composition carries over to give an operation on homotopy classes such that [gl][g2] =
[glg2], and we have the follOWing theorem.
I
THEOREM
If P is a topological group, 1TP is a contravariant functor from
the homotopy category of pointed topological spaces to the category of groups
and homomorphisms. •
We give two examples.
2
51 is an abelian topological group (the multiplicative group of complex
numbers of norm 1). Therefore [X;5 1 ] is an abelian group, and if f: X ~ Y,
then f#: [Y;5 1 ] ~ [X;5 1 ] is a homomorphism.
53 is a topological group (the multiplicative group of quatemions of
3
norm 1). Therefore [X;5 3 ] is a group, and if f: X ~ Y, then f#: [Y;5 3 ] ~ [X;5 3 ]
is a homomorphism.
This group structure on [X;P] was deduced from a group structure on
the set of base-paint-preserving continuous maps from X to P. There are situations in which [X;P] admits a natural group structure, but the set of basepoint-preserving continuous maps from X to P has no group structure. For
example, if P is a pointed space having the same homotopy type as some
topological group P', then 1TP is naturally equivalent to 1TP '. Therefore 1TP can
be regarded as a functor to the category of groups. The following definitions
will be used to describe the additional structure needed on a pointed space P
in order that 1T P take values in the category of groups and homomorphisms.
If f: X ~ Y and g: X ~ Z, we define
(f,g): X
~
Y X Z
to be the map (f,g)(x) = (f(x),g(x)) for x E X.
An H space consists of a pointed topological space P together with a continuous multiplication
p,: P X
P~
P
SEC.
5
H
35
SPACES
for which the (unique) constant map c: P ---7 P is a homotopy identity, that is,
each composite
P ~ PX P
!:"
P
and
P ~ PX P ~ P
is homotopic to I p . The multiplication p, is said to be homotopy associative if
the square
PXPxP~ PxP
1
x 111
PXP
is homotopy commutative, that is, p, (p, X 1) ~ p, (l X p,). A continuous
function qy: P ---7 P is called a homotopy inverse for P and p, if each of
the composites
0
0
and
is homotopic to c: P ---7 P.
A homotopy-associative H space with a homotopy inverse satisfies the
group axioms up to homotopy. Such a pointed space is called an H group.
Clearly, any topological group is an H group.
A multiplication p, in an H space is said to be homotopy abelian if the
triangle
P
where T(Pl,P2) = (P2,Pl), is homotopy commutative. An H group with
homotopy-abelian multiplication is called an abelian H group.
If P and P' are H spaces with multiplications p, and p,', respectively, a
continuous map 0': P ---7 P' is called a homomorphism if the square
is homotopy commutative.
4
THEOREM
A pointed space having the same homotopy type as an
H space (or an H group) is itself an H space (or H group) in such a way that
the homotopy equivalence is a homomorphism.
36
HOMOTOPY AND THE FUNDAMENTAL CROUP
CHAP.
1
PROOF
Let f: P ~ P' and g: P' ~ P be homotopy inverses and let P be an
H space with multiplication f.L: P X P ~ P. If /L': P' X P' ~ P' is defined to
be the composite
P'XP'~PXP~P~P'
then /L' is a continuous multiplication in P' and the composite P' ~
P' X P' -4 P' equals the composite P' -!4 P ~ P X P J4 P -4 P', which
is homotopic to the composite P' -4 P -4 P'. Because fg ~ I p , the map
/L' 0 (I,c') is homotopic to I p . Similarly, the map /L' 0 (c',I) is homotopic to
I p . Therefore P> is an H space. Because the square
P'XP'£P'
gXgl
19
PxP-4P
is homotopy commutative, g is a homomorphism (and so is f). If /L is homotopy
associative or homotopy abelian, so is f.L', and if cp: P ~ P is a homotopy
inverse for P, then fcpg: P' ~ P' is a homotopy inverse for P'. Given an H space P, for any pointed space X there is a law of composi[/L (gl,g2)]' If P is an H group, [X;P]
tion in [X;P] defined by [gl][g2]
becomes a group with this law of composition, and if f: X ~ Y, then
f#: [Y,P] ~ [X;P] is a homomorphism. Therefore we have the following
theorem.
=
0
If P is an H group, TT P is a contravariant functor from the
homotopy category of pointed topological spaces with values in the category
of groups and homomorphisms. If P is an abelian H group, this functor takes
values in the category of abelian groups. S
THEOREM
It is interesting that the following converse of theorem 5 is also valid.
If P is a pointed space such that TTP takes values in the category of groups, then P is an H group (abelian if TTP takes values in the
category of abelian groups). Furthermore, for any pointed space X, the group
structure on 7T P(X) is the same as that given by theorem 5.
6
THEOREM
P and P2: P X P ~ P be the projections, and let
= [Pl] * [P2], where «- is the law
of composition in the group [P X P; PJ. For any maps f, g: X ~ P,
(j,g)#: [P X P; P] ~ [X;P] is a homomorphism and
PROOF
Let Pl: P X P
~
/L: P X P ~ P be a map such that [/L]
[/L
0
(j,g)]
= (j,g)#[/L] = (j,g)#([Pl] * [P2])
= (j,g)#[Pl] * (j,g)#[P2] = [f] * [g]
This shows that the multiplication in [X;P] is induced by the multiplication
map /L.
Let X be a one-point space. The unique map X ~ P represents the
identity element of the group [X;P]. Because the unique map P ~ X induces
SEC.
5 H
37
SPACES
a homomorphism [X;P] --0> [P;P], it follows that the composite P --0> X --0> P,
which is the constant map c: P --0> P, represents the identity element of [P;P].
It follows that f.L (Ip,c) c::-:: Ip and f.L (c,lp) c::-:: I p. Therefore P is an H space.
To prove that f.L is homotopy associative, let ql, q2, q3: P X P X P --0> P
be the projections. Then
0
[f.L
0
0
(1 X f.L)] = (1 X f.L)#[f.L] = (1 X f.L)#[pl] * (1 X f.L)#[p2]
= [ql] * [f.L(q2,q3)] = [ql] * ([q2] * [q3])
Similarly,
Because [P X P X P; P] has an associative multiplication, f.L (1 X f.L) ~
f.L (f.L XI).
To show that P has a homotopy inverse, let <:p: P --0> P be such that
[Ip] * [<:p] = [c]; then f.L(Ip,<:p) c::-:: c. Also, [<:p] * [Ip] = [c], and so f.L(<:p,Ip) c::-:: c.
Therefore <:p is a homotopy inverse for P.
This proves that P is an H group and that the multiplication in 'rTP is induced from that on P. If [P X P; P] is an abelian group, a similar argument
shows that P is an abelian H group. 0
0
The following complement to theorems 5 and 6 is easily established by
similar methods.
7
THEOREM
Let a: P --0> P' be a map between H groups. Then a# is
a natural transformation from 'rT P to 'rT P ' in the category of groups if and only
if a is a homomorphism. -
We describe a particularly useful example of an H group. Let Y be
a pointed topological space with base point yo. The loop space of Y (based at
yo), denoted by SlY [or by Sl(Y,yo)], is defined to be the space of continuous
functions w: (I,i) --0> (Y,yo) topologized by the compact-open topology. SlY is
regarded as a pointed space with base point Wo equal to the constant map of
I to yo. There is a map
defined by
,
{W(2t)
f.L(w,w)(t) = w'(2t _ 1)
To prove that f.L is continuous, let E: SlY X I --0> Y be the evaluation map. By
theorem 2.8 in the Introduction, it suffices to show that the composite
SlY X SlY X I
/L x 1
~
SlY X I
E
--0>
Y
is continuous. The formula which defines f.L shows that this composite is continuous on each of the closed sets SlY X SlY X [0,1;2] and SlY X SlY X [1;2,1].
We construct a number of homotopies to show that SlY is an H group.
38
HOMOTOPY AND THE FUNDAMENTAL CROUP
CHAP.
1
Similar formulas will be used again in Sec. 1. 7 to define homotopies of (nonclosed) paths in a topological space.
To prove that the map w ---> /-t(w,wo) is homotopic to the identity map of
[l Y, define F: [l Y X I ---> [l Y by
F( w,t)(t') =
w(~)
t + 1
O<t,<!22
yo
!22<t'<l
2
-
-
-
2
This formula shows that E(F Xl): ([l Y X I) X 1---> Y is continuous; therefore F is continuous and is a homotopy from the map w ---> /-t( w,wo) to lilY'
Similarly, the map w ---> /-t(wo,w) is homotopic to lilY' Therefore [ly is an
H space with multiplication /-to
To show that /-t is homotopy associative, define
G: [ly X [ly X [lY X 1---> [ly
by the formula
w(~)
t + 1
E(G X l)(w,w',wl/,t,t') =
w'(4t' -
t -
O<t,<t+l
4
1)
WI/(_4_f-,--_2_-_t)
2 - t
~<t'< t+2
4
-
-
4
t+2<t'<1
4 -
Then G: /-t (/-t X lilY) ~ /-t (lilY X /-t), showing that /-t is homotopy associative.
We define a homotopy inverse qy: [ly ---> [ly by qy(w)(t) = w(l - t). Then
we define H: [lY X 1---> [ly by
0
0
yo
O<t'<~
- 2
w(2t' - t)
~<t'<l
2 - 2
w(2 - 2t' - t)
l<t'<l-~
2 2
Yo
1--<t'<1
2 -
E(H X l)(w,t,t') =
t
H is a homotopy from the map w ---> /-t( w,qy( w)) to the constant map of [l Y to
itself. Similarly, there is a homotopy from the map w ---> /-t(qy(w),w) to the constant map of [l Y. Therefore qy is a homotopy inverse for [l Y, and [l Y is an
H group.
If h: Y ---> Y' preserves base points, there is a map
[lh: [l Y ---> [l Y'
SEC.
6
39
SUSPENSION
defined by nh(w)(t) = h(w(t)). Clearly, nh is a homomorphism, and we summarize these remarks about loop spaces as follows.
8
THEOREM
The loop functor n is a covariant functor from the category
of pointed topological spaces and continuous maps to the category of H
groups and continuous homomorphisms. -
The functor n also preserves homotopies. That is, if ho, hI: Y --'.> Y' are
homotopic by a homotopy ht, then nho, nh I : ny --'.> ny' are homotopic by a
homotopy nht. which is a continuous homomorphism for each t E 1.
6
SUSPENSION
This section deals primarily with results dual to those of Sec. 1.5. We consider
pointed spaces Q such that 7TQ is a covariant functor from the homotopy category of pointed spaces to the category of groups and homomorphisms, and
this leads to the concept of H cogroup, dual to that of H group. An important
example of an H cogroup is the suspension of a pOinted space, a concept dual
to that of the loop space. The homotopy groups of a space defined in the section are examples of groups of homotopy classes of maps from suspensions to
the space.
If X and Yare pointed topological spaces, their sum in the category of
pointed topological spaces will be denoted by X v Y. If X has base point Xo
and Y has base point yo, X v Y may be regarded as the subspace X X yo U Xo X Y
of X X Y. If f: X --'.> Z and g: Y --'.> Z, we let (f,g): X v Y --'.> Z be the map defined by the characteristic property of the sum [that is, (f,g) IX = f and
(f,g) I Y = g).
An H cogroup consists of a pointed topological space Q together with a
continuous co multiplication
v:
Q --'.> Q v Q
such that the follOwing properties hold:
Existence of homotopy identity. If c: Q --'.> Q is the (unique) constant
map, each composite
Q~QvQ~Q
Q~QvQ~Q
and
is homotopic to IQ.
Homotopy associativity. The square
QvQ
llvV
QvQ
is homotopy commutative.
QvQvQ
40
HOMOTOPY AND THE FUNDAMENTAL CROUP
CHAP.
I
Existence of homotopy inverse. There exists a map 1/;: Q ~ Q such that
each composite
Q~QvQ~Q
Q~QvQ~Q
and
is homotopic to c: Q ~ Q.
If X is any pointed space and Q is an H cogroup, there is a law of
composition in [Q;X] defined by [fl][fz] = [(fdz) v] which makes [Q;X] a
group.
0
An H cogroup is said to be abelian if the triangle
Q
v/ '\
T'
QvQ~QvQ
where T(ql,qZ) = (qZ,ql) for ql, qz E Q, is homotopy commutative.
If Q and Q' are H cogroups with comultiplications v and v', respectively,
a continuous map f3: Q ~ Q' is called a homomorphism if the square
Q ~ Q vQ
III
lllv Il
Q' ~ Q'vQ'
is homotopy commutative.
The proofs of the following theorems are dual to the proofs of the
corresponding statements about H groups (see theorems 1.5.4, 1.5.5, 1.5.6,
and 1.5.7) and are omitted.
I
THEOREM
A pointed space having the same homotopy type as an
H cogroup is itself an H cogroup in such a way that the homotopy equivalence is a homomorphism. -
2
THEOREM
If Q is an H cogroup, 'TTQ is a covariant functor from the homotopy category of pointed spaces with values in the category of groups and
homomorphisms. If Q is an abelian H cogroup, this functor takes values in the
category of abelian groups. 3
THEOREM
If Q is a pointed space such that 'TTQ takes values in the category of groups, then Q is an H cogroup (abelian if'TTQ takes values in the category of abelian groups). Furthermore, the group structure on 'TTQ(X) is identical
with that determined by the H cogroup structure of Q as in theorem 2. 4
THEOREM
If f3: Q ~ Q' is a map between H cogroups, then f3# is
a natural transformation from 'TTQ' to 'TTQ in the category of groups if and only
if f3 is a homomorphism. -
We describe an example of an H cogroup dual to the loop-space example
of an H group. Let Z be a pointed topological space with base point zoo The
SEC.
6
41
SUSPENSION
suspension of Z, denoted by SZ, is defined to be the quotient space of Z X I
in which (Z X 0) U (zo X 1) U (Z X 1) has been identified to a single point.
This is sometimes called the reduced suspension in the literature, the term
"suspension" being used for the suspension in the category of spaces (no base
points). The latter is defined to be the quotient space of Z X I in which Z X 0 is
identified to one point and Z X 1 is identified to another point.
If (z,t) E Z X I, we use [z,t] to denote the corresponding point of SZ
under the quotient map Z X I ----,) SZ. Then [z,O] = [zo,t] = [z',l] for all z,
z' E Z and tEl. The point [zo,O] E SZ is also denoted by Zo, and SZ is
a pointed space with base point zoo If J: Z ----,) Z', then Sf: SZ ----,) SZ' is defined
by Sf[z,t] = [f(z), t]. Thus S is a covariant functor from the category of
pointed spaces and continuous maps. To show that it is a covariant functor
to the category of H cogroups and homomorphisms, we define a co multiplication
1J:
SZ ----,) SZ v SZ
by the formula
([
1J
]) _ {([z,2t], zo)
z,t - (zo, [z, 2t - 1])
0< t < ~
~ < t < 1
and illustrate it in the diagram (where the dotted lines are collapsed to one
point).
Zo\
5Z
5Zv 5Z
The map 1J provides SZ with the structure of an H cogroup such that if
J: Z ----,) Z', then Sf is a homomorphism. This can be verified directly or
deduced from properties of loop spaces already established. We follow the
latter course.
The functors Q and S defined from the category of pointed spaces and continuous maps to itself are examples of ad;oint functors. This means that for
pointed spaces Z and Y there is a natural equivalence
hom (SZ,Y):::::: hom (Z,QY)
where both sides are interpreted as the set of morphisms in the category of
pointed spaces and continuous maps. This equivalence results from theorem 2.8
in the Introduction, and if g: Z ----,) Q Y, the corresponding g': SZ ----,) Y is defined by g'[z,t] = g(z)(t) for z E Z and tEl. It is obvious that if h: Y ----,) Y',
then (Qh g)' = hog', and if J: Z' ----,) Z, then (g f)' = g' Sf. Therefore
the equivalence g ~ g' comes from a natural equivalence from the functor
hom (S . , .) to the functor hom (. , Q ').
0
0
0
42
HOMOTOPY AND THE FUNDAMENTAL GROUP
CHAP.
I
This natural equivalence passes to morphisms in the homotopy category
of pointed spaces. For pointed spaces a homotopy G: Z X [ ----> Y must map
Zo X [ into yo. Therefore it defines a map F: Z X [/zo X [ ----> Y. Because
S(Z X [/zo X [) can be identified with SZ X [/zo X [by the homeomorphism
[(z,t), t']
(-0'>
z E Z; t, t' E [
([z,t'], t)
it follows that homotopies F: Z X [/zo X [ ----> QY correspond bijectively to
homotopies F': SZ X [/zo X [ ----> Y. Therefore the equivalence above gives
rise to an equivalence
[SZ;Y];:::; [Z;QY]
such that if the maps g: Z ----> Q Y and g': SZ ----> Yare related by g'[ z,t] = g(z)( t),
then [g'] corresponds to [g]. Hence there is a natural equivalence from the
functor [S . ; .] to the functor [. ; Q '].
It follows from these remarks that for a fixed pointed space Z the functor
'lTsz is naturally equivalent to the composite functor 'lTz
Q. Here Q is
regarded as a covariant functor to the homotopy category of H groups and
homomorphisms. Then the composite 'lTz Q takes values in the category of
groups and homomorphisms. By theorem 3, SZ is an H cogroup, and the map
v: SZ ----> SZ v SZ defined above is the one which is the comultiplication in the
H cogroup SZ (or is homotopic to it). In similar fashion, if f: Z ----> Z',
the natural transformation (Sf)# from 'lTsz' to 'lTsz corresponds to the natural
transformation f# from the composite 'lTZ' Q to the composite 'lTz Q. Because
the latter is a natural transformation in the category of groups, so is (Sf)#,
and by theorem 4, Sf is a homomorphism of the H cogroup SZ to the
H cogroup SZ'.
These statements are summarized as follows.
0
0
0
0
5
THEOREM
The suspension functor S is a covariant functor from the
category of pointed spaces and maps to the category of H cogroups and continuous homomorphisms. •
The functor S also preserves homotopies. That is, if fo, f1: Z ----> Z' are
homotopic by a homotopy ft, then Sfo, Sf1 are homotopic by a homotopy Sft,
which is a continuous homomorphism for each t E 1.
We now show that for n ?:: 1 the sphere Sn is homeomorphic to a suspension, and thus obtain an interesting family of H cogroups. The corresponding functors are known as the homotopy group functors and are particularly
important.
6
LEMMA
For n ?:: 0, S(Sn) is homeomorphic to Sn+1.
Let po = (1,0, . . . ,0) be the base point of Sn. We regard Rn+1 as
imbedded in Rn+2 as the set of points in Rn+2 whose (n + 2)nd coordinate
is O. Then Sn is imbedded as an equator in Sn+1.
PROOF
Sn = {z E
Rn+2111zll
and En+1 is also imbedded in En+2:
= 1 and
Zn+2 = O}
SEC.
6
43
SUSPENSION
En+l
= {Z E Rn+Zlllzll ::::
1
and
Zn+Z = O}
Let H+ and H_ be the two closed hemispheres of Sn+l defined by the equator
Sn. Then
H+
= {z E Sn+llzn+z
~
O}
and
H_
= {z E Sn+llzn+2:::: O}
and Sn+l = H+ U H_ and Sn = H+ n H_. Furthermore, the projection map
Rn+Z ~ Rn+l defines projection maps p+: H+ ~ En+1 and p_: H_ ~ En+l,
which are homeomorphisms. A map f: S(Sn) ~ Sn+l is defined by
f[z,t] =
{ p_ -1(2tz + (1 - 2t)po)
p+ -1((2 _ 2t)z + (2t - l)po)
and is verified to be a homeomorphism f: S(Sn) ;:::::; Sn+l.
•
For n ~ 1 the nth homotopy group functor 'TTn is the covariant functor
on the homotopy category of pointed spaces defined by 'TTn = 'TTsn. It follows
from theorems 6 and 5 that these functors take values in the category of
groups and homomorphisms.
In the last two sections of this chapter we give another definition of 'TTl
and study it in more detail. In Chapter 7 we return to the study of the higher
homotopy groups 'TT n.
The following necessary and sufficient condition for a map Sn ~ X to
represent the trivial element of 'TTn(X) is an immediate consequence of
theorem 1.3.12.
7
n
A map a: Sn ~ X represents the trivial element of 'TTn(X) for
1 if and only if a can be continuously extended over En+1. •
THEOREM
~
Before leaving this section let us consider the interplay between two
possible group structures on the set [X; Y] for particular pointed spaces X and
Y (for example, if X is an H cogroup and Y is an H group, this set can
be given a group structure in two ways). It is a fact that under rather general
circumstances two laws of composition on hom (X, Y) in a category are equal,
and we establish this result.
8
THEOREM
Let X and Y be objects in a category and let
laws of composition in hom (X,Y) such that
* and *' be two
(a) * and *, have a common two-sided identity element
(b) * and *' are mutually distributive
Then * and *' are equal, and each is commutative and associative.
PROOF
f:
Statement (a) means there is a map fa: X ~ Y such that for any
X~ Y
f
* fa = fa * f = f = f *' fa = fa *' f
Statement (b) means that for II, fz, gl, gz: X ~ Y
(fl
* fz) *' (gl * gz)
=
(II *' gl) * (fz *' gz)
44
HOMOTOPY AND THE FUNDAMENTAL GROUP
CHAP.
1
If f, g: X ~ Y, then
= (f *, fo) * (fo *, g) = (f * fo) *, (fo * g) = f *, g
and
g * f = (fo *, g) * (f *, fo) = (fo * f) *, (g * fo) = f *, g
Therefore f * g = f *, g = g * f. For associativity we have
(f * g) * h = (f * g) *, (fo * h) = (f *, fo) * (g *, h) = f * (g * h)
f *g
•
9
COROLLARY
If P is an H space and Q is any H cogroup, then [Q;P] is
an abelian group and the group structure is defined by the multiplication map
in P.
This follows on observing that the two laws of composition defined
in [Q;P] by using the comultiplication in Q or the multiplication in P satisfy
the hypotheses of theorem 8. •
PROOF
Note that if P is just an H space (but not an H group), the law of composition in [X;P] defined by the multiplication in P is in general not a group
structure on [X;P]. However, if X is an H cogroup (for instance, a suspension),
it follows from corollary 9 that this law of composition is a group structure on
[X;P], and in this case the resulting group structure on [X;P] is the same no
matter what multiplication map P is given (so long as it is an H space).
10 COROLLARY If P is an H space, '7T n (P) is abelian for all n ~ 1 and the
group structure in '7Tn(P) is defined by the multiplication map in P. •
For a double suspension S(SZ) whose points are represented in the form
[[z,t],t'], with z E Z and t, t' E I, there are two laws of composition in the
set of maps S(SZ) ~ X. Iff, g: S(SZ) ~ X, we define
{f[[z,2t], t']
(f * g)[[z,t,]']
t = g[[z, 2t _ 1], t']
o ~ t ~·Ih
lh~t~l
and
(f
*,
g)[[z,t], t']
2t']
= {f[[z,t],
g[[z,t], 2t' _
1]
o~t'~1h
1h~t'~1
The corresponding operations in [S(SZ);X] satisfy the hypotheses of theorem 8.
Therefore they are equal, and [S(SZ);X] is an abelian group. In particular, we
have the following corollary.
II
COROLLARY
groups.
For n
~
2,
'7T n
is a functor to the category of abelian
•
A similar argument can be applied to the loop space gp, where P is itself
an H space. There is a multiplication map in gp, because it is a loop space,
and another multiplication obtained from the original multiplication in P. The
corresponding laws of composition in [X;gP] satisfy theorem 8. Therefore it
follows that if P is an H space, '7Tfl.P is a contravariant functor to the category
of abelian groups.
SEC.
7
7
45
THE FUNDAMENTAL GROUPOID
THE FUNDAMENTAL GROUPOID
This section concerns paths in a topological space. This leads to another
description (in Sec. 1.8) of the first homotopy group 7T1. introduced in Sec. 1.6.
We shall have occasion to define a number of homotopies between paths in a
topological space. These homotopies are generalizations (to non closed paths)
of those used in Sec. 1.5 to prove that a loop space is an H group and are defined by the same formulas (except that the t and t' arguments are interchanged).
It is clear that this repetition of formulas could have been eliminated by
proving a suitably general result about path spaces instead of merely considering loop spaces in Sec. 1.5. However, each usage has its own value, and
it is hoped that the repetition may be an aid to understanding the formulas.
A groupoid is a small category in which every morphism is an equivalence. We list without proof a number of facts about groupoids which are
easy consequences of general properties of categories.
I
The relation between obiects A and B of a groupoid defined by the condition hom (A,B) =1= 0 is an equivalence relation. •
The equivalence classes of this equivalence relation are called the components of the groupoid. The groupoid is said to be connected if it has just
one component.
2
For any obiect A of a groupoid, the law of composition which sends
f, g: A ~ A to fog: A ~ A is a group operation in hom (A,A). •
3
There is a covariant functor from any groupoid to the category of groups
and homomorphisms which assigns to an obiect A the group hom (A,A) and
to a morphism f: A ~ B the homomorphism
h f : hom (A,A)
defined by h f (g)
= fog
0
f- 1 for g: A
~
~
hom (B,B)
A.
•
Because any morphism f: A ~ B in a groupoid is an equivalence,
hf : hom (A,A) ~ hom (B,B) is an isomorphism. The following statement
describes the collection of isomorphisms obtained by taking all possible morphisms f: A ~ B.
4
If A and B are in the same component of a groupoid, the collection
of isomorphisms {h f If: A ~ B} is a coniugacy class of isomorphisms
hom (A,A) ~ hom (B,B). •
it Let F be a covariant functor from one groupoid 8 to another 8'. Then F
maps each component of 8 into some component of 8', and there is a natural
transformation F* (A) from the covariant functor home (A,A) on 8 to the covariant functor home' (F(A), F(A)) on 8 defined by
F* (A)(f)
= F(f):
F(A) ~ F(A)
f: A ~ A
•
46
HOMOTOPY AND THE FUNDAMENTAL CROUP
CHAP.
I
With these general remarks about groupoids out of the way, we proceed
to define the fundamental groupoid. A path w in a topological space is defined
to be a continuous map w; I --? X [note that the path is the map, not just the
image set w(I)]. The origin of the path is the point w(O), and the end of the
path is the point w(l). We also say that w is a path from w(O) to w(l). A closed
path, or loop, at Xo E X is a path w such that w(O) = Xo = w(l). If wand w'
are paths in X such that end w = orig w', there is a product path w * w' in X
defined by the formula
(w
* w)(t) = {W(2t)
w' (2t
I
_ 1)
=
=
Then orig (w * w')
orig wand end (w * w')
end w'.
We should like to form a category whose objects are the points of
X, whose morphisms from Xl to Xo are the paths from Xo to Xl, and with the
composite defined to be the product path. With these definitions, neither
axiom of a category is satisfied. That is, there need not be an identity morphism for each point, and it is generally not true that the associative law for
product paths holds [that is, w * (w' * w") is usually different from (w * w') * w"].
A category can be obtained, however, if the morphisms are defined not to be
the paths themselves, but instead, homotopy classes of paths.
Two paths wand w' in X are briefly said to be homotopic, denoted
by w ~ w', if they are homotopic relative to i. Thus a necessary condition
that w ~ w' is that w(O) = W'(O) and w(l) = w' (l). For any xo, Xl E X the
relation w ~ w' is an equivalence relation in the set of paths from Xo to Xl.
The resulting equivalence classes are called path classes, and if w is a path in
X, the path class containing it is denoted by [w]. Since two paths in the same
path class have the same origin and the same end, we can speak of the origin
and the end of a path class.
We shall construct a category whose objects are the points of X and
whose morphisms from Xl to Xo are the path classes with Xo as origin and Xl
as end. The following lemma shows that the path class of the product of two
paths depends only on the path classes of the factors, and it will be used to
define the composite in the category.
6
LEMMA
Let [w] and [w'] be path classes in X with end [w] = orig [w'].
There is a well-defined path class [w] * [w'] = [w * w'] with orig ([w] * [w'])
orig [w] and end ([w] * [w']) = end [w'].
=
PROOF
To prove that w ~ Wl and w' ~wi imply w * w' ~ Wl * wi, let
F; I X I --? X be a homotopy relative to j from w to Wl and let F'; I X I --? X
be a homotopy relative to j from w' to wi. A homotopy F * F'; I X I --? X is
defined by the formula
(F
*
F/)( t')
t,
=
{F(2t,tl)
F(2t _ 1, t')
SEC.
7
47
THE FUNDAMENTAL GROUPOID
and illustrated in the diagram
WI
Wi
~
~
F*F'
Then F
* F: w * w' c::o:: WI * WI rel i. •
7
THEOREM
For each topological space X there is a category 0'(X) whose
obiects are the points of X, whose morphisms from Xl to Xo are the path
classes with Xo as origin and Xl as end, and whose composite is the product of
path classes.
PROOF
To prove the existence of identity morphisms, let ex: I --3> X be the
constant map of I to X for any X E X. We show that [ex] = Ix. If W is a path
with w(l) = x, we must prove that w * ex c::o:: w (with a similar property for
paths with origin at x). Such a homotopy F: I X I --3> X is defined by
the formula
F(t,t') =
W(r ~ 1)
O<t<t'+l
2
t'+l<t<l
X
2
-
-
and pictured in the diagram
B
w
w
F
A similar homotopy shows that if w(O) = x, then ex * w c::o:: w.
To prove the associativity of the composite of morphisms, let w, w', and
w" be paths such that end w = orig w' and end w' = orig w". We must prove
that (w * w') * w" c::o:: w * (w' * w"). Such a homotopy G: I X I --3> X is defined
by the formula
G(t,t) =
w(r: J
O<t<t'+l
4
w'(4t - t' - 1)
r+1<t<t'+2
4
4
4t - 2- t')
w" ( -
r+2<t<1
4
-
2 - t'
48
HOMOTOPY AND THE FUNDAMENTAL CROUP
CHAP.
1
and pictured in the diagram
tttJ
w
w
w'
w'
wI!
wI!
G
•
The category 6Jl(X) is called the category of path classes of X, or the
fundamental groupoid of X, the latter because of the following theorem.
6Jl(X) is a groupoid.
Given a path w in X, let w~l: I ~ X be the path defined by w~l(t) =
w(1 - t). To prove that [w~ll = [wl~l in 6Jl(X), we must show that w * w~l c:::::: Ew(O)
[and also that W~l * w c:::::: Ew(l), which follows, however, from the first homotopy, because w = (W~l)~ll. Such a homotopy H: I X I ~ X is defined by
the formula
8
THEOREM
PROOF
H(t,t =
l
-< {2
w(O)
0<
-
w(2t - t')
{< t
w(2 - 2t - tl)
1<t<1-{
2 2
w(O)
1--<t<1
2 -
)
t
2 -
<1
- 2
tl
and pictured in the diagram
H
•
This completes the construction of the fundamental groupoid. The components of the fundamental groupoid are called path components of X. It is
clear that Xo, and Xl are in the same path component of X if and only if there
is a path w in X from Xo to Xl. X is said to be path connected if its fundamental
groupoid is connected. The following is an alternate characterization of the
path components.
9
THEOREM
subspaces of X.
The path components of X are the maximal path-connected
SEC.
7
49
THE FUNDAMENTAL GROUPOID
Let A be a path component of X and let w be a path in X such that
w(o) EA. We show that w is a path in A. For each t E I define a path
Wt: I ----? X by Wt(t') = w(tt') for t' E I. Then Wt is a path in X from w(O)
to w(t). Therefore w(t) is in the same path component of X as xo, namely A.
Since this is so for every t E I, w is a path in A.
A is path connected because if Xo, Xl E A there is a path w in X from
Xo to Xl. By the above result, w is a path in A. Therefore any two points of A
can be joined by a path in A, and A is path connected. Since any path in X
that starts in A stays in A, A is a maximal path-connected subset of X. •
PROOF
I0
LEMMA
A path-connected space is connected.
PROOF
If w is a path in X, then w(I), being a continuous image of the connected space I, is connected. Therefore w(O) and w(l) lie in the same component of X. If X is path connected, any two points of X lie in the same
component, and X is connected. •
The converse of lemma 10 is false, as is shown by the following example.
II
EXAMPLE
X
Let X be the subspace of R2 defined by
= {(x,y)
E R2
I X > 0, y = sin 1X
or
X
= 0,
-1
~ Y~
I}
Then X is connected, but not path connected.
Given a map f: X ----? Y, there is a covariant functor f# from 0l(X) to 0l(Y)
which sends an object X of 0l(X) to the object f(x) of 0l( Y) and the morphism
[w] of 0l(X) to the morphism f#[w] = [f w] of 0l(Y). The functorial properties of f# are easily verified. From the first part of statement 5, or by direct
verification, it follows that f maps each path component of X into some path
component of Y. Therefore there is a covariant functor 7To from the category
of topological spaces and maps to the category of sets and functions such that
7To(X) equals the set of path components of X, and
0
7To(f)
= f#:
7To(X)
----?
7To(Y)
maps the path component of x in X to the path component of f(x) in Y. If
F: fo c:::o fl' then for any x E X there is a path Wx in Y from fo(x) to /1(x) defined by wx(t) = F(x,t) for t E 1. Therefore fo(x) and fl(X) belong to the same
path component of Y, and fo# = /1#. It follows that 7To can be regarded as a
covariant functor from the homotopy category to the category of sets and
functions. This functor characterizes the functor TTX for a contractible space X
as follows.
12 THEOREM If X is a contractible space, then
equivalent functors on the homotopy category.
TTX
and
7To
are naturally
If X and X' have the same homotopy type, then TTX and TTX are
naturally equivalent. It follows from corollary 1.3.11 that if P is a one-point
space, TTX is naturally equivalent to 7Tp. It therefore suffices to prove that 7Tp
PROOF
50
HOMOTOPY AND THE FUNDAMENTAL GROUP
CHAP.
1
is naturally equivalent to 'ITo. 'lTo(P) consists of the single path component P,
and a natural transformation
1/;:
'lTp~
'ITo
is defined by 1/;[f] = f#(P) for [f] E [P;X]. Because XP is in, one-to-one correspondence with X in such a way that homotopies P X I ~ X correspond to
paths I ~ X, it follows that 1/; is a natural equivalence. •
The functor 'ITo is closely related to the functor Ho of example 1.2.6. In
fact, for spaces X whose components and path components coincide, Ho is
the composite of 'ITo with the covariant functor which assigns to every set the
free abelian group generated by that set. In particular, 'ITo could have been used
to obtain the results of Sec. 1.2 that were obtained by using Ho.
8
THE FUNDAMENTAL GROUP
By choosing a fixed point xo E X and considering the path classes in X with
xo as origin and end, a group called the fundamental group is obtained. We
show now that this group is naturally equivalent to the first homotopy group
'lTl, defined in Sec. 1.6. The section closes with a calculation of the fundamental
group of the circle.
Let X be a topological space and let xo E X. The fundamental group of
X based at Xo, denoted by 'IT(X,xo), is defined to be the group of path classes
with Xo as origin and end. It follows from theorem 1.7.8 and statement 1.7.2
that this is a group, and iff: (X,xo) ~ (Y,yo), thenf#is a homomorphism from
'IT(X,xo) to 7T(Y,yo). If, f, f': (X,xo) ~ (Y,yo) are homotopic, then
f#
= f'#:
'IT(X,xo)
~
'IT(Y,yo).
Therefore, we have the following theorem.
I
THEOREM
There is a covariant functor from the homotopy category of
pointed spaces to the category of groups which assigns to a pointed space its
fundamental group and to a map f the homomorphism f#-.
We show that the fundamental group functor 'IT is naturally equivalent to
'IT 1, defined in Sec. 1.6. Let ,\: I ~ S(50) be defined by '\(t) = [-l,t], where
50 consists of the two points -1 and 1 and 1 is its basepoint. Then ,\ induces
a bijection ,\# between the homotopy classes of maps (S(5 0), 1) ~ (X,xo) and
the path classes of closed paths in X at Xo defined by
'\#[g]
= [g'\]
g: (5(5 0 ), 1) ~ (X,xo)
From the definition of the law of composition in [5(50);X] and in 'IT(X,xo), ,\#
is seen to be a group isomorphism. Given a map f: (X,xo) ~ (Y,yo), ,\# commutes with f#. By lemma 1.6.6, S(50) is homeomorphic to 51.
SEC.
2
8
51
THE FUNDAMENTAL GROUP
THEOREM
group functor
7TI
The map 11.# is a natural equivalence of the first homotopy
with the fundamental group functor 7T. -
It will sometimes be convenient to regard the elements of 7T(X,XO) as
homotopy classes of maps (Sl,po) ~ (X,xo), rather than as path classes.
Because any closed path at Xo (and any homotopy between such paths)
must lie in the path component A of X containing Xo, it follows that 7T(A,xo) ::::::
7T(X,XO). Hence the fundamental group can give information only about the
path component of X containing Xo. From general groupoid considerations
(see statements 1.7.3 and 1.7.4), if [w] is a path class in X from Xo to Xl, then
h [wl is an isomorphism from 7T(X,XI) to 7T(X,XO).
3
THEOREM
The fundamental groups of a path-connected space based at
different points are isomorphic by an isomorphism determined up to
coniugacy. -
Even though the fundamental groups based at different points of a pathconnected space are isomorphic, we cannot identify them, because the isomorphism between them is not unique. If the fundamental group at some
point (and hence all points) is abelian, the isomorphism is unique. In general,
the fundamental group need not be abelian; however, the following consequence of theorem 2 and corollary 1.6.10 is a general result about the commutativity of fundamental groups.
4
THEOREM
The fundamental group of a path-connected H space is
abelian, and if wand w' are closed paths at the base point, then
[w]
* [w']
= [,u
0
(w,w')]
where ,u is the multiplication map in the H space.
-
A space X is said to be n-connected for n ;?: 0 if every continuous map
Sk ~ X for k ~ n has a continuous extension over Ek+l. A I-connected
space is also said to be simply connected. Note that if 0 ~ m ~ n, an
n-connected space is m-connected. It follows from theorem 1.6.7 that a space
X is n-connected if and only if it is path connected and 7Tk(X,X) is trivial for
every base point X E X and I ~ k ~ n. From corollary 1.3.13 we have the
following result.
f:
S
LEMMA
A contractible space is n-connected for every n ;?: O.
-
Note that a space is O-connected if and only if it is path connected, and
a space is simply connected if and only if it is path connected, and 7T(X,XO) = 0
for some (and hence all) points Xo E X.
From theorem I we know that two pointed spaces having the same
homotopy type as pointed spaces have isomorphic fundamental groups. To
prove a similar result for two path-connected spaces which have the same
52
HOMOTOPY AND THE FUNDAMENTAL CROUP
CHAP.
1
homotopy type as spaces (no base-point condition) we need some preliminary
results.
LEMMA Let h: I X I ~ X and let ao, a1, /30, and /31 be the paths in X
6
defined by restricting h to the edges of I X I [that is, ai(t) = h(i,t) and
/3i(t) = h(t,i)]' Then (ao * /31) * (a1- 1 * /30- 1 ) is a closed path in X at h(O,O)
which represents the trivial element of 7T(X, h(O,O)).
PROOF Let ao, ai, /30, and /3i be the paths in I X I defined by ai(t) = (i,t)
and fJi( t) = (t,i). Then (ao * /31) * (ai- 1 * /30- 1 ) is a closed path in I X I at
(0,0) and h maps this closed path into (ao * /31) * (a1- 1 * /30- 1 ). Since I X I is
a convex subset of R2, it is contractible, and by lemma 5, it is simply connected. Therefore
and
(ao
* /31) * (a- 1 * /30- 1) = h
c:o::
h
* /31) * (ai- 1 * /3'0- 1))
0
((ao
0
1'(0,0) =
I'h(O,O)
•
7
THEOREM Let f: (X,xo) ~ (Y,yo) and g: (X,xo) ~ (Y,Y1) be homotopic as
maps of X to Y. Then there is a path w in Y from yo to Y1 such that
= h[wJ
f#
0
~:
7T(X,XO) ~ 7T(Y,yo)
Let F: X X I ~ Y be a homotopy from f to g and let w: I ~ Y be
defined by w(t) = F(xo,t). Then w is a path in Y from yo to Y1. If W' is any
closed path in X at Xo, let h: I X I ~ Y be defined by h(t,t') = F(w'(t), t').
Then h(O,t') = F(xo,t') = w(t'), h(t,l) = gw'(t), h(l,t') = w(t'), and h(t,O) =
fW'(t). By lemma 6 we have
PROOF
* gw') * (w- 1 * (fW')-l) c:o:: l'yO
This implies [wJ ~[w'J [WJ-1 = f#[w'J, or (h[wJ g#)[w'J = f#[w'J.
is an arbitrary element of 7T(X,XO), h[wJ ~ = f#. •
(w
0
0
0
Since [w']
0
8
THEOREM Two path-connected spaces with the same homotopy type
have isomorphic fundamental groups.
PROOF Let f: X ~ Y be a homotopy equivalence with homotopy inverse
g: Y ~ X. Let Xo E X and set yo = f(xo), Xl = g(yo), and Y1 = f(X1). Let
fa: (X,xo) ~ (Y,yo) and II: (X,X1) ~ (Y,Y1) be maps defined by f (that is, fo and
II are both equal to fbut are regarded as maps of pairs), and let g': (Y,yo) ~
(X,Xl) be defined by g. Then g' fo: (X,xo) ~ (X,X1) is homotopic, as a map of
X to X, to l(x,xo): (X,xo) C (X,xo), and II g': (Y,yo) ~ (Y,Y1) is homotopic, as
a map of Y to Y, to l(y,yo): (Y,yo) C (Y,yo). It follows from theorem 7
that there are paths w in X from Xl to Xo and w' in Y from Y1 to yo such that
0
0
and
SEc.8
53
THE FUNDAMENTAL GROUP
Therefore we have a commutative diagram
7T(X,XO) ~ 7T(X,X1)
7T(Y,yO) ~ 7T(Y,Y1)
g# is an epimorphism because h[wl is, and it is a monomorphism because
is. Therefore g# is an isomorphism. •
h[w'l
We close with an example of a space with a nontrivial fundamental
group. For this purpose we compute 7T(5 1 ,po) following a method used
by Tucker 1, where 51 = {e i9 } and po = 1.
The exponential map ex: R ~ 51 is defined by ex(t) = eZ'7Tit. Then ex is
continuous, ex(tl + tz) = ex(t1) ex(tz) (where the right-hand side is multiplication of complex numbers), and ex(tt) = ex(tz) if and only if t1 - tz is an
integer. It follows that ex I( -lh, lh) is a homeomorphism of the open interval
(-lh,lh) onto 51 - {e'7Ti}. We let
19: 51 - {e'7Ti}
~
(-lh,lh)
be the inverse of ex I (-lh, lh).
A subset X C Rn will be called starlike from a pOint Xo E X if, whenever
x E X, the closed line segment [xo,x] from Xo to x lies in X.
LEMMA
Let X be compact and starlike from Xo E X. Given any continuous map f: X ~ 51 and any to E R such that ex(to) = f(xo), there exists a
continuous map 1': X ~ R such that f'(xo) = to and ex(f'(x)) = f(x) for all
x E X.
9
Clearly, we can translate X so that it is starlike from the origin; hence
there is no loss of generality in assuming Xo = O. Since X is compact, f is uniformly continuous and there exists e
0 such thl:lt if Ilx - x'il
e, then
I f(x) - f(x') II 2 [that is, f(x) and f(x') are not antipodes in SI]. Since X is
bounded, there exists a positive integer n such that Ilxll/n
e for all x E X.
Then for each 0 :::;: i
n and all x E X
PROOF
>
<
<
<
<
II (f +n l)x - ~ II
=
II~II < e
and so
It follows that the quotient f((f + l)x/n)/f(fx/n) is a point of 51 - {e'7Ti}. Let
~ SI - {e'7Ti} for 0:::;: i
n be the map defined by g;(x) =
gj: X
<
1 See A. W. Tucker, Some topological properties of disk and sphere, Proceedings of the
Canadian Mathematical Congress, 1945, pp. 285-309.
54
f((j
HOMOTOPY AND THE FUNDAMENTAL GROUP
+
CHAP.
1
l)x/n)/f(ix/n). Then, for all x E X, we see that
f(x)
We define 1': X
--;>
I'(x)
= f(O)gO(X)gl(X)
... gn-1(X)
R by
= to + 19(9o(x)) + 19(9l(X)) + ... + 19(9n-1(X))
I' is the sum of n + 1 continuous functions from X to R, so it is continuous.
Clearly, 1'(0) = to and ex(f'(x)) = f(x). •
10
Let X be a connected space and let 1', g': X
LEMMA
0
PROOF
--;>
R be maps such
= ex g' and I'(xo) = g'(xo) for some Xo E X. Then I' = g'.
Let h = I' - g': X
R. Since ex I' = ex g', ex h is the con-
that ex I'
0
--;>
0
0
0
stant map of X to po. Therefore h is a continuous map of X to R, taking only
integral values. Because X is connected, h is constant, and since h(xo) = 0,
h(x) = 0 for all x E X. •
Let 0': I --;> 51 be a closed path at po. Because I is starlike from 0 and
0'(0) = po = ex(O), it follows from lemma 9 that there exists 0": I --;> R such
that 0"(0) = 0 and ex 0" = 0'. By lemma 10, 0" is uniquely characterized by
these properties. Because ex(O"(l)) = po, it follows that 0"(1) is an integer. We
define the degree of 0' by deg 0' = 0"(1).
0
11
LEMMA
deg
0'
= deg
Let 0' and
/3.
/3 be homotopic closed paths in
51 at po. Then
PROOF
Let F: I X I --;> 51 be a homotopy relative to i from 0' to /3. Because
I X I is a starlike subset of R2 from (0,0), it follows from lemma 9 that there
is a map F': I X 1--;> R such that F'(O,O) = 0 and ex F' = F. 5ince F is a
homotopy relative to i, F(O,t') = F(l,t') = po for all t E I. Therefore F'(O,t')
and F'(l,t') take on only integral values for all t' E I. It follows that F'(O,t')
must be constant and F'(l,t') must be constant. Because F(O,O) = 0,
F'(O,t') = 0 for all t' E I. Define 0", /3': 1--;> R by O"(t) = F'(t,O) and /3'(t) =
F'(t,l). Then 0"(0) = 0 and ex 0" = 0'. Therefore deg 0' = 0"(1) = F'(l,O).
Similarly, /3'(0) = 0 and ex /3' = /3, so deg /3 = /3'(1) = F'(l,l). Because
F(l,t') is constant, F'(l,O) = F'(l,l) and deg 0' = deg /3. •
0
0
0
It follows that there is a well-defined function deg from '7T(51,po) to Z defined by
deg [O'J = deg 0'
where
12
0'
is a closed path in 51 at po.
THEOREM
The function deg is an isomorphism
deg: '7T(5 1,po) ;:::::; Z
PROOF
To prove that deg is a homomorphism, let 0' and /3 be two closed
paths in 51 at po and let 0'/3 be the closed path which is their pointwise
SEC.
8
55
THE FUNDAMENTAL GROUP
product in the group multiplication of 51. We know from theorem 4 that
[a] * [,8] = [a,8]. Let a', ,8': I ~ R be such that a'(O) = 0, ex a' = a,
,8'(0) = 0, and ex ,8' = ,8. Then a' + ,8': I ~ R is such that (a' + ,8')(0) = 0
and ex (a' + ,8') = a,8. Therefore
0
0
0
deg ([a]
* [,8]) = deg [a,8] = (a' + ,8')( 1)
= deg a + deg,8 = deg [a] + deg [,8]
showing that deg is a homomorphism.
The map deg is an epimorphism; for if n is an integer, there is a path a~
in R defined by a~(t) = tn. Let an = ex a~. Then clearly, deg [an] = a~(I) = n.
The map deg is a monomorphism; for if deg [a] = 0, there is a closed
path a' in R at 0 such that ex a' = a. 5ince R is simply connected (because
it is contractible, and by lemma 5), a' ~ eo. Then ex a' ~ epo. Therefore
a ~ epo and [a] is the identity element of 'IT(5 1 ,po). •
0
0
0
The method we have used to compute 'IT(51,po) will be generalized
in Chapter 2 to give relations between the fundamental group of a space and
the fundamental groups of its covering spaces.
It follows from theorem 2 that 'IT(51,po) ;::::; [5 1 ,po; 51,po]. Because 51 is a
topological group, the set [51;51] (with no base-point condition) is also a
group under pointwise multiplication of maps, and there is an obvious
homomorphism
y:
13
LEMMA
[5 1 ,po; 51,po]
~
[5 1 ;5 1 ]
~
[5 1 ;51 ]
The homomorphism
y: [51,po; 51,po]
is an isomorphism.
PROOF
To show that y is an epimorphism, let f: 51
for some 0 ::; 0
2'lT. Define a homotopy F: 51 X I
<
~
~
51 and let f(po)
51 by
=e
i8
F(z,t) = f(z)e- it8
Then F is a homotopy from
y[f']po
= [f'] = [f].
f
to a map f' such that f'(po)
= po. Therefore
To show that y is a monomorphism, assume that f: (51,po) ~ (51,po) is
such that y[f]po = [f] is trivial. Then f: 51 ~ 51 is null homotopic. By
theorem 1.3.12, f is null homotopic relative to po. Therefore [f]po is trivial. •
It follows from theorem 12 and lemma 13 that [51; 51] """ Z. The isomorphism
can be chosen so that for each integer n the map z --+ zn from 51 to itself represents
a homotopy class corresponding to n.
56
HOMOTOPY AND THE FUNDAMENTAL GROUP
CHAP.
1
EXERCISES
A CONTRACTIBLE SPACES
I The cone over a topological space X with vertex v is defined to be the mapping cylinder of the constant map X --'> v. Prove that X is contractible if and only if it is
a retract of any cone over X.
2
Prove that Sn is a retract of En+l if and only if Sn is contractible.
3 If CX is a cone over X, prove that (CX,X) has the homotopy extension property with
respect to any space.
4 Prove that a space Y is contractible if and only if, given a pair (X,A) having the
homotopy extension property with respect to Y, any map A --'> Y can be extended over X.
5 Let Y be the comb space of example 1.3.9 and let yo be the point (0,1) E Y. Let Y'
be another copy of Y, with corresponding point yo. Let X be the space obtained by forming the disjoint union of Y and Y' and identifying yo with yo. Prove that X is n-connected
for all n but not contractible. (Hint: Any deformation of X in itself must be a homotopy
relative to Yo.)
B ADJUNCTION SPACES
I Let A be a subspace of a space X and let f: A --'> Y be a continuous map. The adjunction space Z of X to Y by f is defined to be the quotient space of the disjoint union of
X and Y by the identifications x E A equals f(x) E Y for all x E A. Prove that if X and Y
are normal spaces and A is closed in X, then Z is a normal space.
2 A space X is said to be binormal if X X I is a normal space. If X is a binormal space,
Y is a normal space, and f: X --'> Y is continuous, prove that the mapping cylinder of f is
a normal space.
3 Given a continuous map f: A --'> Y, where A is a subspace of a space X, prove that f
can be extended over X if and only if Y is a retract of the adjunction space of X
to Y by f.
4 Let Z be the adjunction space of X to Y by a map f: A --'> Y. Prove that (Z, Y) has
the homotopy extension property with respect to a space W if (X,A) has the homotopy
extension property with respect to W.
C ABSOLUTE RETRACTS AND ABSOLUTE NEIGHBORHOOD RETRACTS
A space Y is said to be an absolute retract (or absolute neighborhood retract) if, given a
normal space X, closed subset A C X, and a continuous map f: A --'> Y, then f can be
extended over X (or f can be extended over some neighborhood of A in X).
I Prove that a normal space Y is an absolute retract (or absolute neighborhood retract)
if and only if, whenever Y is imbedded as a closed subset of a normal space Z, then Y is
a retract of Z (or a retract of some neighborhood of Yin Z).
2 Prove that the product of arbitrarily many absolute retracts (or finitely many
absolute neighborhood retracts) is itself an absolute retract (or absolute neighborhood
retract).
3
Prove that Rn is an absolute retract for all n.
57
EXERCISES
4 Prove that a retract of an absolute retract is an absolute retract and that a retract of
some open subset of an absolute neighborhood retract is an absolute neighborhood
retract.
S Prove that En is an absolute retract and Sn is an absolute neighborhood retract for
all n.
6 Prove that a binormal absolute neighborhood retract is locally contractible (that is,
every neighborhood U of a point x contains a neighborhood V of x deformable to x in U).
7 Prove that a binormal absolute neighborhood retract is an absolute retract if and
only if it is contractible.
o
HOMOTOPY EXTENSION PROPERTY
I Let A be a closed subset of a normal space X, let f: X ~ Y be continuous (where Y
is arbitrary), and let G: A X I ~ Y be a homotopy of f I A. If there exists a homotopy
G': U X I ~ Y of flU which extends G, where U is an open neighborhood of A, show
that there exists a homotopy F: X X I ~ Y of f which extends G.
2 Borsuk's homotopy extension theorem. Let A be a closed subspace of a binormal
space X. Then (X,A) has the homotopy extension property with respect to any absolute
neighborhood retract Y.
3 Let A be a closed subset of a binormal space X and assume that the subspace
A X I U X X 0 C X X I is an absolute neighborhood retract. Then (X,A) has the
homotopy extension property with respect to any space Y.
4 Let A be a closed subset of X and B a subset of Y. Assume that (X,A) has
the homotopy extension property with respect to B and that (X X I, X X i U A X I)
has the homotopy extension property with respect to Y. Prove that if f: (X,A) ~ (Y,B) is
homotopic (as a map of pairs) to a map which sends all of X to B, then it is homotopic
relative to A to such a map.
E
I
COFIBRATIONS
2
Prove that a composite of cofibrations is a cofibration.
Prove that any cofibration is an injective function.
3 For a closed subspace A of X prove that the inclusion map A C X is a cofibration if
and only if X X 0 U A X I is a retract of X X I.
4 If A is a subspace of a Hausdorff space X, prove that if A C X is a cofibration, then
A is closed in X.
S Assume that X is the union of closed subsets Xl and X2 and let A be a subset of X
such that Xl n X2 C A. Prove that if A n Xl C Xl and A n X2 C X2 are cofibrations,
so is A ex.
6
Let A be a closed subspace of a space X. Prove that the following are equivalent: 1
(a) A C X is a cofibration.
(b) There is a deformation D: X X I ~ X rei A [that is, D(x,O)
x and D(a,t)
a
for x E X, a E A, and tEl] and a map cp: X ~ I such that A = cp-l(l) and
D(cp-l(O,l] X 1) C A.
=
=
1 If X is normal, the equivalence of (a) and (c) is proved in C. S. Young, A condition for the
absolute homotopy extension property, American Mathematical Monthly, vol. 71, pp. 896-897,
1964.
58
HOMOTOPY AND THE FUNDAMENTAL GROUP
CHAP.
I
(c) There is a neighborhood U of A deformable in X to A rei A [that is, there is a
homotopy H: U X I ~ X such that H(x,O) = x, H(a,t) = a, and H(x,l) E A for
x E U, a E A, and t E 1] and a map <p: X ~ I such that A = <p-l(l) and <p(x) = 0
if x E X - U.
7 If A C X and BeY are cofibrations with A and B closed in X and Y, respectively,
prove that A X B c X X B U A X Y and X X B U A X Y C X X Yare cofibrations.
F LOCAL SYSTEMS!
I A local system on a space X is a covariant functor from the fundamental groupoid of
X to some category. For any category (' show that there is a category of local systems on
X with values in e. (Two local systems on X are said to be equivalent if they are
equivalent objects in this category.)
2 Given a map f: X ~ Y, show that f induces a covariant functor from the category of
local systems on Y with values in e to the category of local systems on X with values in e.
e,
3 If A is an object of a category
let Aut A be the group of self-equivalences of A
in e If <p: A :.:::: B is an equivalence in e, then show that ip: Aut A ~ Aut B defined by
ip(a) = <p a <p-l is an isomorphism of groups.
0
4
0
If r is a local system on X and Xo E X, show that r induces a homomorphism
fxo: 7T(X,XO)
~ Aut [(xo)
:; If X is path connected, prove that two local systems rand r' on X with values in e
are equivalent if and only if there is an equivalence <p: [(xo) :.:::: r'(xo), such that <p 0 f Xu
is conjugate to f~o in Aut r'(xo).
6 If X is path connected, given an object A E e and a homomorphism a: 7T(X,Xo) ~
Aut A, prove that there is a local system r on X with values in e such that [(xo) = A
and fxo = a.
G
I
THE FUNDAMENTAL GROUP
Prove that the fundamental group functor commutes with direct products.
2 If wand w' are paths in X from Xo to
w * ",,'-1 ~ €XO.
3
U
4
n
Xl,
prove that w
~ w'
if and only if
Let a space X be the union of two open simply connected subsets U and V such that
V is nonempty and path connected. Prove that X is simply connected.
Prove that Sn is simply connected for n
~
2.
:; If there exists a space with a nonabelian fundamental group, prove that the "figure
eight" (that is, the union of two circles with a point in common) has a nonabelian fundamental group (cf. exercise 2.B.4).
6 Let f: I ~ R2 be a continuously differentiable simple closed curve in the plane with
a nowhere-vanishing tangent vector [that is, f(O) = f(I), /'(0) = /'(1), and /'(t) =1= 0 for
1
See N. E. Steenrod, Homology with local coefficients, Annals of Mathematics, vol. 44,
pp. 610-627, 1943.
59
EXERCISES
0::;; t::;; 1]. Let w: 1-> Sl be the closed path defined by wit)
[w] is a generator of '17(Sl).1
= !,(t)/II!,(t)ll. Prove that
7 In R2, let X be the space consisting of the union of the circles Cn, where Cn
has center (l/n,O) and radius lin for all positive integers n. In R3 (with R2 imbedded as
the plane X3 = 0), let Y be the set of points on the closed line segments joining (0,0,1) to
X and let Y' be the reflection of Y through the origin of R3. Then Yand Y' are closed
simply connected subsets of R3 such that Y n Y' is a single point. Prove that Y U Y' is
not simply connected. 2
H
I
SOME APPLICATIONS OF THE FUNDAMENTAL GROUP
2
Prove that Sl and Sn for n ;;:, 2 are not of the same homotopy type.
3
Prove that R2 and Rn for n
Prove that Sl is not a retract of £2.
> 2 are not homeomorphic.
4 Let p(z) = zn + a n-1 zn-1 + ... + a1Z + ao be a polynomial of degree n, having
complex coefficients and leading coefficient 1, and let q(z) = zn. For r
0 let
Cr = {x E R 2 I I x I = r}. Prove that for r large enough, p I Cr and q I Cr are homotopic
maps of Cr into R2 - O.
>
5 Fundamental theorem of algebra. Prove that every complex polynomial has a root.
(Hint: For any r
0 the map q I Cr : Cr -> R2 - 0 is not null homotopic because it induces a nontrivial homomorphism of fundamental groups.)
>
See H. Hopf, Uber die Drehung der Tangenten und Sehnen ebener Kurven, Compositio
Mathematica, vol. 2, pp. 50-62, 1935. For generalizations see H. Whitney, On regular closed
curves in the plane, Compositio Mathematica, vol. 4, pp. 276-284, 1937, and S. Smale, Regular
curves on Riemannian manifolds, Transactions of the American Mathematical Society, vol. 87,
pp. 495-512, 1958.
2 See H. B. Griffiths, The Fundamental group of two spaces with a common point, Quarterly
Journal of Mathematics, vol. 5, pp. 175-19(), 1954.
1
CHAPTER TWO
COVERING SPACES
AND FIBRATIONS
THE THEORY OF COVERING SPACES IS IMPORTANT NOT ONLY IN TOPOLOGY, BUT
also in differential geometry, complex analysis, and Lie groups. The theory is
presented here because the fundamental group functor provides a faithful
representation of covering-space problems in terms of algebraic ones. This
justifies our special interest in the fundamental group functor.
This chapter contains the theory of covering spaces, as well as an introduction to the related concepts of fiber bundle and fibration. These concepts
will be considered again later in other contexts. Here we adopt the view that
certain fibrations, namely, those having the property of unique path lifting,
are generalized covering spaces. Because of this, we shall consider these
fibrations in some detail.
Covering spaces are defined in Sec. 2.1, and fibrations are defined in
Sec. 2.2, where it is proved that every covering space is a fibration. Section 2.3
deals with relations between the fundamental groups of the total space and
base space of a fibration with unique path lifting, and Sec. 2.4 contains a solution of the lifting problem for such fibrations in terms of the fundamental
group functor.
61
62
COVERING SPACES AND FIBRATIONS
CHAP.
2
The lifting theorem is applied in Sec. 2.5 to classify the covering spaces
of a connected locally path-connected space by means of subgroups of its
fundamental group. This entails the construction of a covering space starting
with the base space and a subgroup of its fundamental group. In Sec. 2.6 a
converse problem is considered. The base space is constructed, starting with
a covering space and a suitable group of transformations on it.
In Sec. 2.7 fiber bundles are introduced as natural generalizations of
covering spaces. The main result of the section is that local fibrations are
fibrations. This implies that a fiber bundle with paracompact base space is a
fibration. Section 2.8 considers properties of general fibrations and the concept of fiber homotopy equivalence. These will be important in our later
study of homotopy theory.
I
COVERING PROJECTIONS
A covering projection is a continuous map that is a uniform local homeomorphism. This and related concepts are introduced in this section, along with
some examples and elementary properties.
Let p: X ---.,) X be a continuous map. An open subset U C X is said to be
evenly covered by p if p-1( U) is the disjoint union of open subsets of X each
of which is mapped homeomorphically onto U by p. If U is evenly covered
by p, it is clear that any open subset of U is also evenly covered by p. A continuous map p: X ---.,) X is called a covering projection if each point x E X has
an open neighborhood evenly covered by p. X is called the covering space
and X the base space of the covering projection.
The following are examples of covering projections.
I
Any homeomorphism is a covering projection.
2
If X is the product of X with a discrete space, the projection
is a covering projection.
The map ex: R ---.,) 51, defined by ex(t)
3
a covering projection.
X ---.,)
X
= e 2'7Tit, (considered in Sec 1.8) is
For any positive integer n the map p: 51 ---.,) 51, defined by p(z)
4
covering projection.
= zn, is a
S
For any integer n ?: 1 the map p: Sn ---.,) pn, which identifies antipodal
points, is a covering projection.
6
If G is a topological group, H is a discrete subgroup of G, and G/H is
the space of left (or right) cosets, then the projection G ---.,) G/H is a covering
projection.
A continuous map f: Y ---.,) X is called a local homeomorphism if each
point y E Y has an open neighborhood mapped homeomorphically by f onto
SEC.
1
63
COVERING PROJECTIONS
an open subset of X. If this is so, each point of Y has arbitrarily small neighborhoods with this property, and we have the following lemmas.
7
LEMMA
A local homeomorphism is an open map.
•
8
LEMMA
A covering projection is a local homeomorphism.
PROOF
Let p: X ~ X be a covering projection and let x E X. Let U be an
open neighborhood of p( x) evenly covered by p. Then p-1( U) is the disjoint
union of open sets, each mapped homeomorphically onto U by p. Let [] be
that one of these open sets which contains X. Then [] is an open neighborhood of x such that p I [] is a homeomorphism of [] onto the open subset U
of X. •
A local homeomorphism need not be a covering projection, as shown by
the following example.
EXAMPLE
Let p: (0,3) ~ 51 be the restriction of the map ex: R ~ 51 of
9
example 3 to the open interval (0,3). Because p is the restriction of a local
homeomorphism to an open subset, it is a local homeomorphism. It is also a
surjection, but it is not a covering projection because the complex number
1 E 51 has no neighborhood evenly covered by p.
The following is a consequence of lemmas 7 and 8 and the fact (immediate from the definition) that a covering projection is a surjection.
10
COROLLARY
A covering projection exhibits its base space as a quotient
space of its covering space.
•
For locally connected spaces there is the following reduction of covering
projections to the components of the base space.
If X is locally connected, a continuous map p: X ~ X is a
covering projection if and only if for each component C of X the map
II
THEOREM
p I p-1C: p-1C
~
C
is a covering projection.
PROOF
If P is a covering projection and C is a component of X, let x E C
and let U be an open neighborhood of x evenly covered by p. Let V be the
component of U containing x. Since X is locally connected, V is open in X,
and hence open in C. Clearly, V is evenly covered by p I p-1G. Therefore
p I p-1C is a covering projection.
Conversely, assume that the map p I p-1C: p-1C ~ C is a covering projection for each component C of X. Let x E C and let U be an open neighborhood of x in C evenly covered by p I p-1G. S~nce X is locally connected, Cis
open in X. Therefore U is also open in X and is clearly evenly covered by p.
Hence p is a covering projection. •
In general, the representation of the inverse image of an evenly covered
open set as a disjoint union of open sets, each mapped homeomorphically, is
64
COVERING SPACES AND FIB RATIONS
CHAP.
2
not unique (consider the case of an evenly covered discrete set); however, for
connected evenly covered open sets there is the following characterization of
these open subsets.
12 LEMMA
Let U be an open connected subset of X which is evenly
covered by a continuous map p: X ~ X. Then p maps each component of
p-l( U) homeomorphically onto U.
PROOF
By assumption, p-l( U) is the disjoint union of open subsets, each
mapped homeomorphically onto U by p. Since U is connected, each of these
open subsets is also connected. Because they are open and disjoint, each is
a component of p-l(U). •
13
COROLLARY
Consider a commutative triangle
-
Xl
P
~
-
X2
PJ\ /P2
X
where X is locally connected and Pl and P2 are covering protections.
surjection, it is a covering protection.
If p is
a
If U is a connected open subset of X which is evenly covered by Pl
and P2, it follows easily from lemma 12 that each component of P2 -l( U) is
evenly covered by p. •
PROOF
14 THEOREM If p: X ~ X is a covering protection onto a locally connected
base space, then for any component C of X the map
piC:
C~
p(C)
is a covering protection onto some component of X.
PROOF
Let C be a component of X. We show that p( C) is a component of
X. p( C) is connected; to show that it is an open and closed subset of X, let x
be in the closure of p( C) and let U be an open connected neighborhood of x
evenly covered by p. Because U meets p(C), p-1(U) meets C. Therefore some
component [; of p-l( U) meets C. Since C is a component of X, [; c C.
Then, by lemma 12, p( C) ::J p( 0) = U. Therefore the closure of p( C) is contained in the interior of p( C), which implies that p( C) is open and closed. The
same argument shows that if x E p( C) and U is an open connected neighborhood of x in X evenly covered by p, then U C p( C) and (p I C)-l( U) is the
disjoint union of those components of p-l( U) that meet C. It follows from
lemma 12 that U is evenly covered by piC. Therefore piC: C ~ p( C) is a
covering projection. •
The following example shows that the converse of theorem 14 is false.
15 EXAMPLE Let X = 51 X 51 X ... be a countable product of I-spheres
and for n ?: 1 let Xn = Rn X 51 X 51 X .... Define pn: Xn ~ X by
SEC.
2
65
THE HOMOTOPY LIFTING PROPERTY
Pn(tl, . . . ,tn,
Zl,Z2, . . . )
= (eX(tl),
. . . ,eX(tn),
Zl,Z2, . . . )
Let X = V Xn and define p: X ---7 X so that p I Xn = pn. The components of
X are the spaces Xn and the map p I Xn = pn: Xn ---7 X is a covering projection. However, p is not a covering projection, because no open subset of X is
evenly covered by p.
For later purposes we should like to have the analogues of theorems 11
and 14, in which "component" is replaced by "path component." For this we
need the following definition: a topological space is said to be locally path
connected if the path components of open subsets are open. The following
are easy consequences of this definition.
16 Any open subset of a locally path-connected space is itself locally path
connected. -
I 7 A locally path-connected space is locally connected.
-
18 In a locally path-connected space the components and path components
coincide. 19 A connected locally path-connected space is path connected.
-
From statements 17 and 18 we obtain the following extension of theorems
11 and 14.
20 THEOREM If X is locally path connected, a continuous map p: X ---7 X
is a covering projection if and only if for each path component A of X
p I p-1A: p-1A
---7
A
is a covering projection. In this case, if A is any path component of X, then
p I A is a covering projection of A onto some path component of X. -
2
THE HOMOTOPV LIFTING PROPERTY
The homotopy lifting property is dual to the homotopy extension property. It
leads to the concept of fibration, which is dual to that of co fibration introduced in Sec. 1.4. In this section we define the concept of fibration and prove
that a covering projection is a special kind of fibration. This special class of
fibrations will be regarded as generalized covering projections, and our subsequent study of covering projections will be based on a study of the more general concept. At the end of the chapter we return to the general consideration of fibrations.
We begin with an important problem of algebraic topology, called the
lifting problem, which is dual to the extension problem. Let p: E ---7 Band
f: X ---7 B be maps. The lifting problem for f is to determine whether there is
66
COVERING SPACES AND FIBRATIONS
a continuous map f': X ~ E such that f
arrow in the diagram
=p
0
CHAP.
2
f' -that is, whether the dQtted
E
/"
~P
X~B
corresponds to a continuous map making the diagram commutative. If there
is such a map f', then f can be lifted to E, and we call f' a lifting, or lift, of f
In order that the lifting problem be a problem in the homotopy category, we
need an analogue of the homotopy extension property, called the homotopy
lifting property, defined as follows. A map p: E ~ B is said to have the
homotopy lifting property with respect to a space X if, given maps f': X ~ E
and F: X X I ~ B such that F(x,O) = pf'(x) for x E X, there is a map
F: X X I ~ E such that F'(x,O) = f'(x) for x E X and po F = F. If f' is regarded as a map of X X 0 to E, the existence of F' is equivalent to the
existence of a map represented by the dotted arrow that makes the following
diagram commutative:
X X 0
f'
~
E
Xx I ~ B
If p: E ~ B has the homotopy lifting property with respect to X and
fo, fl: X ~ B are homotopic, it is easy to see that fo can be lifted to E if and
only if it can be lifted to E. Hence, whether or not a map X ~ B can
be lifted to E is a property of the homotopy class of the map. Thus the homotopy lifting property implies that the lifting problem for maps X ~ B is
a problem in the homotopy category.
A map p: E ~ B is called a fibration (or Hurewicz fiber space in the literature) if p has the homotopy lifting property with respect to every space. E
is called the total space and B the base space of the fibration. For
b E B, p-l(b) is called the fiber over b.
If p: E ~ B is a fibration, any path win B such that w(O) E p(E) can be
lifted to a path in E. In fact, w can be regarded as a homotopy w: P X I ~ B
where P is a one-point space, and a point eo E E such that p(eo) = w(O)
corresponds to a map f: P ~ E such that pf(P) = w(P,O). It follows from the
homotopy lifting property of p that there exists a path w in E such that
w(O) = eo and pow = w. Then w is a lifting of w.
I
EXAMPLE
Let F be any space and let p: B X F ~ B be the projection
to the first factor. Then p is a fibration, and for any b E B the fiber over b is
homeomorphic to F.
To prove that a covering projection is a fibration, we first establish the
following unique-lifting property of covering projections for connected spaces.
SEC.
2
67
THE HOMOTOPY LIFTING PROPERTY
2
THEOREM
Let p: X ---7 X be a covering projection and let f, g: Y ---7 X be
liftings of the same map (that is, p f = p g). If Y is connected and f agrees
with g for some point of Y, then f = g.
0
0
PROOF
Let Y1 = {y E Ylf(y) = g(y)}. We show that Y1 is open in Y.
If y E Yl, let U be an open neighborhood of pf(y) evenly covered by p and
let 0 be an open subset of X containing f(y) such that p maps 0 homeomorphically onto U. Then f-1( 0) n g-l( 0) is an open subset of Y containing y
and contained in Y 1 .
Let Y2 = {y E Y I f(y)
g(y)}. We show that Y2 is also open in Y (if X
were assumed to be Hausdorff, this would follow from a general property of
Hausdorff spaces). Let y E Y2 and let U be an open neighborhood of pf(y)
evenly covered by p. Since f(y)
g(y), there are disjoint open subsets 0 1 and
O2 of X such that f(y) E 01 and g(y) E O2 and p maps each of the sets
0 1 and O2 homeomorphically onto U. Then f-1( 0 1) n g-l( O2) is an open
subset of Y containing y and contained in Y2 .
Since Y = Y1 U Y2 and Y1 and Y2 are disjoint open sets, it follows from
the connectedness of Y that either Y1 = 0 or Y1 = Y. By hypothesis,
Y1
0, so Y = Y1 and f = g. •
*-
*-
*-
We are now ready to prove that a covering projection has the homotopy
lifting property.
3
THEOREM
A covering projection is a fibration.
Let p: X ---7 X be a covering projection and let f': Y ---7 X and
F: Y X I ---7 X be maps such that F(y,O) = pf'(y) for y E Y. We show that
for each y E Y there is an open neighborhood Ny of y in Y and a map
F~: Ny X I ---7 X such that F~(y',O) = f'(y') for y' E Ny and pF~ = F I Ny X I.
Assume that we have such neighborhoods Ny and maps F~. If y" E Ny n Ny"
then F~ I y" X I and F~, I y" X I are maps of the connected space y" X I
into X such that for t E I
PROOF
p
0
(F~ I y" X
= F(y",t) = p (F~, I y" X I)(y",t)
= f'(y") = (F~' I y" X I)(y",O), it follows
I)(y",t)
0
Because (F~ I y" X I)(y",O)
from
theorem 2 that F~ I y" X I = F~, I y" X 1. Since this is true for all
y" E Ny n Ny" it follows that F~ I (Ny n Ny') X I = F~, I (Ny n Ny') X I.
Hence there is a continuous map F': Y X I ---7 X such that F' I Ny X I = F~,
and F' is a lifting of F such that F'(y,O) = f'(y) for y E Y. Thus we have
reduced the theorem to the construction of the open neighborhoods Ny and
maps F~.
It follows from the fact that p: X ---7 X is a covering projection (and the
compactness of I) that for each y E Y there is an open neighborhood Ny of
y and a sequence = to
h
tm = 1 of points of I such that for
i = 1, . . . , m, F(Ny X [ti_1,ti]) is contained in some open subset of X
evenly covered by p. We show that there is a map F~: Ny X I ---7 X with the
desired properties. It suffices to define maps
°
< < . .. <
68
COVERING SPACES AND FIB RATIONS
i
= 1,
CHAP.
2
... , m
such that
po Gi
G1(y',O)
G i- 1(y',ti-1)
= FI Ny X [ti_1,t;]
= f'(y')
= Gi(y',ti-1) y'
y' E Ny
E Ny, i = 2, . . . , m
because, given such maps' G i , there is a map Fy: Ny X I ~ X such that
Fy I Ny X [ti-1,t;]
Gi for i
1, . . . , m. Then Fy has the desired properties.
The maps Gi are defined by induction on i. To define Gt, let U be an
open subset of X evenly covered by p such that F(Ny X [to,t1]) c U. Let
{ OJ} be a collection of disjoint open subsets of X such that p-1( U) = U OJ
and p maps OJ homeomorphically onto U for each f. Let Vj = f'-1( OJ). Then
{Vj} is a collection of disjoint open sets covering Ny, and G 1 is defined to be
the unique map such that for each f, G 1 maps Vj X [to,t1] into OJ to be a lifting of F I Vj X [to,lt]. This defines G 1.
Assume G i - 1 defined for 1 < i ~ m. Let U' be an open subset of
X evenly covered by p such that F(Ny X [ti-1,ti]) c U'. Let { Ok} be a collection of disjoint open subsets of X such that p-1( U') = U 0" and p maps Ok
homeomorphicallyonto U' for each k. Let Vk = {y' E Ny I Gi- 1(y',ti-1) E Ok}'
Then {Vk} is a collection of disjoint open sets covering Ny, and Gi is defined
to be the unique map such that for each k, Gi maps V" X [ti-1,ti] into Ok to
be a lifting of F I Vk X [ti-hti]. This defines Gi. •
=
=
A map p: E ~ B is said to have unique path lifting if, given paths wand
w' in E such that pow = pow' and w(O) = w'(O), then w = w'. It follows
from theorem 2 that a covering projection has unique path lifting.
4
LEMMA
If a map has unique path lifting, it has the unique-lifting
property for path-connected spaces.
PROOF Assume that p: E ~ B has unique path lifting. Let Y be path connected
and suppose that f, g: Y ~ E are maps such that p f = p g and f(yo) = g(yo)
for some yo E Y. We must show f = g. Let y E Y and let w be a path in Y
from yo to y. Then f wand g w are paths in E that are liftings of the
same path in B and have the same origin. Because p has unique path lifting,
f 0 w = g 0 w. Therefo.re
0
0
f(y)
= (f
0
0
0
w)(l)
= (g
0
w)(l)
= g(y)
•
The following theorem characterizes fibrations with unique path lifting.
it
THEOREM
A fibration has unique path lifting if and only if every fiber
has no nonconstant paths.
Assume that p: E ~ B is a fibration with unique path lifting. Let w
be a path in the fiber p-1(b) and let w' be the constant path in p-1(b) such
that w'(O)
w(O). Then pow
pow', which implies w w'. Hence w is a
constant path.
PROOF
=
=
=
SEC.
2
69
THE HOMOTOPY LIFTING PROPERTY
Conversely, assume that p: E ----> B is a fibration such that every fiber has
no nontrivial path and let wand w' be paths in E such that pow = pow'
and w(o) = w'(O). For tEl, let WI' be the path in E defined by
"( ') _ {W((I - 2t')t)
- w'((2t' _ I)t)
Wt t
O<t'<V2
V2<t'<1
Then WI' is a path in E from w(t) to w'(t), and pow;' is a closed path in B that
is homotopic relative to j to the constant path at pw(t). By the homotopy lifting property of p, there is a map F': I X 1----> E such that F'(t',O) = w;'(t') and
F' maps
X I U I X 1 U 1 X I to the fiber p-1(pW(t)). Because p-1(pW(t))
has no nonconstant paths, F' maps X I, I X 1, and 1 X I to a single point.
It follows that F'(O,O) = F'(I,O). Therefore w;'(O) = w;'(I) and w(t) = w'(t). -
°
°
We have seen that a covering projection is a fibration with unique path
lifting. It will be shown in Sec. 2.4 that if the base space satisfies some mild
hypotheses, any fibration with unique path lifting is a covering projection.
One reason for studying fib rations with unique path lifting as generalized
covering projections is that the following two theorems are easily proved, but
both are false for covering projections.
6
THEOREM
The composite of fibrations (with unique path lifting) is a
fibration (with unique path lifting). -
7
THEOREM
The product of fibrations (with unique path lifting) is a
fibration (with unique path lifting). -
An example shows that theorem 6 is false for covering projections.
EXAMPLE
Let X and X n , for n ;::: 1, be a countable product of I-spheres.
8
Let Xn = Rn X X and define pn: Xn ----> Xn by
Pn(t1, . . . ,tn,
Zl,ZZ, . . . )
= (ex(t1),
. . . ,ex(tn),
Zl,ZZ, . . • )
Then pn is a covering projection for n ;::: 1. It follows from theorem 2.1.11
that V pn: V Xn ----> V Xn is a covering projection. Since V Xn is the product
of X and the set of positive integers, there is a covering projection V Xn ----> X
(see example 2.1.2). The composite
VX n ----> VX n ----> X
is not a covering projection (cf example 2.1.15).
Similarly, theorem 7 is false for covering projections.
EXAMPLE
For n
9
ex: R ----> 51. Then
>
1, let pn: Xn ----> Xn be the covering projection
is not a covering projection.
It follows from theorem 6 that there is a category whose objects are
topological spaces and whose morphisms are fibrations with unique path
70
COVERING SPACES AND FIB RATIONS
CHAP.
2
lifting. We shall now describe a category, depending on a given base space, which
is of more use in studying covering projections or flbrations. For a given space
X there is a category whose objects are maps p: g -+ X, which are flbrations
with unique path lifting, and whose morphisms are commutative triangles
If Pi: Xi ~ X is an indexed family of objects in this category, let p: V Xi ~ X
be the map such that P I Xi
Pi' Then P is also an object in the category and
is the sum of the collection {Pi} in the category.
To show that this category also has products, given maps Pi: Xi ~ X, let
=
X = {(Xi)
EX
Xi I Pi(Xi) = Pi'(xi') for all t, n
and define p: X ~ X by P((Xi)) = PiXi)' If each Pi is a fibration, so is P, and
if each Pi has unique path lifting, so does p. Hence P is a product of {Pi} in
the category of fibrations with unique path lifting. This map P is called the
fibered product of the maps {Pi}' We consider it in more detail in Sec. 2.8.
There is a similar category whose objects are covering projections with
base space X and whose morphisms are commutative triangles. This category
has finite sums and finite products, but neither arbitrary sums nor arbitrary
products. In fact, for each n let
pn: Rn X Sl X Sl X ...
~
Sl X Sl X ...
be defined by Pn(tl> ... ,tn, Zl,ZZ, . . . ) = (e Z'7Titl, • • • ,e Z'7Titn, Zl> Zz, . . .),
as in example 8. Then the collection {Pn} has neither a sum nor a product in
the category of covering projections with base space X.
3
RELATIONS WITH THE FUNDAMENTAL GROUP
In a fibration with unique path lifting the fundamental group of the total
space is isomorphic to a subgroup of the fundamental group of the base
space. The corresponding subgroup of the fundamental group will lead to a
classification of fibrations with unique path lifting. In fact, we shall see in the
next section that the fundamental group functor solves the lifting problem for
fibrations with unique path lifting. The present section is devoted to consideration of the relation between the fundamental groups of the total space and
the base space of a fibration with unique path lifting.
We begin with a localization property for fibrations which is an analogue
of theorem 2.1.14.
SEc.3
71
RELATIONS WITH THE FUNDAMENTAL GROUP
I
LEMMA
Let p: E ~ B be a fibration. If A is any path component of E,
then pA is a path component of Band p I A: A ~ pA is a fibration.
Since pA is the continuous image of a path-connected space, it
is path connected. It is a maximal path-connected subset of B, for if w is a path
in B that begins in pA, there is a lifting w of w that begins in A. Since A
is a path component of E, w is a path in A. Therefore w = pow is a path in
pA. Hence pA is a maximal path-connected subset of B and, by theorem
1. 7.9, a path component of B.
To show that p I A: A ~ pA has the homotopy lifting property, let
f': Y ~ A and F: Y X I ~ pA be maps such that F(y,O) = pf'(y). Because p
is a fibration, there is a map F': Y X I ~ E such that p F' = F and
F(y,O) = f'(y). For any y E Y, F' must map y X I into the path component
of E containing F'(y,O). Therefore F'(y X 1) c A for all y, and F': Y X I ~ A
is a lifting of F such that F'(y,O) = f'(y). •
PROOF
0
For locally path-connected spaces we have the following analogue of
theorem 2.1.20, which reduces the study of fibrations to the study of fibrations with total space and base space path connected.
2
THEOREM
Let p: E ~ B be a map. If E is locally path connected, p is
a fibration if and only if for each path component A of E, pA is a path component of Band p I A: A ~ pA is a fibration.
PROOF
If p: E ~ B is a fibration and A is a path component of E, it follows
from lemma 1 that pA is a path component of Band p I A: A ~ pA is
a fibration.
To prove the converse, let f': Y ~ E and F: Y X I ~ B be such that
F(y,O) = f'(y). Let {Aj} be the path components of E. Then {Aj} are disjoint
open subsets of E. Let Vj = f-l(A j ). The collection {Vj} is a disjoint open covering of Y. Therefore, to construct a map F': Y X I ~ E such that p F = F
and F'(y,O) = f'(y), it suffices to construct maps Fj: Vj X I ~ E for all i such
that p Fj = F I Vj X I and Fj(y,O) = f'(y,O).
Because F(y X I) is contained in the path component of B containing
F(y,O) = pf'(y), it follows from the fact that pAj is a path component of B
that F(Vj X I) C pAj for all i. Because p I Aj: Aj ~ pAj is a fibration, there is
a map Fj: Vj X I ~ Aj such that pF; = F I Vj X I and F;(y,O) = f'(y) for
y E Vj. Therefore p has the homotopy lifting property. •
0
0
Since every path in a topological space lies in some path component of
the space, it is clear that theorem 2 remains valid if the term "fibration" is
replaced throughout by "fibration with unique path lifting."
The main result on fibrations with unique path lifting is embodied in the
following statement.
3 LEMMA Let p: X ~ X be a fibration with unique path lifting. If wand
w' are paths in X such that w(O) = w'(O) and pow ~ pow', then w ~ w'.
72
COVERING SPACES AND FIB RATIONS
CHAP.
2
PROOF
Let F: I X I ~ X be a homotopy relative to j from pow to pow'
[that is, F(t,O) = pw(t) and F(t,l) = pw'(t), and F(O,t) = pw(O) and F(l,t) =
pw(I)J. By the homotopy lifting property of fibrations, there is a map
F': I X I ~ X such that F'(t,O) = w(t) and p F' = F. Then F'(O X I) and
F'(l X I) are contained in p-l(pw(O)) and p-l(pw(l)), respectively. By
theorem 2.2.5, F'(O X I) and F'(l X I) are single points. Hence F' is a homotopy relative to j from w to some path w" such that w"(O) = w(O) and
pow" = pow'. Since w'(O) = w(O), it follows from the unique-path-lifting
property of p that w' = w" and F': w ~ w' reI i. •
0
It follows from lemma 3 that if p: X ~ X is a fibration with unique path
lifting, then for any two objects Xo and Xl in the fundamental groupoid of X,
p# maps hom (XO,Xl) injectively into hom (p(xO)'P(Xl)). In particular, if
Xo = Xl, we obtain the following theorem.
4
THEOREM
Let p: X ~ X be a fibration with unique path lifting. For
any Xo E X the homomorphism.
P#: 7T( X,xo) ~ 7T(X,Xo)
is a monomorphism.
•
This last result provides the basis for the reduction of problems concerning fibrations with unique path lifting to problems about the fundamental
group. In order that the fundamental group be really representative of the
space in question, we assume that the spaces involved are path connected. It
follows from theorem 2 that this is no loss of generality for locally pathconnected spaces.
5 LEMMA Let p: X ~ X be a fibration with unique path lifting and
assume that X is a nonempty path-connected space. If xo, Xl E X, there is a
path w in X from p( Xo) to p( Xl) such that
P#7T(X,XO)
= h[w]P#7T(X,Xl)
Conversely, given a path w in X from p(Xo) to Xl, there is a point
such that
Xl
E p-l(Xl)
h[w]P#7T(X,Xl) = P#7T(X,Xo)
PROOF
For the first part, let w be a path in X from Xo to
h[w]7T( X,Xl). Therefore
P#7T( X,xo)
Xl.
Then 7T(X,XO)
=
= h[powJP#7T(X,xl)
and so pow will do as the path from p(xo) to P(Xl).
Conversely, given a path w in X from p(xo) to Xl, let w be a path in X
such that w(O) = Xo and pw = w. If Xl = w(l), then
h[wlP#7T( X,Xl) = P#(h[w]7T( X,Xl)) = P# 7T( X,xo)
This easily implies the following result.
•
SEc.3
73
RELATIONS WITH THE FUNDAMENTAL GROUP
6
THEOREM
Let p: X ~ X be a fibration with unique path lifting and
assume that X is a nonempty path-connected space. For Xo E pX the collection {P#7T( X,xo) I Xo E p-l(XO)} is a conjugacy class in 7T(X,XO). If w is a path
in pX from Xo to Xl, then h[w] maps the conjugacy class in 7T(X,Xl) to the conjugacy class in 7T(X,XO). •
Let p: X ~ X be a fibration and let w be a path in X beginning at Xo.
Define a map Fw: p-l(XO) X I ~ X by Fw(x,t) = w(t) and let i: p-l(XO) C X.
Then pi(x) = Fw(x,O) for x E p-l(XO). It follows from the homotopy lifting
property of p that there exists a map G w: p-l(XO) X I ~ X such that
Gw(x,O) = i(x) = x and po Gw = Fw.
Suppose now that p has unique path lifting. We prove that the map
x ~ Gw(x,l) of p-l(XO) to p-l(w(l)) depends only on the path class of w.
If w' ~ w and G~,: p-l(xO) X I ~ X is a map such that G~,(x,O) = x and
p G~, = Fw" then for any x E p-l(XO), let wand w' be the paths in X defined
by w(t) = Gw(x,t) and w'(t) = G~,(x,t). Then wand w' begin at x and
0
pow
= w ~ w' = p
0
w'
It follows from lemma 3 that w ~ w'. Then Gw(x,l) = G~'(x,l) for every
x E p-l(XO). Therefore there is a well-defined continuous map
f[wJ= p-l(w(O))
defined by f[w](x)
w'(O), then f[w]*[w']
~
p-l(w(l))
= Gw(x,l), where Gwis as above. It is clear that if w(l) =
= f[w']
0
f[w].
7 THEOREM Let p: X ~ X be a fibration with unique path lifting. There
is a contravariant functor from the fundamental groupoid of X to the cate-
gory of topological spaces and maps which assigns to X E X the fiber over X
and to [w] the function f[w]. •
The fact that f[w] is a homeomorphism for every [w] leads to the following corollary.
8
COROLLARY
If p: X ~ X is a fibration with unique path lifting and X
is path connected, then any two fibers are homeomorphic. •
If X is path connected and p: X ~ X is a fibration with unique path lifting, the number of sheets of p (or the multiplicity of p) is defined to be the
cardinal number of p-l(X) (which is independent of X E X, by corollary 8).
For a path-connected total space, the multiplicity is determined by the conjugacy class as follows.
9 THEOREM Let p: X ~ X be a fibration with unique path lifting and
assume X and X to be nonempty path-connected spaces. If Xo E X, the multiplicity ofp is the index ofp#7T(X,xo) in 7T(X,p(xo)).
PROOF
By theorem 7, 7T(X,p( xo)) acts as a group of transformations on the right
on p-l(p(XO)) by x [w] = f[w](x) for x E p-l(p(xo)). If xl. X2 E p-l(p(xO)), let
w be a path in X from Xl to X2. Then [p 0 w] E 7T(X,p(Xo)) and Xl 0 [pw] = X2.
0
74
COVERING SPACES AND FIBRATIONS
CHAP.
2
Therefore w(X,p(xo)) acts transitively on p-l(p(XO))' The isotropy group of Xo
[that is, the subgroup of w(X,p(xo)) leaving Xo fixed] is clearly equal to P#w(X,xo).
From general considerations 1 there is a bijection between the set of right
co sets of P#w(X,xo) in w(X,p(xo)) and p-l(p(XO))' •
I 0 EXAMPLE For n ~ 2 the covering p: Sn -,) pn of example 2.1.5 has
multiplicity 2. Because Sn is simply connected, w(pn) :::::: Z2 for n ~ 2.
A fibration p: X -,) X with unique path lifting is said to be regular
if, given any closed path w in X, either every lifting of w is closed or none is
closed.
I I THEOREM Let p: X -,) X be a fibration with unique path lifting. p is
regular if and only if P#w(X,xo) = p#W(X,Xl) whenever p(xo) = P(Xl).
PROOF Assume that p is regular and let W be a closed path in X at xo. Then W is a
closed lifting of pw. Therefore there is a closed lifting WI of pw at Xl. It follows that P#[w] = [pw] = P#[Wl]' Therefore P#w(X,xo) C P#W(X,Xl)' Since the
roles of Xo and Xl can be interchanged, it follows that p#w(X,xo) = P#W(X,Xl)'
Conversely, if P#w(X,xo) = p#W(X,Xl) whenever p(xo) = P(Xl), let w be a
closed path in X at p(xo) having a closed lifting W at xo. Then
[w]
= P#[w]
E P#w(X,xo)
= p#W(X,Xl)
Therefore there is a closed path WI in X at Xl such that PWI ~ w. If WI is a
lifting of w such that wI(O) = Xl, then by the unique-path-lifting property of
p, WI
WI. Therefore WI is a closed lifting of w at Xl and p is regular. •
=
In case X is a nonempty path-connected space, theorems 6 and 11 give
the following result.
12 THEOREM Let p: X -,) X be a fibration with unique path lifting and
assume that X is a nonempty path-connected space. Then p is regular if and
only if for some Xo E X o, P#w(X,xo) is a normal subgroup ofw(X,p(xo)). •
4
THE LIFTING PROBLEM
In this section we show that the fundamental group functor solves the lifting
problem for fibrations with unique path lifting. As a consequence of this, the
fundamental group functor provides a classification of covering projections,
which is discussed in the next section.
Our first result is that any map of a contractible space to the base space
of a fibration can be lifted.
I
LEMMA
Let p: E -,) B be a fibration. Any map of a contractible space
to B whose image is contained in p(E) can be lifted to E.
1 Whenever a group G acts transitively on the right on a set S there is induced a bijection
between the set of right cosets of the isotropy group (of any s E S) in G and the set S.
SEC.
4
75
THE LIFTING PROBLEM
PROOF
Let Y be contractible and let f: Y ~ B be a map such that f( Y) C p(E).
Because Y is contractible, f is homotopic to a constant map of Y to some
point of f(Y). f(Y) C p(E), so this constant map can be lifted to E. The
homotopy lifting property then implies that f can be lifted to E. •
Because we use the fundamental group functor, it will prove technically
simpler to consider the lifting problem for spaces with base points.
2
LEMMA
Let p: (X,xo) ~ (X,xo) be a fibration with unique path lifting.
If yo is a strong deformation retract of Y, any map (Y,yo) ~ (X,Xo) can be
lifted to a map (Y,yo) ~ (X,xo).
PROOF
Let f: (Y,yo) ~ (X,xo) be a map. f is homotopic relative to yo to the
constant map Y ~ Xo. The constant map can be lifted to the constant map
Y ~ xo. By the homotopy lifting property, f can be lifted to a map 1': Y ~ X
such that l' is homotopic to the constant map Y ~ Xo by a homotopy which
maps yo X I to p-l(XO). Because p-l(XO) has no nonconstant path by theorem
2.2.5, !'(yo) = xo· •
We shall apply lemma 2 to a contractible space in order to lift certain
quotient spaces of the contractible space. The usual way to represent a
space as the quotient space of a contractible space is to show it is a quotient
space of its path space. Given yo E Y, the path space P(Y,yo) is the space of
continuous maps w: (1,0) ~ (Y,yo) topologized by the compact-open topology.
There is a function cp: P(Y,Yo) ~ Y defined by cp(w) = w(l). If U is an open
set in Y,
rp-l(U)
= (l;U) = {w
E P(Y,yo) I w(l) E U}
is an open set in P(Y,yo). Therefore rp is continuous.
3
LEMMA
The constant path at yo is a strong deformation retract of the
path space P(Y,yo).
A strong deformation retraction F: P(Y,yo) X I
stant path at yo is defined by
PROOF
F(w,t)(t')
= w((l
- t)t')
~
P(Y,yo) to the con-
w E P(Y,yo); t, t' E I •
We have shown that rp is a continuous map of the contractible path space
P(Y,yo) to Y. If Y is path connected, rp is clearly surjective. If Y is also locally
path connected, the following theorem shows that rp is a quotient projection.
4
THEOREM
A connected locally path-connected space Y is the quotient
space of its path space P(Y,yo) by the map rp.
PROOF
We know that rp is continuous, and because a connected locally pathconnected space is path connected, it is surjective. To complete the proof it suffices to show that rp is an open map. Let w E P(Y,yo) and let W = n l<i<n(K i ; Ui)
be a neighborhood of w, where Ki is compact in I and Ui is op~n-in Y. We
enumerate the K's so that for some 0 ::::; k ::::; n, 1 E Kl n '" n Kk and
76
COVERING SPACES AND FIB RATIONS
CHAP.
2
I ~ K k+1 U ... U Kn. Because w(l) E UI n ... n Uk, there is a pathconnected neighborhood V of w(l) contained in UI n ... n Uk. Choose
0< t'
I such that [t',!] n (Kk+1 U ... U Kn) = 0 and w([t',l]) C V.
To prove that rp( W) ::J V, which completes the proof, let y' E V and let
w' be a path in V from w(t') to y'. Define w: 1---7 Y by
<
o :; t :;
w(t)
w(t) =
For i
> k, W(Ki)
W(Ki)
1w'G =~ )
t'
t'<t<l
= W(Ki) CUi. For i :; k,
= w(Ki
= Ui
n [O,t']) U w(K i n [t',l]) C W(Ki) U w'(I) C Ui U V
Therefore w E Wand rp( w)
= y'. Hence rp( W)
::J
V.
•
We can put these results together to obtain the following result, called
the lifting theorem.
5
THEOREM
Let p: (X,xo) ---7 (X,xo) be a fibration with unique path lifting. Let Y be a connected locally path-connected space. A necessary _and
sufficient condition that a map f: (Y,yo) ---7 (X,xo) have a lifting (Y,yo) ---7 (X,xo)
is that in 7T(X,XO)
f#7T(Y,yo) C P#7T(X,Xo)
PROOF
Iff': (Y,yo)
---7
(X,xo) is a lifting of f, then f = po f' and
f#7T(Y,yo) = P#f#7T(Y,yo) C P#7T(X,Xo)
which shows that the condition is necessary.
We now prove that the condition is sufficient. It follows from lemmas 3
and 2 that if Wo is the constant path at yo, the composite
(P(Y,yo), wo)
'P
(Y,yo)
---7
f
---7
(X,xo)
can be lifted to a map f: (P(Y,yo), wo) ---7 (X,xo). We show that if f#7T(Y,yO) C
P#7T(X,XO) and if w, w' E P(Y,yo) are such that rp(w) = rp(w'), then f(w) = j(w').
Let wand w' be the paths in P(Y,yo) from Wo to wand w', respectively,
defined by w(t)(t')
w(tt') and w'(t)(t')
w'(tt'). Then
wand f W' are
paths in X from Xo to J(w) and f(w'), respectively, such that
=
f
=
pofow=forpow=fow
and
p
f
0
0
0
0
w' = f
0
w'
Because w * W'~I is a closed path in Yat yo and f#7T(Y,yO) C P#7T(X,Xo), there
is a closed path w in X at Xo such that (f w) * (f w')-I c::::: pow. Then
0
po (f w) = f
0
0
w c::::: (p
0
w)
* (f
0
0
w') = p
0
(w
* (f
0
w'))
By lemma 2.3.3, f w c::::: W * (f w'). In particular, the endpoint of f w,
which is j(w), equals the endpoint of f w', which is j(w').
It follows that there is a function f': (Y,yo) ---7 (X,xo) such that f' rp = f,
0
0
0
0
0
SEC.
4
77
THE LIFTING PROBLEM
and using theorem 4, we see that f' is continuous. Because
pOf'ocp=pof=focp
and cp is surjective, p
0
f' = f.
Therefore f' is a lifting of f.
•
Let p: E ~ B be a fibration. A section of p is a map s: B ~ E such that
p s = IB (thus a section is a right i.nverse of p). It follows easily from the
homotopy lifting property that there is a section of p if and only if [p 1has a
right inverse in the homotopy category. Because a section is a lifting of the
identity map B C B, the following is an immediate consequence of theorem 5.
0
6
COROLLARY
Let p: (X,xo) ~ (X,xo) be a fibration with unique path lifting. If X is a connected locally path-connected space, there is a section
(X,xo) ~ (X,xo) of P if and only if P#7T(X,Xo) = 7T(X,XO). •
7
COROLLARY
Let p: X ~ X be a fibration with unique path lifting. If X
is a nonempty path-connected space and X is connected and locally path
connected, then p is a homeomorphism if and only if for some Xo E X,
P#7T(X,Xo)
= 7T(X,p(Xo)).
PROOF
If P is a homeomorphism, P#7T(X,XO) = 7T(X,p(Xo)). Conversely, if
P#7T(X,Xo) = 7T(X,p(Xo)), then by theorem 2.3.9, p is a bijection. By corollary 6,
it has a continuous right inverse. Therefore p is a homeomorphism. •
If p: X ~ X is a covering projection and X is path connected, a necessary and sufficient condition that p be a homeomorphism is that P#7T(X,Xo) =
7T(X,p(Xo)) for some Xo E X. This condition on the fundamental groups implies
that p is a bijection, and by lemmas 2.1.8 and 2.1.7, p is open; hence for covering projections corollary 7 is valid without the assumption that X be locally
path connected. This is definitely false for fibrations with unique path lifting
if X is not locally path connected, because p need not be open. The following
example shows this.
8
EXAMPLE
four sets
Let X be the subspace of RZ defined to be the union of the
Al = {(x,y) Ix = 0, -2 ~ Y ~ I}
Az = {(x,y) I 0 ~ x ~ 1, Y = -2}
A3 = {(x,y)lx = 1, -2 ~ Y ~ O}
A4 = {(x,y) I 0
x ~ 1, Y = sin 27T/X}
<
illustrated in the diagram
(0,1)
(1,0)
A,
A,
(1,-2)
78
COVERING SPACES AND FIB RATIONS
CHAP.
2
Let X be the half-open interval [0,4) and define p: X ~ X to map [O,IJ linearly
onto AI, [1,2] linearly onto A 2 , [2,3J linearly onto A 3 , and [3,4) homeomorphically onto A4 by the map t ~ (t - 3, sin(2'7T/(t - 3))). Then X and X are
path connected and p: X ~ X is a fibration with unique path lifting. However,
p is not a homeomorphism, although X and X are both simply connected.
For locally path-connected spaces the lifting theorem provides the following criterion for determining whether an open path-connected subset of
the base space is evenly covered by a fibration.
9
LEMMA
Let p: X ~ X be a fibration with unique path lifting. Assume
that X and X are locally path connected and let U be an open connected subset of X. Then U is evenly covered by p if and only if every lifting to X of a
closed path in U is a closed path.
PROOF
If U is evenly covered by p and w is a path in p~l( U), then w is a
path in some component U of p~l(U). By lemma 2.1.12, p maps U homeomorphically onto U. Therefore, if pow is a closed path in U, w is a closed
path in U. Hence the condition is necessary.
It is also sufficient, because if Xo E U and Xo E p~l(XO), the hypothesis
that every lifting of a closed path in U at Xo is a closed path in X implies that
in '7T(X,xo)
where i: (U,xo) C (X,xo)
By theorem 5, there is a lifting iio: (U,xo) ~ (X,xo) of i. The collection
{iio(U) I Xo E p~l(XO)} consists of path-connected sets which, by lemma 2.2.4,
are disjoint. We show that their union equals p~l(U). If x E p~l(U), let w be
a path in U from p( x) to Xo and let w be a lifting of w such that w(O) = x. Then
w(l) E p~l(XO), and therefore w is a path in is(1)(U). Hence x E is(I)(U) and
{i~o( U) I Xo E p~l(XO)} is a partition of p~l( U) into path-connected sets. Since
p~l( U) is open and X is locally path connected, i.fo( U) is open in X for each
Xo E p~l(XO). Clearly, p is a homeomorphism of iio( U) onto U for each
Xo E p~l(XO), and U is evenly covered by p. •
A space X is said to be semilocally I-connected if every point Xo E X has
a neighborhood N such that '7T(N,xo) ~ '7T(X,xo) is trivial.
10 THEOREM Every fibration with unique path lifting whose base space is
locally path connected and semilocally I-connected and whose total space is
locally path connected is a covering protection.
It follows from lemma 9 and the definition of semilocally I-connected
space that each point of the base space has an open neighborhood evenly
covered by the fibration. •
PROOF
SEC.
5
5
79
THE CLASSIFICATION OF COVERING PROJECTIONS
THE CLASSIFICATION OF COVERING PROJECTIONS
This section contains a classification of covering projections over a connected
locally path-connected base space. It is based on the lifting theorem and reduces the problem of equivalence of covering projections to conjugacy of
their corresponding subgroups of the fundamental group of the base space.
A large part of the section is devoted to constructing a covering projection
corresponding to a given subgroup of the fundamental group of the base
space.
Let X be a connected space. The category of connected covering spaces
of X has objects which are covering projections p: X ~ X, where X is
connected, and morphisms which are commutative triangles
-
Xl
f
--?
Pl\ I
-
Xz
pz
.A
If X is locally path connected and p: X ~ X is an object of this category,
then, by lemma 2.1.8, p is a local homeomorphism and X is also locally path
connected. We show that in this case every morphism in this category is a
covering projection.
I
LEMMA
In the category of connected covering spaces of a connected
locally path-connected space every morphism is itself a covering proiection.
Consider a commutative triangle
PROOF
where Pi and pz are covering projections and X is locally path connected. It
follows from corollary 2.1.13 that f is a covering projection if it is surjective.
Because Xz is connected and locally path connected, it is path connected.
Let Xl E Xl and Xz E Xz be arbitrary and let Wz be a path in Xz from f(xl)
to X2· Because Pi is a fibration, there is a path Wi in Xl beginning at Xl such
that Pi Wi = pz W2. By the unique path lifting of P2,j Wi = W2. Therefore
0
0
0
f(wl(l)) = wz(l) = :\:z
proving that f is surjective.
•
The next result determines when there is a morphism from one object to
another in the category of connected covering spaces of X.
2
THEOREM
Let Pi: Xl ~ X and pz: X 2 ~ X be obiects in the category
80
COVERING SPACES AND FIBRATIONS
CHAP.
2
of connected covering spaces of a connected locally path-connected space X.
The following are equivalent:
(a) There is a covering projection f: Xl ~ X2 such that P2 f = Pl.
(b) For all l:l E Xl and X2 E X2 such that Pl(X\) = P2(X2), Pl#7T(X1.Xl) is
conjugate in 7T(X,Pl(Xl)) to a subgroup of P2#7T(X2,X2)'
(c) There exist Xl E Xl and X2 E X2 such that Pl(Xl) = P2(X2) and
Pl#7T(X l ,Xl) is conjugate in 7T(X,Pl(Xl)) to a subgroup of P2#7T(X2,X2).
0
=
(a)
(b) Given f: Xl ~ X2 such that P2
X2 E X2 are such that Pl(Xl) = P2(X2), then
PROOF
Pl#7T(Xl ,Xl)
= P2#
0
f#7T(Xl ,Xl)
c
0
f
= PI,
if Xl E Xl and
P2#7T(X2,f(xl))
Because f(xl) and X2 lie in the same fiber of P2: X2 ~ X, it follows from
theorem 2.3.6 that P2#7T(X2,f(Xl)) and p2#7T(X2,X2) are conjugate in 7T(X,Pl(Xl)).
(b)
(c) The proof is trivial.
(c)
(a) Assume that Xl E Xl and X2 E X2 are such that Pl(Xl) = P2(X2)
and that Pl#7T(X1.Xl) is conjugate in 7T(X,Pl(Xl)) to a subgroup of p2#7T(X 2,X2).
By theorem 2.3.6, there is a point X2 E X2 such that P2(X2) = P2(X2) and such
that Pl#7T(Xl ,Xl) C P2#7T(X2,X2)
=
=
Because Xl is a connected locally path-connected space, the lifting theorem
implies the existence of a map f: (Xl,Xl) ~ (X 2,X2) such that P2 f = Pl. •
0
3
COROLLARY
Two objects in the category of connected covering spaces
of a connected locally path-connected space X are equivalent if and only if
their fundamental groups (at some two points over the same point of X) map
to conjugate subgroups of the fundamental group of X (at this point). •
We give two examples.
4
Because every nontrivial subgroup of 7T(5 l ) ;::::: Z is infinite cyclic, by
corollary 3 every connected covering space X ~ 51 is equivalent to
ex: R ~ 51 or to the map 51 ~ 51 sending z to zn for some positive integer n.
:; For n 2 2, 7T(pn) ;::::: Z2, and every connected covering space X ~ pn is
equivalent to the double covering 5n ~ pn or to the trivial covering pn C pn.
A universal covering space of a connected space X is an object p: X ~ X
of the category of connected covering spaces of X such that for any object
p': X' ~ X of this category there is a morphism
X ~ X'
p\ Ip'
X
in the category. It can be shown (see the paragraph following theorem 13 below)
that a universal covering space is a regular covering space. The next result follows
from this, theorem 2 and corollary 3.
6
COROLLARY
Two universal covering spaces of a connected locally pathconnected space are equivalent. •
SEC.
5
81
THE CLASSIFICATION OF COVERING PROJECTIONS
Another result also follows from theorem 2.
7
COROLLARY
A simply connected covering space of a connected locally
path-connected space X is a universal covering space of x. •
Having reduced the comparison of connected covering spaces of X to a
comparison of their corresponding subgroups of the fundamental group of X,
we shall determine which subgroups of the fundamental group correspond
to covering spaces. This necessitates the construction of covering spaces. Let
X be a space and let CU be an open covering of X. If Xo E X, let w(CU,xo) be
the subgroup of w(X,xo) generated by homotopy classes of closed paths having
a representative of the form (w * w') * w-l, where w' is a closed path lying in
some element of CU and w is a path from Xo to w'(O). The following statements
are easily verified.
8
If 'Y is an open covering of X that refines 611, then w('Y,xo) C w(ql,xo).
9
w(G/l,xo) is a normal subgroup of w(X,Xo).
10 If w is a path in X, then h[w]w(GU.,w(I))
•
•
= w(G/l,w(O)).
•
The connection of the groups w(G/l,xo) with covering projections is
explained by the following result.
I I LEMMA Let p: X ~ X be a covering proiection and let CU be a covering
of X by open sets each evenly covered by p. For any Xo E X
PROOF
If w' is a closed path lying in some element of GU., then, by lemma 2.4.9,
any lifting of w' is a closed path in X. Hence any path of the form (w * w') * w- 1 ,
where w' is a closed path lying in some element of GU., can be lifted to
a closed path (namely, to (w * w') * w-l, where wand w' are suitable liftings
of wand w', respectively]. Hence any element of w("Il,p(fo)) has a representative which can be lifted to a closed path at xo. •
The following theorem characterizes those flbrations with unique path
lifting which are covering projections.
12 THEOREM Let p: X ~ X be a fibration with unique path lifting, where
X and X are connected locally path-connected spaces. Then p is a covering
protection if and only if there is an open covering "Il of X and a point Xo E X
such that
If P is a covering projection, the desired result follows from lemma II.
Conversely, if there is such an open covering "Il and point Xo E X, it follows
from statements 9 and 10 that for any point Xo E X, w(G/l,p(xo)) c p#w(X,xo).
Using lemma 2.4.9, it follows that every element of "Il is evenly covered
by p. •
PROOF
82
COVERING SPACES AND FIB RATIONS
CHAP.
2
Lemma 11 gives a necessary condition for a subgroup of '7T(X,xo) to
correspond to a covering space. The next result proves that this necessary
condition is also sufficient.
13 THEOREM Let X be a connected locally path-connected space and let
Xo E X. Let H be a subgroup of '7T(X,xo) and assume that there is an open covering CYl of X such that '7T(Gil,xo) C H. Then there is a covering projection
p: (X,xo) ~ (X,xo) such that P#'7T(X,xo) = H.
PROOF
Suppose such a covering projection exists, and suppose, moreover,
that the space X is path connected. The projection <p: (P(X,xo),wo) ~ (X,xo)
of the path space of (X,xo) can then be lifted to a map <pI: (P(X,xo),wo) ~ (X,xo),
which is surjective. If wand WI are elements of P(X,xo), then <p1(W) = <p1(W I )
if and only if <p( w) = <p( WI) and [w * wI-I] E P#'7T(X,xo) = H. Therefore, for
path connected X there is a one-to-one correspondence between the points of
X and equivalence classes of P(X,xo) identifying w with WI if w(l) = wl(l)
and [w * WI-I] E H (the group properties of H imply that this is an
equivalence relation). Hence it is natural to try to construct X by suitably
topologizing these equivalence classes of P(X,xo). We could start with the
compact-open topology on P(X,xo) and use the quotient topology on the set
of equivalence classes, but it seems no simpler than merely topologizing the
set of equivalence classes directly, as is done below.
We consider the set of all paths in X beginning at Xo. If wand WI are two
such paths, set w - WI if w(l) = wl(l) and [w * wI-I] E H. This is an equivalence
relation, and the equivalence class of w will be denoted by <w). Let X be the
set of equivalence classes. There is a function p: X ~ X such that p( <w») =
w(l). If U is an open subset of X and w is a path beginning at Xo and ending
in U, <w, U) will denote the subset of X consisting of all the equivalence
classes having a representative of the form w * WI, where WI is a path in U
beginning at w(l).
We prove that the collection {< w, U) } is a base for a topology on X. If
<WI) E <w, U), then WI - w * w" for some path w" lying in U. If w is any path
in U beginning at wl(l), then
WI
* W-
(w
* w") * W -
w
*
(w"
*
w)
showing that <wl,U) C (w,U). Since w - WI * w"-I, (w) E (WI,U). The same
argument shows that (w,U) C (wl,U), and so (w,U) = <WI,U). Therefore, if
w" E (w,U) n (WI,UI ), then (w", U nUl) C (w,U) n (WI,UI ), and so the
collection {( w, U) } is a base for a topology on X.
Let X be topologized by the topology having {( w, U) } as a base. Then p
is continuous; for if p( (w») E U, then p( ( w, U») C u. p is also open, because
p( (w, U») clearly equals the path component of U containing w( 1), and this is
open because X is locally path connected.
Let GLL be an open covering of X such that '7T(GLL,xo) CHand let V be an
open path-connected subset of X contained in some element of GiL We show
that V is evenly covered by p, which will imply that p is a covering projection.
SEC.
5
THE CLASSIFICATION OF COVERING PROJECTIONS
83
If (W) E p-I(V), then (w,Y) C p-I(V). The sets {(w,Y) I (w) E p-I(V)}
are open and their union equals p-I(V). If (w,Y) n (w''y) =F 0, let
(w") E (w'y) n (w''y). Then (w"'y) = (w,V) and (w",V) = (w',V).
Hence the sets {(w'y) I (w) E p-I(V)} are either identical or disjoint. To
prove that V is evenly covered by p, it suffices to show that p maps each set
(w, V) bijectively to V (because p has already been shown to be continuous
and open). If x E V, let w' be a path in V from will to x. Then (w * w') E (w,Y)
and p( (w * w'») = x, showing that p is surjective. Assume p( w * WI) =
p(w * W2). Then wI(l) = w2(1), so (w * WI) * (w * w2tI is a closed path in X
at Xo. Also, [(w * WI) * (w * w2tl] = [(w * (WI * W2- 1 )) * w- I ]
Since WI * W2- I is a path in V and V is contained in some element of U,
[(w * (WI * W2- 1 )) * w- I ] E "7T(01,xo) C H. Therefore w * WI - W * W2 and
(w * 'WI) = (w * W2), showing that p is injective.
We have shown that p: X ----> X is a covering projection. Let Xo = (wo),
where Wo is the constant path in X at Xo. It remains only to verify that
p#"7T(X,xo) = H. For this we need an explicit expression for the lift of a path
in X that begins at Xo. Let w be a path in X beginning at xo, and for t E 1, define a path Wt in X beginning at Xo by Wt(t') = w(tt'). Let w: 1 ----> X be defined
by wit) = (Wt). We prove that w is continuous. If w(to) E (w',U), then
pw(to) = w(to) E U and (w',U) = (Wto'U), Let N be any open interval in 1
containing to such that wiN) C U. If tEN, then Wt - Wlo * Wto,l, where
Wto,I(t') = w(to + t'(t - to)). Therefore, for tEN
wit) = (WI) = (Wlo * Wlo,t) E (Wto,U) = (w',U)
and so w is continuous, Furthermore, pw(t) = wt(l) = wit). Hence w is a lift
of w beginning at w(O) = Xo and ending at will = (w).
If [w] E H, then w - Wo and (w) = xo. Therefore the lift w of w constructed above is a closed path in X at xo, proving that H C p#"7T(X,xo). On
the other hand, if w' is a closed path in X at Xo and pw' = w, let w be
the path in X constructed above. Since w is a lift of w beginning at xo,
it follows from the unique path lifting of p that w = w'. Therefore will =
w'(l) = fo. Since will = (w), w - wo, showing that p#"7T(X,xo) C H. •
Note that if p: J( --> X is a universal covering space it is a regular covering.
In fact, if Xo E X and ~ is a covering of X by open sets evenly covered by p
than by 2.5.11 "7T( ~p(io)) c p#1T(X,xo) c 1T(X,p(xo))
By 2.5.13 there exists a connected covering q: CY, y) --> (X, p(xo)) such that
q# 1T( Y, y) = 1T( ~ p(xo)). Since p: X --> X is universal there is a map f:X --> Y such
that qf = p. By 2.5.2, p#1T(X,xo) is conjugate in 1T(X,p(Xo)) to a subgroup
of 1T( ~ p(xo)). By 2.5.9, 1T( ~ p(xo)) is normal so we must have p 1T(X, xo) C
1T(UZ;p(Xo) and so p#1T(X,xo) = 1T(~p(Xo)) is normal.
#
A space X is semilocally I-connected (defined in Sec. 2.4) if and only if
there is an open covering 0[[ of X such that "7T(0[l,xo) = O. Hence we have the
following result.
14 COROLLARY A connected locally path-connected space X has a simply
connected covering space if and only if X is semilocally I-connected. •
From corollaries 14 and 6 and theorem 2 we obtain the next result.
15 COROLLARY Any universal covering space of a connected locally pathconnected semilocally I-connected space is simply connected. •
84
COVERING SPACES AND FIBRATIONS
CHAP.
2
Not every connected locally path-connected space has a universal covering space. We give two examples.
16 An infinite product of I-spheres has no universal covering space.
17 Let X be the subspace of R2 equal to the union of the circumferences of
circles en, with n ~ 1, where en has center at (lin, 0) and radius lin. Then
X is connected and locally path connected but has no universal covering
space.
It is possible for a connected locally path-connected space to have a universal covering space that is not simply connected. We present an example.
18 EXAMPLE Let Y1 be the cone with base X equal to the space of example 17
[Y1 can be visualized as the set of line segments in R3 joining the points of X to
the point (0,0,1)] and let Yl be the point at which all the circles of X are tangent. Let (Y2,Y2) be another copy of (Yl,Yl). Let Z = Y1 V Y2. Then Z
is connected and locally path connected but not simply connected (cf. exercise
l.G.7, a closed path oscillating back and forth from Y1 to Y2 around the
decreasing circles en is not null homotopic). However, Y1 and Y2 are each closed
contractible subsets of Z. By the lifting theorem, each of them can be lifted
to any covering space of Z, so that Yl is lifted arbitrarily and Y2 is lifted arbitrarily. Therefore any covering projection with base Z has a section. It follows
that any connected covering space of Z is homeomorphic to Z.
In the category of fibrations with unique path lifting over a fixed pathconnected base space (and with path-connected total spaces) there is always
a universal object (that is, an object which has morphisms to any other object
in the category). We sketch a proof of this fact. Let X be a path-connected
space and let ~(X) be the collection of topological spaces whose underlying
sets are cartesian products of X and the set of right co sets of some subgroup
of the fundamental group of X. It follows from theorem 2.3.9 that any fibration whose base space is X and total space is path connected is equivalent to
a fibration X --,) X, where X E ~X). Since 'X(X) is a set, those fibrations
X --,) X with unique path lifting, where X is a path-connected space in ':,,"\:(X),
constitute a set. We may form the fibered product of this set (as in Sec. 2.2).
Any path component of this fibered product is then the desired universal fibration
with unique path lifting.
If X is a connected locally path-connected space, it follows from theorem 13
that for any open covering G[l of X there is a path-connected covering space
of X whose fundamental group is isomorphic to '1T("I1,xo). This implies that if X
is a universal object in the category of path-connected fibrations over X with
unique path lifting, then '1T(X,xo) is isomorphic to a subgroup of n-u '1T("Il,xo).
In particular, if n~l'1T(G[L,xo) = 0, then X has a simply connected fibratioi" with
unique path lifting that is a universal object in the category. Thus the spaces
in examples 16 and 17 both have universal fibrations with unique path lifting
that are simply connected. The space Z of example 18 is its own universal
fibration with unique path lifting.
SEC.
6
6
85
COVERING TRANSFORMATIONS
COVERING TRANSFORMATIONS
In this section we consider a problem inverse to the one of the last section, in
which we constructed covering projections with given base space; we ask for
covering projections with given covering space. On any regular covering
space we prove that there is a group of covering transformations. The covering projection is then equivalent to the projection of the covering space onto
the space of orbits of the group of covering transformations.
Let p: X ---7 X be a fibration with unique path lifting. It is clear that
there is a group of self-equivalences of this fibration (a self-equivalence is a
homeomorphism f: X ---7 X such that p f = pl. We denote this group by
G( X I X). In case p: X ---7 X is a covering projection, G( X I X) is also called the
group of covering transformations of p. In general, there is a close analogy of
G( X I X) with the group of automorphisms of an extension field leaving a
subfield pointwise fixed.
If X is path connected, it follows from lemma 2.2.4 that two selfequivalences of p: X ---7 X that agree at one point are identical. Hence we
have the following lemma.
0
I
LEMMA
Let p: X ---7 X be a fibration with unique path lifting. If X is
path connected and Xo E X, then the function f ---7 f(xo) is an iniection of
G(X I X) into the fiber of p over p(xo). •
Theorem 2.3.9 established a bijection from the set of right cosets of
P#7T(X,Xo) in 7T(X,p(Xo)) to the fiber of p over p(xo). Combining the inverse of
this bijection with the function of lemma 1 yields an injection f from G( X I X)
to the set of right co sets of p#7T(X,Xo) in 7T(X,p(Xo)). f is defined explicitly as
follows. For any f E G(X I X) let w be a path in X from Xo to f(xo). Then pow is
a closed path in X at p(xo), and the right coset (p#7T(X,Xo)) [p w] is independent of the choice of w. The function f assigns to f this right coset.
Given Xo E X, let N(p#7T(X,Xo)) be the normalizer of p#7T(X,Xo) in 7T(X,p(Xo)).
Thus N(p#7T(X,Xo)) is the subgroup of 7T(X,p(Xo)) consisting of elements
[w] E 7T(X,p(xo)) such that p#7T(X,XO) is invariant under conjugation by [w].
N(p#7T(X,XO)) is the largest subgroup of 7T(X,p(Xo)) containing p#7T(X,XO) as a
normal subgroup.
0
2 THEOREM Let p: X ---7 X be a fibration with unique path lifting. Let X
be path connected and let Xo E X. Then f is a monomorphism of G( X I X) to
the quotient group N(p#7T(X,XO))/P#7T(X,XO). If X is also locally path connected,
f is an isomorphism.
We already know that f is an injection. We show that f is a function
from G( XI X) to the set of right co sets of P#7T( X,xo) by elements of N(p#7T(X,XO)).
PROOF
86
COVERING SPACES AND FIB RATIONS
CHAP.
2
If W is a path in X from Xo to f(xo), there is a commutative square
7T(X,p(XO)) ~ 7T(X,p(Xo))
Since J: (X,xo) ~ (X,f(xo)) is a homeomorphism,
f#7T(X,XO) = 7T(X,f(XO))
and since P#f# = P#,
h(po.,JP# 7T(X,:fo) = h(po.,JP#f# 7T(X,:fo)
= P# h(.,J7T(X,f(:fo))
=
=
h(po.,JP# 7T(X,f(:fo))
P# 7T(X,f(:fo))
Hence [p w] E N(p#7T(X,XO)). Because tf;(f) is equal to the right coset
(p#7T(X,XO)) [p W], tf; is an injection of G(X I X) into the set of right cosets of
P#7T(X,XO) by elements of N(p#7T(X,XO)).
We now verify that tf; is an homomorphism. If h h E G(X I X) let WI and
W2 be paths in X from Xo to fl(XO) and h(xo), respectively. Then iI W2
is a path from fl(XO) to fd2(XO), and WI * (iI (2) is a path from Xo to fd2(XO).
Therefore tf;(fd2) is the right coset
0
0
0
0
(p#7T(X,XO))[(p
WI)
0
* (p
0
fl
0
(2)] = (p#7T(X,Xo))[p
0
WI]
* [p
0
W2]
and this equals tf;(iI)tf;(f2).
Finally, we show that if X is locally path connected, tf; is an epimorphism
to the set of right cosets of P#7T( X,xo) in N(P#7T( X,xo)). Assume that [w] E
7T(X,p(Xo)) belongs to N(p#7T(X,Xo)). Let W be a lifting of wending at Xo and
let x = w(O). Then
P#7T(X,Xo)
= h[wJ(p#7T(X,XO)) = P#(h[wJ7T(X,xo)) = P#7T(X,X)
Because X is connected and locally path connected, the lifting theorem
implies the existence of maps J: (X,xo) ~ (X,x) and g: (X, x) ~ (X,xo) such
that p f = p and p g = p. From the unique-lifting property (lemma 2.2.4),
it follows that fog = Ii and go f = Ii. Therefore f E G(X I X) and tf;(f)
equals the right coset (p#7T(X,Xo))[w]-I. •
0
0
Combining theorem 2 with theorem 2.3.12, we have the following
corollary.
3
COROLLARY
Let p: X ~ X be a fibration with unique path lifting. If X
is connected and locally path connected and Xo E X, then p is regular if and
only if G(X I X) is transitive on each fiber of p, in which case
tf;: G(X I X) :::::: 7T(X,p(XO))/P#7T(X,XO)
•
SEC.
6
COVERING TRANSFORMATIONS
87
If X is simply connected, any fibration p: X ~ X is regular, and we also
have the next result.
COROLLARY
Let p: X ~ X be a fibration with unique path lifting,
where X is simply connected, locally path connected, and nonempty. Then
the group of self-equivalences of p is isomorphic to the fundamental group of
4
X.
-
If p: X ~ X is a regular covering projection and X is connected and
locally path connected, then X is homeomorphic to the space of orbits of
G(X I X) (an orbit of a group of transformations G acting on a set S is
an equivalence class of S with respect to the equivalence relation Sl - S2 if
there is g E G such that gSl = S2). We are interested in the converse problem
-that is, in knowing what conditions on a group G of homeomorphisms of a
topological space Y will ensure that the projection of Y onto the space of
orbits YIG is a regular covering projection whose group of covering transformations is equal to G.
A group G of homeomorphisms of a topological space Y is said to be discontinuous if the orbits of G in Yare discrete subsets of Y. G is properly
discontinuous if for y E Y there is an open neighborhood U of y in Y such
that if g, g' E G and g U meets g' U, then g = g'. G acts without fixed points
if the only element of G having fixed points is the identity element. The
following are clear.
:. A properly discontinuous group of homeomorphisms is discontinuous
and acts without fixed points. 6
A finite group of homeomorphisms acting without fixed points on a
Hausdorff space is properly discontinuous. -
If G is the group of covering transformations of a covering projection,
then a simple verification shows that G is properly discontinuous. We now
show that any properly discontinuous group of homeomorphisms defines a
covering projection.
7 THEOREM Let G be a properly discontinuous group of homeomorphisms
of a space Y. Then the protection of Y to the orbit space YI G is a covering
protection. If Y is connected, this covering protection is regular and G is its
group of covering transformations.
PROOF
Let p: Y ~ YIG be the projection. Then p is continuous. It is
an open map, for if U is an open set in Y, then p-1(p(U)) = U {gU I g E G}
is open in Y, and therefore pUis open in YI G. Let U be an open subset of Y
such that whenever gU meets g'U, then g = g'. We show that p(U) is evenly
covered by p. The hypothesis on U ensures that {gU I g E G} is a disjoint collection of open sets whose union is p-1(p(U)). It suffices to prove that p I gU
is a bijection from gU to p(U). If Y E U, then p(gy) = p(y), so p(gU) = p(U).
If P(gY1) = P(gY2), with Y1, Y2 E U, there is g' E G such that gY1 = g'gY2'
88
COVERING SPACES AND FIBRATIONS
CHAP.
2
Therefore gU meets g'gU, and g = g'g. Hence g' = ly and gYl = gY2' We
have proved that p is a homeomorphism of gU onto p( U). Since G is properly
discontinuous, the sets p( U) evenly covered by p constitute an open covering
of YIG.
Because p(gy) = p(y), we see that G is contained in the group of
covering transformations of p. Since G is transitive on the fibers of p, it
follows from theorem 2.2.2 that if Y is connected, G equals the group of covering transformations. Since the group of covering transformations is transitive on each fiber, the covering projection is regular. U
COROLLARY
Let G be a properly discontinuous group of homeomorphisms
of a simply connected space Y. Then the fundamental group of the orbit
space YI G is isomorphic to G.
By theorem 7, G is the group of covering transformations of the regular covering projection p: Y -c1> YI G. By theorem 2, If; is a monomorphism of
G into the fundamental group of YI G. Because G is transitive on the fibers of
p, If; is an isomorphism. -
PROOF
9
EXAMPLE
Let S3 = {(ZO,Zl) E C211zo12 + IZll2 = I} and let p and q be
relatively prime integers. Define h: S3 -c1> S3 by
h(zo,zl)
= (e27TiIPzo,e27TQiIPZl)
Then h is a homeomorphism of S3 with period p (that is, hP
on S3 by
n(ZO,Zl)
= 1), and Zp acts
= hn(ZO,Zl)
where n denotes the residue class of the integer n modulo p. In this way Zp
acts without fixed points on S3. The orbit space of this action of Zp on S3 is
called a lens space and is denoted by L(p,q). By statement 6 and corollary 8,
the fundamental group of L(p,q) is isomorphic to Zp.
10 EXAMPLE Let S2n+l = {(ZO,Zl, . . . ,zn) E Cn+l I ~ IZil2 = I} and let
ql, . . . ,qn be integers relatively prime to p. Define h: S2n+1 -c1> S2n+l by
Then, as in example 9, h determines an action of Zp on S2n+1 without fixed
points; the orbit space is called a generalized lens space and is denoted by
L(P,ql, . . . ,qn). Its fundamental group is isomorphic to Zp.
It is possible to use theorem 7 to show that the projection Y -c1> YIG is
a regular fibration with unique path lifting even when it may not be a covering projection. Note that if G acts on Y without fixed points, so does any subgroup of G, and if G' is a normal subgroup of Y, then GIG' acts without fixed
points on YI G'.
I I THEOREM Let G be a group of homeomorphisms acting without fixed
points on a path-connected space Y and assume that there is a decreasing
sequence of subgroups
SEC.
7
89
FIBER BUNDLES
such that
(a ) n G n = {!y}
(b) G n+1 is a normal subgroup of Gn for n ~ 0
(c) Gn/G n+1 is a properly discontinuous group of homeomorphisms on
Y/G n+1 and the proiection Y -3> Y/G n is a closed map for n ~ 0
(d) Any orbit of Y under Gn for n ~ 0 is compact
Then the proiection p: Y -3> Y/G is a regular fibration with unique path
lifting whose group of self-equivalences is G.
Since Y/G n
7 that the projection
PROOF
= (Y/Gn+I)/(Gn/Gn+I)' it follows from (c)
and theorem
pn+l: Y/Gn+1 -3> Y/Gn
is a regular covering projection for n
~
O. Let
Y = {(Yn) E X (Y/Gn) I Pn+I(Yn+l) = Yn for n ~ O}
p: Y-3> Y/G by p((Yn)) = yo. It is easy to verify that P is a fibra-
and define
tion with unique path lifting (it is the fibered product of the maps
{plo ... Pi}).
For n ~ 0 there is a continuous closed projection map cpn: Y -3> Y/Gn
such that pn+1 CPn+1 = CPn. Therefore there is a continuous closed map
cP: Y -3> Y defined by cp(y) = (CPn(Y)) and such that P cP = p. To prove that cP
is a homeomorphism, it suffices to show that it is a bijection. If cp(y) = cp(Y'),
then for n ~ 0 there is gn E Gn such that Y = gnY'· Then gnY' = gmY' for all
m and n, and because G acts without fixed points, gm = gn for all m and n.
Therefore gn E G m for all m, and by (a), gn = !y. It follows that Y = y', and
hence that cP is injective.
If (Yn) E Y, then CPn -IYn is an orbit of Y under Gn. By (d), CPn -IYn is
compact. Since
0
0
0
CPn -IYn = CP;;~IP;;~IYn :J CP;;~IYn+l
the collection {CPn -IYn} consists of compact sets having the finite-intersection
property. Therefore n CPn -IYn =1= 0. If yEn CPn -IYn, then cp(y) = (Yn),
showing that cP is surjective.
We have shown that cP: Y -3> Yis a homeomorphism. Therefore p: Y -3> Y/ G
is a fibration with unique path lifting. Since each element of G is a selfequivalence of p, the group of self-equivalences of p is transitive on each
fiber. By corollary 3, p is a regular fibration and G is the group of selfequivalences of p. •
7
FIBER BUNDLES
A covering space is locally the product of its base space and a discrete space.
This is generalized by the concept of fiber bundle, defined in this section,
because the total space of a fiber bundle is locally the product of its base
90
COVERING SPACES AND FIBRATIONS
CHAP.
2
space and its fiber. The main result is that the bundle projection of a fiber
bundle is a fibration. l
A fiber bundle ~ = (E,B,F,p) consists of a total space E, a base space B,
a fiber F, and a bundle projection p: E ----'> B such that there exists an open
covering {U} of B and, for each U E {U}, a homeomorphism CPu: U X F----,>
p-l( U) such that the composite
UX F
<Pc
~
p-l(U)
p
----'>
U
is the projection to the first factor. Thus the bundle projection p: E ----'> Band
the projection B X F ----'> B are locally equivalent. The fiber over b E B is defined to equal p-l(b), and we note that F is homeomorphic to p-l(b) for every
b E B. Usually there is also given a structure group C for the bundle consisting
of homeomorphisms of F, and we define this concept next.
Let C be a group of homeomorphisms of F. Given a space F' and a collection <P = {cp} of homeomorphisms cp: F ----'> F', define cpg: F ----'> F' for cp E <P
and g E C by cpg(y) = cp(gy) for y E F. The collection <P is called a C structure on F'if
(a) Given cp E <P and g E C, then cpg E <P
(b) Given CPl, cpz E <P, there is g E C such that CPl = CPzg
Condition (a) implies that C acts on the right on <P, and condition (b) implies
that this action of C is transitive on <P. A fiber bundle (E,B,F,p) is said to have
structure group C if each fiber p-l(b) has a C structure <p(b) such that there
exists an open covering {U} of B and, for each U E {U}, a homeomorphism
CPu: U X F ----'> p-l( U) such that for b E U, the map F ----'> p-l(b) sending x to
cpu(b,x) is in <p(b). It is clear that a given fiber bundle can always be given the
structure of a fiber bundle with structure group the group of all homeomorphisms of F. It is also clear that a given fiber bundle can sometimes be given
the structure of a fiber bundle with two different structure groups of homeomorphisms of F.
An n-plane bundle, or real vector bundle, is a fiber bundle whose fiber is
Rn and whose structure group is the general linear group CL(Rn), which consists of all linear automorphisms of Rn. A complex n-plane bundle, or complex
and whose structure group
vector bundle, is a fiber bundle whose fiber is
is CL(cn).
We give some examples.
en
I
For spaces Band F the product bundle is the fiber bundle (B X F, B, F, p),
where p: B X F ----'> B is projection to the first factor (it has the trivial group
as structure group).
2 Given that p: g ---t X is a covering projection and X is a connected and
locally path connected space, if Xo E X, then (X,X,p-l(Xo),p) is a fiber bundle
(and if X is path connected, it can be given the structure of a fiber bundle with
1 For the general theory of fiber bundles see N. E. Steenrod, The Topology of Fibre Bundles,
Princeton University Press, Princeton, N.J., 1951.
SEC.
7
FIBER BUNDLES
91
structure group 7T(X,Xo), where 7T(X,Xo) acts on p-1(Xo) by [w]£ = £[W]-1, with the
right-hand side as in the proof of theorem 2.3.9).
3
Given that M is a differentiable n-manifold and T(M) is the set of all tangent vectors to M, there is a fiber bundle (T(M ),M,Rn,p), where p: T(M) ---7 M
assigns to each tangent vector its origin. This is called the tangent bundle
and is denoted by r(M). Because it can be given the structure group GL(Rn),
it is an n-plane bundle, and if M is a complex manifold of complex dimension
m, then r(M) is a complex m-plane bundle.
4
Given that H is a closed subgroup of a Lie group G and that GIH is the
quotient space of left cosets and p: G ---7 G I H the projection, then (G, GI H,H,p)
is a fiber bundle (having structure group H acting on itself by left translation).
5 Represent 5n as the union of closed hemispheres E"- and E~ with intersection 5n - 1 and let G be a group of homeomorphisms of a space F. Given a
map cp: 5n - 1 ~ G such that the map 5n - 1 X F ~ F sending (x,y) to cp(x)y is
continuous, let Eq; be the space obtained from (E"- X F) v (E'l- X F) by identifying (x,y) E E"- X F with (x,cp(x)y) EE~ X F for x E 5n - 1 and y E F. These
identifications are compatible with the projections E"- X F ~ E"- and
E':- X F ~ E~. Therefore there is a map pq;: Eq; ~ 5n such that each of the
composites
P. 5
and
E~ X F ---7 Eq; ----7 n
is projection to the first factor. Then (E<p,5 n,F,p<p) is a fiber bundle (having
structure group G) which is said to be defined by the characteristic map cpo
6
Let Pn(C) be the n-dimensional complex projective space coordinatized
by homogeneous coordinates. If Zo, Zl, . . . , Zn E C are not all zero, let
[zo,Zl, . . . ,znl E Pn(C) be that point of Pn(C) having homogeneous coordinates Zo, Zl, . . . , Zn. Regard 52n +1 as the set {(ZO,Zl, . . . ,zn) E Cn+1 I
~ IZil2 = I} and define p: 52n +1 ---7 Pn(C) by P(ZO,Zl, ... ,zn) = [zo,Zl, ... ,znl
If Ui C Pn(C) is the subset of points having a nonzero ith homogeneous
coordinate, it is easy to see that p-1( Ui ) is homeomorphic to Ui X 51. Therefore there is a fiber bundle (5 2n +1,Pn(C),5 1,p) (having structure group 51 acting
on itself by left translation), and this is called the Hopf bundle.
7
If Q is the division ring of quaternions, there is a similar map
p: 54n +3 ---7 Pn(Q) and a quaternionic Hopf bundle (54n+3,Pn(Q),53,p) (having
structure group 53 acting on itself by left translation).
The structure group will not be important for our purposes. Thus we
define an n-sphere bundle to be a fiber bundle whose fiber is 5n [usually it is
also required that it have as structure group the orthogonal group O(n + 1) of
all isometries in GL(Rn+1 )]. If ~ is an n-sphere bundle, we shall denote its
total space by E~. The mapping cylinder of the bundle projection E~ ---7 B is
the total space E~ of a fiber bundle (E~,B,En+l,p~), where P( E~ ---7 B is the retraction of the mapping cylinder to B (and p< I E~: E< ---7 B is the original
bundle projection).
92
COVERING SPACES AND FIB RATIONS
CHAP.
2
If ~ = (E,B,Rn+l,p) is an (n + I)-plane bundle having structure group
O(n + 1), it is possible to introduce a norm in each fiber p-l(b). The subset
E' C E of all elements in E having unit norm is the total space of an n-sphere
bundle (E', B, Sn, pIE') called the unit n-sphere bundle of ~. If the base space
B of an (n + I)-plane bundle is a paracompact Hausdorff space, the bundle
can always be given O(n + 1) as structure group. In particular, there is a unit
tangent bundle of a paracompact differentiable manifold.
Two fiber bundles (El,B,F,Pl) and (E 2,B,F,P2) with the same fiber and
same base are said to be equivalent if there is a homeomorphism h: El ~ E2
such that P2 h = Pl. If they both have structure group G, they are equivalent over G if there is a homeomorphism h as above, with the additional
property that if cp E <l>l(b), then h cp E <I>2(b) for b E B. A fiber bundle is
said to be trivial if it is equivalent to the product bundle of example 1 (or,
equivalently, if it can be given the trivial group as structure group).
In view of example 2, fiber bundles are related to covering spaces in
much the same way that fibrations are related to fibrations with unique path
lifting. The rest of this section is devoted to a proof of the fact that in a fiber
bundle (E,B,F,p) whose base space B is a paracompact Hausdorff space the
map p is a fibration.
A map p: E ~ B is called a local fibration if there is an open covering
{U} of B such that p I p-l( U): p-l( U) ~ U is a fibration for every U E {U}.
It is clear that a fibration is a local fibration 1 and that any bundle projection
is a local fibration.
Given a map p: E ~ B, we define a subspace BeE X BI by
0
0
B
= ((e,w)
E E X BI I w(O)
There is a map p: EI ~ B defined by p(w)
A lifting function for p is a map
A:
= p(e)}
= (w(O), pow)
for
w:
I ~ E.
B ~ EI
which is a right inverse of p. Thus a lifting function assigns to each point
e E E and path w in B starting at p(e) a path A(e,w) in E starting at e that is a
lift of w. The relation between lifting functions and fibrations is contained in
the following theorem.
8
THEOREM
A map p: E
lifting function for p.
~
B is a fibration if and only if there exists a
PROOF
The proof involves repeated use of theorem 2.8 in the Introduction.
If p is a fibration, letf': B ~ E and F: ii X I ~ B be defined by f'(e,w) = e
lOur proof of the converse for paracompact Hausdorff spaces B can be found in W. Hurewicz,
On the concept of fibre space, Proceedings of the National Academy of Sciences, U.S.A., vol.
41, pp. 956-961 (1955). Another proof can be found in W. Huebsch, On the covering homotopy theorem, Annals of Mathematics, vol. 61, pp. 555-563 (1955). Generalizations and
related questions are treated in A. Dold, Partitions of unity in the theory of fibrations, Annals
of Mathematics, vol. 78, pp. 223-255 (1963).
SEC.
7
93
FIBER BUNDLES
and F((e,w), t) = w(t). Then
F((e,w),O)
= w(O) = p(e) = (p
0
f')(e,w)
By the homotopy lifting property of p, there is a map F': B X I ~ E such
that F'((e,w), 0) = f'(e,w) = e and p F' = F. F' defines a lifting function A
for p by 11.( e,w)( t) = F'( (e,w), t).
Conversely, if Ais a lifting function for p, let f': X ~ E and F: X X I ~ B
be such that F(x,O) = pf'(x). Let g: X ~ BI be defined by g(x)(t) = F(x,t).
There is a map F': X X I ~ E such that F'(x,t) = A(f'(x),g(x))(t). Because
F'(x,O)
f'(x) and p F' F, P has the homotopy lifting property. •
0
=
0
=
iT be defined by
Let p: E ~ B and let W be a subset of BI. Let
W = {(e,w,s)
E E X W X I I w(s)
= p(e)}
An extended lifting function over W is a map
A: W~ EI
such that p(A(e,w,s)(t)) = w(t) and A(e,w,s)(s) = e. Thus an extended lifting
function is a function which lifts paths to paths that pass through a given
point of E at a given parameter value. It is reasonable to expect the following
relation between the existence of lifting functions and extended lifting
functions.
9
LEMMA
A map p: E ~ B has a lifting function if and only if there is
an extended lifting function over BI.
If A is an extended lifting function over BI, a lifting function Afor p is
defined by A(e,w) = A(e,w,O).
To prove the converse, given a path w in B, let Ws and Ws be the paths in
B defined by
PROOF
ws(t)
w(s - t)
= ( w(O)
S(t) _ (w(s + t)
w
- w(1)
O:=;t:=;s
s:=;t:=;1
0:=;t:=;1-s
1-s<t<1
The maps (w,s) ~ Ws and (w,s) ~ WS are continuous maps BI X I ~ BI.
Given a lifting function A: B ~ EI for p, we define an extended lifting function A over BI by
A(e,ws)(s - t)
A(e,w,s)(t) = ( A(e,wS)(t _ s)
The main step in proving that a local fibration is a fibration is the fitting
together of extended lifting functions over various open subsets of BI. For this
we need an additional concept. A covering {W} of a space X is said to be
numerable if it is locally finite and if for each W there is a function
fw: X ~ [0,1] such that W
{x E X I fw(x) =1= O}.
=
94
COVERING SPACES AND FIBRATIONS
CHAP.
2
10 LEMMA Let p: E ~ B be a map. If there is a numerable covering {Wj}
of B1 such that for each i there is an extended lifting function over Wj, then
there is a lifting function for p.
PROOF
Let the indexing set be J = U} and for each i let fj: B1 ~ I be a map
such that Wj
{w E B1 Ifj(w) =1= O}. For any subset a C !let W"
Uh " Wj
and define f,,: BI ~ R by
=
=
f,,(w)
= L.jE"fj(W)
(this is a finite sum and is continuous because {Wj} is locally finite). Then
f,,(w) ~ 0 for w E B1 and
W"
= {w
E B1 I f,( w) =1= O}
We define B" = {(e,w) E B I w E W,,},
Consider the set of pairs (a,A,,), where a C J and A,,: B" ~ E1 is a lifting
function over B" [that is, A,,(e,w)(O)
e and pA,,(e,w)(t)
w(t)). We define a
partial order S in this set by (a,A,,) S ({3,Ap) if a C {3 and A,,(e,w) = Ap(e,w)
whenever (e,w) E B" and f,,(w)
fp(w) [so if (e,w) E B" and A,,(e,w) =1= Ap(e,w),
then w E W j for some i E {3 - a).
To prove that every simply ordered subset {ai,A"I} has an upper bound,
let {3 = U ai. We shall define Ap: Bp ~ E1 so that (ai,A"I) S ({3,Ap) for all i.
Let U be any open subset of Wp meeting only finitely many W j with i E {3,
say Wj" ... , Wjr (Wp can be covered by such sets U). Choose i so that
it, . . . , ir all belong to ai· Then if ai C ak, f"l I U = f"k I U. Because
(ai,A"I) S (ak,A"k)' it follows that A"j(e,w) = A"k(e,w) for (e,w) E B"l' with
w E U. Therefore there exists a map Ap: Bp ~ EI such that Ap(e,w) = A"j(e,w)
for ai sufficiently large. We now show that (ai,A"I) S ({3,Ap). If (e,w) E B"l
and A"l(e,w) =1= Ap(e,w), there exists ak such that (ai,A"I) S (ak,A"k) and
A"l(e,w) =1= A"k(e,w). This implies w E Wj for some i E ak - ai. Therefore
w E Wi for some i E {3 - ai, hence (ai,A"i) S ({3,Ap).
By Zorn's lemma, there is a maximal element (a,A,,). To complete the
proof we need only show that a
J. If a =1= J, let io E J - a and let
{3 a U {to}. Define g: Wp ~ R by g(w) f,,(w)/fp(w). Then 0 S g(w) S 1,
g(w) =1= 0 ¢=:> W E W", and g(w) =1= 1 ¢=:> W E W io ' Define f.L: Bio ~ E by
=
=
=
=
=
f.L(e,w) =
[~,,(e'W)(2g(W) A,,(e,w)(g(w))
=
%)
Os g(w) S ¥oJ
S g(w) S %
% S g(w) S 1
¥oJ
Then f.L is continuous. Let A be an extended lifting function over Wjo and
define Ap: Bp ~ E1 by
Ap(e,w)(t)
=
A(e,w,O)(t)
A,,(e,w)(t)
Os t S 2g(w) A(f.L(e,w), w, 2g(w) - %)(t) 2g(w) - % S t S 1
A,,(e,w)(t)
Os t s g(w)
}
A(f.L(e,w), w, g(w))(t)
g(w) S t S 1
%}
Os g(w) S ¥oJ
¥oJ
S g(w) S %
% S g(w) S 1
Then Ap is a well-defined lifting function over Wp. Moreover, for (e,w) E 13",
SEC.
7
95
FIBER BUNDLES
if Aa(e,w) =1= Aj3(e,w), then g(w) =1= 1 and w E Wjo' Since io E f3 - a, this
means that (a,A a) (f3,Aj3), contradicting the maximality of (a,Aa). In case p has unique path lifting, lemma lO would hold for any open
covering {Wj} of BI such that there is a lifting function over Wj for each i
(because the uniqueness of lifted paths enables the extended liftings to be
amalgamated to a lifting for p). This was used in the proof of the theorem
that a covering projection is a fibration (theorem 2.2.3), which was valid without any assumption on the base space.
<
I I LEMMA Given a map p: E ---? B and subsets Ub .
that there is an extended lifting function over U1I, U2I , .
the subset of BI defined by
W
= {w E BI I w ([ i
Uk of B such
Ui, let W be
.,
,
k 1 , ~J) C U for i = 1, . . . , k}
i
Then there is an extended lifting function over W.
PROOF
Let Ai be an extended lifting function over U/ for i = 1, . . . , k.
Given a path w E W, let Wi be the path equal to w on [(i - l)/k, ilk] and
constant on [0, (i - l)/k] and on [ilk, 1]. Given (e,w,s) E W such that
(n - l)/k :::;; s :::;; nlk, define ei E E for i = 0, . . . , k inductively so that
en-l
en
= An(e,wn,s) ( n k 1 )
= An(e,wn,s)( ~)
and
An extended lifting function A over W is defined by
Ai (ei,wi'
A(e,w,s)(t) =
1
)(t)
An(e,wn,s)(t)
i-1<t<i.<n-1
k
-kk
n-1<t<~
k
- - k
~<i<t<i+1
k - k-
-
k
We are now ready for the main result on the passage from a local fibration to a fibration.
12 THEOREM Given a map p: E ---? B and a numerable covering GIL of B such
that for U E GIL, P I p-l(U): p-l(U) ---? U is a fibration, then p is a fibration.
PROOF
Let GIL = {Uj } and for k 2: 1, given a set of indices
be the subset of BI defined by
fl, . . . ,ik, let
With ... ik
W hh ... ik
= {w E BI I w([ i k 1 , ~])
C Uji> i
= 1,
... , k}
96
COVERING SPACES AND FIBRATIONS
CHAP.
2
It is then clear that the collection {Whh ... h} (with k varying) is an open
covering of BI, and by lemma 11, each set Whiz ... ik has an extended lifting
function. For k fixed the collection {Whh ... ik} is locally finite. In fact, if
W E BI, for each i = 1, ... , k there is a neighborhood Vi of w([(i - l)lk, ilk])
meeting only finitely many Vj. Then n1,,;i,,;k <[(i - l)lk, ilk], Vi) is a neighborhood of W meeting only finitely many {Whh .. ik}'
For each 1let fr B ~ I be a continuous map such that jj(b) =I=- 0 if and
only if b E Vj. Define
h: BI ~ I by
h, ...
h,,,.h(W) =inf{Aw(t)I
ikl :=::;t:=::; ~,i=l, ... ,k}
Thenjj, .. . h(w) =I=- 0 if and only if wE W h ... h'
Unfortunately, the collection {Whiz ... h} (all k) is not locally finite,
otherwise the proof would be complete by lemma 10. This difficulty is circumvented by modifying the sets Whiz ... ik' Since for fixed m the collection
{Wh ... h} with k
m is locally finite, the sum of the functions h, ... ik with
k
m is a continuous real-valued function gm on BI. Define
<
<
!i! ... jm
= inf(sup(O,.ii, ... jm
- mgm), 1)
Then!i, ... jm: BI ~ I and we define Wi! ... jm = {w E BI l!i! ... jm(w) =I=- OJ.
Clearly, Wi! ... jm C Wj, ... jm; therefore there is an extended lifting function
over Wi, ... jm' To complete the proof, it follows from lemma 10 that we need
only verify that {Wi, ... ik} (with k varying) is a locally finite covering of BI.
For w E BI, let m be the smallest integer such that
jm(w) =I=- 0 for
some it, ... ,1m. Then gm(w)
0 and!i, .. . jm(w) h, .. .jm(w) =I=- o. Therefore w E Wi, ... jm' proving that {Wi! .. , jm} is a covering of BI. To show that it
is locally finite, assume chosen so that
m and
im(W)
liN. Then
~(w)
liN and NgMW)
1. Hence N~(W')
1 for all Wi in some neighborhood Vof w. Therefore all functions f il ... ik with k 2 N vanish on V. But this
means that the corresponding set Wi! ... ik is disjoint from V. Since the collecN is locally finite, the collection {Wi! . " ik} (all k) is
tion {Wi, ... ik} with k
locally finite. -
=
>
N
>
=
N>
>
hI'"
.ii! ...
>
<
The fact that any open covering of a paracompact Hausdorff space has a
numerable refinement, leads to our next theorem.
13 THEOREM If B is a paracompact Hausdorff space, a map p: E ~ B is a
fibration if and only if it is a local fibration. A bundle projection is a local fibration. Therefore, we have the following
corollary.
14 COROLLARY If (E,B,F,p) is a fiber bundle with base space B paracompact and Hausdorff, then p is a fibration. -
8
FIURATIONS
This section contains a general discussion of fibrations. We establish a relation
between cofibrations and fibrations which allows the construction of fibrations
from cofibrations by means of function spaces. We also prove that every map
is equivalent, up to homotopy, to a map that is a fibration (this dualizes a
SEC.
8
97
FIB RATIONS
similar result concerning cofibrations). The section contains definitions of the
concepts of fiber homotopy type and induced fibration and a proof of the
result that homotopic maps induce fiber-homotopy-equivalent fibrations.
We begin with an analogue of theorem 2.7.8 for cofibrations. Given a
map f: X' ~ X, let X be the quotient space of the sum (X' X I) v (X X 0),
obtained by identifying (x',O) E X' X I with (f(x'),O) E X X 0 for all x' EX'.
We use [x',t] and [x,O] to denote the points of X corresponding to (x',t) E X' X I
and (x,O) E X X 0, respectively. Then [x',O] = [f(x'),O]. There is a map
i:
X~
X X I
defined by
i[x',t]
i[x,O]
= (f(x'),t)
= (x,O)
x' EX', tEl
x EX
A retracting function for f is a map
p: X X I ~
X
which is a left inverse of [. In case f is a closed inclusion map, so is [, and a retracting function for f is a retraction of X X I to t!le subspace X' X I U X X O.
I
THEOREM A map f: X'
retracting function for f.
~
X is a cofibration if and only if there exists a
PROOF
If f is a co fibration, let g: X ~ X and G: X' X I ~
defined by g(x) = [x,O] and G(x',t) = [x',t]. Because
G(x',O)
X be the
maps
= [x',0] = [f(x'),O] = gf'(x)
it follows from the fact that f is a co fibration that there exists a map p: X X I ~ X
such that p(x,O) = g(x) and p(f(x'),t) = G(x',t). Then p is a retracting function
for f.
Conversely, given maps g: X ~ Y and G: X' X I ~ Y such that
G(x',O) = gf(x') for x' E X', define
G:X~Y
=
by G[x',t]
G(x',t) and G[x,O] = g(x). If p: X X I ~ X is a retracting function for f, the map F = Gop: X X I ~ Y has the properties F(x,O) = g(x)
and F(f(x'),t) = G(x',t), showing that f is a cofibration. This leads to the following construction of fibrations from cofibrations.
2
THEOREM
Let f: X' ~ X be a cofibration, where X' and X are locally
compact Hausdorff spaces, and let Y be any space. Then the map p: yx ~ yX'
defined by p(g) = g f is a fibration.
0
Let p: X X I ~ X be a retracting function for
theorem 1). Then p defines a map
PROOF
f
(which exists by
p': yx ~ yXXJ
such that p'(g) = gop for g: X ~ Y. Because X' and X are locally compact
98
COVERING SPACES AND FIB RATIONS
CHAP.
2
Hausdorff spaces, so is X, and by theorem 2.9 in the introduction, yXXI::::; (yX)I
and
yX::::; {(g,G) E yx X (yx)ll go f
= G(O)}
Therefore p' corresponds to a lifting function for p: yx
theorem 2.7.8, p is a fibration. •
~ yx,
and by
3
COROLLARY
For any space Y let p: yI ~ Y X Y be the map p(w)
(w(O),w(l)) for w: I ~ y. Then p is a fibration.
=
PROOF
Because i X I U I X 0 is a retract of I X I, the inclusion map i C I
is a co fibration [equivalently, the pair (I,~ has the homotopy extension property with respect to any space]. The result follows from theorem 2 and the
observation that yi is homeomorphic to y X Y under the map g ~ (g(O),g(l))
for g: i ~ Y. •
Let f: B' ~ Band p: E ~ B be maps and let E' be the subset of B' X E
defined by
E' = {(b',e) E B' X E I f(b') = p(e)}
E'is called the fibered product of B' and E (more precisely, the fibered product
of f and p; cf Sec. 2.2). Note that there are maps p': E' ~ B' and 1'; E' ~ E
defined by p'(b',e) = b' and f' (b',e) = e. E' and the maps p' and f' are
characterized as the product of f: B' ~ Band p: E ~ B in the category
whose objects are continuous maps with range B and whose morphisms are
commutative triangles
Xl
h
~
Xz
gl\ Igz
B
The following properties are easily verified.
4
If pis in;ective (or sur;ective), so is p'.
•
:; If p: B X F ~ B is the trivial fibration, then p': E'
the trivial fibration B' X F ~ B'. •
~
B' is equivalent to
6
If P is a fibration (with unique path lifting), so is p'.
7
If P is a fibration, f can be lifted to E if and only if p' has a section.
•
•
Note that since the fibered product is symmetric in Band E (or rather,
in f and p), there is a similar set of statements where p and p' are replaced by
f andf'.
If p: E ~ B is a fibration (or covering projection) and f: B' ~ B is a map,
then, by property 6 (or property 5), p': E' ~ B' is a fibration (or covering
projection) and is called the fibration induced from p by f (or covering pro;ection induced from p by f). If ~ = (E,B,F,p) is a fiber bundle and f: B' ~ B
is a map, it follows from property 5 that there is a fiber bundle (E',B' ,F,p').
This is called the fiber bundle induced from ~ by f and is denoted by f*~. In
the case of an inclusion map i: B' C B we use E I B' to denote the fibered
SEC.
8
99
FIBRATIONS
product of B' and E, and if ~ is a fiber bundle with base space B, ~ I B' will
denote the fiber bundle with base space B' induced by i. Observe that ~ I B'
is equivalent to (p-l(B'), B', F, P I p-l(B' )).
8
COROLLARY
For any space Y and point yo E Y, let p: P(Y,Yo) ~ Y be
the map sending each path starting at yo to its endpoint. Then p is a fibration whose fiber over yo is the loop space ny.
Let f: Y ~ Y X Y be defined by f(y) = (Yo,Y) and let p: yI ~ Y X Y
be the fibration of corollary 3. The fibration induced by f is equivalent to the
map p: P(Y,Yo) ~ Y, where p(w) = w(l), and p-l(yO) the fiber over yo, is by
definition, the loop space Q Y. •
PROOF
It follows from corollary 3 that the map p': yI ~ Y defined by p' (w) = w(O)
[or by p'(W) = w(l)] is a fibration, because it is the composite of fibrations
yI ~ Y X Y ~ Y. If p: E ~ B is any map and p': BI ~ B is the fibration
defined by p'(W) = w(O), then the fibered product of E and BI is just the
space B used to define the concept of lifting function for p.
These remarks about fibered products and induced fibrations have analogues for cofibrations. Given maps fl: X ~ Xl and fz: X ~ X 2 , the cofibered
sum of Xl and X2 is the quotient space X' of Xl v X2 obtained by identifying
ft(x) with fz(x) for all x EX. There are maps il: Xl ~ X' and i 2: X2 ~ X', and
these characterize X' as the sum of ft and fz in the category whose objects
are maps with domain X and whose morphisms are commutative triangles. If
fl: X ~ Xl is a cofibration, so is i2: X2 ~ X', and this is called the cofibration
induced from fl by fz.
The map ho: X' ~ X' X I defined by ho(x') = (X',O) is a co fibration for
any space X', and if f: X' ~ X is any map, the cofibered sum of X' X I and X
is just the space X used to define the concept of retracting function for f.
Let p: E ~ B be a fibration. Maps fo, ft: X ~ E are said to be fiber
homotopic, denoted by fo p fl, if there is a homotopy F: fo ~ ft such that
pF(x,t)
pfo(x) for x E X and tEl (in which case p fo
po /1). This is an
equivalence relation in the set of maps X ~ E. The equivalence classes are
denoted by [X;E ]p, and if f: X ~ E, [f]p denotes its fiber homotopy class.
The concept of fiber homotopy is dual to the concept of relative homotopy.
We use induced fibrations to prove that any map is, up to homotopy
equivalence, a fibration. Let f: X ~ Y and let p': yI ~ Y be the fibration defined by p'(W) = w(O). Let p: Pf ~ X be the fibration induced from p' by f.
It is called the mapping path fibration of f and is dual to the mapping cylinder.
There is a section s: X ~ P, of p defined by s(x) = (X,wf(xj), where wf(xj is the
constant path in Y at f(x). There is also a map pI!: Pf ~ Y defined by
pl!(x,w) = w(l). We then have the following dual of theorem 1.4.12.
=
9
THEOREM
0
Given a map f: X
~
=
Y, there is a commutative diagram
X ~ P,
f\ Ip"
Y
100
COVERING SPACES AND FIBRATIONS
CHAP.
2
such that
(a) lp{ p sop
(b) p" is a fibration
The triangle is commutative by the definition of the maps involved.
(a) Define F: P, X I ~ P, by F((x,w), t) = (X,Wl_t), where Wl_t(t') =
w((l - t)t'). Then F is a fiber homotopy from lp{ to sop.
(b) Let g: W ~ Pf and G: W X I ~ Y be such that G(w,O) = p"g(w)
for wE W. Then there exist maps g': W ~ X and g": W ~ YI such that
g"(w)(O) = fg'(w) and g(w) = (g'(w),g"(w)) for wE W. We define a lifting
G': W X I ~ P, of G beginning with g by G'(w,t) = (g'(w), g(w,t)), where
g( w,t) E yI is defined by
PROOF
°:s;:s;
g(wt)(t') = {g"(w)(2t'/(2 - t))
,
G(w,2t' + t - 2)
1
2t'
:s; 2 - t :s; 2, w E W
:s; 2t' :s; 2, w E W
2 - t
Since p" has the homotopy lifting property, it is a fibration.
-
It follows that the fibration p": P, ~ Y is equivalent (by means of
s: X ~ Pf and p: P, ~ X) in the homotopy category of maps with range Y to
the original map f: X ~ Y. In replacing f by an equivalent fibration, we
replaced X by a space P, of the same homotopy type, whereas in Sec. 1.4,
when f was replaced by an equivalent co fibration, the space Y was replaced
by a space Z, of the same homotopy type.
Two fibrations Pl: El ~ B and P2: E2 ~ B are said to be fiber homotopy
equivalent (or to have the same fiber homotopy type) if there exist maps
f: El ~ E2 and g: E2 ~ El preserving fibers in the sense that P2 f = Pl
and Pl g = fz and such that g f::::::
IE, and fog ~
IE.,.- Each of the
PI
p,
maps f and g is called a fiber homotopy equivalence. The rest of this section
is concerned with fiber homotopy equivalence.
We begin with the following result concerning liftings of homotopic maps.
0
0
0
°
°
10 THEOREM Let p: E ~ B be a fibration and let Fo, F l : X X I ~ E be
maps. Given homotopies H: p Fo ~ P Fl and G: Fo I X X ~ Fl I X X
such that H(x,O,t) = pG(x,O,t), there is a lifting H': X X I X I ~ E of H
which is a homotopy from Fo to Fl and is an extension of G.
0
=
PROOF
Let A
f: X X
A~Eby
0
(I X 0) U (0 X l) U (I X 1)
c
I X I and define
f(x,t,O) = Fo(x,t)
f(x,O,t) = G(x,O,t)
f(x,t,l) = Fl(x,t)
Then H I X X A = P 0 f. Because there is a homeomorphism of I X I with
itself taking A onto I X 0, there is a homeomorphism of X X I X I with
itself taking X X A onto X X I X 0. It follows from the homotopy lifting
property of p that there is a lifting H': X X I X I ~ E of H such that
H'I X X A =f
-
SEC.
8
101
FIB RATIONS
Taking Hand G to be constant homotopies, we obtain the following
corollary.
I I COROLLARY Let p: E ---7 B be a fibration and let Fo, F 1 : X X I ---7 E be
liftings of the same map such that Fo I X X 0 = Fl I X X O. Then Fo V Fl
reI X X o. •
Let p: E ---7 B be a fibration and let w: I ---7 B be a path in its base space.
By the homotopy lifting property of p, there exists a map F: p-l(W(O)) X 1---7 E
such that pF(x,t) = w(t) and F(x,O) = x for x E p-l(W(O)) and t E 1. Let
f: p-l(w(O)) ---7 p-l(w(l)) be the map f(x) = F(x,l). It follows from theorem 10
that if w c::-= w' are homotopic paths in B and if F, F': p-l(w(O)) X 1---7 E are
such that pF(x,t) = w(t), pF'(x,t) = w'(t), and F(x,O) = x = F'(x,O) for
x E p-l(W(O)) and t E I, then the maps f, f': p-l(w(O)) ---7 p-l(w(l)) defined by
f(x) = F(x,l) and f(x) = F'(x,l) are homotopic. Hence there is a well-defined
homotopy class [fl E [p-l(W(O));p-l(W(l))] corresponding to a path class [w]
in B. We let h[w] = [fl.
The following is the form theorem 2.3.7 takes for an arbitrary fibration.
12 THEOREM Let p: E ---7 B be a fibration. There is a contravariant functor
from the fundamental groupoid of B to the homotopy category which assigns
to b E B the fiber over b and to a path class [w] the homotopy class h[ w].
If Wb is the constant path at b, let F: p-l(b) X 1---7 E be the map
F(x,t) = x. The corresponding map f: p-l(b) ---7 p-l(b) defined by f(x) = F(x,l)
is the identity map. Hence
PROOF
h[wb] = [lp-l(b)]
showing that h preserves identities.
Let wand w' be paths in B such that w(l) = w'(O). Given a map
F: p-l(w(O)) X 1---7 E such that F(x,O)
x and pF(x,t)
w(t) for x E p-l(w(O))
and t E I, and given F': p-l(w(l)) X 1---7 E such that F'(x',O) = x' and
pF'(x',t) = w'(t) for x' E p-l(w'(O)) and t E I, let f: p-l(W(O)) ---7 p-l(w'(O)) be
defined by f(x) = F(x,l) and let F": p-l(w(O)) X 1---7 E be defined by
=
=
0
,,{F(x,2t)
= F'(f(x), 2t _ 1)
F (x,t)
1f2
~
t
~
S tS
1fz, x E p-l(W(O))
1, x E p-l(W(O))
Then pF"(x,t) = (w * w')(t) and F"(x,O) = x for x E p-l(w(O)) and t E 1. Let
f: p-l(w'(O)) ---7 p-l(w'(l)) be defined by f(x') = F'(x',I). Then F"(x,l) = f(f(x))
for x E p-l(W(O)), which shows that
h[w * w'] = h[w'] * h[w]
Therefore h is a contravariant functor.
•
This yields the following analogue of corollary 2.3.8 for an arbitrary
fibration.
13 COROLLARY If p: E ---7 B is a fibration with a path-connected base space,
any two fibers have the same homotopy type. •
102
COVERING SPACES AND FIB RATIONS
CHAP.
2
The following result asserts that homotopic maps induce fiber-homotopyequivalent fibrations.
14 THEOREM Let p: E ~ B be a fibration and let fo, h: X ~ B be homotopic. The fibrations induced from p by fo and by f1 are fiber homotopy
equivalent.
PROOF
Let po: Eo ~ X and P1: E1 ~ X be the fibrations induced from p by
fo and h, respectively, and let fo: Eo ~ E and Ii: E1 ~ E be the corresponding maps such that p fo = fo po and p Ii = h Pl. Given a homotopy
F: X X I ~ B from fo to h, there are maps Fo: Eo X I ~ E and
Fl.: E1 X I ~ E such that p Fo = F (po X II) and p F1 = F (P1 X II)
and Fo I Eo X 0 = fO and F11 E1 X 1 = Ii· Let go: Eo ~ E1 and g1: E1 ~ Eo
be the fiber preserving maps defined by the property F~ (x,I) = J; go(x) for x E
Eo and F ~ (y,O) = f~gl(Y) for y E E1 • Then
0
0
0
p
0
Fo
0
(gl X II)
=F
0
0
0
0
(po X II)
0
0
(g1 X II)
=F
0
0
(P1 X II)
and
It follows from theorem 10 that Fl ':::: Fo (gl X II). In a similar fashion
P
Fo '::::
F1 (go X II). This implies that gOg1 PI
':::: lEI and glg0 ~
lEo.
•
P
Po
0
0
Clearly, a constant map induces a trivial fibration, and we have the
following result.
15 COROLLARY If p: E ~ B is a fibration and B is contractible, then p is
fiber homotopy equivalent to the trivial fibration B X p-1(b o) ~ B for any
b o E B. •
Let B be a space which is the join of some space Y with So. Then
B = C_ Y U C+ Y, where C_ Y and C+ Yare cones over Yand C_ Y n C+ Y = Y.
Let yo E Y and let p: E ~ B be a fibration with fiber Fo
p-1(yO). It follows
=
from corollary 15 that there are fiber homotopy equivalences f-: C_ Y X Fo ~
p-1(C_ Y) and g+: p-1(C+y) ~ C+Y X Fo. A clutching function fl: Y X Fo ~ Fo
for p is a function fl defined by the equation
y E Y,
Z
E Fo
wheref_: C_Y X Fo ~ p-1(C_Y) and g+: p-1(C+y) ~ C+Y X Fo are fiber
homotopy equivalences. If C_ Y and C+ Yare contractible to yo relative to Yo,
it follows from theorem 10 that f- and g+ can be chosen so that z ~ f-(yo,z)
is homotopic to the map Fo C p-1(C_ Y) and z ~ g+(z) is homotopic to the
map z ~ (Yo,z) of Fo to C+ Y X Fo. In this case the clutching function fl corresponding to f- and g+ has the property that the map z ~ fl(Yo,Z) is homotopic to the identity map Fo C Fo.
Let EqJ be the fiber bundle over Sn defined by a characteristic map cp:
Sn-1 ~ G, as in example 2.7.5 (where G is a group of homeomorphisms of the
fiber F). Then E":. = C_Sn-1 and E'f- = c+sn-l, and it is easy to verify that
f- and g+ can be chosen so that the corresponding clutching function
fl: Sn-1 X F ~ F is the map fl(x,z) = cp(x)z.
103
EXERCISES
EXERCISES
A
I
LOCAL CONNECTEDNESS
Prove that a space X is locally path connected if and only if for any neighborhood
U of x in X there exists a neighborhood V of x such that every pair of points in V can be
jOined by a path in U.
2 If X is a space, let X denote the set X retopologized by the topology generated by
path components of open sets of X. Prove that X is locally path connected and that the
identity map of X is a continuous function i: X ~ X having the property that for any
locally path-connected space Y a function f: Y ~ X is continuous if and only if
i f: Y ~ X is continuous.
0
3
For any space X let X and i: X ~ X be as in exercise 2. Prove that i#: 7T(X,XO) :::::; 7T(X,Xo).
B COVERING SPACES
I Let X be the union of two closed simply connected and locally path-connected subsets A and B such that A n B consists of a single point. Prove that if p: X ~ X is a nonempty path-connected fibration with unique path lifting, then p is a homeomOIphism.
2
Let X = {(x,y) E R2[ x or y an integer} and let
X = SI
SI =
V
{(Zlh)
E Sl X Sl [ ZI = 1 or
Prove that the map p: X ~ X such that p(x,y)
3
=
(e 2 '7TiX,e 2 '7Tiv)
Z2
= I}
is a covering projection.
With p: X ~ X as in exercise 2 above, let Y C X be defined by
Y = {(x,y) EX [0 :s:; x :s:; 1,0 :s:; Y :s:; I}
Prove that Y is a retract of X and that (p [ Y)# maps a generator of 7T(Y) to the commutator of the two elements of 7T(X) corresponding to the two circles of X.
4
Prove that 7T(Sl v SI) is nonabelian.
C THE COVERING SPACE ex: R ~ 51
I For an arbitrary space X prove that a map
f
X ~ R such that f
f: X ~ S1 can
= ex f if and only if f is null homotopic.
be lifted to a map
0
2 Let X be a connected locally path-connected space with base point Xo E X. Prove
that the map
[X,xo; S1,l]
which assigns to
~
Hom (7T(X,XO), 7T(S1,l))
[fl the homomorphism
f#: 7T(X,XO) ~ 7T(S1,l)
is a monomorphism (the set of homotopy classes being a group by virtue of the group
structure on SI).
3 Prove that any two maps from a simply connected locally path-connected space to SI
are homotopic.
4
Prove that any map of the real projective space pn for n ;::: 2 to Sl is null homotopic.
104
5
COVERING SPACES AND FIB RATIONS
CHAP.
2
Prove that there is no map f: Sn -> Sl for n :::: 2 such that f( - x) = - f(x).
6 Borsuk-Ulam theorem. Prove that if f: S2 -> R2 is a map such that f( -x) = -f(x),
then there exists a point Xo E S2 such that f(xo) = O.
D
COVERING SPACES OF TOPOLOGICAL GROUPS
I Let H be a subgroup of a topological group and let G/H be the homogeneous space
of right cosets. Prove that the projection G -> G/ H is a covering projection if and only
if H is discrete.
2 Prove that a connected locally path-connected covering space of a topological group
can be given a group structure that makes it a topological group and makes the projection map a homomorphism.
A local homomorphism cp from one topological group G to another G' is a continuous map from some neighborhood U of e in G to G' such that if gl, gz, glg2 E U,
then CP(glg2)
CP(gl)cp(gZ). A local isomorphism from G to G' is a homeomorphism cp from
some neighborhood U of e to some neighborhood U' of e' such that cp and cp-I are both
local homomorphisms (in which case G and G' are said to be locally isomorphic).
=
3 Prove that a continuous homomorphism cp: G -> G' between connected topological
groups is a covering projection if and only if there exists a neighborhood U of e in G
such that cp I U is a local isomorphism from G to G'.
4 Let cp be a local homomorphism from a connected topological group G to a topological
group G' defined on a connected neighborhood U of e in G. Let G be the subgroup
of G X G' generated by the graph of cp (that is, generated by {(g,g') E G X G' I g' = cp(g),
g E U}). G is topologized by taking as a base for neighborhoods of (e,e') the graph of
cp I N as N varies over neighborhoods of e in U. Prove that G is a connected topological
group, the projection PI: G -> G is a covering projection, and the projection P2: G -> G'
is continuous.
5 Prove that two connected locally path-connected topological groups are locally isomorphic if and only if there is a topological group which is a covering space of each of
them.
6 If G is a simply connected locally path-connected topological group and cp is a local
homomorphism from G to a topological group G', prove that there is a continuous homomorphism cp': G -> G' which agrees with cp on some neighborhood of e in G.
E
FIB RATIONS
I
If p: E -> B is a fibration, prove that p(E) is a union of path components of B.
2 If a fibration has path-connected base and some fiber is path connected, prove that
its total space is also path connected.
3 Let p: E -> B be a fibration and let X be a locally compact Hausdorff space. Define
p': EX -> EX by p'(g) = P g for g: X -> E. Prove that p' is a fibration.
0
4 Let p: E -> B be a fibration and let bo E p(E), F = p-1(b o). Let X be a space
regarded as a subset of some cone CX. Prove that the map
P#: [CX,X; E,F] -> [CX,X; B,bol
is a bijection.
=
5 Let p: E -> B be a fibration and let eo E E, bo p(eo), and F
simply connected, prove that 7T(F,eo) -> 7T(E,eo) is an epimorphism.
= p-I(bo).
If B is
EXERCISES
105
6 Let p: E -7 B be a fibration and let eo E E and bo = p(eo). If p-l(b o) is simply connected, prove that
p#: 7T(E,eo) ::::; 7T(B,bo)
7 Let p: E -7 B be a fibration and bo E p(E). If E is simply connected, prove that
there is a bijection between 7T(B,b o) and the set of path components of p-l(bo).
CHAPTER THREE
POLYHEDRA
IN CHAPTER TWO THE FUNDAMENTAL GROUP FUNCTOR WAS USED TO CLASSIFY
covering spaces. We now consider the problem of computing the fundamental
group of a specific space. We shall show that the fundamental groups of many
spaces (the class of polyhedra) can be described by means of generators and
relations.
A polyhedron is a topological space which admits a triangulation by a
simplicial complex. Thus we start with a study of the category of simplicial
complexes. A simplicial complex consists of an abstract scheme of vertices and
simplexes (each simplex being a finite set of vertices). Associated to such a
simplicial complex is a topological space built by piecing together convex cells
with identifications prescribed by the abstract scheme. Since the topological
properties of these spaces are determined by the abstract scheme, the study
of simplicial complexes and polyhedra is often called combinatorial topology.
A compact polyhedron admits a triangulation by a finite simplicial complex. Thus these spaces are effectively described in finite terms and serve as a
useful class of spaces for questions involving computability of functors.
Sections 3.1 and 3.2 are devoted to definitions and elementary topological
107
108
POLYHEDRA
CHAP.
3
properties of polyhedra. Section 3.3 introduces the concept of subdivision of
a simplicial complex, and it is shown that a compact polyhedron admits arbitrarily fine triangulations. This result is used in Sec. 3.4 to prove the simplicialapproximation theorem, which asserts that continuous maps from compact
polyhedra to arbitrary polyhedra can be approximated by simplicial maps.
The technique of simplicial approximation is used in Sec. 3.5 to prove
that the set of homotopy classes of continuous maps from a compact polyhedron
to an arbitrary polyhedron can be described abstractly in terms of triangulations of the polyhedra. In Sec. 3.6 this result provides an abstract description
of the fundamental group of a polyhedron as the edge-path group of a triangulation, which is used in Sec. 3.7 to obtain a system of generators and relations for the fundamental group of a polyhedron. It is also shown in Sec. 3.7
that the fundamental group functor provides a faithful representation of the
homotopy category of connected one-dimensional polyhedra. Section 3.8 consists of applications of the results on the fundamental group, some examples
of polyhedra, and a description of the fundamental group of an arbitrary
surface.
I
SIMPLICIAL COMPLEXES
This section contains definitions of the category of simplicial complexes and
of covariant functors from this category to the category of topological
spaces.
A simplicial complex K consists of a set {v} of vertices and a set {s} of
finite nonempty subsets of {v} called simplexes such that
(a) Any set consisting of exactly one vertex is a Simplex.
(b) Any nonempty subset of a simplex is a simplex.
A simplex s containing exactly q + 1 vertices is called a q-simplex. We
also say that the dimension of s is q and write dim s = q. If s' C s, then s' is
called a face of s (a proper face if s' =1= s), and if s' is a p-simplex, it is called
a p-face of s. If s is a q-simplex, then s is the only q-face of s, and a face s'
q. It is clear that any simplex has
of s is a proper face if and only if dim s'
only a finite number of faces. Because any face of a face of s is itself a face
of s, the simplexes of K are partially ordered by the face relation (written
s' ~ s if s' is a face of s).
It follows from condition (a) that the O-simplexes of K correspond bijectively to the vertices of K. It follows from condition (b) that any simplex is
determined by its O-faces. Therefore K can be regarded as equal to the set of
its simplexes, and we shall identify a vertex of K with the O-simplex corresponding to it.
<
SEC.
1
109
SIMPLICIAL COMPLEXES
We list some examples.
I
The empty set of simplexes is a simplicial complex denoted by 0.
2
For any set A the set of all finite nonempty subsets of A is a simplicial
complex.
3
If s is a simplex of a simplicial complex K, the set of all faces of s is a
simplicial complex denoted by s.
4
If s is a simplex of a simplicial complex K, the set of all proper faces of s
is a simplicial complex denoted by s.
it If K is a simplicial complex, its q-dimensional skeleton Kq is defined to
be the simplicial complex consisting of all p-simplexes of K for p :::; q.
6
Given a set X and a collection GlJf = {W} of subsets of X, the nerve of GlJf,
denoted by K(01J)), is the simplicial complex whose simplexes are finite nonempty subsets of GlJ\ with nonempty intersection. Thus the vertices of K(GlJf)
are the non empty elements of GlJ\.
7
If K1 and K2 are simplicial complexes, their join K1
complex defined by
K1
* K2
= K1
V
Thus the set of vertices of K1
and the set of vertices of K 2 •
K2
U
{Sl v s21
Sl
* K2 is the simplicial
E K1, S2 E K2}
* Kz is the set sum of the set of vertices of K1
8
There is a simplicial complex whose set of vertices is Z and whose set of
simplexes is
{{ n} 1 n E Z} U {{ n, n
+
I} 1 n E Z}
9
For n 2:: 1 regard Zn as partially ordered by the ordering of its coordinates (that is, given x, x' E Zn, then x :::; x' if for the ith coordinates
Xi :::; Xi in Z). There is a simplicial complex whose set of vertices is Zn and
whose simplexes are finite nonempty totally ordered subsets {XO, . . . ,xq }
of Zn (that is, XO :::; Xl :::; ••• :::; xq ) such that for all 1 :::; i ~ n, Xi q - Xio
0
or 1.
=
If K is a simplicial complex, its dimension, denoted by dim K, is defined
to equal -1 if K is empty, to equal n if K contains an n-simplex but no (n + 1)simplex, and to equal 00 if K contains n-simplexes for all n 2:: o. Thus
sup {dim sis E K}. K is said to be finite if it contains only a finite
dim K
number of simplexes. If K is finite, then dim K
00; however, if dim K
00,
K need not be finite (example 8 is an infinite simplicial complex whose
dimension is 1).
A simplicial map <p: K1 ~ K2 is a function <p from the vertices of K1 to
the vertices of K2 such that for any simplex s E K1 its image <p(s) is a simplex
of K2. For any K there is an identity simplicial map lK: K ~ K corresponding
=
<
<
110
POLYHEDRA
CHAP.
3
to the identity vertex map. Given simplicial maps K1 ~ K2 -'4 K 3, the composite simplicial map lj; qJ: K1 ---? K3 corresponds to the composite vertex map.
Therefore there is a category of simplicial complexes and simplicial maps.
A subcomplex L of a simplicial complex K, denoted by L C K, is a subset of K (that is, s E L => s E K) that is a simplicial complex. It is clear that
a subset L of K is a subcomplex if and only if any simplex in K that is a face
of a simplex of L is a simplex of L. If L C K, there is a simplicial inclusion
map i: L C K.
A subcomplex L C K is said to be full if each simplex of K having all its
vertices in L itself belongs to L. There is a subcomplex N of K consisting of
all simplexes of K with no vertex in L. Clearly, N is the largest subcomplex
of K disjoint from L. If s ::;: {VO,V1' . . . ,Vq} is any simplex of K, then either
no vertex of s is in L (in which case sEN), or every vertex belongs to L (in
which case, if L is full, s E L), or the vertices can be enumerated so that
Vi E L if i ::;; P and Vi Et L if i
p, where 0 ::;; p
q. In the latter case,
s ::;: s' Us", where s' = {vo, ... ,vp} is in L, if L is full, and s" = {Vp+1' ... ,Vq}
is in N. Therefore we have the following result.
0
<
>
10 LEMMA If L is a full subcomplex of K and N is the largest subcomplex
of K dis;oint from L, any simplex of K is either in N, or in L, or of the form
s' U s" for some s' ELand s" E N. •
There is a category of simplicial pairs (K,L) (that is, K is a simplicial
complex and L is a subcomplex, possibly empty) and simplicial maps qJ:
(Kl,L 1) ---? (K 2,L 2) (that is, qJ is a simplicial map K1 ---? K2 such that qJ(L 1) C L2).
The category of simplicial complexes is a full subcategory of the category of
simplicial pairs. There is also a category of pointed simplicial complexes K
(that is, K is a simplicial complex together with a distinguished base vertex)
and simplicial maps preserving base vertices which is a full subcategory of the
category of simplicial pairs. Following are some examples.
I I For any q the q-dimensional skeleton Kq is a subcomplex of K, and if
p ::;; q, Kp is a subcomplex of Kq.
12 For any s E K there are subcomplexes
s esc
K.
13 If {Lj}jEJ is a family of subcomplexes of K, then nLj and U L j are also
subcomplexes of K.
14 Given that A C X, "2lS = {W} is a collection of subsets of X, and KA("2lS)
is the collection of finite nonempty subsets of "ill whose intersection meets A
in a non empty subset, then K A ("2ll') is a subcomplex of the nerve K('1lJ').
We now define a covariant functor from the category of simplicial complexes and simplicial maps to the category of topological spaces and continuous
maps. Given a nonempty simplicial complex K, let IKI be the set of all functions a from the set of vertices of K to I such that
(a) For any a, {v E K I a(v) -=1= O} is a simplex of K (in particular, a(v) -=1= 0
SEC.
1
SIMPLICIAL COMPLEXES
111
for only a finite set of vertices).
(b) For any a, ~VEKO'(v) = l.
If K = 0, we define IKI = 0.
The real number O'(v) is called the vth barycentric coordinate of O'. There
is a metric d on IKI defined by
d(O',f3) = V~VEK [O'(v) - f3(v)J2
and the topology on IKI defined by this metric is called the metric topology.
The set IKI with the metric topology is denoted by IKld.
We shall define another topology on IKI. For s E K the closed simplex lsi
is defined by
lsi = {a E IKII O'(v) =1= 0 => v Es}
q-simplex, lsi is in one-to-one correspondence
If s is a
with the set
{x E Rq+1 I 0 ~ Xi ~ 1, ~ Xi = I}. Furthermore, the metric topology on IKld
induces on lsi a topology that makes it a topological space Isld homeomorphic
to the above compact convex subset of Rq+1. If Sl, S2 E K, then clearly
Sl n S2 is either empty (in which case IS11 n IS21 = 0) or a face of Sl and of
S2 (in which case IS1 n s21 = IS11 n IS21). Therefore, in either case IS11d n IS21d
is a closed set in hid and in IS2Id, and the topology induced on this intersection from IS11d equals the topology induced on it from IS2\d. It follows from
theorem 2.5 in the Introduction that there is a topology on IKI coherent with
{Isld I s E K}. This topology will be called the coherent topology. The space
of K, also denoted by IKI, is the set IKI with the coherent topology. (What we
call here the coherent topology is known in the literature as the weak topology.)
Note that lsi = Isld; we shall also use lsi to denote the space lsi.
Because a subset A C IKI is closed (or open) in the coherent topology if
and only if A n lsi is closed (or open) in lsi for every s E K, we have the following theorem and its corollary.
15 THEOREM A function f: IKI ----> X, where X is a topological space, is continuous in the coherent topology if and only if f Ilsl: lsi ----> X is continuous
for every s E K. •
16 COROLLARY A function f: IKI ----> X is continuous in the coherent topology if and only if f I IKql: IKql ----> X is continuous for every q ~ o. •
It follows from theorem 15 that the identity map of the set IKI is a continuous map IKI ----> IK\d. Note that L C K => ILl C IKI and ILld is a closed
subset of IKld (which implies that ILl is a closed subset of IKI). Furthermore,
if {Li}i EJ is a collection of subcomplexes of K, then U ILil = IU Lil and
n ILil = In Lil.
The coherent topology has the following property.
17 THEOREM For any simplicial complex K, its space IKI is a normal
Hausdorff space.
112
POLYHEDRA
CHAP. 3
Because IKld is a Hausdorff space and i: IKI ~ IKld is continuous,
is a Hausdorff space. To prove that IKI is normal it suffices to show that
if A is a closed subset of IKI, any continuous map f: A ~ I can be continuously extended over IKI. By theorem 15, the existence of such an extension of
f is equivalent to the existence of an indexed family of continuous maps
{fs: lsi ~ I Is E K} such that
PROOF
IKI
(a) If s' is a face of s, then fs Ils'l
(b) fs I (A nisi) = fl (A nisi)
= fs'
The existence of the family {fs} is proved by induction on dim s. If s is
a O-simplex, lsi is a single point, and either lsi E A, in which case we define
fs = f I lsi, or lsi ¢ A, in which case we define fs arbitrarily.
Let q
0 and assume fs defined for all simplexes s with dim s
q to
satisfy conditions (a) and (b). Given a q-simplex s, define J:: lsi U (A n lsI) ~ I
by the conditions
>
<
fs Ils'l = fs'
s' a face of s
f; I (A nisI) = f I (A nisi)
Because {fs' }dim S'5,q satisfies conditions (a) and (b), f; is a continuous map of
the closed subset lsi U (A n lsi) of lsi to I. By the Tietze extension theorem,
there exists a continuous extension fs: lsi ~ I of f;. •
The same technique can be used to prove that IKI is perfectly normal
(that is, every closed subset of IKI is the set of zeros of some continuous realvalued function on IKI) and paracompact.
For s E K the open simplex (s) C IKI is defined by
(s)
= {a E IKII a(v) =j= 0 <=> v E s}
Although a closed simplex is a closed set in IKI, an open simplex need not be
open in IKI. However, the open simplex (s) is an open subset of lsi because
(s)
lsi - lsi. Every point a E IKI belongs to a unique open simplex (namely,
the open simplex (s), where s = {v E K I a(v) =j= O}). Therefore the open
simplexes constitute a partition of IKI.
If A is a nonempty subset of IKI that is contained in some closed simplex
lsi, there is a unique smallest simplex s E K such that A C lsi. This smallest
simplex is called the carrier of A in K. If A C (s), then the carrier of A is
necessarily s. In particular any point a of IKI has as carrier the simplex s such
that a E (s).
=
18 LEMMA Let A C IKI; then A contains a discrete subset (in the coherent
topology) that consists of exactly one point from each open simplex meeting A.
For each s E K such that A n (s) =j= 0 let as E A n (s) and let
Because any closed simplex can contain at most a finite subset of
A', it follows that every subset of A' is closed in the coherent topology and A'
is discrete. •
PROOF
A'
= {as}.
SEC.
1
113
SIMPLICIAL COMPLEXES
Because a compact subset of any topological space can contain no infinite
discrete set, we have the following result.
19 COROLLARY Every compact subset of IKI is contained in the union of a
finite number of open simplexes. -
A finite simplicial complex has a compact space. The converse follows
from corollary 19.
20
COROLLARY
compact.
A simplicial complex K is finite if and only if
IKI
is
-
We establish the folloWing analogue of theorem 15 for homotopies.
21 THEOREM A function F: IKI X I ~ X is continuous if and only if
F I (lsi X 1): lsi X I ~ X is continuous for every s E K.
Because IKI has the topology coherent with the collection of its closed
simplexes, and each closed simplex is a closed compact subset of IKI, it follows
that IKI is compactly generated. By theorem 2.7 in the Introduction, IKI X I
is also compactly generated. It follows from corollary 19 that every compact
subset of IKI X I is contained in ILl X I for some finite subcomplex L C K.
Therefore IKI X I has the topology coherent with the collection {ILl X I I
L C K, L finite}. It is clear that this topology is identical with the topology
coherent with {lsi X I I s E K} (because if L is finite, ILl X I has the topology
coherent with {lsi X I I s E L}). •
PROOF
If IP: Kl ~ K2 is a simplicial
lIPid: IKlid ~ IK21d defined by
IIPld(a)(v' )
map, then there is a continuous map
= ~q>(V)=v' a(v)
The same formula defines a continuous map
commutative square
IKII
1'Pll
IK21
~
v' E K2
IIPI: IKII
~
IK21,
and there is a
IKlid
11'Pld
~ IK21d
An easy verification shows that II and lid are covariant functors from the
category of simplicial complexes to the category of topological spaces, and
IKI ~ IKld is a natural transformation between them. These functors can also
be regarded as defined on the category of simplicial pairs to the category of
pairs of topological spaces.
A triangulation (K,f) of a topolOgical space X consists of a simplicial
complex K and a homeomorphism f: IKI ~ X. If X has a triangulation, X is
called a polyhedron. Similarly a triangulation ((K,L), f) of a pair (X,A) consists of a simplicial pair (K,L) and a homeomorphism f: (IKI, ILl) ~ (X,A). If
114
POLYHEDRA
CHAP.
3
(X,A) has a triangulation, (X,A) is called a polyhedral pair. In general, a given
polyhedron will have triangulations (K1,h) and (K z,/2), for which Kl and Kz
are not isomorphic simplicial complexes.
Following are some examples.
22 For any n ~ 1, (En+l,Sn)is homeomorphic to (lsi, lsi), where s is an (n
simplex. Therefore (En+l,Sn) is a polyhedral pair.
+
1)-
23 Given that K is the simplicial complex of example 8 and f: IKI ---7 R is
defined so that f(1 {n} I) = nand f II {n, n + I} I is a homeomorphism of
I{n, n + I} I onto the closed interval [n, n + 1], then (K,f) is a triangulation
of R, and R is a polyhedron.
24 For n ~ 1, given that K is the Simplicial complex of example 9 and
f: IKI ---7 Rn is defined by the equation (f(a))i = ~XEZn a(x)(x)i, then (K,f) is
a triangulation of Rn, and Rn is a polyhedron.
Given a vertex v E K, its star is defined by
st v = {a E
[KII a(v) 7"= O}
Because a ---7 a(v) is a continuous map from IKld to I, st v is open in IKld, and
hence also in IKI. It is immediate from the definition that
a E st v
~
~
Therefore st v
= U {<s)
carrier a has v as vertex
a E <s)
where s has v as vertex
I v is vertex of s}.
25 LEMMA Let L C K and let Vo, Vl, . . . , Vq be vertices of K. Then
vo, Vb . . . ,Vq are vertices of a simplex of L if and only if
nO"i.,:q
st Vi
n ILl 7"=
0
PROOF
If there is a simplex s E L with vertices vo, ... ,vq, then <s) C st Vi
for every i, and <s) C ILl. Therefore n st Vi n ILl 7"= 0. Conversely, if
n st Vi n ILl 7"= 0, let a E n st Vi n ILl. Then a(vi) 7"= 0 for 0 <:;; i <:;; q, and
carrier a is a simplex s of L whose vertices include Vo, . . . ,Vq. Then the set
{vo, . . . ,Vq} is a face of s and must belong to L, because L is a complex. -
This yields the following relation between K and the open covering of
of vertex stars.
IKI
26 THEOREM Let 01 = {st v I v E K}. The vertex map cp from K to K(0l) defined by cp(v) = st v is a simplicial isomorphism cp: K ::::; K(01), and for any
L C K, cp IL: L::::; KILI(01). -
2
UNEARITV IN SIMPUCIAL CO;\IPI.EXES
The linear structure in the set of all functions from any set to R defines linearity in the space of a simplicial complex. This section is devoted to a study
SEC.
2
115
LINEARITY IN SIMPLICIAL COMPLEXES
of such linearity. We show that a closed simplex lsi is homeomorphic to the
cone with base lsi. This implies that a closed simplex can be parametrized by
"polar coordinates," which are convenient for the construction of maps. We
use them to prove that a polyhedral pair has the homotopy extension property
with respect to any space.
We also consider linear imbeddings in euclidean space of the space of a
simplicial complex; this entails a discussion of locally finite simplicial complexes. Such complexes are characterized by the property that their spaces
are locally compact or the equivalent property that the coherent and metric
topologies coincide on their spaces.
Let K be a simplicial complex and let aI, . . . , a p be points of a closed
simplex lsi. Given real numbers t l , . . . , tp such that 0 S ti S 1 for
i = 1, . . . , p and such that L:ti = 1, the function a = L:tiai is again a
point of lsi. Therefore each closed simplex has a linear structure such that
convex combinations of its points are again points of the closed simplex.
Conversely, if a = L:tiili has a simplex s as carrier (so that a E (s»), then
each ai E lsi. Therefore we have the following lemma.
I
LEMMA
A convex combination of points of IKI is again a point of IKI
if and only if the points all lie in some closed simplex. •
We shall find it convenient to identify the vertices of K with their characteristic functions. That is, if v is a vertex of K, we regard v as also being
the function from vertices v' E K defined by
v(v') =
{~
v
v
7'= v'
= v'
If a E IKI, then we can write a = L:vEK a(v)v, the sum on the right being a
convex combination of points of IKI.
Let X be a topological space which is a subset of some real vector space.
We assume that X has a topology coherent with its intersections with finitedimensional subspaces each such intersection being topologized as a subspace of the finite-dimensional topological linear space in which it lies. For
example, X is euclidean space or X is the space of a simplicial complex.
A continuous map f: IKI --> X is said to be linear on K if it is linear in terms of
barycentric coordinates. That is, f is linear if for every a E IKI, L: v E K a( v)f( v)
is a point of X and
f(a) = L:vEK a(v)f(v)
It is then clear that a linear map is uniquely determined by the vertex map fa
from vertices of K to X such that fo(v)
f(v). Conversely, a vertex map fa
from vertices of K to X may be extended to a linear map f: IKI --> X if and
only if for every simplex s E K all convex combinations of elements in fo(s)
lie in X.
=
116
POLYHEDRA
If cp: Kl
~
K2 is a simplicial map, then the definition of
Icpl(a)
Icpl
CHAP.
3
shows that
= L a(v)lcpl(v)
Therefore Icpl is linear.
Let X be a topological space. The cone X * w with base X and vertex w
is defined to be the mapping cylinder of the constant map X ~ w. The points
of X * ware parametrized by [x,t] with x E X and tEl, where x E X is
identified with [x,O] and [x,l] is identified with w for all x E X. Because w is
a strong deformation retract of X * w, a cone is contractible.
2
LEMMA
For any simplex
s of K
lsi * w is homeomorphic to lsi.
define a map f: lsi * w ~ lsi by
the cone
Choose a point Wo E (s) and
+ (1 - t)a. Then f is continuous (because the linear operations in lsi are continuous). To show that f is injective, assume f([a,t]) =
f([,8,t']) for a, ,8 E lsi and t, t' E 1. Then
PROOF
f([a,t)) = two
two
+ (1
- t)a = t'wo
+ (1
- t'),8
Let s have vertices vo, Vl, . . . , Vq and suppose that C': = Laivi, [3 = L,8iVi,
and Wo = L YiVi. Because a, [3 E lsi, there is i such that aj = 0 and there is k
such that ,8k = O. Then
(t - t')yj = (1 - t')[3j
and
Because Yj =1= 0, t ;::0: t'. Similarly, tyk + (1 - t)ak = t'Yk and so t' ;::0: t. Therefore t = t'. It follows then that (1 - t) a = (1 - t),8, and if t =1= 1, a = [3.
Therefore either t = t' and a = ,8 or t = t' = 1. In either case [a,t] = [,8,t'],
and f is injective.
We now show that f is surjective. Clearly, f([a,O)) = a and f([ a,l)) =
wo, and so f maps onto lsi and woo To show that every point of (s) - Wo is
on a unique line segment from Wo to some point of lsi, let a E (s), with
a =1= wo, and suppose that a = Laivi. Consider the function cp(t') =
(1 + t')a - t' woo cp(O) = a E (s), and as t' increases, the barycentric coordinates
of cp(t') change continuously. Because a ,e. wo, there is some i such that ai < ri.
Therefore
cp(t')(Vi) = ai - t'(Yi - ai)
is a monotonically decreasing function of t'. By continuity, there exists a
unique t'
0 such that cp(t')(Vi) = O. Hence there exists a to
0 which is
the smallest t' for which cp(tO)(Vi) = 0 for any 0 ::::: i ::::: q. Then cp(to) E lsi and
>
>
a = 1
to
+ to
Wo
+
1
1
( ')
+ to cp to
shows that a = f([cp(to), t'0/(1 + t6)]), andfis surjective.
Because f is a continuous bijection from a compact space to a Hausdorff
space, it is a homeomorphism. •
SEC.
2
117
LINEARITY IN SIMPLICIAL COMPLEXES
The barycenter b(s) of the simplex s = {VO,Vl'
the point
b(s)
,Vq} is defined to be
= ~O<i<Q
_1_ Vi
- - q + 1
Clearly, b(s) E <s), and so the carrier of b(s) is s. By lemma 2, lsi is homeomorphic to lsi * w in such a way that w corresponds to b(s). If a E lsi and
tEl, the point tb(s) + (1 - t)a of lsi will be parametrized by polar coordinates [a,t], where [a,t] denotes the point of lsi * w corresponding to the
given point of lsi. Then [a,O]
a and [a,I]
b(s) for all a E lsi. We use
polar coordinates for the following homotopy.
=
lsi
3 LEMMA For any simplex s,
tract of lsi X 1.
=
X 0 U
lsi
X I is a strong deformation re-
=
If s is a O-simplex, lsi
0 and we know the point lsi X 0 is a strong
deformation retract of the closed interval lsi X I. If dim s
0, we define a
deformation retraction
PROOF
F:
to
lsi
X 0 U
lsi
F([a,t], t', t")
lsi
>
XI XI
--7
lsi
X I
X I by the formula in polar coordinates
=
([ a, (1 - t")t
J
+ t"(: ~
+ t"~~-t' t')
([a, (1 - t")t], (1 - t")t'
(1 - t")t')
t'
t2t))
2t
< 2t
~
t'
and diagram it for the cases of a I-simplex and a 2-simplex:
oE------j---------"
--
lsi
I
For any subcomplex L C K the subspace IKI X 0 U
is a strong deformation retract of IKI X 1.
4
COROLLARY
=
ILl
•
XI
PROOF
Let Xn
IKI X 0 U IKn U LI X I for n Z -1. We first show that
for each n Z 0 the space Xn-l is a strong deformation retract of Xn. For each
n-simplex s E K - L let Fs: lsi X I X I --7 lsi X I be a strong deformation
retraction of lsi X I to lsi X 0 U lsi X I (which exists, by lemma 3). For
n Z 0 define a map
118
POLYHEDRA
CHAP.
3
by the conditions
Fn
Iisl
X I X I
Fn(x,t)
= Fs
=x
for an n-simplex s E K - L
x E Xn-l, tEl
Then Fn is well-defined and continuous (because for every simplex s the
restriction Fn Iisl X I X I is continuous), and Fn is a strong deformation
retraction of Xn to Xn-l.
Let fn: Xn ---? Xn-l be the retraction defined by fn(x) = Fn(x,l) for
x E Xn. Let an = lin for n :::: 1, and define G n: Xn X I ---? Xn by induction
on n so that
= 1~o(x,
Go(x,t)
o s:: t s::
t - a2 )
1 - a2
a2
a2
s:: t s:: 1
and for n :::: 1
x
By induction on n, it is easily verified that G n is a strong deformation retraction of Xn to X-l such that G n I Xn-l X I = G n- 1 . Therefore there is a map
G:
IKI
X I X I
---?
IKI
X I
such that G I Xn X I = G n. Then G is a strong deformation retraction of
IKI X I to IKI X 0 U ILl X I. •
5
COROLLARY
A polyhedral pair has the homotopy extension property
with respect to any space.
PROOF
It suffices to show that if L C K, then (IKI, ILl) has the homotopy
extension property with respect to any space Y. Given g: IKI ---? Y and
G: ILl X I ---? Y such that G(a,O) = g(a) for a E ILl, let f: IKI X 0 U ILl X I ---? Y
be defined by f(a,O) = g(a) for a E IKI and f(a,t) = G(a,t) for a E ILl and
t E 1. Because ILl is closed in IKI, fis continuous. By corollary 4, IKI X 0 U
ILl X I is a retract of IKI X I. Therefore f can be extended to a continuous
map F: IKI X I ---? Y. Then F(a,O) = g(a) for a E IKI and F IILI X 1= G. •
Let us now consider linear imbeddings of
IKI
in euclidean space.
6
LEMMA A linear map f: lsi ---? Rn is an imbedding if and only if it maps
the vertex set of s to an affinely independent set in Rn.
Let f(Vi) = pi, where s = {v;}. We show that the set {p;} is affinely
dependent if and only if f is not injective. {p;} is affinely dependent if and
only if there exist ai not all zero such that '2:.aiPi = 0 and '2:.ai = O. Assume
the points Pi enumerated so that ai :::: 0 for i
io and ai 0 for i io.
PROOF
s::
<
>
2
SEC.
119
LINEARITY IN SIMPLICIAL COMPLEXES
Then Li<;io aipi = Li>io ( - ai)pi' If a = Li<;io ai = Li>io - ai, then
Li<;jo(ai/a)Pi = Li>io( -ai/a)pi. It follows from the linearity of f that
f(Li<;io(ai/a)vi) = f(Li>io( - ai/ a)vi), showing that f is not injective.
Conversely, if f is not injective, then f(Laivi) = f(LfJivi), where ajo =1= fJjo
for some io. Then L(ai - fJi)Pi = 0 and L(ai - fJi) = O. Because aio - fJjo =1= 0,
the set {p;} is affinely dependent. •
A simplicial complex K is said to be locally finite if every vertex v of K
belongs to only finitely many simplexes of K.
7
If K is locally finite, every point of IKld has a neighborhood of
ILld, where L is a finite subcomplex of K.
LEMMA
the form
PROOF
Let a E IKk Then a E st v for some vertex v of K. Because v is a
vertex of only finitely many simplexes {s;} of K, st v is contained in the compact set UISil. Let L = {s E K I s is a face of Si for some i}. Then L is a finite
subcomplex of K, and a E st v C ILld. •
8
THEOREM
For a simplicial complex K, the following are equivalent:
(a) K is locally finite.
(b) IKI is locally compact.
(c) IKI ~ IKld is a homeomorphism.
(d) IKI is metrizable.
(e) IKI satisfies the first axiom of countability.
PROOF
(a) = (b). By lemma 7, if a is a point of IKld, there is a finite subcomplex L C K such that a is in the interior of ILk Then a is in the interior
of ILl in IKI. Therefore ILl is a compact neighborhood of a in IKI.
(b) = (c). To show that IKi ~ IKld is an open map, let U be an open subset of IKI with compact closure 0 in IKI. It suffices to show that U is open in
IKk Because 0 is compact, there is a finite subcomplex L C K such that
C ILl (by corollary 3.1.19). Let Kl be the subcomplex of K defined by
o
Kl
If
sE K
(s)
- K 1 , then
lsi n
= {s E K Iisl
n U
=
0}
U is a nonempty open subset of
lsi.
Therefore
n U =1= 0 and (s) n ILl =1= 0. The fact that the open simplexes of K
form a partition of K implies that s E L, and we have shown that K = Kl U L.
Now, IKld - IKlid is an open subset of IKk Because L is finite, ILl ~ ILld is
a homeomorphism. Therefore U is open in ILld, and so it is open in ILld - IK1I d.
Because ILld - IKlid = IKld - IK1I d, U is open in IKk
(c) = (d). Because IKld is metrizable, if IKI and IKld are homeomorphic,
then IKI is also metrizable.
(d)
(e). Every metrizable space satisfies the first axiom of countability.
(e)
(a). Assume that K is not locally finite and let v be a vertex of an
infinite set of simplexes {Sdi=1,2, ... of K. Assume that v has a countable base
of neighborhoods {U;}i=1,2, ... in IKI. Without loss of generality, we may
=
=
120
POLYHEDRA
CHAP.3
assume Ui => Ui+1 for all i ~ 1. For each i, (Si) n Ui =1= 0, because v, being
a vertex of Si, is in the closure of (Si). Let ai E (Si) n Ui. Then the sequence
{ ai} has v as a limit point (because each Ui contains all aj with i ~ i), but in
the coherent topology the set {ad is discrete, because it meets every closed
simplex lsi in a finite set. •
A realization of a simplicial complex K in Rn is a linear imbedding of IKI
in Rn. The following theorem characterizes those complexes K which have
realizations in some euclidean space.
If K has a realization in Rn, then K is countable and locally
finite, and dim K :s; n. Conversely, if K is countable and locally finite, and
dim K :s; n, then K has a realization as a closed subset in R2n+1.
9
THEOREM
Let f: IKI ~ Rn be a linear imbedding. If K is uncountable, it follows
from lemma 3.1.18 that IKI contains an uncountable discrete set A'. Then
f(A') is an uncountable discrete subset of Rn, which is impossible because Rn
is separable. Therefore K is countable. Clearly IKI is metrizahle and, by
theorem 8, K is locally finite. It follows from lemma 6 and theorem 5.3 in the
Introduction that dim K :s; n.
To prove the converse statement, let {Pi} be a sequence of points
in R2n+1 such that
PROOF
(a) Every set of 2n + 2 of the points Pi is affinely independent.
(b) If C is any compact subset of R2n+1, there exists i such that C is disjoint from the convex subset of R2n+1 generated by the set {Pi I i ~
n.
For example, let HI => H2 => ... be a decreasing sequence of closed halfq,
spaces of R2n+1 such that nHi = 0, and assuming Pi defined for i
inductively choose pq to be a point of Hq not lying on any of the finite
number of affine varieties determined by 2n + 1 or fewer points of the set
(p;l 1 ::;; i ::;; q - II.)
Assume that K is countable and locally finite and dim K :s; n, and let
{Vdi=I,2, . .. be an enumeration of the vertices of K. Define f: IKI ~ R2n+1 to
be the linear map such that f(Vi) = Pi. Because of condition (a), it follows
that for any s E K, f Iisl is a linear imbedding of lsi in IKI, and if sand
s' E K, then
<
f(lsl
n
Is'l)
= f(lsl) n f(ls'l)
Therefore fis injective. Because of condition (b), if C is any compact subset of
R2n+1, there is ; such that f-I(C) C U {st Vi Ii :s; f}. Since K is locally finite,
this implies that f-I(C) C ILl for some finite subcomplex L C K. Therefore
f-I(C) is compact in IK1. If A is closed in IKI and C is compact in R2n+1, then
f(A) n C = f(A n f-I(C)) is closed in C [because A n f-I(C) is a closed subset of the compact subset f-I(C) of IKI and f I f-I(C) is a homeomorphism of
SEC.
3
121
SUBDIVISION
f-l(C) to f(f-l(C))]. Therefore f is a closed map and is a linear imbedding of
IKI as a closed subset in R2n+l. •
3
SUBDIVISION
Our main interest in simplicial complexes is in the polyhedra they describe.
To study a polyhedron it is important to consider its different triangulations
and their interrelationships. This section is devoted to proving the existence
of "small" triangulations of a polyhedron, which are used in the next section
in proving that arbitrary continuous maps between polyhedra can be approximated by simplicial maps.
Let K be a simplicial complex. A subdivision of K is a simplicial complex
K' such that
(a) The vertices of K' are points of IKI.
(b) If s' is a simplex of K', there is some simplex s of K such that s' C lsi
(that is, s' is a finite non empty subset of lsI).
(c) The linear map IK'I -,) IKI mapping each vertex of K' to the corresponding point of IKI is a homeomorphism.
Note that conditions (a) and (b) assert that every simplex s' of K' has a
carrier s E K. If K' is a subdivision of K, we identify IK'I and IKI by the linear
homeomorphism of condition (c). The following fact is immediate from the
definition.
I
Any subdivision of a subdivision of K is itself a subdivision of K.
•
The next fact is also true (but somewhat more difficult to prove).
2
If K' and K" are subdivisions of K, there is a subdivision K"' of K that
is a subdivision of K' and of K". •
Thus, statements 1 and 2 assert that the subdivisions of K form a directed
set with respect to the partial ordering defined by the relation of subdivision.
3
LEMMA
Let K and K' be simplicial complexes satisfying conditions (a)
and (b). If s E K is the carrier of s' E K', then (s') C (s).
PROOF
Let vb, . . . , v; be the vertices of s' and let vo, . . . , Vq be the
vertices of the carrier s of s'. Because s' C lsi, for 0 :::; i :::; p, vi = ~(l'ijVj.
Because s is the smallest such simplex, for 0 :::; i :::; q there exists 0 :::; i :::; P
such that (l'ij =I=- O. Let f3 E (s'). Then
and because
f3i
(s') C (s).
•
>0
for all i, ~f3i(l'ij
,
>0
for all
i.
Therefore
f3
E (s) and
122
POLYHEDRA
CHAP.
3
4
THEOREM
Let K' and K be simplicial complexes satisfying conditions
(a) and (b). Then K' is a subdivision of K if and only if for s E K the set
{(s') Is' E K', (s') C (s)} is a finite partition of (s).
PROOF
Assume that K' and K satisfy conditions (a) and (b) and the condition
that {(s') Is' E K', (s') C (s)} is a finite partition of (s) for s E K. Because
any simplex s E K has only a finite number of faces, it follows that
K'(s)
= {s'
E K' I there exists a face
Sl
of s such that (s') C (Sl)}
is a finite subcomplex of K', and the linear map hs: IK'(s)1 ~ lsi that maps
each vertex of K'(s) to itself is a homeomorphism. Therefore there is a continuous map g: IKI ~ IK'I such that g Iisl = h s - 1 for s E K, which is an inverse
of the linear map h: IK'I ~ IKI. Therefore h is a homeomorphism, and K' and K
satisfy condition (c).
Conversely, if K' is a subdivision of K, then {s' I s' E K'} is a partition of
IK'I = IKI· For s E K, consider the sets (s') n (s) for s' E K'. By lemma 3,
either (s') n (s) = 0 or (s') C (s). Therefore {(s') Is' E K', (s') C (s)}
is a partition of (s). Because lsi is compact, it follows from .corollary 3.1.19
that this set is a finite partition of (s). •
We use this result to show that any subdivision of K simultaneously
subdivides every subcomplex of K.
5
COROLLARY
Let K' be a subdivision of K and let L be a subcomplex
of K. There is a unique sub complex L' of K' which is a subdivision of L.
If L' is a subcomplex of K' that is a subdivision of L, then
L' = {s' E K' I (s') C ILl}, which proves the uniqueness of L'. To prove the
existence of L', we prove that {s' E K'I (s') C ILl} has the desired properties.
It is clear that this set is a subcomplex L' of K' and that L' and L satisfy conditions (a) and (b) above. We use theorem 4 to show that L' is a subdivision
of L. If s E L, by theorem 4 the set {(s') Is' E K', (s') C (s)} is a finite partition of (s). By definition of L',
PROOF
{(s') Is' E K', (s') C (s)}
= {(s')
Is' E L', (s')
Therefore, by theorem 4, L' is a subdivision of L.
C (s)}
•
The subdivision L' of L in corollary 5 is called the subdivision of L
induced by K' and is denoted by K' I L.
From the definition of subdivision two facts are immediate.
6
If J: IKI ~ X is linear on K and K' is a subdivision of K, then f is also
linear on K'. •
7
If ((K,L), f) is a triangulation of (X,A) and K' is a subdivision of K, then
((K' ,K' I L), f) is also a triangulation of (X,A). •
F'or any simplicial complex we construct a particular subdivision, called
the barycentric subdivision. For this we need the following lemma, which
shows how to extend a subdivision of s to a subdivision of § for any simplex s.
SEC.
3
123
SUBDIVISION
8
LEMMA
Let s be a simplex of some complex and let K' be a subdivision
of s. For any Wo E <s), K' * Wo is a subdivision of s.
In the statement of lemma 8, Wo is regarded as a simplicial complex
having a single vertex and K' * Wo is the join defined in example 3.1. 7. It is
clear that K' * Wo satisfies requirements (a) and (b) for a subdivision of s. It
follows from lemma 3.2.2 that any point of lsi either equals wo, belongs to lsi,
or belongs to a unique open simplex of the form <s' U {wo}), where s' E K'.
Therefore the open simplexes of IK' * wol constitute a finite partition of lsi,
and by theorem 4, K' * Wo is a subdivision of s. •
PROOF
The subdivision of
a 2-simplex s.
s obtained by applying lemma 8 is pictured below for
K' = pictured subdivision of
the boundary of the triangle
s = triangle and
its faces
K' * Wo = pictured
triangles and their faces
We are now ready to prove the existence of the barycentric subdivision.
Let K be a simplicial complex. We define sd K to be the simplicial complex
whose vertices are the barycenters of the simplexes of K and whose simplexes
are finite nonempty collections of barycenters of simplexes which are totally
ordered by the face relation in K. Thus the simplexes of sd K are finite sets
{b(so), . . . ,b(sq)} such that Si-l is a face of Si for i = 1, . . . , q. We shall
always assume the vertices of a simplex of sd K to be enumerated in this order.
It is clear that sd K is a simplicial complex and that if L is a subcomplex
of K, then sd L is a subcomplex of sd K. Furthermore, if b(sq) is the last
vertex of a simplex s' Esd K, then s' C ISql, and since Sq is the carrier of b(sq),
Sq is the carrier of s'. Therefore sd K and K satisfy conditions (a) and (b).
9
THEOREM
sd K is a subdivision of K.
PROOF
We show that sd K and K satisfy the hypotheses of theorem 4.
If s E K, then, by lemma 3 and the remarks above,
<s) } = {s' E sd K I last vertex of s' = b(s) }
= {s' E sd s I <s') C <s)}
Therefore we need only show that sd s is a subdivision of s for any s E K. We
{s' E sd K
I <s')
C
do this by induction on dim s. If dim s = 0, sd 8 = 8 is a subdivision of 8. For
q
0, assume that sd 81 is a subdivision of 81 for every simplex S1 with
dim Sl
q, and let s be a q-simplex. By the inductive assumption, sd s is a
subdivision of s. The definition of the barycentric subdivision shows that
sd s = sd s * b(s). By lemma 8, this is a subdivision of s. •
>
<
124
POLYHEDRA
CHAP.
3
The subdivision sd K is called the barycentric subdivision of K. Tile
iterated barycentric subdivisions sdn K are defined for n 2 0 inductively,
so that
sd O K
sdn K
10
LEMMA
=K
= sd (sd n - 1 K)
n 21
If L is a subcomplex of K, sd L is a full subcomplex of sd K.
PROOF
Let {b(so), . . . ,b(sq)} be a simplex of sd K all of whose vertices
belong to sd L. Then Si-1 is a face of Si for i
1
. . , q and each Si E L.
Therefore {b(so), . . . ,b(sq)) E sd L. •
=
I I COROLLARY Let (X,A) be a polyhedral pair. Then A is a strong deformation retract of some neighborhood of A in X.
Because of statement 7 and lemma 10, it suffices to consider the case
(X,A) = (IKI,ILI), where L is a full subcomplex of K. Let N be the largest
subcomplex of K disjoint from L. We prove that ILl is a strong deformation
retract of IKI - INI. If a E IKI - INI, then, by lemma 3.1.10, either a E ILl
or there exist vertices Vo, . . . , vp ELand vertices Vp+1, . . . , Vq E N,
with 0 ::; p and p + 1 ::; q, such that a E <Vo, . . . ,vq). In the latter case,
a
~O~i~qaiVi' with ai
0, and we define a
~o~i~pai. Then 0
a
1
and we let ai
a;/a for 0 ::; i ::; P and ai'
a;/(1 - a) for p + 1 ::; i ::; q.
Then a = aa' + (1 - a)a", where a' = ~o~i~paivi is in ILl and a" =
~p+1~i~qa~'Vi is in INI. A strong deformation retraction F: (IKI - INI) X I ~
IKI - INI of IKI - IN] to ILl is defined by
PROOF
=
=
F(a,t)
>
= {~a' + (1
=
- t)a
=
< <
a E ILl, t E I
a E IKI - (INI U ILl), t E I
F is continuous because F IILI X I is continuous, and for any simplex of K of
the form s' Us", where s' ELand s" E N, F I [Is' U s"l n (IKI - IN!)] X I
is continuous. •
Let X be a polyhedron and let 621 be an open covering of X. A triangulation (K,f) of X is said to be finer than 621 if for every vertex v E K there is
U E 621 such that f(st v) C U. A simplicial complex K is said to be finer than
an open covering 621 of IKI if the triangulation (K,1IKI) of IKI is finer than 621
(that is, for each vertex v E K there is U E 621 such that st v C U). We show
that if 621 is any open covering of a compact polyhedron, there are triangulations finer than 621.
A metric on IKI is said to be linear on K if it is induced from the norm
in Rn by a realization of K in Rn. Any finite simplicial complex has linear
metrics, and if K' is any subdivision of K, a metric that is linear on IKI is also
linear on IK'I.
12
LEMMA
Given a metric linear on an m-simplex s, then for any s' E sd s
diam Is'l ::;
m:
1 diam lsi
SEC.
3
125
SUBDIVISION
Let {Pi lOs; i S; m} be points of Rn and assume that y is a convex
combination of {Pi} (that is, y = ~tiPi' where ~ti = 1 and ti ~ 0) and let
x ERn. Then
PROOF
Ilx - yll S; Ilx - ~tJPill = II~ti(x - Pi)11 S; ~tillx - Pill
Therefore Ilx - yll S; sup Ilx - Pill. If x is also a convex combination of {Pi},
then Ilx - yll S; sup Ilpi - Pill.
Regard lsi as imbedded linearly in Rn, with vertices po, P1, . . . ,pm'
Then, by the above result, diam lsi S; sup Ilpi - Pill, and if s' is a simplex of
sd 8, diam Is'l S; sup {lip' - pI/III p', p" E s'}. Therefore we need only show
that if p' = (Po + ... + pq)/(q + 1) and p" = (po + ... + pr)/(r + 1),
where q S; r, then lip' - p"ll S; [m/(m + 1)] sup Ilpi - Pill. Again by the
result above,
lip' - pI/II
S; sup
{Ilpi - p"lll 0
~ Piii
S;
S; i S; q}
and also, for 0 S; i S; q,
Ilpi - p"ll
= Ilpi -
r
+1 1
0
5J<;T
1
+1 1
0
~ Ilpi - Pill
<;J<;r
Therefore
lip' - p"ll
S; r
~ 1 sup {Ilpi - Pill I 0 S; i S; q, 0 S; i S; r}
< _r_ diam lsi
-r+1
Because r S; m, r/(r + 1) S; m/(m + 1) and diam
Given a metric on
IKI,
COROLLARY
S; [m/(m + 1)] diam
lsi. •
we define mesh of K by
mesh K
13
Is'l
= sup {diam lsi I s E K}
If K is an m-dimensional complex and
linear on K, then
mesh (sd K)
<
- m
m
+
1
mesh K
IKI
has a metric
•
This gives us the important result toward which we have been heading.
14 THEOREM Let GiL be an open covering of a compact polyhedron X. Then
X has triangulations finer than G[,
PROOF
Let (K,f) be a triangulation of X. We shall show that there exists an
integer N such that if n ~ N, then (sd n K, f) is finer than GiL. Let IKI be provided with a metric linear on K and let E
0 be a Lebesque number of the
open covering f-1GiL = {f- 1U I U E GiL} with respect to this metric [thus, if
>
126
POLYHEDRA
CHAP.
3
c IKI and diam A :S; e, then f(A) is contained in some element of ''II]. Such
0 exists because IKI is compact. Let m = dim K and choose
a number e
N so that [m/(m + l)]N mesh K :S; e/2 (such an N exists because limn~x
[m/(m + l)]n = 0). If n ~ N, then, by corollary 13, mesh sd n K :S; e/2. If v'
is any vertex of sd n K, diam (st v') :S; 2 mesh sdn K :S; e. Therefore f(st v') is
contained in some element of 01, and (sd n K, f) is finer than 0i1 if n ~ N. •
A
>
This last result is true even if X is not compact. More precisely, if (K,f)
is a triangulation of a polyhedron X and "It is an open covering of X, there
exist subdivisions K' of K such that (K',f) is finer than "ILl However, when X
is not compact K' cannot generally be chosen to be an iterated barycentric
subdivision of K, and so the proof for this case is more complicated than the
proof of theorem 14. We need only the form proven in theorem 14, however,
and so omit further consideration of the more general case.
4
SIMPLICIAL APPROXIMATION
A continuous map between the spaces of simplicial complexes can be suitably
approximated by simplicial maps. This section contains a definition and
characterization of the approximations and a proof of their existence for maps
of a compact polyhedron into any polyhedron. Finally, we apply the result
obtained to deduce some connectivity properties of spheres.
Let K1 and K2 be simplicial complexes and let f: IK11 ---7 IK21 be continuous. A simplicial map cp: Kl ---7 K2 is called a simplicial approximation to f if
f(a) E (S2) implies Icp/(a) E /S2/ (or, equivalently,f(a) E /s21 implies /cp/(a) E IS21)
for a E /Kl/ and S2 E K 2. Note that if v is a vertex of K1 such that f(v) is a
vertex of K 2, then /cp/(v) = f(v). Therefore we obtain the following result.
I
LEMMA Let f: IKll---7 IK21 be a map and suppose that for some subcomplex L1 C K 1, f IIL11 is induced by a simplicial map L1 ---7 K2. If
cp: K1 ---7 K2 is a simplicial approximation to f, then IcplI ILll = f IILll. •
In particular, the only simplicial approximation to a map Icpl: IK11 ---7 IK21
induced by a simplicial map cp: K1 ---7 K2 is cp itself. One sense in which
a simplicial approximation is an approximation is the following.
LEMMA
Let cp: K1 ---7 K2 be a simplicial approximation to a map f:
IKll ---7 IK21 and let A C IKll be the subset of IK11 on which Icpl and f agree.
Then /cpl ~ frel A.
PROOF A homotopy relative to A from Icpl to f is defined by the equation
2
F(a,t)
= tf(a) + (1
- t)(lcpl(a))
See theorem 35 in J. H. C. Whitehead, Simplicial spaces, nucleI, and m-groups, Proceedings of
the London Mathematical Society, vol. 45, pp. 243-327 (1939).
1
SEC.
4
127
SIMPLICIAL APPROXIMATION
The right-hand side is well-defined, because if f(a) E <S2), then 1<pI(a) E IS21,
and so F(a,t) E IS21 for tEl. The continuity of F is easily verified. Clearly, if
a E A, then F(a,t) = f(a) for all t E 1. Therefore F: 1<p1 = frel A. •
The following theorem is a useful characterization of simplicial approximations.
3
to
f:
A vertex map <p from Kl to K2 is a simplicial approximation
IKII ~ IK21 if and only if for every vertex v E Kl
THEOREM
f(st v) C st <p(v)
PROOF
Assume that <p is a simplicial approximation to f. Let a E st v and
suppose f(a) E <S2). Then a(v) :::j= 0 and 1<pI(a) E IS21. Because <p is simplicial,
1<pI(a)(<p(v)) :::j= O. Therefore <p(v) is a vertex of IS21, and f(a) E st <p(v). Since
this is so for every a E st v, f(st v) C st <p(v).
Conversely, assume that <p is a vertex map such that f(st v) C st <p(v) for
every vertex v E K l . We show that <p is a simplicial map. If {vd are vertices
of a simplex of K l , then n st Vi :::j= 0 (by lemma 3.1.25) and
o :::j= f( n
st Vi)
c
n f(st Vi) c n
st <P(Vi)
By lemma 3.1.25, {<P(Vi)} are vertices of some simplex of K 2. Therefore <p is
a simplicial map Kl ~ K 2.
To show that <p is a Simplicial approximation to f, assume a E <Sl) and
f(a) E (S2) and let v be any vertex of Sl. Then a E st v and, by hypothesis,
f(a) E st <p(v). Therefore <p(v) is a vertex of S2. This is so for every vertex v of
Sl. Because <p is simplicial, 1<p1(lsl!) C IS21. Hence 1<pI(a) E IS21, and <p is a
simplicial approximation to f •
We are also interested in simplicial approximations <p: (K 1 ,L 1 ) ~ (K 2 ,L 2 )
to maps f: (IKll,ILll) ~ (IK 21,IL21). The folloWing corollary shows that any
simplicial approximation Kl ~ K2 to a map f: (IKll,IL l !) ~ (lK21,IL21) is
automatically a simplicial approximation when regarded as a map of pairs.
4
COROLLARY
Let f: IKII ~ IK21 be a map such that f(ILll) C IL21 for
Ll C Kl and L2 C K2 and let <p: Kl ~ K2 be a simplicial approximation to
f. Then <p I Ll maps Ll to L2 and is a simplicial approximation to fllLll.
PROOF
By theorem 3, it suffices to show that if v is a vertex of L 1 , then <p(v)
is a vertex of L2 such that
f(st v n ILl!) c st <p(v) n IL21
Since <p is a simpliCial approximation to f, f(st v) C st <p(v), and if v is a vertex
of L 1 , then f(v) E <S2) for some S2 E L2 [because f(ILll) C IL 2Il. Therefore
<p(v) is a vertex of L2 and
f(st v n ILl!) C f(st v) n IL21 C st <p(v) n IL21
•
It follows from corollary 4 that any simplicial approximation to a map
128
POLYHEDRA
f:
(IKII,ILII) ---7 (IK 21,IL 21) is a simplicial map cp: (K1,L 1)
lemma 2, it follows that f '::::: Icpl as a map of pairs.
---7
CHAP.
3
(K 2,L2). From
5
COROLLARY
The composite of simplicial approximations to maps is a
simplicial approximation to the composite of the maps.
PROOF
Let cp: KI ---7 K2 be a simplicial approximation to f: IKII ---7 IK21 and
let 1/;: K2 ---7 K3 be a simplicial approximation to g: IK21 ---7 IK 3 1. Then;
by theorem 3, for a vertex v E KI
gf(st v) C g(st cp(v)) C st 1/;cp(v)
and 1/;cp: Kl
---7
K3 is thus a simplicial approximation to gf: IKII
---7
IK 3 1.
•
Theorem 3 leads to the following necessary and sufficient condition for
the existence of a simplicial approximation to a map.
S
THEOREM
A map f: IKII ---7 IK21 admits simplicial approximations
KI ---7 K2 if and only if KI is finer than the open covering {f-I(st v) I v is a
vertex of K 2 }.
PROOF
By theorem 3, there exist simplicial approximations to f if and only
if for each vertex VI E KI there is a vertex V2 E K2 such that st VI C f-l(st V2).
This is equivalent to the condition that Kl is finer than (f-l(st v)} v E K 2 • •
If K' is a subdivision of K, then for vertices v' E K' and v E K
Combining this fact with theorem 3 yields the following corollary.
7
COROLLARY
Let K' be a subdivision of K. A vertex map cp from K' to K
is a simplicial approximation to the identity map IK'I C IKI if and only if
v' E st cp(v') for every vertex v' E K'. •
In particular, if K' is a subdivision of K, there exist simplicial approximations K' ---7 K to the identity map IK'I C IKI. Combining theorems 6 and
3.3.14 and corollary 4, we obtain the following simplicial-approximation
theorem.
8
THEOREM
Let (KbLI) be a finite simplicial pair and let f: (IKII,ILIi) ---7
(IK 21,IL 2i) be a map. There exists an integer N such that if n ~ N there are
simplicial approximations (sd n Kb sdn L I ) ---7 (K 2,L2) to f. •
As remarked at the end of Sec. 3.3, theorem 3.3.14 is also valid for an
arbitrary polyhedron X. Therefore, if KI is arbitrary and f: IKII ---7 IK21 is a
map, there exists a subdivision Ki of KI and a simplicial approximation
Ki ---7 K2 to f: IKil ---7 IK 21. If KI is not finite, however, Ki cannot generally
be taken to be an iterated barycentric subdivision of K I .
9
EXAMPLE
If 8 is the complex consisting of all proper faces of a 2-simplex s,
then 181 is homeomorphic to 51, and therefore [181;181] is an infinite set.
SEC.
5
129
CONTIGUITY CLASSES
Because 8 is a finite complex, there are only a finite number of simplicial maps
sd n 8 ~ 8 for any n. Therefore for any n there exist maps 181 ~ 181 having no
simplicial approximation sd n 8 ~ 8.
10
EXAMPLE
fine f:
181
~
181
Let 8 be as in example 9 and let its vertices be Vo, V1, Vz. Deto be the map linear on sd 8 such that
f(vo) = b{ VO,V1}
f(b{ vo,vd) = V1
f(V1)
f(b{ VbVZ})
= b{ V1,VZ}
= Vz
f(vz)
f(b{vz,vo})
= b{ vz,vo}
= Vo
Then f c:::: 11.sl, but there is no simplicial approximation 8 ~ 8 to f. There are
exactly eight simplicial approximations cp: sd S --+ s to f [cp is unique on b{vo,th},
b{ V1,VZ}, and b{ vz,vo}, and <p(vo) = Vo or Vb <P(V1) = Vl or Vz, and <p(vz) = Vz
or vol.
As an application of the technique of simplicial approximation, we
deduce the following useful result.
II
THEOREM
5n is (n - I)-connected for n 2': 1.
<
By theorem 1.6.7, it suffices to prove that if m
n, any map
Sl be an (m + I)-simplex and Sz an (n + 1)simplex. Then 5m and 5n are homeomorphic, respectively, to 1811 and 18zl. By
theorem 8 and lemma 2, it suffices to show that if <p: sdi 81 ~ 82 is any
simplicial map, then 1<p1 is null homotopic. Because dim (sd i 81) = m
n, <p
maps sdi 81 into the m-dimensional skeleton of 82. Therefore there is some
0' E 1821 such that
PROOF
5m
~
5n is null homotopic. Let
<
Because 18z1 - 0' is homeomorphic to 5n minus a point, which is homeomorphic to Rn, it is contractible. Therefore 1<p1 is null homotopic. •
In particular, we have the following result.
12
COROLLARY
For n
> 1, 5n is simply connected.
•
Because 5n is locally path connected, corollary 12 and the lifting theorem
imply that any continuous map f: 5n ~ 51 can be factored through the covering
map ex: R ~ 51. Since R is contractible, this implies the following corollary.
13
:;
COROLLARY
For n
> 1 any continuous map 5n ~ 51 is null homotopic.
•
CONTIGUITY CLASSES
In the last section it was shown that any continuous map between the spaces
of simplicial complexes has simplicial approximations defined on sufficiently
fine subdivisions of the domain complex. In general, simplicial approximations
to a given continuous map are not unique, and in this section we investigate
this nonuniqueness.
130
POLYHEDRA
CHAP.
3
We shall define an analogue of homotopy, called contiguity, in the category of simplicial pairs and simplicial maps. Different simplicial approximations to the same continuous map will be shown to the contiguous. The main
result of the section is the existence of a bijection between the set of homotopy classes of continuous maps (from the space of a finite simplicial complex
to the space of an arbitrary complex) and the direct limit of a certain sequence of
contiguity classes of simplicial maps.
Let (KI,L I ) and (K 2 ,L 2 ) be simplicial pairs. Two simplicial maps qJ,
qJ': (KI,L I ) ----> (K 2 ,L z ) are contiguous if, given a simplex s E KI (or sELl),
cp(s) U cp'(s) is a simplex of K2 (or of L2). Obviously, this is a reflexive and
symmetric relation in the set of simplicial maps (KI,L I ) ----> (Kz,L z), but
in general it is not transitive. There is, however, an equivalence relation,
denoted by qJ ~ qJ', in this set of simplicial maps that is defined by qJ ~ qJ' if
and only if there exists a finite sequence qJo, qJI, . . . , qJn such that qJo = qJ
and qJn = qJ' and such that qJi-1 and qJi are contiguous for i = 1, 2, . . . , n.
The corresponding equivalence classes are called contiguity classes, and the
set of contiguity classes of simplicial maps from (KI,LI) to (K 2 ,L z ) is denoted
by [KI,LI; Kz,Lz]. If qJ: (KbLI) ----> (Kz,L z ) is a simplicial map, its contiguity
class is denoted by [qJ].
We shall see that contiguity classes are algebraic analogues of homotopy
classes. We begin by showing that contiguity classes can be composed.
I
LEMMA
Composites of contiguous simplicial maps are contiguous.
Assume that qJ, qJ': (K I,Ll) ----> (K z,L 2 ) are contiguous and 1/;, f:
(Kz,Lz) ----> (K3,L3) are contiguous. If s is a simplex of KI (or LI)' qJ(s) U qJ'(s)
is a simplex of K2 (or L2). Therefore
PROOF
l/;(qJ(s) U qJ'(s)) U 1/;'(qJ(s) U qJ'(s))
is a simplex of K3 (or L3). This implies that the subset I/;qJ(s) U l/;'qJ'(s) is
a simplex of K3 (or L 3) and that I/;qJ, l/;'qJ': (KI,L I ) ----> (K 3,L3) are contiguous. •
It follows easily from lemma 1 that if qJ ~ qJ' and I/; ~ 1/;', then
I/;qJ' ~ l/;'qJ'. Therefore there is a well-defined composite of contiguity
classes
I/;qJ
~
[I/;]
0
[qJ] = [l/;qJ]
for (KI,LI) ~ (K z,L 2 ) -'4 (K3,L3). Thus there is a contiguity category whose
objects are simplicial pairs and whose morphisms are contiguity classes
of simplicial pairs. There are full subcategories of the contiguity category
determined by the pairs (K, 0) or by the pointed simplicial complexes.
2
LEMMA
Contiguous simplicial maps which agree on a subcomplex
define continuous maps which are homotopic relative to the space of the
subcomplex.
SEC.
5
131
CONTIGUITY CLASSES
PROOF
Assume that cp, cp': (K 1,L 1 ) ~ (K 2 ,L 2 ) are contiguous and agree on
L C Kl. Define a homotopy F: (iKll X J, ILll X 1) -+ (lK2 1,IL 2 1) reI ILl from
Icpl to Icp'l by the equation
F(a,t)
= (1
- t)(lcpl(a))
+ t(lcp'l(a))
Since homotopy is an equivalence relation, if cp - cp', then Icpl
Therefore we have the following result.
~
Icp'l.
3
COROLLARY
There is a covariant functor from the contiguity category
of simplicial pairs to the homotopy category of topological pairs which
assigns to (K,L) the pair (IKI,ILI) and to [cp] the homotopy class [Icpl]. -
The next result considers different simplicial approximations to the same
continuous map.
4
LEMMA
Two simplicial approximations (K1,L 1 )
continuous map are contiguous.
~
(K 2 ,L 2 ) to the same
PROOF
Let cp, cp': (K 1 ,L 1 ) ~ (K 2 ,L2 ) be simplicial approximations to f:
(IK 1 1,IL1 1) ~ (IK21,IL21) and let {v;} be a simplex of K 1. Then
st Vi =1= 0,
and by theorem 3.4.3,
n
o =1= f(n st Vi)
C nf(st
Vi) c
n (st CP(Vi) n
st CP'(Vi))
Therefore {cp( Vi)} U {cp' (Vi)} is a simplex of K 2. If {v;} is a simplex of L 1, a
similar argument shows that {cp( Vi)} U {cp' (Vi)} is a simplex of L 2. Therefore
cp and cp' are contiguous. Since it was necessary to subdivide in order to obtain simplicial approximations to arbitrary continuous maps, we should also expect to subdivide to
make contiguity classes correspond to homotopy classes. An example will
illustrate the relation between homotopy and contiguity.
=
=
:.
EXAMPLE Let s be a 2-simplex with vertices Vo, V1, V2 and let K1
K2 S.
Any vertex map from K1 to K2 is a simplicial map. Therefore there are
exactly 27 simplicial maps K1 ~ K 2. Of these 27, there are 21 which map K1
into a proper subcomplex of K 2 , and these constitute one contiguity class. Of
the remaining 6, each is the only element of its contiguity class, the 3 even
permutations of the vertices defining homotopic continuous maps corresponding to one generator of the group
[IK11;IK21J :::::; [5 1 ;51] :::::; Z
and the 3 odd permutations corresponding to the other generator of this
group. Therefore [K 1 ;K 2 ] consists of 7 elements, and the image
[K 1 ;K 2 ] ~ [IK 1 1;IK21J
consists of 3 elements.
This example shows that simplicial maps which define homotopic continuous maps need not be in the same contiguity class. The next result shows
132
POLYHEDRA
CHAP.
3
that a finite simplicial complex can be subdivided so that homotopic simplicial
maps from it to some other complex can be simplicially approximated on the
subdivision by maps in the same contiguity class; it is the analogue for
homotopy of the simplicial-approximation theorem.
THEOREM
Let I, f': (lKll,ILli) ~ (IKzl,ILzi) be homotopic, where Kl is
6
finite. Then there exists N such that I and f' have simplicial approximations
cP, cP': (sd N Kl, sdN L l ) ~ (Kz,Lz)
respectively, in the same contiguity class.
Let F: (IKll X I, ILll X I) ~ (IKzl,ILzj) be a homotopy from Itof'.
Because IKll is compact, there exists a sequence 0 = to
tl
tn = 1
of points of I such that for a E IKll and i = 1, 2, . . . ,n there is a vertex
v E Kz such that F(a,ti_l) and F(a,ti) both belong to st v. Let k (IKll,ILll) ~
(IKzl,ILz!) be defined by Ii(a) = F(a,ti)' Then 1=10 and f' = In, and for
i = 1,2, . . . ,n the set
PROOF
< < ... <
qli =
{fi-l(st v) n Ii-=-~(st v) I v E K z }
is an open covering of IKll. Let N be chosen large enough so that sd N Kl is
finer than "Ill, G/lz, ... , GIln (which is possible, by theorem 3.3.14). For
i = 1, 2, . . . , n let CPi be a vertex map from sdN Kl to Kz such that
Ii(st v) U Ii-l(st v)
c
st CPi(V)
for each vertex v E Kl (such a vertex map CPi exists because sdN Kl is finer
than GIl;). By theorem 3.4.3,
CPi: (sdN Kl, sdN L l )
~
(Kz,Lz)
is a simplicial approximation to Ii and to Ii-1. Because CPi and CPi+l are
simplicial approximations to Ii, it follows from lemma 4 that CPi and CPHI are
contiguous for i = 1, 2, . . . , n - 1. Therefore CPl ~ cpn, and also CPl is a
simplicial approximation to 10 = I and CPn is a simplicial approximation to
In = f'. •
Unlike the simplicial-approximation theorem, this last result is definitely
false if Kl is not a finite simplicial complex. That is, given homotopic maps
I, f': IKll ~ IKzl, there need not be a subdivision Kl of Kl such that I and f'
have simplicial approximations Kl ~ K2 in the same contiguity class.
7
EXAMPLE
Let Kl = K2 equal the simplicial complex of example 3.1.8,
with space homeomorphic to R. Let cP: Kl ~ K2 be the identity simplicial
map and cP': Kl ~ K2 be the constant simplicial map sending every vertex of
Kl to the vertex 0 of K 2. Since R is contractible, Icpl ~ Icp'l. However, if Kl is
any subdivision of K 1 , a simplicial approximation 1/;: Kl ~ K2 to Icpl must be
surjective to the vertices of K2 and a simplicial approximation 1/;': Kl ~ K2
to Icp'l must be the constant map to O. Since two contiguous maps Kl ~ K2
either both map onto a finite set of vertices or neither does,1/; and 1/;' are not
in the same contiguity class.
SEC.
5
133
CONTIGUITY CLASSES
We show that if KI is finite the set of homotopy classes of maps
[IKII,ILII; IK 21,IL2Uis the direct limit of the set of contiguity classes
[sd n K I , sd n L I ; K 2 ,L2 ]
Note that simplicial approximations (sd Kl, sd L I ) ~ (KI,L I ) to the identity
map (Isd KII, Isd LII) c (IKII,ILII) exist, by corollary 3.4.7, and any two are'
contiguous, by lemma 4. Because the composites of contiguous simplicial
maps are contiguous by lemma 1, there is a well-defined map
sd: [KI,L I ; K2,L 2] ~ [sd K I, sd L I ; K 2,L2]
defined by
sd[<p]
= [<pA]
where A: (sd K I, sd L I ) ~ (Kl,L I ) is any simplicial approximation to the.
identity (Isd KII, Isd LIJ) C (IKII,ILII) and <p: (KI,LI) ~ (K 2 ,L2 ) is an arbitrary
simplicial map. By iteration there is obtained a sequence
...
[sdn K I, sdn L I ; K 2,L2]
~
~
[sdn+1 K I, sdn+1 L I ; K 2,L2] ~ ...
which begins with [KI,L I ; K 2 ,L 2 ] on the left and extends indefinitely on the
right. The direct limit lim~ {[sdn K I , sdn L I ; K 2 ,L2 ]} is a functor of two arguments contravariant in (Kl,L I ) and covariant in (K 2,L 2). For finite KI this
functor is naturally equivalent to the functor [IKII,ILll; IK21,IL 21].
8
If KI is a finite simplicial complex, there is a natural
THEOREM
equivalence
lim~
{[sdn K 1, sdn LI; K2,L2]} :::::: [IKll,ILll; IK21,IL2 1]
PROOF
A function from the direct limit to [IKII,ILII; IK21,IL21] consists of a
sequence of functions
fn: [sd n K I , sdn L I; K 2 ,L2] ~ [IKII,ILll; IK 2 1,IL2 1]
for n ~ 0 such that fn = fn+1 sd for n ~ O. Such a sequence fn is defined
by fn[<p] = [I<pl] for <p: (sdn K I , sdn L 1 ) ~ (K 2 ,L 2 ), because if
0
An: (sdn+1 Kl, sdn+1 L 1 )
~
(sdn K1 , sdn L 1 )
is a simplicial approximation to the identity map
(lsdn+IKII, Isdn+1L 1J) C (lsdnK11, IsdnL11)
then, by lemma 3.4.2, IAnl
c:::::
1, and
/n+l sd[lf]
= [IIfAnl] = [Irpl] = fn[rp]
The sequence {fn} defines a natural transformation
f: lim~ {[sdn K1, sdn L1; K2,L2]} ~ [IKll,ILll; IK 2 1,IL 2 J]
and we show that f is a bijection.
It follows easily from the simplicial-approximation theorem that Un}
satisfies (a) of theorem 1.3 of the Introduction; for if g: (IK11,IL1J) ~ (IK 21,IL 21)
134
POLYHEDRA
is a map and ([J: (sd n Kl, sd n L l )
then I([JI ~ g, and
~
fn[([J]
CHAP.
3
(K 2 ,L 2 ) is a simplicial approximation to g,
= [I([JI] = [g]
To show that {fn} satisfies (b) of theorem 1.3 of the Introduction, assume
([J, <p': (sd n K l , sd n L l )
~
(K 2 ,L 2 )
are such that 1<p1 ~ l<p'l. By theorem 6, there exists m
1<p'1 have simplicial approximations
1/;,1/;': (sd m K 1 , sdm L 1 )
~
~
n such that
I([JI
and
(K 2 ,L 2 )
in the same contiguity class. Let
Am.n: (sd m K l , sdm L 1)
~
(sd n K 1 , sd n L 1)
be the composite Am,n = An An+l ... Am-I. Then Am,n is a simplicial approximation to the identity map, and because <p is a simplicial approximation
to 1<p1, <pAm,n is also a simplicial approximation to 1<p1, by corollary 3.4.5. By
lemma 4, <pAm,n is contiguous to 1/;. Similarly, <P'Am,n is contiguous to 1/;'. Since
I/; and 1/;' are in the same contiguity class, so are <pAm,n and <p'Am,n. This means
that sdm-n[<p] = sdm-n[<p'] in [sd m K 1 , sd m L 1 ; K 2 ,L 2 ]. •
For finite Kl the last reslllt furnishes an algebraic description of the set
As an application, note that if K2 is a countable complex,
there are only a countable number of simplicial maps (sd n K l , sd n L 1 ) ~
(K 2,L 2) for n ~ O. Therefore [sd n K 1 , sd n L l ; K 2,L 2] is countable for n ~ O.
Because the direct limit of a sequence of countable sets is countable, we
obtain the following result.
[!Kll,ILll; IK21,IL21].
COROLLARY
Let (X,A) and (Y,B) be polyhedral pairs with X compact
and Y the space of a countable complex. Then [X,A; Y,B] is a countable set. •
9
6
THE EDGE ·PATH GROIJPOID
It was shown in the last section that for finite Kl. [IKll;IK21] is describable as
a limit in which Kl is subdivided but K2 is not. In particular, for any simplicial
complex K the set of path classes of IKI from Vo to VI is determined by the
simplicial structure of K. This is made explicit in the present section, where
we define a simplicial analogue of the fundamental groupoid of a space. In
the next section the fundamental group of a polyhedron is presented in terms
of generators and relations.
An edge of a simplicial complex K is an ordered pair of vertices (v,v')
which belong to some simplex of K. The first vertex V is called the origin of
the edge, and the second vertex v' is called the end of the edge. An edge path
~ of K is a finite nonempty sequence ele2 ... er of edges of K such that end
SEC.
6
135
THE EDGE-PATH GROUPOID
ei = orig ei+l for i = 1, . . . , r - 1. We define orig ~ = orig el and end ~ =
end er • A closed edge path at Vo E K is an edge path ~ such that orig ~ =
Vo = end ~. If ~ 1 and ~ 2 are edge paths of K such that end ~ 1 = orig ~ 2, we
define the product edge path ~ l~ 2 to be the edge path consisting of the
sequence of edges of ~ 1 followed by the sequence of edges of ~ 2. Then
orig ~ l~ 2 = orig ~ 1 and end ~ l~ 2 = end ~ 2. It is clear that if end ~ 1 = orig ~ 2
and end ~2 = orig ~3, then ~1(~2~3) = (~1~2)~3. The product of edge paths thus
satisfies associativity; however, there are no left or right identity elements for
the product. To obtain a category (as was done for paths in a topological
space) it is necessary to define an equivalence relation in the set of edge paths
of K.
Two edge paths ~ and t' in K are simply equivalent if there exist vertices
v, v', and v" of K belonging to some simplex of K such that the unordered
pair {Sot'} equals one of the following:
The unordered pair {(v,v"), (v,v')(v',v")}
The unordered pair gl(V,V"), ~l(V,v')(v',v")} for some edge path ~l in K
with end ~l = v
The unordered pair {(v,v")h (v,v')(v',v")~z} for some edge path ~2 in K
with orig ~ 2 = v"
The unordered pair gl(v,v")h ~1(V,V')(v',V")~2} for some edge paths
~l and ~2 in K with end ~l = v and orig ~2 = v".
Two edge paths ~ and t will be said to be equivalent, denoted by ~ - t,
if there is a finite sequence of edge paths ~o, ~l' . . . , ~n such that ~ = ~o
and
= ~n, and ~i-l and ~i are simply equivalent for i = 1, . . . , n. This
is an equivalence relation, and the following statements are easily verified.
r
t'
implies that
I
~ -
2
~l
3
If orig ~
-
~1' ~2
-
~
and
t'
~2 and end ~l
have the same origin and the same end.
= orig ~2 imply ~1~2 -
= V1 and end ~ = V2, then (V1,V1)~ -
m
~ -
~1~2'
~(V2,V2)'
•
•
•
If ~ is an edge path,
denotes its equivalence class. It follows from
statement 1 that "orig
and end [~] are well-defined (by orig
= orig ~
and end
= end
From statement 2 there is a well-defined composite
[~1] [~2] = [~1~ 2] defined if end ~ 1 = orig ~ 2. We then have the following
simplicial analogue of theorem 1.7.7.
m
n
m
m
0
4
THEOREM
There is a category &;(K) whose objects are the vertices of K
and whose morphisms from V1 to Vo are the equivalence classes
with
orig
= Vo and end = V1 and whose composite is [~1] [~2]'
m
m
m
0
PROOF
The existence of identity morphisms follows from statement 3, and
the associativity of the composite is a consequence of the associativity of the
product of edge paths. •
We now show that E9(K) is a groupoid. If e = (v,v') is an edge of K, we
136
POLYHEDRA
define e- I
r- I = er - I
= (v',v),
...
=
and if r
el ... e r is an edge path in K, we define
el- l . The following statements are then easily verified.
6
=r •
orig r- I = end r and end r- I = orig r
7
rl - r2 implies rl- 1
8
If orig r
:;
CHAP.3
(r-I)-I
r2- 1 .
-
•
•
= VI and end r = V2, then rr- I - (VI, VI) and r-Ir - (V2,V2). •
follows that in f9(K), [r- I] = L~rl, and so &~(K) is a groupoid. This
It
groupoid is called the edge-path groupoid of K. If Vo is a vertex of K, the
operation m 0 [t'] in the set of elements of C(K) with origin and end at Vo
gives a group denoted by E(K,Vo) and is called the edge-path group of K with
base vertex Vo·
To compare 6~(K) [and E(K,vo)] with <3'(IKI) [and '7T(IKI,vo)], for r ;;::: 1 let
Ir denote the subdivision of I into r equal subintervals; that is, Ir is the
simplicial complex
Given an edge path r = el ... er in K with r edges, let
plicial map defined by
CPr ( r
~) _ {Orig ei+1
-
end ei
Then ICPrl: I ~ IKI is a path in
statements hold.
IKI,
O~i~r-l
1
~ i ~ r
and it is easily seen that the following
ICPrl = orig r and end Icpli = end r •
lOr - r implies ICPrl ~ Icpr! reI t •
I I Ifendrl = origr2, then ICPrlr21 ~ ICPrll * ICPr21
9
CPr: Ir ~ K be the sim-
orig
reU.
•
It follows that there is a natural transformation p from 6~(K) to <3'(IKI)
which assigns to V E K the point v E IKI and to a morphism
in f9(K) the
morphism [ICPrl] in <3'(IKI). We shall prove that for vertices vo, VI E K, P is a
bijection
m
p:
hom~;
(VI.Vo) ;::::; hom6j> (VI,VO)
This can be obtained from theorem 3.5.8, but there is also a direct proof
(which seems no longer than a proof based on theorem 3.5.8).
12
LEMMA
For any Vo, VI E K the function
p: hom~; (VI,VO) ~ bom~p (VI,VO)
is surjective.
PROOF
Given a path w: I ~
IKI
from Vo to
VI.
because I =
Ihl,
it follows
SEc.6
137
THE EDGE-PATH GROUPOID
from theorem 3.4.8 that there is a simplicial map
cp:
sdn 11
---7
K
which is a simplicial approximation to w. Since sdn 11 = I 2n, there is an edge
path r = el ... e2n defined by ei = (cp((i - 1)/2n), cp(i/2n)) such that cP = CPI
for this Because cp(O)
w(O) and cp(l)
w(I), it follows from lemma 3.4.2
that Icpl ~ w reI i. Therefore [w] = [Icpl] = [lcpll] = p[n •
r
=
=
We shall need some preliminary lemmas before showing that p is injective.
13 LEMMA For any simplex s two edge paths in
the same end are equivalent.
PROOF
end r
s with the same origin and
It suffices to prove that if r is any edge path in s from orig r = v and
= v', then r is equivalent to the edge path consisting of the single edge
(v,v'). This is proved by induction on the number of edges of
r •
14 LEMMA Let rand t be edge paths in K, each with r edges, such that
CPI' CPr: Ir ---7 K are contiguous. Then
r - t.
PROOF
Let r = el ... e r, where ei = (Vi-1.Vi), and let t = el ... e~, where
ei = (VLhVi). For 0 ::::; i ::::; r let ei = (Vi,vi) (this is an edge of K because
CPI and CPr are contiguous). From the definition of equivalence
Because CPI and CPr are contiguous, for each 1 ::::; i ::::; r there is some simplex
Si of K such that ei, ei, ei-l, and ei all are edges of Si. By lemma 13, elel - el
and ej-=-\eiei - ei for 2 ::::; i ::::; r - 1, and e;}ler - e;. Therefore
r
15 LEMMA Let = el ... e r be an edge path of K and let A: I2r ---7 Ir be a
simplicial approximation to the identity map II2rl C IIrl. Then CPIA = CPr:
I2r ---7 K for some t = el ... eZ r and
t.
r-
=
=
PROOF
Let ei
(Vi-1.Vi) for 0 ::::; i ::::; r. Then e:H-le2i
(Vi-l,Vi)(Vj.Vi) for
a vertex Vi which equals either Vi-lor Vi, By lemma 13, ~i-1~t - fq and
t -r •
We are now ready for the main result on the edge-path groupoid.
16 THEOREM
For vertices vo, VI E K the function
p: hom& (Vl,VO)
---7
hom6j> (Vl,VO)
is a bijection.
PROOF
In view of lemma 12, it only remains to prove that p is injective.
Assume that rand t are edge paths from Vo to VI such that Icpll c::o Icpr! reI i.
By juxtaposing trivial edges (VI, VI) at the end of r or t sufficiently often, we
can replace rand t by equivalent edge paths haVing an equal number of
138
POLYHEDRA
CHAP.3
edges. Hence there is no loss of generality in assuming sand S' both to have r
edges. Then <PI' <Pr: Ir ~ K are such that !<PI! ~ !<pr! rel i. By theorem 3.5.6,
there exists m such that if A.: sd m Ir ~ Ir is a simplicial approximation to the
identity, then <PIA. and <PrA. are in the same contiguity class. Now <PIA. = <PI!
and <PrA. = <Pn for edge paths S1 and
in K. By lemma 15 (and induction
Therefore
on m), S - Sl and S'- si. From lemma 14 it follows that Sl -
n
n.
S -S'. •
If <P: K1
<P#:
~
t~(K1) ~ &~(K2)
K2 is a simplicial map, there is a covariant functor
defined by
<p#[sl
= [<pm1
where, if S = (VO,V1)(V1,V2) ... (V r -1,V r ), then <p(s) = (<p(vO),<P(V1)) ...
(<P(V r _1),<P(V r )). It is trivial to verify that commutativity holds in the square
G~(K1) ~ &~(K2)
Therefore we have the following result.
17 COROLLARY On the category of pointed simplicial complexes K with
base vertex Vo, P is a natural equivalence of the covariant functor E(K,vo)
with the covariant functor '7T(IKI,vo). •
Note that fi,(K) is determined by the 2-skeleton of K; that is, the edge
paths of K are determined by pairs of vertices of K which are vertices of a
simplex, and the equivalences between edge paths are determined by triples
of vertices which are vertices of a simplex. Hence &~(K2) ~ 6~(K).
18 COROLLARY For any pointed simplicial complex (K,vo), the inclusion
map K2 C K induces an isomorphism
'7T(IK21,vo) ~ '7T(IKI,vo)
•
If s is a simplex of K, any two of its vertices belong to the same component of 0(K). Therefore the components {Ej} of &~(K) define a partition
of K into subcomplexes {Kj}, called the components of K, defined by
Kj = {s E K I s has some vertex in Ej}. K is said to be connected if it contains
exactly one component.
19 THEOREM If {Kj} are the components of K, then {!Kjl} are the path
components of IKI·
PROOF If v is a vertex of K, then st v is path connected and so belongs to
the same path component of IKI as v. It follows from theorem 16 that two vertices
of K are in the same component of 0'(IKI) if and only if they are in the same
component of 0(K). Therefore, if {Ed is the set of components of t~(K), the
SEC.
7
139
GRAPHS
path components of IKI are the sets {Pd defined by Pj = U {st v I v E Ej }.
Clearly, P j = IKjl. •
7
GRAPHS
We show how a system of generators and relations for the edge-path group
E(K, vol can be determined. This provides a method for finding generators
and relations of the fundamental group of a polyhedron. Since every edge
path of K is an edge path of the one-dimensional skeleton of K, we start with
a description of the edge path group of a one-dimensional complex.
A one-dimensional simplicial complex is called a graph. A tree is defined
to be a simply connected graph.
I
LEMMA
A graph is a tree if and only if it is contractible.
PROOF
Since a contractible space is simply connected, a contractible graph
is a tree. Conversely, let K be a tree and let CiO be a point of IKI. We ~hall define a homotopy F: IKI X I ~ IKI from the identity map 1 of IKI to the constant map c of IKI to CiO. Since IKI is path connected, for each vertex v of K
there is a path Wv in IKI from v to CiO. We now define F on v X I by F(v,t) =
w,,(t). For every I-simplex s of K the map F is defined on the subset
(lsi X 0) U (lsi X 1) U (lsi X 1) of lsi X I. Since IKI is simply connected and
(lsi X I, (lsi X 0) U (lsi X 1) U (lsi X I)) is homeomorphic to (E2,SI), it
follows that F can be extended over lsi X I. In this way we obtain a map
F: IKI X I ~ IKI whose restriction to each lsi X I is continuous. Then F is
continuous and F: 1 ~ c. •
2
LEMMA
Let K be a connected simplicial complex. Then K contains a
maximal tree, and any maximal tree contains all the vertices of K.
PROOF
The collection of trees contained in K is partially ordered by inclusion. Let {Kj} be a simply ordered set of trees in K and let T = U Kj. We
show that T is a tree. Since K j is one-dimensional, T is one-dimensional.
Since {Kd is a simply ordered set of trees, it follows that any finite subcomplex of T is contained in some K j • To show that T is simply connected, let f:
Si ~ IT I, where i = 0 or 1. Then f(Si) is compact and is therefore contained
in IKjl for some ;. Since IKjl is simply connected, the map f: Si ~ IKjl can be
extended to a map!': Ei+l ~ IKjl c ITI, and ITI is simply connected.
As a result, every simply ordered set of trees in K has a tree as upper
bound. Zorn's lemma can be applied to yield a maximal tree in K. If T is a
maximal tree of K and does not contain all the vertices of K, it follows from
the connectedness of K that there is a I-simplex {vl,vd E K with VI E T and
V2 ~ T. Let Tl = T U {{ V2}, {vbvd}. Since VI is a strong deformation retract of I {VI,V2}1, ITI is a strong deformation retract of ITII. Therefore ITll is
contractible, and so TI is a tree strictly larger than T, contradicting the maximality of T. •
140
POLYHEDRA
CHAP.3
It follows from lemma 2 that if K is a connected complex and T is
a maximal tree in K, then K - T consists of simplexes of dimension ~ 1.
Because IT I is contractible, any edge path in K is determined by its part in
K - T, as we shall see below. This motivates the following definition.
Let T be a maximal tree of the connected complex K. Let G be the
group generated by the edges (v,v') of K with the relations
(a) If (v,v') is an edge of T, then (v,v') = 1.
(b) If V, v', and v" are vertices of a simplex of K, then (v,v')(v',v")
3
THEOREM
= (v,v").
With the notation above, E(K,vo) ;::::; G.
PROOF
Since T is connected and contains the vertices of K, for each vertex
v of K there is an edge path rv in T such that orig rv = Vo and end rv = v. If
(v,v') is an edge of K, the edge path rv(v,v')rv,-l is a closed edge path in K at
Vo. Therefore there is a homomorphism a of the free group F generated by
the edges of K into E(K,vo) defined by a(v,v') = [rv(v,v')rv,-l].
We show that a can be factored through G. If (v,v') is an edge of T, then
rv(v,v')rv,-l is a closed edge path in T. Because T is simply connected,
[rv(v,v')rv,-l] = 1 and a sends relations of type (a) into 1. If v, v' and v" are
vertices of a simplex of K, then
[rv(v,v')rv,-l]
0
[rv,(v',v")rv,,-l]
= [rv(v,v')(v',v")rv,,-l]
= [rv(v,v")rv,,-l]
Therefore a((v,v')(v',v")) = a(v,v"), and so there is a homomorphism
a': G ~ E(K,vo) such that a'(v,v') = a(v,v') = [rv(v,v')rv,-l].
To prove that a' is an isomorphism we construct an inverse 13': E(K,vo) ~ G
as follows. For each closed edge path r = el ... er let 13m = el ... er,
where the right-hand side is interpreted as a product in G. If rand
are simply equivalent, then because of the relations of type (b), 13m = f3W).
Therefore there is a homomorphism 13': E(K,vo) ~ G such that f3'm = 13m.
We show that a' and 13' are inverses of each other. Given an edge path
r
(VO,Vl)(V1.V2) ... (vr,vo), then a'f3'm
WJ, where
r
=
=
r = rvo(vo,Vl)rVl-1rvl(Vl,02)rV"-1 ... rvr(vr,vo)rv;l
- r vo(VO,Vl)(Vl,V2) ... (vr,vo)rvo- 1
Since rvo is a closed edge path in the simply connected complex T, rvo - 1
and
Therefore a'f3' is the identity on E(K,vo).
Consider f3'a'(v,v') = f3(rv)(v,v')f3(rv,-1). Since rv and rv,-l are paths in
T, they are products of edges in T. Hence f3(rv) and f3(rv,-1) are both equal to
1 by relations of type (a). Therefore f3'a'(v,v')
(v,v'), and since {(v,v')}
generate G, 13'a' = 1 on G. •
r - r.
=
In case K is finite, there are only a finite number of edges of K, and G
is finitely generated. Similarly, there are only a finite number of relations of
type (a) or (b). G is thereby represented as the quotient group of a finitely
generated free group by a finitely generated subgroup. Hence we have the
following corollary.
SEC.
7
141
GRAPHS
4
COROLLARY
If K is a finite connected simplicial complex, then E(K,vo)
is finitely presented. COROLLARY
If K is a connected graph, E(K,vo) is a free group, and if T
is a maximal tree in K, a set of generators of E<K,vo) is in one-to-one correspondence with the I-simplexes of K - T.
:;
Consider the group C. Because of relations of type (a), we need only
consider edges of K not in T. Every such edge e corresponds to an order of
the vertices of some I-simplex of K - T. The relations of type (b) imply that
the oppositely ordered edge equals e- 1 in C. Therefore the group C is generated by edges one for each I-simplex of K - T. There are no relations
on these generators of C, for if v, 0', and v" are vertices of a simplex of K,
then at least two of them are equal. If v = v' or v' = v", the corresponding
relation of type (b) is the trivial relation I(v',v") = (v',v") or (v,v')1 = (v,v').
If v = v", the corresponding relation is (v,v')(v',v) = 1, which, in terms of
our generators, becomes ee- 1 = 1. PROOF
6
EXAMPLE
Let J = U} be a set and let X be the pOinted space which is
the sum (in the category of pointed spaces) of pointed I-spheres {Sj1 }iEJ.
Each Si 1 can be triangulated by Sj, where Sj is a 2-simplex Sj = {vj,vj,vi'} and
Vj corresponds to the base point of Sj 1. Then X can be triangulated by the
complex K with vertices
{v} U {vj,vnhJ
and I-simplexes
{ {v,vj} }iEJ U {{ v,vj'} }jEJ U {{ vj,vj'} }iEJ
Let T be the maximal tree in K such that K - T consists of the collection
{ {vj,v'f} }iEJ. By corollary 5, E(K,v) is a free group on generators in one-to-one
correspondence with /. Therefore there is an isomorphism of 7T(X,XO), where
xo corresponds to v, with the free group generated by /.
7
EXAMPLE
Let X be the complement in R2 of a set of p disjoint closed
disks or points. Then X has the same homotopy type as the sum of p pointed
I-spheres. Therefore the fundamental group of X is a free group with p
generators.
For connected graphs the fundamental group functor is a faithful representation of the category of their underlying spaces and homotopy classes by
means of groups and homomorphisms. This is summarized in the following
theorem.
8
THEOREM
Let Kl and K2 be connected graphs and let Vo be a vertex of
K 1 . Then
(a) Any continuous map f: IK11 ~ IK21 is homotopic to a continuous
map f': IKll ~ IK21 such that f'(vo) is a vertex of K 2.
(b) If Vo is any vertex of K2 and h: 7T(IK 1 1,vo) ~ 7T(IK 21,vo) is an
arbitrary homomorphism, there exists a continuous map f: (IK 1 1,vo) ~
142
POLYHEDRA
CHAP.
3
(IK21,vb) such that h
= f#.
(c) Let Vo and Vo be vertices of K2 and assume that II, h IKII ---7 IK21
are maps such that h(vo) = Vo and fz(vo) = Vo. Then h ':':' fz if and only
if there is a path W in IK21 from Vo to vo'such that the following triangle
is commutative:
PROOF
Since K2 is connected, it is path connected, and (a) follows from the
fact that the pair (IK 1 1,vo) has the homotopy extension property with respect
to IK21 (by corollary 3.2.5).
To prove (b), let T be a maximal tree in K 1 . Let {Sj} be the simplexes of
Kl - T and for each i let ej = (Vj,vj) be an edge whose vertices are the
vertices of Sj in some order. For each vertex v in Kl there is an edge path Sv
in T from Vo to v. For each i let
Wj =
Is
Vj
ejS;} I
Then [Wj] E 7T(IK 1 1,vo), and by corollaries 5 and 3.6.17, the set {wd is a system of free generators of 7T(IK 1 1,vo). For each j let wi he a closed path in IK21
at vb such that h[wj] = [wj]. We define a continuous map
f:
by f( ITI)
(IK 1 1,vo)
---7
(IK 2 1,vo)
= vo, and for each i we define f I ISj I by
f(tvi + (1 - t)Vj) = wj(t)
where the points of ISjl are written in the form tvi + (1 - t)Vj for t E 1.
is continuous because its restriction to IT I and to each ISjl is continuous. Clearly, f#[Wj] = [wj] = h[wjJ; therefore f# = h.
To prove (c), note that if fl ':':' f2' there is a path W in IK21 from
Vo to Vo such that, by theorem 1.8.7, h# = h[wlfz#. Conversely, if h# =
h[wlfz#, let T be a maximal tree in Kl and for each vertex v of Kl
let Sv be an edge path in T from Vo to v. We shall define F: IKll X [ ---7 IK21,
a homotopy from h to fz, in several stages. First we set F(x,O) = h(x) and
F(x,l) = fz(x) for x E IK 1 1. Then Fhas been defined on (IK 1 1X 0) U (IK 1 1XI).
If v is a vertex of K 1 , we defin~ F(v,t) = ((h(lsv -11) * w) * fzISvl)(t) for t E [.
Then F(v,O)
h(v) and F(v,l)
fz(v), and F is thus defined on IK 101 X [to
be consistent with its previous definition on (IK 1 1X 0) U (IK 1 1X 1). It only
remains to extend F over lsi X [ for each I-simplex S E K 1 . Let v and v' be
the vertices of S in some order. Then lsi X [ is a square with the following
product, arbitrarily associated, as boundary
f
=
=
SEc.8
143
EXAMPLES AND APPLICATIONS
(l(v,v')1 X 0)
* (v'
X I)
* (l(v',v)1
X 1)
* (v
X I)-I
F can be extended over lsi X I if and only if F maps this product into a null
homotopic path of IK21. By the definition of F, the above path is sent into a
path homotopic to the following product associated arbitrarily
hl(v,v')1
* (fll~v,-11 * (.0 * hl~v,1) * fzl(v',v)1 * (fzl~v-ll * (.0-1 * hl~vl)
c::::hl(v,v')1 * fll~v,-11 * ((.0 * h(l~v'l * l(v',v)1 * l~v-ll) * (.0-1) * fll~vl
c:::: hl(v,v')1 * f11~v,-11 * h(l~v'l * 1(v',v)1 * I~v -11) * hl~vl
Therefore F can be extended over lsi X I, and the resulting map F: IKll X I ~
[Kzl will be continuous, because for each closed simplex lsi of Kl its restriction to lsi X I is continuous. Then F: h c:::: fz, •
It follows from theorem 8b that if f: (IK11,vo) ~ ([Kzl,vb) induces an isomorphism f#: 7T([K 11,vo):::::; 7T(IKzl,vo), then there is a continuous map g:
(IKzl,vb) ~ (IKll,vo) such that ~ = (f#)-1. By theorem 8c, it follows that
gf c:::: l1K!1 and fg c:::: l1K21' Hence we have the next result.
S
COROLLARY
Let K1 and Kz be connected graphs with Vo a vertex of K1
and vb a vertex of K z. A continuous map f: (IK11,vo) ~ (IKzl,vb) is a homotopy equivalence if and only if f induces an isomorphism f#: 7T(IK11,vo) :::::;
7T(IKzl,vb).
•
The step-by-step extension procedure used to construct the homotopy F
to prove theorem 8c is a standard method for constructing continuous maps
on the space of a complex. The map is constructed on one skeleton at a time
and extended over the next skeleton.
8
EXAMPLES AND APPLlC."-TiONS
This section contains assorted results concerning the fundamental group. We
begin with some applications to the theory of free groups; in particular, we
show that any subgroup of a free group is free. Next we consider the effect
on the fundamental group of attaching 2-cells to a space. We use the result
obtained to prove that any group is isomorphic to the fundamental group of
some space. Finally, we describe how the fundamental group of a surface can
be represented by means of generators and relations.
If K is a simplicial complex and IX E IKI has carrier s (that is, IX E <s»),
then for any subdivision K' of s the simplicial complex K' * IX is a subdivision
of s (by lemma 3.3.8). It follows that a modified barycentric subdivision of K
can be constructed whose vertices are IX and the barycenters of simplexes of
K other than s. Therefore there is a subdivision of K having 0' as a vertex,
and we have the following result.
144
I
POLYHEDRA
LEMMA
vertex.
2
If a E
IKI,
CHAP.
3
there is a subdivision K' of K having a as a
•
THEOREM
A polyhedron is locally contractible.
PROOF
In view of lemma 1, it suffices to prove that if v is a vertex of
a simplicial complex K, every neighborhood U of v in IKI contains a neighborhood V of v deformable in U to v. Let U be a neighborhood of v and let
A = st v. Define F: A X I ~ IKI by
F(a,t) = tv
+ (1 - t)a
Then F is a deformation of A in IKI to the point v, and F(v X I) = v E U.
Therefore there is some neighborhood Vof v in A such that F(V X I) C U.
Because A = st v is open in IKI, V is a neighborhood of v in IKI. Since
F I V X I is a deformation of V in U to v, IKI is locally contractible. •
It follows from theorem 2 that the theory of covering projections applies
to polyhedra, and corresponding to any subgroup of the fundamental group
of a polyhedron there is a covering projection. We show that any covering
projection of a polyhedron corresponds to a simplicial map.
a
THEOREM
Let p: X ~ X be a covering proiection, where X is a polyhedron. Then X is a polyhedron, of the same dimension as X, in such a way
that p corresponds to a simplicial map.
Assume that p: X ~ IKI is a covering projection. For any simplex
the closed simplex lsi is simply connected. It follows from the lifting
theorem that the inclusion map lsi C IKI can be lifted to a map lsi ~ X, and
it follows from the unique lifting theorem that two such liftings are either
identical or have disjoint images. Hence there are as many liftings of lsi as
sheets of X over lsi.
Define a simplicial complex K to have the collection {p-1(V) I v is a
vertex of K} as vertex set and to have simplexes {S}, where s = {vo, ... ,vq}
is a simplex of K if and only if there is a simplex s = {vo, ... ,v q } in K and
a lifting is: lsi ~ X of lsi such that h(Vi) = Vi for 0 ::;; i ::;; q [in which case
s = p(s) and h are both unique]. Then K is a simplicial complex and has the
same dimension as K. If 81 is a face of s, then P(Sl) is a face of p(s) and
is IIp(sl) I = h,· Therefore the collection {fds di defines a continuous map
f: IKI ~ X such that
PROOF
sE K
j(2:. a iv i)
= h(2:. a iP(Vi))
Let cp: K ~ K be the simplicial map cp(v)
tive triangle
~aivi E
= p(v). Then there is a commuta-
JKJ~ X
1'P1\
/p
JKJ
lsi
SEc.8
145
EXAMPLES AND APPLICATIONS
To complete the proof it suffices to prove that (K,f) is a triangulation of
(that is, that f is a homeomorphism). If v is a vertex of K, then st v,
being contractible, is evenly covered by p. For v E p-l(V) let Uv be the
component of p-l(st v) containing 0. Then p I Uv is a homeomorphism of Uv
onto st v. By the definition of K and cp, Icpll st i5 is a homeomorphism of st 0
onto st v for D E p-l(V). From the commutativity of the above triangle,
fl st i5 is a homeomorphism of st v onto Uv for v E p-l(v). Since Icpl-l(st v) =
U{st 131 DE p-l(V)}, fllcpl-l(st v) is a homeomorphism of Icpl-l(st v) onto
p-l(st v). Since this is so for every vertex v of K, fis a homeomorphism of IKI
onto
X
x. •
The following corollary is an interesting application of these results.
4
COROLLARY
Any subgroup of a free group is free.
PROOF
Let F be a free group. It follows from example 3.7.6 that there is a
polyhedron (in fact, a wedge of I-spheres) X with base point Xo such that
'7T(X,xo) :::::: F. Let F' be any subgroup of F. Under the above isomorphism F'
corresponds to some subgroup H C '7T(X,xo). Let p: X -!> X be a covering projection such that X is path connected, p( xo) = Xo, and p#'7T( X,xo) = H. By
theorem 3, X is homeomorphic to the space of a connected graph. By corollary 3.7.5, '7T(X,xo) is a free group. •
If K is a finite connected graph, it follows from corollary 3.7.5 that
E(K,vo) is a free group on I - no + nl generators, where no is the number
of vertices of K and nl is the number of I-simplexes of K. If p: X -!> IKI is a
covering projection of multiplicity m, the number of q-simplexes in the corresponding triangulation (K,f) of X (given by theorem 3) equals mnq, where nq
is the number of q-simplexes of K. Therefore the method used to prove
corollary 4 also yields the following result.
S
COROLLARY
Let F be a free group on n generators and let F be a subgroup of F of index m. Then F is a free group on 1 - m + mn generators. •
We now investigate the effect on the fundamental group of the process
of attaching cells. Let A be a closed subset of a space X. X is said to be
obtained from A by adjoining n-cells {ejn}, where n ::::: 0, if
(a) For each j, ejn is a subset of X.
(b) If ejn = ejn n A, then for i =1= i', ejn - ejn is disjoint from ern - ern.
(c) X has a topology coherent with {A,f1n) and X = A U Uj f1n.
(d) For each j there is a map
ff
(En,Sn-l)
-!>
(ejn,ejn)
such that jj(En) = et, fj maps En - Sn-l homeomorphically onto ejn - ejn,
and ejn has the topology coinduced by jj and the inclusion map ejn C ejn.
Note that if n = 0, X is the topological sum of A and a discrete space.
A map fj: (En,Sn-l) -!> (ejn,l?jn) satisfying condition (d) above is called a
146
POLYHEDRA
CHAP.
3
characteristic map for ejn, and fj I Sn-1: Sn-1 ~ A is called an attaching map
for ejn. X is characterized by A and the collection {fj I sn-1} of attaching
maps. Given A and an indexed collection of maps {gj: Sn-1 ~ A}, there is a
space X obtained from A by attaching n-cells {ejn} by the maps gj. X is
defined as the quotient space of the topological sum V Ejn v A, where
Ejn = En for each j, by the identifications z E Sjn-1 equals gj(z) E A. Then the
inclusion map (Ejn,Sjn-1) C (V Ejn v A, V St- 1 v A) followed by the projection to (X,A) is a characteristic map jj: (Ejn,Sjn-1) ~ (X,A) for an n-cell
ejn = jj(Ejn).
Following are two examples.
6
If K is a simplicial complex, IKql is obtained from IKq- 1 by adjoining
q-cells {lsi I s is a q-simplex of K}.
1
°
7
For i = 1, 2, or 4 let Fi be R, C, or Q, respectively, and for q 2:: let
Pq(Fi) be the real, complex, or quaternionic projective space of dimension q.
Pq(Fi) is imbedded in Pq+1(Fi ) by the map [to, tb . . . ,tqJ ~ [to, t1, . . . ,tq,OJ
for tj E h Then Pq+1(F i ) is obtained from Pq(Fi) by adjoining a single (q + l)icell. If E(q+1)i is identified with the space {(to,t1' . . . ,tq) E Fiq+1 12:ltjl2 S; I},
then a characteristic map f: (E(q+1)i,S(q+1)i-1) ~ (Pq+1(F i ),Pq(F i )) for this
single cell is defined by the equation
f(to,t1, . . . ,tq) = [to,t1, . . . ,tq, 1 - 2: Itjl2J
Let X be obtained from A by adjoining n-cells for n 2:: 2. Then
for any point Xo E A the inclusion map i: (A,xo) C (X,xo) induces an
epimorphism
8
LEMMA
i#: 7T(A,xo)
~
7T(X,XO)
Let X be obtained from A by adjoining the n-cells {ejn}, and for each
j let Yj E ejn - cjn and let B j be a neighborhood of Yj in ejn - cjn homeomorphic to En. Let w: (1,1) ~ (X,xo) be a closed path at Xo. We show that w is
homotopic to a path in U = X - {yj} j. By the compactness of I, we can subdivide I by points = to < t1 < ... < tn = 1 such that for S; i < neither
W[ti,ti+1J C U or W[ti,ti+1J C Bj for some j. If W[ti,ti+1l u W[ti+bti+2l C Bj,
we can omit the point ti+1 from the subdivision of I to obtain another subdivision of I with the same property. Continuing in this way we can obtain a
subdivision such that if W[ti,ti+1J C Bj, then neither W[ti-1,til nor W[ti+1,ti+2J
is contained in Bj. It follows that W(ti) =1= Yj and W(ti+1) =1= Yj. For each such i,
because Bj - Yj is path connected and Bj is simply connected, wi [ti,ti+1J is
homotopic reI {ti,ti+d to a path contained in Bj - Yj. Since altogether there
are only a finite number of such subintervals of I, w ~ w', where w'(I) C U.
Because Sn-1 is a strong deformation retract of En minus a point, it follows
that cjn is a strong deformation retract of ejn - Yj. Therefore A is a strong deformation retract of U and w' ~ w", where w"(I) CA. Then i#[w"l = [wJ. •
PROOF
°
°
SEc.8
9
147
EXAMPLES AND APPLICATIONS
COROLLARY
For all n
~ 0,
Pn(C) and Pn(Q) are simply connected.
PROOF
Because Po(C) and Po(Q) are each one-point spaces, the result follows
by induction on q, using lemma 8 and the fact that Pq + 1 (C) is obtained from
Pq(C) by adjoining a 2(q + I)-cell and Pq+ 1 (Q) is obtained from Pq(Q) by
adjoining a 4(q + I)-cell. •
We want to compute the kernel of i# for the case n = 2. Given any map
g: 51 ~ A, where A is path connected, and given a point Xo E A, a normal
subgroup of 7T(A,xo) is determined as follows. If g(po) = Xl and w is a path in
A from Xo to Xl, then h[wl&( 7T(51,po)) is a cyclic subgroup of 7T(A,xo), and for a
different choice of w we obtain a conjugate subgroup in 7T(A,xo). Therefore
the normal subgroup of 7T(A,xo) generated by h[wlg#( 7T(51,po)) is independent
of the choice of the path w. Similar statements apply to a collection of maps
{g{ 51 ~ A}. There is a well-defined normal subgroup of 7T(A,xo) determined
by these maps.
10 THEOREM Let A be a connected polyhedron and let X be obtained from
A by attaching 2-cells to A by maps {g( 51 ~ A}. If N is the normal subgroup of 7T(A,xo) determined by the maps {&}, then
i#: 7T(A,xo)
~
7T(X,XO)
is an epimorphism with kernel N.
PROOF
By lemma 8, i# is a surjection. Let p: A ~ A be a covering projection
such that A is path connected, p(xo) = Xo, and P#(7T(A,XO)) = N. Because N is
normal in 7T(A,xo), p is a regular covering projection. Because N is the subgroup
determined by the maps {gj}, each map gj lifts to a map [!,( 51 ~ A. Let X
be the space obtained from A by attaching 2-cells for all the lifted maps {[!,j}
and extend p to a map pi: X ~ X such that pi maps each 2-cell of X homeomorphically onto its corresponding 2-cell of X. Then pi is easily seen to be a
covering projection.
We know from the definition of N that i#(N) = 1. Assume that
[w] E 7T(A,xo) is in the kernel of i#. Let w be any lifting of w in A such that
w(O) = xo. Then w is a lifting of w in X. Because w is null homotopic in X,
wis a closed path in X. Therefore wis a closed path in A, and so
[w] = p#[ w] EN.
Note that for the proof of theorem 10 it was not necessary that A be a
connected polyhedron. It would have been sufficient to assume A path
connected, locally path connected, and semilocally I-connected.
II
COROLLARY
For any group G there is a space X with 7T(X,XO) :::::; G.
PROOF
Represent G as the quotient group of a free group F and a normal
subgroup N. There is a polyhedron A such that 7T(A,xo) :::::; F (in fact, as in
example 3.7.6, A can be taken to be a wedge of I-spheres). For each A E N
148
POLYHEDRA
CHAP.3
let gA: (Sl,po) ~ (A,xo) be a map such that [gAl corresponds to A under the
isomorphism '1T(A,xo) ;::::; F. Let X be the space obtained from A by attaching
2-cells by the maps {gAl. By theorem 10, there is an isomorphism '7T(X,Xo) :=:> G. •
We now specialize to the case of a surface. These are the spaces of finite
two-dimensional pseudomanifolds without boundary. An n-dimensional
pseudomanifold without boundary (or absolute n-circuit) is a simplicial complex K such that
(a) Every simplex of K is a face of some n-simplex of K.
(b) Every (n - I)-simplex of K is the face of exactly two n-simplexes of K.
(c) If sand s' are n-simplexes of K, there is a finite sequence
s
S1. S2, . . • ,Sm
s' of n-simplexes of K such that Si and Si+l have
an (n - I)-face in common for I S; i
m.
=
=
<
We define a surface to be the space of a finite two-dimensional pseudomanifold without boundary in which the star of every vertex is homeomorphic to R2. It can be shown l that every surface has a normal form consisting
of a polygon in the plane with identifications of its edges. These fall into
classes, those with h :::::: 0 handles and those with k crosscaps. The surface
with 0 handles is the polygon with identifications of its edges pictured as
o
a
a
Surface with 0 handles
Topologically it is homeomorphic to the 2-sphere S2. For h
with h handles is pictured as
> 0 the surface
-----Surface with h
> 0 handles
The surface with one handle is topologically the torus.
1 See S. Lefschetz, Introduction to Topology, Princeton University Press, Princeton, N.J., 1949,
and H. Seifert and W. Threlfall, Lehrbuch der Topologie, B. C. Teubner, Verlagsgesellschaft,
Leipzig, 1934.
EXERCISES
For k
n
149
2:: 1, the surface with k crosscaps is pictured as
1
Ck
,
Cl
C2
------ ,
Surface with k crosscaps
The surface with one crosscap is topologically the real projective plane P'2,
and the surface with two cross caps is topologically the Klein bottle.
The normal form represents a surface with h 2:: 1 handles as a wedge of
2h I-spheres with a single 2-cell attached by a suitable map. If A is the wedge
of 2h I-spheres, then '17(A) is a free group on 2h generators, which generators
we denote by ai and bi, where 1 :::;; i :::;; n. If X is the surface with h handles,
X is obtained from A by attaching a single 2-cell to A by a map g: 51 ~ A
such that g# maps a generator of '17(5 1) to the element alblal-lb1-1 ...
ahbhah -lb h-1 E '17(A). Theorem 10 then provides a description of '17(X) in
terms of generators and relations. Similar remarks apply to a surface with
k 2:: 1 crosscaps. The result is summarized below.
I 2 The fundamental group of a surface is
(a) Trivial for the surface with no handles.
(b) A group with generators aI, b l , . . . ,ah, b h and the single relation
a1b1a1-1bl-1 ... ahbhah -lb h-1 = 1 for a surface with h 2:: 1 handles.
(c) A group with generators C1, Cz, . . . , Ck and the single relation
C12C22 ... Ck Z = 1 for a surface with k 2:: 1 crosscaps. -
EXERCISES
A TOPOLOGICAL PROPERTIES OF POLYHEDRA
I Prove that a compact polyhedron is an absolute neighborhood retract. (Hint: Assume
X = IKI and let K be a subcomplex of a simplex s. Use induction on the number of simplexes in s - K and the fact that a retract of an open subset of an absolute neighborhood
retract is an absolute neighborhood retract.)
2 Give an example of a space X and closed subset A C X such that A and X are both
polyhedra but (X,A) is not a polyhedral pair.
3
Prove that an open subset of a compact polyhedron is a polyhedron. [Hint: Since
IKI - U is a G a, there exists a sequence of open subsets Vi of IKI such that
n Vi
IKI - u. By induction on n, construct a sequence of subdivisions Kn and subcumplexes Ln C Kn such that (a) Kn is finer than the covering {U, Vn }, (b) Ln is the
largest subcomplex of Kn such that ILnl C U, and (c) Kn+l is a subdivision of Kn containing Ln as subcomplex. Then L
U Ln is a simplicial complex such that ILl
IKI - U.l
=
=
=
POLYHEDRA
150
CHAP.
3
" Let Y be an n-connected space and K be a simplicial complex. Prove that any continuous map IKI ~ Y is homotopic to a map which sends IKnl to a single point. If
fo, h: (IKI,IKnl) ~ (Y,yo) are homotopic, prove that they are homotopic relative to IKn- 1 1·
:; Let Y be a space which is n-connected for every n and let (X,A) be a polyhedral pair.
Prove that two maps X ~ Y which agree on A are homotopic relative to A.
6 Prove that a polyhedron is contractible if and only if it is n-connected for every n.
If it has finite dimension m, it is contractible if and only if it is m-connected.
B
I
EXAMPLES
Prove that pn is a polyhedron for all n.
2
Let K be the simplicial complex consisting of vertices V1, V2, . . . , vp and simplexes
and {Vp,V1} and let I be the simplicial complex with
o and I as vertices and {0,1} as I-simplex. Then K * I is a simplicial complex with
vertices V1, . . . , v p , 0, and 1. If q is an integer relatively prime to p and Vi is defined
for all integers i to be equal to Vj if i
f mod p, then let X be the space obtained from
IK * II by identifying the 2-simplex {Vi,Vi+1,0} linearly with the 2-simplex {vi+q,vi+Q+1,I}
for all i. Prove that X is homeomorphic to the lens space L(p,q) and that X is a polyhedron.
{V1,V2}, {V2,V3}, . . . , {Vp_1,Vp },
=
3
Prove that the generalized lens space L(p, q1, . . . ,qn) is a polyhedron.
" If X and Yare polyhedra and one of them is locally compact, prove that X * Y and
X X Yare also polyhedra.
C PSEUDOMANIFOLDS
A simplicial complex is said to be homogeneously n-dimensional if every simplex is a face
of some n-simplex of the complex. An n-dimensional p8eudomanifold is a simplicial
complex K such that
(a) K is homogeneously n-dimensional.
(b) Every (n - I)-simplex of K is the face of at most two n-simplexes of K.
(e) If 8 and 8 f are n-simplexes of K, there is a finite sequence 8 = 81, 82, . . . , 8 m = 8 f
of n-simplexes of K such that 8i and 8i+1 have an (n - I)-face in common for
I:::;; i m.
<
The boundary of an n-dimensional pseudo manifold K, denoted by K, is defined to be
the subcomplex of K generated by the set of (n - I)-simplexes which are faces of exactly
one n-simplex of K. (If K is empty, then K is an n-dimensional pseudomanifold without
boundary, as defined in Sec. 3.8.)
I Prove that an n-simplex is an n-dimensional pseudomanifold whose boundary, as a
pseudomanifold, is s.
2 If K is a pseudomanifold and L is a subdivision of K, prove that L is a pseudomanifold and t = L I K.
3 If K is a finite I-dimensional pseudomanifold, prove that K is either empty or consists of exactly two vertices.
" Give an example of an n-dimensional pseudo manifold K such that K is neither
empty nor an (n - I)-dimensional pseudomanifold.
D
SIMPLICIAL MAPS
In the first four exercises K will be a finite n-dimensional pseudo manifold, where n
> 0,
151
EXERCISES
with nonempty boundary K, K' will be a simplicial subdivision of K, and cp: K' ---7 K will
be a simplicial map such that cp I K' maps K' to K and is a simplicial approximation to
the identity map IK'I C IKI. Furthermore, sn-I will be a fixed (n - I)-simplex of K and
sn will be the unique n-simplex of K having sn-I as a face.
I For each n-simplex s' of K' let a(s') be the number of (n - I)-faces of s' mapped
onto sn-I by cpo Prove that a(s') = 1 if and only if cp maps s' onto sn and that a(s') = 0
or 2 otherwise.
2 Prove that the number of n-simplexes of K' mapped onto sn by cp has the same parity
as the number of (n - I)-simplexes of K' mapped onto sn-I by cpo [Hint: They both have
the same parity as L a(s'), the summation being over. all n-simplexes s' of K'.]
3 Spemer lemma. Prove that the number of n-simplexes of K' mapped onto sn by cp is
odd. (Hint: Use induction on n.)
4
Prove that
IKI is not a retract of IKI.
:; Brouwer fixed-point theorem. Prove that every continuous map of En to itself has a
fixed point.
E
SIMPLICIAL MAPPING CYLINDERS
Let cp: K ---7 L be a simplicial map between simplicial complexes whose vertex sets are
disjoint. We assume that the vertices of K are simply ordered. The simplicial mapping
cylinder M of cp is the simplicial complex whose vertex set is the union of the vertex sets
of K and L and whose simplexes are the simplexes of K and of L and all subsets of sets
of the form {vo, . . . ,Vk, cp(Vk)' ... ,cp(vp )}, where {VO,VI, . . . ,vp } is a simplex of K
and Vo
VI
vp in the simple ordering of the vertices of K.
< < . .. <
I Prove that the inclusion maps i: K C M and i: L C M are simplicial maps. If
p: M ---7 L is defined by p(v) = cp(v) for va vertex of K and p(v') = v' for v' a vertex of L,
then prove that p is a simplicial map such that cp = poi and poi = I L .
2
If K is finite, prove that
i
0
p and 1M are contiguous.
3 Prove that ILl is a deformation retract of IMI.
F
EDGE-PATH GROUPS
I Prove that if K is a simplicial complex, there is a one-to-one correspondence between
equivalence classes of local systems on IKI with values in
and natural equivalence
classes of covariant functors from the edge-path groupoid of K to
e
e.
2 Van Kampen's theorem for simplicial complexes.! Let K be a connected simplicial
complex with connected subcomplexes LI and L2 such that LI n L2 is connected and
K = LI U L 2• Let Vo be a vertex of LI n L2 and let it: (LI n L 2 , vol C (LI,vo) and
i2 : (Ll n L 2 , vol C (L 2 ,vo). Prove that E(K,vo) is isomorphic to the quotient group of the
free product of E(L1,vo) with E(L 2 ,vo) by the normal subgroup generated by the set
{(i1#[m
0
(i2#m-l) I m E E(Li n L 2, vol}
3 If G is a finitely presented group, prove that there is a finite connected two-dimensional simplicial complex K whose edge-path group is isomorphic to G.
1 For the topological case see P. Olum, Non-abelian cohomology and Van Kampen's theorem,
Annals of Mathematics, vol. 68, pp. 658-668, 1958.
152
POLYHEDRA
CHAP.
3
'" Let X be a space with base pOint Xo E X. Prove that there exists a polyhedron Y,
with base point Yo E Y, and a continuous map f: (Y,Yo) ~ (X,xo) such that
f#: 7T(Y,yO) ;::::: 7T(X,Xo).
G NERVES OF COVERINGS
If Gil = {U} in an open covering of a space X and K(Gil) is its nerve, a canonical map
f: X ~ IK(GlI)1 is a continuous map such that f-l(st U) C U for every U E Gil.
I If Gil is a locally finite open covering of X, prove that there is a one-to-one correspondence between canonical maps X ~ IK(Gll)1 and partitions of unity subordinate to GIL
2 If GIl is a locally finite open covering of X, prove that any two canonical maps
X ~ IK(Gll)I are homotopic.
If Gil and C1{ are open coverings of X, with 'Ya refinement of Gil, a canonical proiection from
to ~ is a function 'f' which assigns to each V E r an element 'f'( V) E ~ such that
V C <p(V).
r
3 Prove that a canonical projection from C1{ to Gil defines a simplicial map K(V) ~ K(GlI)
and any two canonical projections from 'Y to Gil define contiguous simplicial maps
K('Y) ~ K(Gil).
'" If <p: K('Y) ~ K(GIl) is a canonical projection and f: X ~ IK('Y)I is a canonical map,
prove that the composite 1<p1 f: X ~ IK(Gll)I is a canonical map.
0
a Let X be a paracompact space and let g: X ~ IKI be a continuous map (where K is
a simplicial complex). Prove that there exists a locally finite open covering G)1 of X and a
simplicial map <p: K(G)I) ~ K such that for any canonical map f: X ~ IK(G)I)1 the composite 1<p1 f is homotopic to g. [Hint: Choose Gil to be any locally finite open refinement
of the open covering {g-l(st v) I v a vertex of K}, and for U E Gil choose <pi U) a vertex
of K such that U C g-l(st <p(U)).l
0
6 Let X be a compact Hausdorff space and let K be a simplicial complex. Prove that
there is a bijection
lim_ {[K(Gil);K]} ;::::: [X;IKll
where the direct limit is with respect to the family of finite open coverings of X directed
by refinement with maps induced by canonical projections and the bijection is induced
by canonical maps.
H DIMENSION THEORY
A topological space X is said to have dimension ~ n, abbreviated dim X ~ n, if every
open covering of X has an open refinement whose nerve is a simplicial complex of
dimension ~ n. If dim X ~ n but dim X i n - 1, then X is said to have dimension n,
denoted by dim X = n. If dim Xi n for any n, we write dim X = 00.
I
If A is a closed subset of X, prove that dim A
~
dim X.
2
If K is a finite simplicial complex with dim K
~
n, prove that dim IKI ~ n.
If s is an n-simplex, prove that dim lsi = n. (Hint: Let Gil be the open covering of
of stars of the vertices of s and assume that there is a refinement 'Y of Gil such that
dim K('Y) ~ n - 1. Let K' be a subdivision of s finer than "If'. There are simplicial maps
K' ~ K('Y) ~ s whose composite A is a simplicial approximation to the identity map
IK'I C lsi·)
3
lsi
153
EXERCISES
4
m
Let X be a paracompact space with dim X :::;: n. Prove that any map X ~ Sm, with
> n, is null homotopic.
:; Let X be a compact metric space and let C be the space of maps
topologized by the metric
f:
X~
R2n+l
d(f,g) = sup {llf(x) - g(x) II I x E X}
Prove that C is a complete metric space, and if
Cm
= {f E C I diam f-l(Z) < ~ for all z E R2n+l}
then show that Cm is an open subset of C for every positive integer m and
set of homeomorp~isms of X into R2n+1.
ti
C m is the
6 If X is a compact metric space of dimension:::;: n, prove that Cm is a dense subset
of C for every positive integer m. [Hint: Let Gil be a finite open covering of X by sets of
diameter < 11m such that dim K(01) :::;: n and let h: IK(01)1 ~ R2n+l be a realization of
K(c'll). If f: X ~ IK(01)1 is any canonical map, then h f E Cm. Given g: X ~ R2n+l and
given e> 0, show that it is possible to choose Gil and h as above, so that d(h f, g)
e.]
0
0
<
If X is a compact metric space of dimension :::;: n, prove that X can be embedded in
R2n+l (in fact, the set of homeomorphisms of X into R2n+l is dense in C).
7
CHAPTER FOUR
HOMOLOGY
THIS CHAPTER INTRODUCES THE CONCEPT OF HOMOLOGY THEORY, WHICH IS OF
fundamental importance in algebraic topology. A homology theory involves a
sequence of covariant functors Hn to the category of abelian groups, and we
shall define homology theories on two categories-the singular homology theory
on the category of topological pairs and the simplicial homology theory on the
category of simplicial pairs. The former is topologically invariant by definition
and is formally easier to work with, while the latter is easier to visualize
geometrically and by definition is effectively computable for finite simplicial
complexes. The two theories are related by the basic result that the singular
homology of a polyhedron is isomorphic to the simplicial homology of any of
its triangulating simplicial complexes.
The functor Hn measures the number of "n-dimensional holes" in the
space (or simplicial complex), in the sense that the n-sphere Sn has exactly one
n-dimensional hole and no m-dimensional holes if m =1= n. A O-dimensional
hole is a pair of points in different path components, and so Ho measures
path connectedness. The functors Hn measure higher dimensional connectedness, and some of the applications of homology are to prove higher dimensional
155
156
HOMOLOGY
CHAP.
4
analogues of results obtainable in low dimensions by using connectedness
considerations.
Sections 4.1 and 4.2 are devoted to the definition of the category of chain
complexes and to an appropriate concept of homotopy in this category.
Homology theory is introduced as a sequence of covariant functors naturally
defined from the category of chain complexes to the category of abelian groups.
Simplicial homology theory is defined by means of a covariant functor
from the category of simplicial complexes to the category of chain complexes.
We study it in detail in Sec. 4.3, where it is shown that two different definitions (one based on oriented simplexes, the other on ordered simplexes) are
isomorphic. In similar fashion, singular homology theory is defined via a
covariant functor from the category of topological spaces to the category of
chain complexes. Its basic properties are considered in Sec. 4.4, where it is
shown that "small" singular simplexes suffice to define singular homology.
Section 4.5 introduces the concept of exact sequence. All the homology
functors Hn occur together in the exact sequences of homology, and it is for
this reason that we consider all these functors Simultaneously, rather than one
at a time. Section 4.6 is devoted to the exact Mayer-Vietoris sequences connecting the homology of the union of two spaces (or simplicial complexes),
the homology of the spaces, and the homology of their intersection. We use
these to prove the isomorphism of the simplicial homology groups of a simplicial
complex with the singular homology groups of its corresponding space.
Section 4.7 contains some applications of homology theory. We prove
that euclidean spaces of different dimensions are not homeomorphic. We also
prove the Brouwer fixed-point theorem and the more general Lefschetz fixedpoint theorem. Finally, we prove Brouwer's generalization of the Jordan curve
theorem (that an (n - I)-sphere imbedded in Sn separates Sn into two components), and we establish the invariance of domain. Section 4.8 contains a
discussion of the axiomatic characterization of homology given by Eilenberg
and Steenrod, as well as some related concepts.
I
CHAIN COMPLEXES
This section introduces the category of chain complexes and chain maps and
the homology functor on this category. We also define covariant functors from
the category of simplicial complexes and from the category of topological
spaces to the category of chain complexes. The composites of these and the
homology functor define homology functors on the category of simplicial
complexes and on the category of topological spaces.
A differential group C consists of an abelian group C and an endomorphism a: C ~ C such that aa
O. The endomorphism a is called the differential, or boundary operator of C. There is a category whose objects are
differential groups and whose morphisms are homomorphisms commuting
with the differentials.
=
SEC.
1
157
CHAIN COMPLEXES
For a differential group C there is a subgroup of cycles Z(C) = ker 0 and
a subgroup of boundaries B(C) = im o. Because 00 = 0, B(C) C Z(C). The
homology group H( C) is defined to be the quotient group
H(C) = Z(C)/B(C)
The elements of H( C) are called homology classes. If z is a cycle, its homology
class in H( C) is denoted by {z}. Two cycles Z1 and Z2 are homologous, denoted
by Z1 - Z2, if their difference is a boundary, that is, if {zt} = {Z2}.
If 7': C ---7 C' is a homomorphism of differential groups commuting with
the differentials, then 7' maps cycles of C to cycles of C' and boundaries of C
to boundaries of C'. Therefore 7' induces a homomorphism
7'* :H(C)
---7
H(C')
such that 7'* {z} = {7'(z)} for z E Z( C). Because (7'17'2)* = 7'1* 7'2*, there is a
covariant functor from the category of differential groups to the category of
groups which assigns to a differential group C its homology group H( C) and
to a homomorphism 7' its induced homomorphism 7'* .
A graded group C = {C q } consists of a collection of abelian groups C q
indexed by the integers. Elements of C q are said to have degree q. A homomorphism 7': C ---7 C' of degree d from one graded group to another consists of a
collection 7' = {7'q: Cq ---7 C~+d} of homomorphisms indexed by the integers. We
shall omit the subscript in 7' q where there is no likelihood of confusion. It is
obvious that the composite of homomorphisms of degrees d and d' is a homomorphism of degree d + d', and that there thus is a category of graded
groups and homomorphisms (with each homomorphism having some degree).
It has a subcategory of graded groups and homomorphisms of fixed degree O.
Because the sum of two homomorphisms from C to C' of degree 0 is again a
homomorphism from' C to C' of degree 0, hom (C,C') is an abelian group
[hom (C,C') being the set ()f morphisms in the category whose morphisms are
homomorphisms of degree 0].
A differential graded group (sometimes abbreviated to DC group) is a
graded group that has a differential compatible with the graded structure
(that is, the differential is of degree r for some r). A chain complex is a differential graded group in which the differential is of degree -1. Thus a chain
complex C consists of a sequence of abelian groups Cq and homomorphisms
Oq: Cq ---7 Cq- 1
indexed by the integers such that the composite
Cq+1
0.+1
~
Cq
0.
---7
C q_ 1
is the trivial homomorphism. The elements of C q are called q-chains of the
complex. Most of the chain complexes we consider will have the additional
property that Cq = 0 for q
O. Such a complex is said to be nonnegative.
A free chain complex is a chain complex in which C q is a free abelian group
for every q.
<
158
HOMOLOGY
CHAP.
4
For. a chain complex the group of cycles Z( C) is a graded group consisting of the collection {Zq(C) = ker Oq}, and the group of boundaries B(C) is a
graded group consisting of {Bq( C) = im oq+d. The homology group H( C) is
a graded group consisting of {Hq( C) = Zq( C) I Bq( C) }.
A chain map T: C ~ G' (also called a chain transformation) between
chain complexes is a homomorphism of degree 0 commuting with the differentials. Thus T is a collection {Tq: Cq ~ C~} such that commutativity holds in
each square
Cq
Gq
~
Tql
C~
Cq- 1
1
'fq_l
0'q
~
C~_1
It is clear that there is a category of chain complexes whose objects are chain
complexes and whose morphisms are chain maps. It is also clear that if
C and G' are two objects in this category, hom (C,G') is an abelian group.
If T: C ~ G' is a chain map, its induced homomorphism
T*: H(C)
~
H(G')
is the homomorphism of degree 0 such that (T*)q{Z} = {Tq(Z)} for Z E Zq(C).
The following theorem is easily verified.
I
THEOREM
There is a covariant functor from the category of chain complexes to the category of graded groups and homomorphisms of degree 0
which assigns to a chain complex C its homology group H( C) and to a chain
map T its induced homomorphism T*. For any two chain complexes the map
T ~ T* is a homomorphism from hom (C,G') to hom (H(C),H(G')). •
A subcomplex G' of a chain complex C, denoted by G' C C, is a chain
complex such that C~ C Cq and o~ = Oq I C~ for all q. There is then an inclusion map i: G' C C consisting of the collection of inclusion maps {C~ C C q }.
There is also a quotient chain complex CI G' = {Cql C~} with boundary operator induced from that of C by passing to the quotient. The collection of projections {Cq ~ CqIC~} is the proiection chain map C ~ CIG'.
To describe a covariant functor from the category of simplicial complexes
to the category of free chain complexes, let K be a simplicial complex. An
oriented q-simplex of K is a q-simplex s E K together with an equivalence
class of total orderings of the vertices of s, two orderings being equivalent if
they differ by an even permutation of the vertices. If va, VI, . . • , Vq are the
vertices of s, then [VO,V1, . . . ,vq ] denotes the oriented q-simplex of K
consisting of the simplex s together with the equivalence class of the ordering
Va
VI
Vq of its vertices.
For q
0 there are no oriented q-simplexes. For every vertex V of K
there is a unique oriented O-simplex [v], and to every q-simplex, with q :::: 1,
there correspond exactly two oriented q-simplexes. Let Cq(K) be the abelian
group generated by the oriented q-simplexes (Jq with the relations (J1 q + (Jzq = 0
< < ... <
<
SEC.
1
159
CHAIN COMPLEXES
if 01 q and ozq are different oriented q-simplexes corresponding to the same
0, and for q ~ 0 Cq(K) is a free
q-simplex of K. Then Cq(K) = 0 for q
abelian group with rank equal to the num her of q-simplexes of K. If K is
empty, Cq(K) = 0 for all q.
We define homomorphisms Oq: Cq(K) ~ Cq_1 (K) for q ~ I by defining
them on the generators by
<
(a)
where [VO,Vl, . . . ,1\, ... ,vq] denotes the oriented (q - I)-simplex obtained
by omitting Vi. If 01 q + ozq = 0 in Cq(K), then it is easily verified that
Oq(Olq) + Oq(ozq)
0 in Cq _ 1 (K). Therefore Oq extends to a homomorphism
from Cq(K) to Cq_1 (K). For q ~ 0 we define Oq to be the trivial homomorphism
from Cq(K) to Cq_ 1 (K). lt is not difficult to show that OqOq+l = 0 for all q.
Therefore there is a free nonnegative chain complex C(K) = {Cq(K),oq},
which is called the oriented chain complex of K. lts homology group, denoted
by H(K), is a graded group {Hq(K) = Hq(C(K))}, called the oriented homology
group of K. Hq(K) is called the qth oriented homology group of K.
If K is realized in some euclidean space, the oriented q-simplexes of K
are q-simplexes of K together with orientations, in the sense of linear algebra,
of the affine varieties spanned by them. The boundary of an oriented q-simplex
is the sum of its oriented (q - I)-faces, with each face oriented by the orientation compatible with that of the q-simplex, as shown in the diagrams.
=
Vo~~------------------~
An oriented q-cycle z of K is a "closed" collection of oriented q-simplexes,
with each (q - I)-simplex lying in the boundary of z the same number of
times with each orientation. Hq(K) is the group of equivalence classes of these
q-cycles, two cycles being equivalent if their difference is a boundary. Thus
Hq(K) corresponds intuitively to the group generated by the q-dimensional
"holes" in IKI.
lt is convenient to add more generators and more relations to the chain
groups Cq(K). If Vo, VI, . . . , Vq are vertices (not necessarily distinct) of
some simplex of K, we define [VO,Vl, . . . ,vq] E Cq(K) to be 0 if the vertices
are not distinct and to be the oriented q-simplex as defined above if they are
distinct. Then equation (a) remains correct for these added generators (that is,
if the vertices VO,Vl, . . . ,Vq are not all distinct, the left-hand side of equation (a) is 0 and the right-hand side can also be verified to be 0).
160
HOMOLOGY
CHAP.
4
If cP: KI ~ K2 is a simplicial map, there is an associated chain map
C(cP): C(KI) ~ C(K2) defined by
(b)
C(CP)([VO,VI' . . . ,Vq])
= [cp(vo), CP(VI),
. . . ,cp(vq)]
Note that if va, Vb . . . ,Vq are distinct vertices of some simplex of K I , then
cp(vo), CP(VI), . . . , cp(vq) are vertices of some simplex of K2 but are not
necessarily distinct. Therefore the right-hand side of equation (b) above would
not be defined unless we had defined [VO,Vb . . . ,Vq] as an element of Cq,
whether or not the terms Vi are distinct. It is easy to verify that C( cp) is
a chain map.
2
THEOREM
There is a covariant functor C from the category of simplicial
complexes to the category of chain complexes which assigns to K its oriented
chain complex C(K). -
The composite of the functor C and the homology functor is a covariant
functor, called the oriented homology functor, from the category of simplicial
complexes to the category of graded groups. To a simplicial complex K
it assigns the graded group H(K) = {Hq(K) = Hq(C(K))}, and to a simplicial
map cP: KI ~ K2 it assigns the homomorphism cP*: H(KI) ~ H(K2) of degree
o induced by C(cP): C(Kl) ~ C(K2)' If L is a subcomplex of K, and i: L C K,
then C(i): C(L) ~ C(K) is a monomorphism by means of which we identify
C(L) with a subcomplex of C(K).
We next describe the singular chain functor from the category of
topological spaces to the category of chain complexes. Let po, PI, P2, . . . be
an infinite sequence of different elements fixed once and for all. For q 2': 0
let !1q be the space of the simplicial complex consisting of all nonempty subsets of {PO,Pb . . . ,pq} (therefore !1q is the closed simplex IpO,PI, . . . ,pql).
For q 2': 0 and 0 ~ i ~ q + 1 let
be the linear map defined by the vertex map
j<i
j'?i
. ()
eJ+1
Pi = {Pi
Pi+l
Then eJ+I(M) is the closed simplex IpO,PI," . ,pi, ... 'PHIl in !1q+1 opposite the vertex pi, and direct computation shows that
3
If 0 ~
i<i
~ q
+ 1, then eb+2e3+1 = e&+2eb:;:i.
Let X be a topological space. For q
defined to be a continuous map
a: !1q
~
-
2': 0 a singular q-simplex a of X is
X
>
For q
0 and 0 ~ i ~ q the ith face of a, denoted by
the singular (q - I)-simplex of X which is the composite
a(i)
=a
0
eqi : !1q-l
~
!1q
~
X
a(i),
is defined to be
SEC.
I
161
CHAIN COMPLEXES
It follows from statement 3 that
4
If q > 1 and 0
~
i <i
~
q, then (a(i»)(j)
= (a0»)(H).
-
The singular chain complex of X, denoted by fl(X), is defined to be the
free nonnegative chain complex fl(X) = {flq(X),a q}, where flq(X) is the free
abelian group generated by the singular q-simplexes of X for q 2': 0 [and
flq(X) = 0 for q < 0], and for q 2': 1, aq is defined by the equation
aq(a) =
2: (-l)ia(i)
O",i<:q
This is a chain complex because aq aq + 1 = 0 is an immediate consequence
of statement 4. If X is empty, flq(X) = 0 for all q.
If f: X ~ Y is continuous, there is a chain map
fl(f): fl(X)
~
fl(Y)
defined by fl(f)(a) = f a for a singular q-simplex a: flq
a chain map, and we have the following result.
0
~
X. Then fl(f) is
:;
THEOREM
There is a covariant functor fl from the category of topological spaces to the category of chain complexes which assigns to X its singular
chain complex t.(X). -
The composite of the functor t. and the homology functor is a covariant
functor, called the Singular homology functor, from the category of topolOgical spaces to the category of graded groups. To a space X it assigns the
graded group H(X) = {Hq(X) = Hq(fl(X))} and to a map f: X ~ Y it assigns
the homomorphism
f*: H(X)
~
H(Y)
of degree 0 induced by fl(f): fl(X) ~ fl(Y). Hq(X) is called the qth Singular
homology group of x. If A is a subspace of X and i: A C X, then the map
fl(i): t.(A) ~ t.(X) is a monomorphism by means of which we identify t.(A)
with a subcomplex of fl(X).
The category of chain complexes has arbitrary sums and products of
indexed collections. That is, if {OLEJ is an indexed collection of chain complexes, there is a sum chain complex EB 0 and a product chain complex X 0
in which (EB O)q = EB Oq and (X O)q = X Oq for every q. It follows that
Zq(EB 0) = EB Zq(O) and Bq(EB 0) = EB Bq(O) and Zq(X 0) = X Zq(O)
and Bq(X 0) = X Bq(O) for all q. Therefore H(EB 0) = EB H(O) and
H(X 0) = X H(O).
6
THEOREM
On the category of chain complexes the homology functor
commutes with sums and with products. -
The category of chain complexes also has direct and inverse limits (whose
qth chain groups are appropriate limits of the qth chain groups of the factors).
162
7
HOMOLOGY
THEOREM
CHAP.
4
The homology functor commutes with direct limits.
Let {Ca,'Ta fl } be a direct system of chain complexes and let {C,ia} be
the direct limit of this system (that is, ia: Ca ~ C, and if a ::;: [3, then
ia = ifl 'Ta fl : Ca ~ GIl ~ C). Then {H(ca),'Ta~} is a direct system of graded
groups, and we show that {H(C),i a *} is the direct limit of this system.
We show that 1.3a of the Introduction is satisfied. Let {z} E Hq(C). Then
z = iac" for some ca E (ca)q. Since
PROOF
0
there is [3 with a ::;: [3 such that 'TaflaqaCa = o. Then Taflca is a cycle of (Cfl)q
and ifl'T aflca = iaca = z. Therefore i fl * { 'T aflca} = {z}.
We show that 1.3b of the Introduction is also satisfied. Because we are
dealing with the direct limit of groups, it suffices to show that if {za} E Hq( Ca)
is in the kernel of ia*, then there is y with a ::;: y such that {za} is in the
kernel of 'T]*. If ia*{za} = 0, then iaza = aq+1c for some c E Cq+1. Because
c = iflcfl for some [3, we have iaza = ifla~+lCfl. Choose y' so that a, [3 ::;: y'.
Then iy'('T"Y'za - Tfly'ag+1c fl ) = O. Therefore there is y with y' ::;: y such
that 'Ty,y('TaY'za - 'Tfly'ag+1cfl ) = o. Then 'TaYZ a = aqY+l('TflYcfl), so 'Ta~ {za} = O. •
It is false that the homology functor commutes with inverse limits. We
present an example to show this.
EXAMPLE
For any integer n 2 1 let Cn be the chain complex with
(Cn)q = 0 if q =I=- 0 or 1 and (Cnh ~ (Cn)o equal to Z ~ Z, where the homomorphism is multiplication by 2. For each n let 'Tn: Cn+1 ~ Cn be the chain
map which is multiplication by 3 on each chain group, and for n ::;: m define
'Tn m: Cm ~ Cn to be the composite Tn m = 'Tn'T n+1 ... 'T m- 1. Then {Cn,'T nm} is
an inverse system whose inverse limit is the trivial chain complex. Therefore
Hq(lim~{Cn,'Tnm}) = 0 for all q. However, Ho(Cn) = Z2 for all nand
'Tnr::: Ho(C m) ;::::; Ho(Cn) for all n ::;: m. Therefore lim~ {Ho(Cn), 'Tn~} ;::::; Z2.
8
2
CHAIN HOMOTOPY
This section deals with homotopy in the category of chain complexes. For
free chain complexes we prove that contractibility is equivalent to triviality
of all the homology groups. This leads to discussion of a method for constructing chain maps and homotopies by a general procedure known as the method
of acyclic models. The section closes with a definition of mapping cone of a
chain map and its relation to the chain map.
Let 'T, 'T': C ~ C' be chain maps. A chain homotopy D from 'T to 'T', denoted by D: 'T ~ 'T', is a homomorphism D = {Dq} from C to C' of degree 1
such that for all q
SEC.
2
163
CHAIN HOMOTOPY
If there is a chain homotopy from T to T', we say that T is chain homotopic to
T' and write T ~ T'. It is trivial that chain homotopy is an equivalence relation in the set of chain maps from C to C' . The corresponding set of equivalence
classes is denoted by [C; C'], and if T: C --+ C' is a chain map, its equivalence
class is denoted by [T].
I
LEMMA
The composites of chain-homotopic chain maps are chain
homotopic.
PROOF
Assume D:
T
~ T', where T, T': C ---;. C', and
D:
i ~ i', where
C' ---;. C". Then
if, if':
is of degree 1 and is a chain homotopy from iT to f' T'.
•
It follows that there is a category whose objects are chain complexes and
whose morphisms are chain homotopy classes. A chain map T: C -,) C' is
called a chain equivalence if [T1is an equivalence in the homotopy category
of chain complexes. If there is a chain equivalence from C to C', we say that
C and C' are chain equivalent.
2
THEOREM
If T, T': C -,) C' are chain homotopic, then
T*
PROOF
Assume D:
T ~ T'.
= T*:
For any
H(C) -,) H(C')
Z
E ZiC)
= Tq(Z) showing that Tq(Z) ~ T~(Z) and T* {z} = T* {z}.
06+1Dq(Z)
T~(Z)
•
A chain contraction of a chain complex C is a homotopy from the identity chain map Ie to the zero chain map Oe of C to itself. If there is a chain
contraction of C, C is said to be chain contractible. C is said to be acyclic if
H(C) = 0 (that is, Hq(C) = 0 for all q).
3
COROLLARY
Assume that C is a chain complex such that Ie c::::: Oe. By theorem 2,
However, (l e )* = IH(C) and (0 0 )* = OH(C). Therefore
= OH(C), which can happen only if H(C) = O. •
PROOF
(l e )*
I H (C)
A contractible chain complex is acyclic.
= (0 0 )*.
The converse of corollary 3 is false.
4
EXAMPLE
Let C be the chain complex with Cq = 0 for q =1= 0, 1, 2 and
with Cz ~ C1 ~ Co equal to Z -4 Z ~ Z2, where ll'(n) = 2n, f3(2m) = 0, and
f3(2m + 1) = 1. Then C is acyclic but not contractible. In fact, if D: Ie ~ Oe
were a chain contraction of C, then the homomorphism f3 would have a right
inverse Do: Z2 -,) Z, but any homomorphism Z2 ---;. Z is trivial.
If C is assumed to be a free chain complex, there is a converse of
corollary 3.
164
:.
HOMOLOGY
THEOREM
CHAP.
4
A free chain complex is acyclic if and only if it is contractible.
We show that if C is an acyclic free chain complex, it is contractible.
For each q the map aq is an epimorphism of Cq to Bq-l(C) = Zq-l(C).
Because Cq- l is free, so is Za-l(C), and there is a homomorphism
PROOF
Sq_l: Zq-l(C)
~
Cq
which is a right inverse of aq. Then lCq - Sq_la q maps Cq to Zq(C), and we
define {Dq} by
Then
aq+lDq + Dq_laq = lCq
-
Sq_laq +
sq_l(l~_l -
which shows that {Dq} is a chain contraction of C.
Sq_2a q_l)a q = lCq
•
The method of proof of theorem 5 is a standard one used to construct
chain maps and homotopies from a free chain complex to an acyclic chain
complex. We now extend it to obtain a general method of constructing chain
maps and chain homotopies, called the method of acyclic models. Repeated
application of this method will be made in subsequent discussions. We consider a special version of the method of acyclic models which suffices for our
applications. l
A category with models consists of a category e and a set 0fL of objects
of 2 called models. Given a covariant functor G from a category e with
models 0fL to the category of abelian groups, a basis for G is an indexed collection {& E G(Mj ) }iEJ, where M j E 0fL such that for any object X of the
indexed collection
e
{G(f)(&) }iEJ,fE hom (Mj.xl
e
is a basis for G(X). If G has a basis, it is called a free functor on with models 0lL
In this case, if h E hom (X, Y), then G(h) maps each basis element of G(X)
to some basis element of G(Y). Hence G is the composite of the covariant
functor which assigns to X the set {G(f)(gj) liE J, f E hom (Mj,X)} with the
covariant functor of example 1.2.2, which assigns to every set the free abelian
group generated by it.
Let G be a covariant functor from a category with models 0fL to the
category of chain complexes. G is said to be free if G q is a free functor to the
category of abelian groups.
e
6
EXAMPLE
Let K be a simplicial complex and let e(K) be the category
defined by the partially ordered set of subcomplexes of K (as in example 1.1.11).
Let 0fL(K) = {s I S E K} be models for e(K). We show that the covariant
functor C which assigns to each subcomplex of K its oriented chain complex
is a free nonnegative functor on e(K) with models 0fL(K) to the category of
1 A general treatment can be found in S. Eilenberg and S. Mac Lane, Acyclic models, American Journal of Mathematics, vol. 79, pp. 189-199 (1953).
SEC.
2
165
CHAIN HOMOTOPY
chain complexes. For each model s of dimension q choose once and for all an
oriented q-simplex o(s) which generates Cq(s). Then the indexed collection
{o( s) I dim s = q} S E K is a basis for Cq . Hence Cq is free witb models G)lL(K).
7
EXAMPLE
Let (' be the category of topological spaces with models
GJlL = {~q I q ::::: O} and let 6, be the singular chain functor. Then ~ is free and
nonnegative on (' with models GJlL. In fact, if ~q: 6,q C 6,q, then the singleton
{~q E ~q(6,q)} is a basis for 6,q.
Let G be a covariant functor on a category (' to the category of chain
complexes. Then there are covariant functors Hq(G), for all q, from (' to the
category of abelian groups that assign to an object X the group Hq( G(X)). If
(' is a category with models GJll, a functor G from (' to the category of chain
complexes is said to be acyclic in positive dimensions if Hq(G(M)) = 0
0 and M E :llL We now establish the main result dealing with
for q
the construction of chain maps and homotopies.
>
8
THEOREM
Let (' be a category with models ')ll and let G and G' be covariant functors from (' to the category of chain complexes such that G is free
and nonnegative and G' is acyclic in positive dimensions. Then
(a) Any natural transformation Ho(G) -,) Ho(G') is induced by a natural
chain map T: G -,) G'.
(b) Two natural chain maps T, T': G -,) G' inducing the same natural
transformation Ho(G) -,) Ho(G') are naturally chain homotopic.
For every object X of Cl we must define a chain map T(X): G(X) -,)
G'(X) [or a chain homotopy D(X): T(X) c::::' T'(X)] such that if h: X -,) Y is a
morphism in Cl, then
PROOF
T(Y)G(h) = G'(h)T(X)
[or D(Y)G(h) = G'(h)D(X)]
For q ::::: 0 let {gj E Gq(Mj)}jEJq be a basis for G q , where Mj E GJll for
each j E Jq. Then Gq(X) has the basis
{G q( f)(gj)} iE Jq,{ E hom (Mj,xl
It follows that Tq(X) [or Dq(X)] is determined by the collection {Tq(Mj)(gj)}iEJq
and the equation
(b)
Dq(X)(L.nijGq(fij)(~))
=
'2:nijGq+l(fii)Dq(Mi)(~)
We shall define Tq(X) by induction on q so that
(c)
OTq(X) = Tq_l(X)O
and define Dq(X) by induction on q so that
(d)
166
HOMOLOGY
Having defined Ti [or D i ] for i
for i E Iq so that
< q,
where q
> 0, it
CHAP.
4
suffices to define
Tq(Mj)(~)
(e)
= Tq_l(Mj)(o~)
OTq(Mj)(~)
and to define Dq(Mj)(~) for
(f)
oDq(Mj)(~)
i E Iq so that
= Tq(Mj)(~)
- T&(Mj)(gj) -
Dq_l(Mj)(o~)
since Tq(X) [and Dq(X)] are then determined by equation (a) [or by (b)]. It will
then be true that Tq(X) [and Dq(X)] are natural and will satisfy equation (c)
[and (d)].
Given a natural transformation rp: Ho(G) ~ Ho(G'), the inductive definition
of T proceeds as follows. For q
0 we define To(Mj)(gj) for i E 10 to be any element of G&(Mj) such that {To(Mj)(~)} = rp(Mj){~}. We use equation (a) to
define TO (X) for all X. Then, for g E Go(X), {TO(X)(g)} = rp(X){g}. In
particular, for i E 11> To(Mj)(o~) is a boundary in G&(Mj). Hence we can define
Tl(Mj)(~) E G1(Mj) so that OTl(Mj)(~) = To(Mj)(o~). We then use equation (a)
to define Tl(X) for all X. Assuming Ti defined for i
q, where q > I, so that
equation (c) is satisfied, we observe that the right-hand side of equation (e) is
a cycle of G~_l(Mj). Because q
1, Hq_1(G'(Mj)) = 0, and we define Tq(Mj)(~)
to satisfy equation (e). We next define Tq(X) for all X to satisfy equation (a).
This completes the definition of T.
Given T, T': G ~ G' such that T and T' induce the same natural transformation Ho(G) ~ Ho(G'), we define Do(Mj)(~) for i E 10 to be any element of
G1(Mj) whose boundary equals To(Mj)(~) - T&(Mj)(~). Then Do(X) is defined
for all X by equation (b). Assuming Di defined for i
q, where q 0, so that
equation (d) is satisfied, we observe that the right-hand side of equation (f)
is a cycle of G~(Mj). Because q
0, Hq(G'(Mj)) = 0, and this cycle is a
boundary. We define Dq(Mj)(~) E Gq+1(Mj) to satisfy equation (f), use equation (b) to define Dq(X) for all X, and complete the definition of D. •
=
<
>
<
>
>
The last result provides another proof of theorem 5 for nonnegative complexes. In fact, if C is a free nonnegative chain complex, let e be the category
consisting of one object X and one morphism Ix and let C be regarded as a
covariant functor on e with model {X}. Then C is a free nonnegative functor,
and if C is an acyclic chain complex, the functor C is acyclic in positive
dimensions. In this case, because Ie and Oe are chain transformations of C
inducing the same homomorphism of Ho( C) = 0, it follows from theorem 8
that Ie ~ Oe, and C is contractible.
There is a useful algebraic object (related to the mapping cylinder of
Sec. 1.4) which we now describe. Let T: C ~ C' be a chain map. The mapping
cone of T is the chain complex C {Cq,a q} defined by Cq Cq- 1 EEl Cq and
=
Ciq(c,c') = (-Oq_l(C), T(C)
=
+ o~(c'))
c E Cq- 1 ,
C'
E C~
The following result is trivial to verify.
9
so is
LEMMA
C. •
C is a chain complex, and if C and C' are free chain complexes,
SEC.
3
167
THE HOMOLOGY OF SIMPLICIAL COMPLEXES
The next theorem is the main reason for introducing mapping cones.
10 THEOREM A chain map is a chain equivalence if and only if its mapping cone is chain contractible.
PROOF
Assume that T: C ~ C' is a chain equivalence. There exist T': C' ~ C
and D: C ~ C and D': C' ~ C' such that D: T'T ~ lc and D': TT' ~ Ie-.
Define D: C ~ C by D(c,c') = (Cl,C2), where
Cl
Cz
= D(c) + T'D'T(C) - T'TD(c) + T'(C')
= D'TD(c) - D'D'T(C) - D'(c')
A straightforward computation shows that D is a chain contraction of C.
Conversely, assume that D is a chain contraction of C. Define T': C' ~ C
and D: C ~ C and D': C' ~ C' by the equations
(T'(C'), -D'(c')) = D(O,c')
(D(c),o) = D(c,O)
Direct verification shows T' to be a chain map and D:
lc', so T is a chain equivalence. -
T' T
~ lc
and D':
TT'
~
Combining this with theorem 5 and lemma 9 yields the following result.
I I COROLLARY A chain map between free chain complexes is a chain
equivalence if and only if its mapping cone is acyclic. -
3
THE 1I0MOLOGY OF SIMPLICIAL COMPLEXES
This section begins with a discussion of augmented chain complexes and
their reduced homology groups. Next we define the ordered chain complex of
a simplicial complex and prove that it is chain equivalent to the oriented
chain complex. We use this result to show that simplicial maps in the same
contiguity class induce chain-homotopic chain maps. We also compute Ho(K)
in terms of the components of K. At the end of the section the relative
homolugy groups and the Euler characteristic of a simplicial pair are defined.
In the category of nonempty simplicial complexes any simplicial complex
P consisting of a single vertex is a terminal object. If K is a nonempty simpliciai compiex, the simpliCial map K -> P has a right inverse. Therefore the
induced homology map H(K) ~ H(P) has a right inverse. Because Hq(P) = 0
if q =1= 0 and Ho(P) ;::::: Z, it follows that there is an epimorphism Ho(K) ~ Z.
Since Ho(K) = CO(K)/a l Cl(K), there is an epimorphism E: Co(K) ~ Z such
that Ea l = O. Similarly, in the category of nonempty topological spaces X any
one-point space is a terminal object. The same kind of considerations yield an
epimorphism E: ~o(X) ~ Z such that Ea l = O. This motivates the following
definition of augmentation.
An augmentation (over Z) of a chain complex C is an epimorphism
E: CO ~ Z such that EOl: C l ~ Co ~ Z is trivial. An augmented chain complex
168
HOMOLOGY
CHAP.
4
is a nonnegative chain complex C with augmentation. An augmentation e of
a chain complex can be regarded as an epimorphic chain map of C to the
chain complex (also denoted by Z) whose only nontrivial chain group is Z in
degree O. For this chain complex Z, it is clear that Hq(Z) = 0 for q 0:/= 0 and
that Ho(Z) = Z. Therefore e induces an epimorphism e*: Ho(C) ~ Z. Hence
an augmented chain complex has a nontrivial homology group in degree O.
The oriented chain complex C(K) of a nonempty simplicial complex K is
augmented by the homomorphism e: Co(K) ~ Z defined by e([v]) = 1 for
every vertex f) of K. The singular chain complex Ll(X) of a nonempty space X
is augmented by the homomorphism e: Llo(X) ~ Z defined by e(a) = 1 for
every singular O-simplex of X.
A chain map T: C ~ C' between augmented chain complexes preserves
augmentation if e' T = e: Co ~ Z. Note that T preserves augmentation if
and only if T* does-that is, if and only if e,;, T* = e*: Ho(C) ~ Z. There is
a category of augmented chain complexes and chain maps preserving
augmentation. A chain homotopy in this category is any chain homotopy
between chain maps in the category.
We want to divide out the functorial nontrivial part of Ho(C)
of an augmented chain complex C. The reduced chain complex C of
an augmented chain complex C is defined to be the chain complex defined
by Cq = C q if q 0:/= 0, Co = ker e, and aq = aq [note that a1( ( 1) c Co
because ea 1 = 0]. Thus Cis the kernel of the chain map e: C ~ Z. If T: C ~ C'
is a chain map preserving augmentation, T induces a chain map C ~ C'
between their reduced chain complexes. The homology group H( C) is called
the reduced homology group of C and is denoted by H(C). For a non empty
simplicial complex K we define H(K) = H(C(K)), and for a nonempty topological space X we define H(X) = H(Ll(X)). Because the chain complex of an
empty simplicial complex or an empty topological space has no augmentation,
the reduced groups are not defined in this case. For that reason some of the
arguments, which otherwise involve the reduced groups, require a special
remark in the case of empty complexes or spaces.
Clearly, there is an inclusion chain map C C C.
0
0
I
LEMMA
If C is an augmented chain complex, then
qo:/=O
q=O
Because Z is a free group, Co:::::; Co E8 Z. Then Zq( C)
q 0:/= 0, Zo( C) :::::; Zo( C) E8 Z, and Bq( C) = Bq( C) for all q. •
PROOF
= Zq( C)
if
It is clear that if T: C ~ C' is an augmentation-preserving chain map, the
isomorphism of lemma 1 commutes with T *. It is also obvious that if C is a
free augmented chain complex, C is a free chain complex.
It follows from lemma 1 that if C is an augmented chain complex,
Ho( C) 0:/= O. Hence an augmented chain complex is never acyclic. The most
that can be hoped for is that E will be acyclic.
SEC.
3
THE HOMOLOGY OF SIMPLICIAL COMPLEXES
169
2
LEMMA
If C is an augmented chain complex, C is chain contractible if
and only if the augmentation e is a chain equivalence of C with the chain
complex Z.
Let C be the mapping cone of the chain map e: C ~ Z. Then
and Cq = Cq- 1 if q
0, and 1 = e and aq = - Oq-1 for q> 1.
By theorem 4.2.10, e is a chain equivalence if and only if C is chain
contractible.
We show that C is chain contractible if and only if C is chain contractible.
If D: C ~ C is a chain contraction of C, define D: C ~ C by Dq _ 1 =
- Dq I Cq- 1 • Then D is a chain contraction of C. Conversely, if D is a chain
contraction of C, define D: C ~ C so that Do: Z ~ Co is a right inverse of
e: Co ~ Z, D1 : CO ~ C1 is 0 on Do(Z) and equal to - Do on Co, and for
q
1, Dq : Cq _ 1 ~ Cq is equal to - Dq _ 1 • Then D is a chain contraction of C. •
PROOF
Co
=Z
>
a
>
Let 2 be a category with models 01L A functor G' from C! to the category
of augmented chain complexes (and chain maps preserving augmentation) is
said to be acyclic if G'(M) is acyclic for M E ':)fL. For augmented chain complexes there is the following form of the acyclic-model theorem.
3
THEOREM
Let 2 be a category with models 0R and let G and G' be
covariant functors from t' to the category of augmented chain complexes
such that G is free and G' is acyclic. There exist natural chain maps preserving
augmentation from G to G', and any two are naturally chain homotopic.
PROOF Let {~ E Go(Mj)}jEJO be a basis for Go. By lemma 1, e': Ho(G'(Mj));::::: Z,
and there is a unique Zj E Ho(G'(Mj)) such that e'(zj) = e(~). A natural transformation Ho(G) ~ Ho(G') is defined by sending p:niPo(fij)(~)} E Ho(G(X))
to L.nijG&(fij)Zj E Ho(G'(X)) for i E Jo and fij E hom (Mj,X) (where X is any
object of 8), and this is the unique natural transformation Ho(G) ~ Ho(G')
commuting with augmentation. The theorem now follows from theorem
4.2.8. •
With the hypotheses of theorem 3 there is a unique natural transformation from H(G) to H(G') commuting with augmentation. It is the homomorphism induced by any natural chain map preserving augmentation from G to G'.
'"
COROLLARY
Let G and G' be free and acyclic covariant functors from
a category C! with models LJR to the category of augmented chain complexes.
Then G and G' are naturally chain equivalent; in fact, any natural chain
map preserving augmentation from G to G' is a natural chain equivalence.
PROOF
Let '1': G ~ G' be a natural chain map preserving augmentation
(which exists, by theorem 3). Also by theorem 3, there is a natural chain map
'1": G' ~ G preserving augmentation and there are natural chain homotopies
D: '1" 0 '1' c::=: 1a and D': '1' 0 '1" c::=: 1a,. •
We are ultimately interested in comparing the chain complex C(K) of a
simplicial complex K with the singular chain complex .l(IKI) of the space of K.
170
HOMOLOGY
CHAP.4
For this purpose we introduce a chain complex !::.(K) intermediate between
them. Let K be a simplicial complex. An ordered q-simplex of K is a sequence
Va, VI, . . . ,Vq of q + 1 vertices of K which belong to some simplex of K.
We use (VO,Vb . . . ,Vq) to denote the ordered q-simplex conSisting of the
sequence Va, Vb . . . ,Vq of vertices. For q
0 there are no ordered q-simplexes. An ordered O-simplex (v) is the same as the oriented O-simplex [v].
An ordered I-simplex (v,v') is the same as an edge of K.
We define a free nonnegative chain complex, called the ordered chain
complex of K, by !::.(K) = {!::.q(K),a q}, where !::.q(K) is the free abelian group
generated by the ordered q-simplexes of K [and !::'q(K) = 0 if q
0] and aq
is defined by the equation
<
<
Then !::.(K) is a chain complex, and if K is nonempty, !::.(K) is augmented by
the augmentation e(v)
1 for any vertex v of K. If cp: KI ~ K2 is a simplicial
map, there is an augmentation-preserving chain map
=
!::.(cp): !::.(KI)
such that !::.(cp)(VO,VI' ... ,Vq)
the following theorem.
~
!::.(K2)
= (cp(vo), CP(VI),
... ,cp(vq)). Therefore we have
:.
THEOREM
There is a covariant functor!::. from the category of nonempty
simplicial complexes to the category of free augmented chain complexes
which assigns to K the ordered chain complex !::.(K). •
If L is a subcomplex of K and i: L C K, then !::.(i): !::.(L) ~ !::.(K) is
a monomorphism by means of which we identify !::.(L) with a subcomplex of
!::.(K). If <:3{K) is the category defined by the partially ordered set of subcomplexes of K and 011(K) = {S I s E K}, then!::. is a free functor on <:3{K) with
models 0ll(K).
For any simplicial complex K there is a surjective chain map (preserving
augmentation if K is nonempty)
JL: !::.(K)
~
C(K)
such that JL(vo,Vl. . . . ,Vq) = [VO,VI . . . ,Vq]. Then JL is a natural transformation from!::' to C on the category of simplicial complexes. We shall show
that it is a chain equivalence for every simplicial complex. The following
theorem will be used to show that!::. and C are acyclic functors on <:3{K) with
models 0ll(K).
6
THEOREM
Let K be a simplicial complex and let w be the simplicial
complex consisting of a single vertex. Then 3.(K * w) and C(K * w) are chain
contractible.
PROOF
Since the proofs are analogous, we give the details only in the ordered
complex. According to lemma 2, it suffices to prove that e: !::.(K * w) ~ Z is a
SEC.
3
171
THE HOMOLOGY OF SIMPLICIAL COMPLEXES
chain equivalence. Define a homomorphism 7: Z -) 6. o(K * w) by 7(1) = (w)
and regard it as a chain map 7: Z -) 6.(K * w). Then e 7 = l z . To show that
It.(K*w) ~ 7 0 e, define a chain homotopy D: It.{K*w) ~ 7 0 e by the equation
0
D(vo,vI, ... ,Vq) = (W,VO,Vl, ... ,Vq)
-
Because a q-simplex is the join of a (q - I)-face with the opposite vertex,
we have the next result.
7
8
J.L:
COROLLARY
For any simplex s E K, Li(8) and C(8) are acyclic.
-
THEOREM
For any simplicial complex K the natural chain map
6.(K) -) C(K) is a chain equivalence.
If K is empty, 6.(K) = C(K) and J.L is the identity, so the result is true in
this case. If K is nonempty, it follows from corollary 7 that 6. and C are free
acyclic functors on f2{K) with models 'VR.(K) = {8 Is E K}. By corollary 4, J.L is
a natural chain equivalence of 6. with Con f2{K). In particular, J.L: 6.(K) -) C(K)
is a chain equivalence. PROOF
The next result is that the functors 6. and C convert contiguity of simplicial maps into chain homotopy of chain maps. This result could also be
proved by the method of acyclic models.
THEOREM
Let <p, <p': Kl -) K2 be in the same contiguity class. Then
6.(<p), 6.(<p'): 6.(Kl) -) 6.(K 2) are chain homotopic, and in similar fashion
C(<p), C(<p'): C(Kl) -) C(K2) are chain homotopic.
9
PROOF
Because chain homotopy is an equivalence relation, it suffices to prove
the theorem for the case that <p and <p' are contiguous. An explicit chain
homotopy D: 6.(<p) ~ 6.(<p') is defined by the formula
D(vo,vl. ... ,Vq)
=O:o;t:o;q
~ (-I)i(<p'(vo),
... ,<P'(Vi), <P(Vi), ... ,<p(vq))
That C(<p) and C(<p') are chain homotopic follows from the fact that 6.(<p) and
6.( <p') are chain homotopic and from theorem 8. 10 THEOREM The homology groups of a complex are the direct sums of the
homology groups of its components.
PROOF
If {Kj} are the components of K, then EBC(Kj)
follows from theorem 4.1.6. -
= C(K).
The result
If {K,,} is the collection of finite subcomplexes of K directed by inclusion, then C(K) ::::: lim~ {C(K,,)}. From theorem 4.1.7 we have the next result.
1 1 THEOREM The homology groups of a simplicial complex are isomorphic
to the direct limit of the homology groups of its finite subcomplexes. -
We are now ready to compute Ho(K).
12
LEMMA
If K is a nonempty connected simplicial complex, then Ho(K) =
o.
172
HOMOLOGY
CHAP.4
PROOF
Let Vo be a fixed vertex of K. For any vertex v of K there is an edge
path ele2 ... er of K with origin at Vo and end at v. Then el + e2 + . .. + er
is a I-chain Cv E Ll1(K) such that oC v v - Vo. Since E(~nvv)
~nv, we see
that if ~nvv is any O-chain of lo(K), then ~nv = 0 and
=
o(~nvcv)
=
= ~nvv - ~nvvo = ~nvv
Therefore Ho(Ll(K)) = 0, and by theorem 8, Ho(K) = O.
•
13 COROLLARY For any simplicial complex K, Ho(K) is a free group whose
rank equals the number of nonempty components of K.
If K is empty, Ho(K) = 0, and the result is valid in this case. If K is
nonempty and connected, it follows from lemmas 12 and 1 that Ho(K) ~ Z.
The general result then follows from theorem 10. •
PROOF
If L is a subcomplex of K, there is a relative oriented homology group
H(K,L) = {Hq(K,L) = Hq(C(K)/C(L))} of K modulo L. If L is empty,
H(K, 0) = H(K) is called the absolute oriented homology group of K. Similarly, there is a relative ordered homology group H(Ll(K)/Ll(L)) of K modulo
L that generalizes the absolute ordered homology group H(Ll(K),Ll( 0 )). The
relative homology groups H(K,L) and H(Ll(K),Ll(L)) are covariant functors
from the category of simplicial pairs to the category of graded groups.
If Hq(K,L) is finitely generated (which will necessarily be true if K - L
contains only finitely many simplexes), it follows from the structure theorem
(theorem 4.14 in the Introduction) that Hq(K,L) is the direct sum of a free
group and a finite number of finite cyclic groups Znl EEl Zn2 ® ... EEl Zn k ,
where ni divides ni+l for i = 1, . . . , k - 1. The rank p(Hq(K,L)) is called
the qth Betti number of (K,L), and the numbers nl, n2, . . . , nk are called
the qth torsion coefficients of (K,L). The qth Betti number and the qth
torsion coefficients characterize Hq(K,L) up to isomorphism.
A graded group C is said to be finitely generated if Cq is finitely generated for all q and Cq = 0 except for a finite set of integers q. It is obvious that
if C is a finitely generated chain complex, H(C) is a finitely generated graded
group. Given a finitely generated graded group C, its Euler characteristic
(also called the Euler-Poincare characteristic), denoted by X(C), is defined by
X(C)
14
THEOREM
= ~(-I)qp(Cq)
Let C be a finitely generated chain complex. Then
x(C) = x(H(C))
By definition, Zq(C) C Cq and the quotient group Cq/Zq(C)
By theorem 4.12 in the Introduction,
PROOF
p(Cq)
Similarly, Hq(C)
~
Bq-1(C).
= p(Zq(C)) + p(Bq-1(C))
= Zq(C)/Bq(C), and again by theorem 4.12 of the Introduction,
p(Zq(C))
= p(Hq(C)) + p(Bq(C))
SEC.
4
173
SINGULAR HOMOLOGY
Eliminating p(Zq(C)), we have
p(Cq)
= p(Hq(C)) + p(Bq(C)) + p(Bq-l(C))
Multiplying this equation by ( -1)q and summing the resulting equations over
q yields the result. •
If H(K,L) is finitely generated, its Euler characteristic, called the Euler
characteristic of (K,L), is denoted by X(K,L).
15
COROLLARY
If K - L is finite and if lXq equals the number of q-simplexes
of K - L, then
X(K,L)
= ~(-1)qlXq
PROOF
If K - L is finite, Cq(K)/Cq(L) is a free group of rank lXq. The result
follows from theorem 14. •
4
SINGULAR HOMOLOGV
In this section we define a natural transformation from the ordered chain
complex to the singular chain complex of its space. This will be shown
in Sec. 4.6 to be a chain equivalence for every simplicial complex K. We also
give a proof, based on acyclic models, that homotopic continuous maps induce chain-homotopic chain maps on the singular chain complexes. There is
then a computation of Ho(X) in terms of the path components of X. The final
result is that the subcomplex of the singular chain complex generated by
"small" singular simplexes is chain equivalent to the whole singular chain
complex. l
Let K be a simplicial complex. Given an ordered q-simplex (VO,Vl, . . • ,vq )
of K, there is a singular q-simplex in IKI which is the linear map 6.q ~ IKI
sending Pi to Vi for 0 ::::: i ::::: q. This imbeds 6.(K) in 6.(IKI), and we define an
augmentation-preserving chain map
p; 6.(K)
~
6.(IKI)
to send (VO,Vl, • • • ,v q ) to the linear singular simplex defined above. Then p
is a natural chain map from the covariant functor 6.(.) to the covariant
functor 6.(1 • I) on the category of simplicial complexes. It will be shown in
Sec. 4.6 that p is a natural chain equivalence. We prove now that it is a chain
equivalence for the complex s of an arbitrary simplex s.
1 LEMMA Let X be a star-shaped subset of some euclidean space. Then
the reduced singular complex of X is chain contractible.
1 Our treatment is similar to that in S. Eilenberg, Singular homology theory, Annals of Mathematics, vol. 45, pp. 407-447 (1944).
174
HOMOLOGY
CHAP.4
PROOF
Without loss of generality, X may be assumed to be star-shaped from
the origin. We define a homomorphism 7: Z ~ ilo(X) with 7(1) equal to the
singular simplex il o ~ X which is the constant map to 0. Then EO 7
I z . We
define a chain homotopy D: il(X) ~ il(X) from It.(X) to 7 E. If a: ilq ~ X is a
singular q-simplex in X, let D(a): M+I ~ X be the singular (q + I)-simplex in
X defined by the equation
=
0
D(a)(tpo
+ (I
- t)a)
= (I
- t)a(a)
and t E
If q > 0, then (D(a))(O) = a,
°for a E q,IpI, . . . ,pq+11= D(a(i»).
If q = 0, then (D(a))(O) = a and
1.
~
i ~
(D(a))(HI)
Therefore
aD + Da
and D: It.(X) ~
'i
0
E.
and for
(D(a))(I)
= It.(X) -
7
0
= 7(1).
E
By lemma 4.3.2, Li(X) is chain contractible.
-
2
COROLLARY
For any simplex s the chain map v induces an isomorphism
of the ordered homology group of s with the singular homology group of lsi.
PROOF
Because v preserves augmentation, v induces a homomorphism p*
from H(il(s)) to H(lsl), and under the isomorphism of lemma 4.3.1, v* =
v* EB I z . By corollary 4.3.7, H(il(s)) = 0. By lemma I and corollary 4.2.3,
ii(lsj) 0. Therefore v* is an isomorphism. -
=
We use lemma I to prove that if fo, it: X ~ Y are homotopic, then
il(fo) , il(!t): il(X) ~ il(Y) are chain homotopic. We prove this first for the
maps ho, hI: X ~ X X I, where ho(x) = (x,O) and hl(x) = (x,I).
3 THEOREM
chain maps
The maps ho, hI: X ~ X X I induce naturally chain-homotopic
il(ho)
~
il(hl): il(X)
~
il(X X 1)
=
PROOF
Let il'(X)
il(X X I). Then il and il' are covariant functors from
the category of topological spaces to the category of augmented chain complexes and il(ho) and il(h l ) are natural chain maps preserving augmentation
from il to il'. Since il is free with models {M} and
~'(ilq)
= Li(ilq X 1)
is acyclic, by lemma I, it follows from theorem 4.3.3 that il(h o) and il(h l ) are
naturally chain homotopic. This special case implies the general result.
4
COROLLARY
If fo,!t: X
~
il(fo)
PROOF
!t
= Fh
~
Yare homotopic, then
il(fl): il(X)
~
il(Y)
Let F: X X I ~ Y be a homotopy from fo to!t. Then fo
Therefore, using theorem 3,
l .
= Fho and
SEC.
4
175
SINGULAR HOMOLOGY
Since f:1q is path connected for every q, any singular simplex a: /1q ---'? X
maps f:1q to some path component of X. Hence, if {Xj} is the set of path components of X, then /1(X) = ffi/1(Xj). By theorem 4.1.6, we have the following
theorem.
it THEOREM The singular homology group of a space is the direct sum of
the singular homology groups of its path components. -
Because /1q is compact, every singular simplex a: /1q ---'? X maps /1q into
some compact subset of X. Hence, if {X,,} is the collection of compact subsets of X directed by inclusion, then /1(X) = lim~/1(X,,). By theorem 4.1.7, we
have our next result.
THEOREM
The singular homology group of a space is isomorphic to the
direct limit of the singular homology groups of its compact subsets. -
6
We now compute the O-dimensional homology group of a space.
7
LEMMA
Ho(X)
= O.
If X is a nonempty path-connected topological space, then
PROOF
Let xo be a fixed point of X. For any point x E X there is a path Wx
from Xo to x. Because /1 1 is homeomorphic to I, WX corresponds to a singular
x and ap)
Xo. A singular O-simplex
I-simplex ox: /1 1 ---'? X such that ax(O)
in X is identified with a point of X. Therefore a O-chain (that is, a O-cycle) of
X is a sum ~nxx, where nx = 0 except for a finite set of x's. Since e(~nxx) =
~nx, we see that if e(~nxx) = 0 [that is, if ~nxx E Eo(X)], then
=
a(~nxax)
Therefore Ho(X)
= ~nxx -
=
(~nx)xo
= ~nxx
= o. -
8
COROLLARY
For any topological space X, Ho(X) is a free group whose
rank equals the number of nonempty components of X.
PROOF
If X is empty, Ho(X) = 0, and the result is valid in this case. If X is
nonempty and path connected, it follows from lemmas 7 and 4.3.1 that
Ho(X) ;::::; Z. The general result now follows from theorem 5. -
If A is a subspace of X, there is a relative singular homology group
H(X,A) = {Hq(X,A) = Hq(/1(X)j /1(A))} of X modulo A. H(X, 0) = H(X) is
called the absolute singular homology group of x. The relative homology
group is a covariant functor from the category of topological pairs to the
category of graded groups. We show that this functor can be regarded as defined on the homotopy category of pairs.
9
THEOREM
If fo, fl: (X,A)
fo*
PROOF
fo
---'?
(Y,B) are homotopic, then
= h*: H(X,A) ---'? H(Y,B)
Let F: (X X I, A X I)
---'?
(Y,B) be a homotopy from fo to h Then
I, A X I) are defined by
= Fho and h = Fhl. where ho, hI: (X,A) ---'? (X X
176
HOMOLOGY
=
CHAP.
4
=
ho(x)
(x,O) and k1(X)
(x,l). By theorem 3, there is a natural chain ~omo­
topy D: !l(ho) ~ !l(h1)' where ho, h 1: X ~ X X I are maps defined by ho and
h 1. Because D is natural, D(!l(A)) C !l(A X 1). For i = 0 or 1 there is a commutative diagram
!l(A)
!l(X)
C
~
!l(X)/!l(A)
~(h')l
!l(A X I) C !l(X X 1) ~ !l(X X 1)/!l(A X 1)
and a chain homotopy V: !l(ko) ~ !l(h1) is obtained by passing to the quotient with D. By theorem 4.2.2,
ko*
= h1*: H(X,A) ~ H(X X I, A
X 1)
Then
fo*
= F*h o* = F*k 1* = f1*
•
If Hq(X,A) is finitely generated, its rank is called the qth Betti number of
(X,A) and the orders of its finite cyclic summands given by the structure
theorem are called the qth torsion coefficients of (X,A). If H(X,A) is finitely
generated, its Euler characteristic is called the Euler characteristic of (X,A),
denoted by X(X,A).
The remainder of this section is directed toward a proof that the subcomplex of the singular chain complex generated by small singular simplexes
is chain equivalent to the singular chain complex. We begin by defining
a subdivision chain map in singular theory. A singular simplex a: flq ~ !In is
said to be linear if a(L.tiPi) = L.tia(Pi) for ti E I with L.ti = 1. If a is linear,
so is a<i) for 0 ~ i ~ q. Therefore the set of linear simplexes in !In generates
a subcomplex !l'(!ln) C !l(!ln).
A linear simplex a in !In is completely determined by the points a(Pi). If
Xo, Xl, . . . ,Xq E !In, we write (XO'X1' . . . ,Xq) to denote the linear simplex
a: !lq ~ !In such that a(Pi) = Xi. With this notation, it is clear that
o(xo, . . . ,Xq)
= L.( -l)i(xo,
Furthermore, the identity map
(PO,Pb . . . ,Pn)'
~n:
... ,Xi. ... ,Xq)
!In C !In is the linear simplex
Let bn be the barycenter of !In (that is, bn
a homomorphism
= L.(l/(n + l))pi.
~n
=
For q ;:::: 0
is defined by the formula
f3n(XO, . . . ,Xq) = (bn,xo, . . . ,Xq)
Let '7': Z ~ !lo(!ln) be defined by '7'(1)
10
= (b n). Direct computation shows that
SEC.
4
177
SINGULAR HOMOLOGY
For every topological space X we define an augmentation-preserving
chain map
sd:
~(X) ~ ~(X)
D:
~(X) ~ ~(X)
and a chain homotopy
from sd to ll>(xJ, both of which are functorial in X. That is, if f: X ~ Y, there
are commutative squares
~(X) ~ ~(X)
~(X) ~ ~(X)
!l(f)
t
t Ll(f)
t !l(f)
!l(nt
~(Y) ~ ~(Y)
~(Y) ~ ~(Y)
Both sd and D are defined on q-chains by induction on q. If c is a O-chain, we
define sd(c) = c and D(c) = O. Assume sd and D defined on q-chains for
o ~ q n, where n 2': 1. We define sd and D on the universal singular
n-simplex ~n: ~n C ~n by the formulas
<
sd(~n)
= f3n(sd a(~n))
D(~n) =
For any singular n-simplex a:
f3n(sd
~n ~
sd(a)
D(a)
(~n)
-
~n
-
Da(~n))
X we define
= ~(a)(sd(~n))
= ~(a)(D(~n))
Then sd and D have all the requisite properties.
If X is a metric space and c = ~naa is a singular q-chain of X, we define
mesh c
II
LEMMA
Let
~n
= sup {diam a( M) I na =1= O}
have a linear metric and let c be a linear q-chain of ~n.
Then
mesh (sd c)
< -q- mesh c
+1
- q
PROOF
The proof is based on induction on q, using the inductive definition
of sd. It suffices to show that if a = (XO,Xl, . • • ,Xq) is a linear q-simplex of ~n,
then mesh (sd (1) ~ (q/(q + 1)) mesh a. If b = ~ (l/(q + l))xi, a computation
similar to that of lemma 3.3.12 shows that the distance from b to any convex
combination of the points Xo, Xl, . . . , Xq is less than or at most equal to
(q/(q + 1)) mesh (xo, . . . ,xq). Therefore
mesh (sd a)
By induction
~ sup
C!
1 mesh a, mesh (sd aa))
178
HOMOLOGY
mesh (sd oa)
:s; q ; 1
CHAP.
4
mesh oa
< -q-mesha
-q+1
which yields the result.
-
We next define augmentation-preserving chain maps
for m
2:: 0 by induction
sdo
= 1a
(X)
and
sdm = sd(sdm- 1)
Then, from lemma 11, we obtain the following result.
12
COROLLARY
Let iln have a linear metric and let c E
mesh (sdm c)
il~(iln).
Then
:s; [q/(q + l)]m mesh c -
Let G(1 = {A} be a collection of subsets of a topological space X and let
il(.:i)l) be the subcomplex of il(X) generated by singular q-simplexes a: ilq ~ X
such that a(ilq) C A for some A E G(1 [if a(ilq) C A, then a(i)(M-l) C A, and
so il(Gll) is a subcomplex of il(X)]. Because sd and D are natural, sd(il('1l)) C il(c~L)
and D(il(G(1)) C il(G(1).
13 LEMMA Let '11 = {A} be such that X = U {int A I A E'1l}. For any
singular q-simplex a of X there is m 2:: 0 such that sdm a E il('1l).
PROOF
Because X = U{int A I A E '1L}, M = U{a-1(int A) I A E '1l}. Let
ilq be metrized by a linear metric and let A.
0 be a Lebesgue number for
the open covering {a-1(int A) I A E '1L} of M relative to this metric. Choose
m 2:: 0 so that [q/(q + l)]m diam M :s; A.. By corollary 12, mesh (sd m ~q) :s; A..
Therefore every singular simplex of sdm ~q maps into a-1(int A) for some
A E .:i)l. Then sdm a = il(a) sdm ~q is a chain in il(.:i)l). -
>
We are now ready to prove the chain equivalence mentioned earlier.
14 THEOREM Let '1L = {A} be such that X = U {int A I A E '1l}. Then the
inclusion map il('1L) C il(X) is a chain equivalence.
PROOF For each singular simplex a in X let m(a) be the smallest nonnegative
integer such that sdm(a)a E il(G(1). Such an integer m(a) exists by lemma 13, and
it is clear that m(a) = 0 if and only if a E il("Il). Furthermore, m(a(i») :s; m(a)
for 0 :s; i :s; deg a.
Define D: il(X) ~ il(X) by D(a) = LO,oh;m(a)-l D sdi(a). Then D(a) = 0
if and only if a E il('1l). Also
= Lsdi+1(a) -
Lsdi(a) - LDsdi(oa)
Li (-l)iD sdi(a(i»)
tJo( a) = Li ( -l)i LO,oj,om(a(;~)~lD sdj(ali))
oD(a)
= sdm(a)(a) - a -Lo<j<m(a)-l
SEC.
5
179
EXACTNESS
Therefore
a
+ afJ( a) + lJa( a)
= Li ( -l)i Lm(a(i)l$j~ m(a)-lD scfj(a(i»)
+ sdm(a)( a)
is in il(Gil). Define r: il(X) ~ il(Gil) by r(a) = a + afJ(a) + fJa(a). Then r is a
chain map preserving augmentation. Clearly, if i: il(Gil) C il(X), then
r i = 1,,("11) and fJ: i r ~ 1,,(X). Therefore [r) = [i)-I, and i is a chain
equivalence. •
0
0
5
EXACTNESS
In this section we consider the relations among the homology groups of C', C,
and C/G', where C' is a subcomplex of C. A concise way of summarizing
these relations is by means of the concept of exact sequence. The basic result
is the existence of an exact sequence connecting the homology of G', C, and
C/G'.
A three-term sequence of abelian groups and homomorphisms
G' ~ G ~ Gil
is said to be exact at G if ker f3 = im 0'. A sequence of abelian groups and
homomorphisms indexed by integers (which mayor may not terminate at
either or both ends)
is said to be an exact sequence if every three-term subsequence of consecutive groups is exact at its middle group. Note that an exact sequence terminating at one end with a trivial group can be extended indefinitely on that end
to an exact sequence by adjoining trivial groups and homomorphisms.
A short exact sequence of abelian groups, written
o~
G' ~ G ~ Gil ~ 0
is a five-term exact sequence whose end groups are trivial. In such a short
exact sequence 0' is a monomorphism and f3 is an epimorphism whose kernel
is 0'( G'). Therefore 0' is an isomorphism of G' with the subgroup 0'( G') C G,
and f3 induces an isomorphism from the quotient group G/O'(G') to Gil. The
group G is called an extension of G' by Gil.
Given an exact sequence
... ~ G n + 1 ~ G n ~ G n let G~ = ker
sequences
an
= im
O'n+l.
1
~
...
Then the given sequence gives rise to short exact
for every G n not on one or the other end of the original sequence, and the
180
HOMOLOGY
CHAP.
4
composite G n ~ G~-l ~ Gn- 1 equals an·
A homomorphism y from one sequence {G n --.:'~ G n - 1 } to another
{Hn ~ Hn-d with the same set of indices (that is, of the same length) is a
sequence {Yn: G n ~ Hn} of homomorphisms such that the following diagram
is commutative:
~
G n+1
lXn+l
Yn+l1
~
Gn
------C>
y"
Hn+l
f3n+l
------C>
an
-----')
1
1
Hn
Gn- 1
~
...
Yn-l
13"
~
Hn- 1
~
...
There is a category of exact sequences with the same set of indices. In particular, there is a category of short exact sequences, and also a category of
exact sequences (indexed by all the integers).
Note that a sequence of abelian groups and homomorphisms
is a chain complex if and only if im a n+ 1 C ker an for all n. This is half of the
condition of exactness at Cn. For a chain complex C, the group Hn(C) =
ker an/im an+1 is a measure of the nonexactness of the sequence at Cn. Thus
a chain complex is an exact sequence if and only if its graded homology
group is trivial. In any case, the fact that the homology group measures the
nonexactness of the chain complex suggests that there should be some relation between homology and exactness, and this is indeed so.
A short exact sequence of chain complexes, written
O~C'~C~C"~O
is a five-term sequence of chain complexes and chain maps such that for all q
there is a short exact sequence of abelian groups
O
~
f3
C"
0
' aq Cq~
Cq---+
q~
Q
A homomorphism from one short exact sequence of chain complexes to another consists of a commutative diagram of chain maps
O~C'~C~C"~O
There is a category of short exact sequences of chain complexes and homomorphisms.
I
EXAMPLE
Let C' be a subcomplex of a chain complex C and let
i: C' C C and ;: C ~ C/C' be the inclusion and projection chain maps,
respectively. There is a short exact sequence of chain complexes
o~
C' ~ C ~ C/C' ~ 0
SEC.
5
181
EXACTNESS
Given a subcomplex C' C C and a chain map T: C -!> C such that T(C') C
there is a homomorphism
C,
0 -!> C'~ C~ C/C' -!>O
T1
T'l
-
1T"
j
0 -!> C' -!> c-!> C/C' -!>O
I
where T' = TIC' and T" is induced from T by passing to the quotient.
2
EXAMPLE
If C is an augmented chain complex, there is a short exact
sequence of chain complexes
O-!>C-!>C~Z-!>O
There is a covariant functor C from the category of simplicial pairs to
the category of short exact sequences of chain complexes which assigns to
(K,L) the short exact sequence
o -!> C(L) -!> C(K) -!> C(K)/C(L) -!> 0
Similarly, there is a covariant functor ~ from the category of topological pairs
to the category of short exact sequences of chain complexes which assigns to
(X,A) the short exact sequence
o -!> ~(A) -!> ~(X) -!> ~(X)/~(A) -!> 0
There is also a covariant functor ~ from the category of simplicial pairs to the
category of short exact sequences of chain complexes which assigns to (K,L)
the short exact sequence
o -!> ~(L) -!> ~(K) -!> ~(K)/ ~(L) -!> 0
Then fl is a natural transformation from ~ to C and II is a natural transformation from ~ to ~(I . I) (both natural transformations in the category of short
exact sequences of chain complexes).
We define covariant functors H', H, and H" from the category of short
exact sequences of chain complexes
o -!> C' ~ C !!..o, C" -!> 0
to the category of graded groups such that H', H, and H" map the above
sequence into H(C'), H(C), and H(C"), respectively.
3
LEMMA
On the category of short exact sequences of chain complexes
o -!> C'
~ C
!!.."
C" -!> 0
there is a natural transformation 0*: H" -!> H' such that if {z"} E H(C"),
then 0* {z"} = {ad o,8-1z"} E H(C').
182
PROOF
HOMOLOGY
CHAP.4
There is a commutative diagram
o ---? C~+l ~
o ---?
o ---?
d
C~
a't
Cq+ 1
at
~ Cq
at
~ C~+l
---? 0
L
---? 0
tao
C~
tao
'
<xcq-l ---?
PC"
Cq-l
---?
q-l---? 0
in which each row is a short exact sequence of groups. If z" is a q-cycle of C",
let c E Cq be such that f3(c)
z". Then
=
f3(ac) = a"f3(c) = a"z" = 0
Therefore there is a unique c' E C~-l such that a(c')
= ac. Then
a(a'c') = aa(c') = aac = 0
Because a is a monomorphism, a'c' = O. Hence c' is a (q - I)-cycle of C'.
We show that the homology class of c' in C' depends only on the homology class of z" in C", which will prove that there is a well-defined homomorphism a* {z"} = {c'}. Let Cl E Cq be such that f3(Cl) - z". Then there is
d" E C~+l such that f3(Cl) = f3(c) + a"d". Choose dE Cq + l such that
f3( d) = d". Then
f3(Cl)
= f3(c) + a"f3(d) = f3(c + ad)
Therefore there is a d' E q such that Cl = c
aCl
+ ad + a(d'),
and
= ac + aa(d') = a(c') + a(a'd') = a(c' + a'd')
= c' + a'd' - c' and {a-l(aCl)} = {a- (ac)}, showing that
1
Hence a-l(aCl)
a* is well-defined.
To prove that a* is a natural transformation, assume given a commutative diagram of chain maps
o ---?
C' ~ C ~ C" ---? 0
where the horizontal rows are short exact sequences. Then
T~a*{Z"}
= T~{a-laf3-lz"} = {T'a- l af3- l z"}
= {a- l Taf3- l z"} = {a-lap-lT"Z"} =
(3* T* {z"}
•
The natural transformation a* is called the connecting homomorphism
for homology because of its importance in the following exactness theorem.
4
THEOREM
There is a covariant functor from the category of short exact
sequences of chain complexes to the category of exact sequences of groups
which assigns to a short exact sequence
SEC.
5
183
EXACTNESS
o
-7
C' ~ G
!!.
G"
-7
0
the sequence
~ Hq(C') ~ Hq(C) ~ HiG") ~ Hq_1(C') ~ ...
PROOF
The sequence of homology groups is functorial on short exact
sequences because 0* is a natural transformation. It only remains to verify
that it is an exact sequence. This entails a proof of exactness at Hq(C'), Hq(C),
and Hq( G"), each exactness requiring two inclusion relations. Therefore the
proof of exactness has six parts. We shall prove exactness at Hq(G") and leave
the other parts of the proof to the reader.
(a) im 13* C ker 0*. Let {z} E Hq(C). Then
0* 13* {z}
= 0* {f3(z)} = {a- 1of3- 1f3(z)} = {a-1oz} = {a-1(O)} = 0
(b) ker 0* C im 13*. Let {z"} E ker 0*. Then there is c E Gq such that
f3(c) = Z" and (ldO(C) = o'(d') for some d' E G~. The difference c - a(d') E Gq
is such that
o(c - a(d')) = OC - a(o'd') = 0
Hence {c - a(d')} E Hq(C) and
f3*{c - a(d')} = {f3(c) - f3a(d')} = {Z"}
•
Combining theorem 4 with example 2, we again obtain lemma 4.3.1. As
an example of the utility of exactness, note that the following corollary
is immediate from theorem 4.
5
COROLLARY
Given a short exact sequence of chain complexes
0-7C'~G~G"-70
(a) C' is acyclic if and only if 13*: H(C) ;::::; H(G").
(b) G is acyclic if and only if 0*: H(G") ;::::; H(C').
(c) Gil is acyclic if and only if a*: H(C') ;::::; H(C).
•
In (b) above it should be noted that 0* has degree -1. It follows from
corollary 5 that if two of the chain complexes C', G, and Gil are acyclic, so is
the third.
6
COROLLARY
Given an exact sequence of abelian groups
and a subsequence
(that is, G~ C Gn and
the quotient sequence
a~
= an I G~),
. . . -7
is exact.
Gn/G~
-7
the subsequence is exact if and only if
Gn-dG~-l
-7 . . .
184
HOMOLOGY
CHAP.
4
Let C be the chain complex consisting of the original exact sequence
and let C' be the subcomplex consisting of the subsequence. Then the quotient chain complex C/G' is the quotient sequence. Because C is an exact sequence, C is acyclic, and 0*: Hq(C/G'):::::: Hq_1(G'). Therefore G' is exact
[that is, H(C')
0] if and only if C/C' is exact [that is, H(C/G')
0]. •
PROOF
=
7
THEOREM
=
The direct limit of exact sequences is exact.
PROOF
Each exact sequence is an acyclic chain complex. The direct limit is
also a chain complex, and it is acyclic, by theorem 4.1.7. Therefore the limit
sequence is exact. •
This result is false if direct limit is replaced by inverse limit, because the
homology functor fails to commute with inverse limits.
Let K be a simplicial complex and let Ll C L2 C K. By the Noether isomorphism theorem, there is a short exact sequence of chain complexes
o-?
C(L 2 )/C(L 1 ) ~ C(K)/C(Ll) ~ C(K)/C(L2)
-?
0
By theorem 4, there is an exact sequence
... ~ Hq(L 2 ,L 1) ~ Hq(K,L 1) ~ Hq(K,L 2) ~ H q _ 1 (L 2 ,L 1 ) ~
where i* is induced by i: (L 2 ,L 1) C (K,L 1 ),;* is induced by;: (K,L 1 ) C (K,L 2),
and 0* is the connecting homomorphism. This sequence is called the homology
sequence of the triple (K,L 2,L1 ). It is functorial on triples. If Ll = 0, the resulting exact sequence
.. , ~ Hq(L 2) ~ Hq(K) ~ Hq(K,L 2) ~ Hq- 1 (L 2) ~ ...
is called the homology sequence of the pair (K,L 2 ). It is functorial on pairs.
Because there is an inclusion map of the triple (K,L 2 , 0) into the triple
(K,L2,Ll)' the next result follows.
8 LEMMA The connecting homomorphism 0*: Hq(K,L 2 )
the triple (K,L 2 ,L1 ) is the composite
-?
H q _ 1(L 2,L1) of
Hq(K,L 2) ~ Hq_1(L 2) ~ H q - 1 (L 2 ,L 1 )
of the connecting homomorphism of the pair (K,L2) followed by the homomorphism induced by k: (L 2 , 0) C (L 2 ,L 1 ). •
If L is a nonempty subcomplex of a simplicial complex, C(L) C C(K),
and by the Noether isomorphism theorem, C(K)/C(L) :::::: C(K)/C(L). Therefore there is a short exact sequence of chain complexes
o -?
C(L) ~ C(K) ~ C(K)/C(L)
-?
0
The corresponding exact sequence
. .. ~ Hq(L) ~ Hq(K) ~ Hq(K,L) ~ Hq_1(L) ~
is called the reduced homology sequence of the pair (K,L). It is not defined
if L = 0, because C(L) has no augmentation in this case.
SEC.
5
185
EXACTNESS
In the same way, there is a singular homology sequence of a triple
(X,A,B) and of a pair (X,A). If A is nonempty, there is also a reduced homology
sequence of (X,A). All these sequences are exact, and the analogue of lemma 8
is valid relating the connecting homomorphism of a triple to the connecting
homomorphism of a pair.
9
LEMMA
Let s be an n-simplex. Then
o
q =1= n
q=n
Hq(s,s)::::::; {Z
Cq(.s) = Cq(s) if q =1= n. Therefore [C(s)/C(s)]q
[C(s)/C(s)]n::::::; Z. •
PROOF
= ° if
q =1= n, and
Because H(s) = 0, by corollary 4.3.7, it follows from the exactness of the
reduced homology sequence of (s,s) that 0*: Hq(s,s) ::::::; Hq_l(s) for all q.
Therefore we have the next result.
10
COROLLARY
If s is an n-simplex, then
q=l=n-l
q=n-l
•
We conclude by proving the following five lemma (so named because of
the five-term exact sequences involved in its formulation).
I I
LEMMA
Given a commutative diagram of abelian groups and homomor-
phisms
"'.' G4~
"'4 G 3~
"'3 G2~
"" G I
G5 ~
Y5I
H5
y'l
y'l
Y3I
(3, H (3, H (33 H (3,
~
4
~
3
~
2
~
1
Yl
HI
in which each row is exact and YI, Y2, Y4, and Y5 are isomorphisms, then Y3
is an isomorphism.
The proof is straightforward. To show that Y3 is a monomorphism,
assume Y3(g3) = 0. Then Y2£X3(g3) = f33Y3(g3) = 0. Therefore £X3(g3) = 0.
Hence there is g4 E G 4 such that £X4(g4) = g3. Then !34Y4(g4) = 0, and there
is h5 E H5 such that !35(h5) = Y4(g4). There is g5 E G 5 with Y5(g5) = h 5. Then
Y4(£X5(g5)) = Y4(g4), and so g4 = £X5(g5). Then g3 = £X4£X5(g5) = 0.
To show that Y3 is an epimorphism let h3 E H 3. There is g2 E G 2 such
that Y2(g2) = !33(h3)' Then YI£X2(g2) = !32!33(h3) = 0. Therefore £X2(g2) = 0,
and there is g3 E G 3 such that £X3(g3) = g2. Then !33(h3 - Y3(g3)) = 0, and
there is h4 E H4 such that !34(h4) = h3 - Y3(g3). Let g4 E G 4 be such that
Y4(g4) = h4. Then g3 + £X4(g4) E G 3 and Y3(g3 + £X4(g4)) = Y3(g3) + f34(h 4) =
PROOF
h3 ·
•
Note that to prove Y3 a monomorphism we merely needed Y2 and Y4 to
be monomorphisms and Y5 to be an epimorphism, and to prove Y3 an epimorphism we merely needed Y2 and Y4 to be epimorphisms and YI to be a
186
HOMOLOGY
CHAP.
4
monomorphism. This type of proof is called diagram chasing and will be
omitted in the future.
We shall have several occasions to use the five lemma. We mention the
following as a typical example. For any simplicial pair (K,L) the natural transformation fL from the ord~red homology theory induces a homomorphism of
the corresponding exact sequences
----'>
...
By theorem 4.3.8, fL* is an isomorphism on the absolute groups. It follows
from the five lemma that it is also an isomorphism on the relative groups.
12 COROLLARY For any simplicial pair (K,L) the natural transformation fL
induces an isomorphism from the ordered homology sequence of (K,L) to the
oriented homology sequence of (K,L). •
6
MAYER-VIETORIS SEqUENCES
There is an exact sequence which relates the homology of the union of two
sets to the homology of each of the sets and to the homology of their intersection. This sequence provides an inductive procedure for computing the
homology of spaces which are built from pieces whose homology is known.
We shall define this exact sequence as well as its analogue involving relative
homology groups, and use them to prove that the natural transformation 1!
from 6(K) to 6(IKI) is a chain equivalence for any simplicial complex K.
Let Kl and K2 be subcomplexes of a simplicial complex K. Then
Kl n K2 and Kl U K2 are subcomplexes of K, and C(Kl)' C(K2) C C(K).
Clearly C(KI n K2) = C(Kl) n C(K2) and C(Kl) + C(K2) = C(KI U K2)'
Let i 1: Kl n K2 C K 1, i 2: Kl n K2 C K 2, h: Kl C Kl U K 2, and
;2: K2 C Kl U K 2. Then we have a short exact sequence of chain complexes
o ----'> C(KI n K 2) ~ C(K 1) EB C(K 2) ~ C(KI U K 2) ----'> 0
i(c) = (C(i1)c, - C(i2)C) and ;(Cl,C2) = C(h)Cl + C(i2)C2. The
where
sponding exact sequence of homology groups
corre-
... ~ Hq(Kl n K 2) ~ Hq(Kl) EB Hq(K 2) ~ Hq(Kl U K 2) ~
Hq_1(K 1 n K 2) ~
is called the Mayer- Vietoris sequence of the sub complexes Kl and K 2 • The
homomorphisms i* and;* in the Ma yer-Vietoris sequence are described by
means of homomorphisms induced by inclusion maps by
and
SEc.6
187
MAYER-VIETORIS SEQUENCES
for Z E H(Kl n K2), Zl E H(Kl)' and Z2 E H(K2)'
If Kl n K2 =1= 0, there is a commutative diagram of abelian groups and
homomorphisms
z
z®z
z
o~
~O
where a(n) = (n,-n) and f3(n,m) = n + m. Since the rows are exact and the
vertical homomorphisms are epimorphisms, it follows from corollary 4.5.6 that
there is an exact sequence of the kernels
o~
-
CO(Kl
n K 2) ~i
-
-
j-
CO(Kl) ® CO(K 2) ~ CO(Kl U K2) ~ 0
and so there is a short exact sequence of chain complexes
o~
n K 2) ~
C(Kl
C(Kl) ® C(K2) ~ C(Kl U K2) ~ 0
The corresponding exact sequence of reduced homology groups
. .. ~ Hq(Kl
n K 2) ~
Hq(Kl) ® Hq(K2) ~ Hq(Kl U K2) ~
is called the reduced Mayer- Vietoris sequence of Kl and K 2 •
If (K1,L 1) and (K 2 ,L2 ) are simplicial pairs in K, there is also a short exact
sequence
which is a subsequence of the short exact sequence
o~
C(Kl
n
K 2) ~ C(K 1) EB C(K2)
~
C(Kl U K2) ~ 0
It follows from corollary 4.5.6 that the quotient sequence is a short exact
sequence of chain complexes
o~
C(Kl
n K 2)/C(L1 n
L 2) ~ C(K1)/C(L1) ® C(K2)/C(L2) ~
C(Kl U K 2)/C(L1 U L 2) ~ 0
The corresponding exact sequence of homology groups
... ~ Hq(Kl n K 2 , Ll n L 2) ~ Hq(K1,L 1) ® Hq(K 2,L 2) ~
Hq(Kl U K 2, Ll U L 2) ~
is called the relative Mayer- Vietoris sequence of (K1,L 1) and (K 2 ,L 2 ).
The relative Mayer-Vietoris sequence specializes to the exact sequence of
a triple or a pair. In fact, given a triple (K,L~,L2)' the relative Mayer-Vietoris
sequence of (K,L 2 ) and (L1,L 1) is easily seen to be the homology sequence of
the triple (K,L 1,L 2) as defined in Sec. 4.5. In case L2 = 0, the relative
Mayer-Vietoris sequence of (K, 0) and (L1,L 1) is the homology sequence of
the pair (K,L 1 ).
An inclusion map (KbLl) C (K 2,L 2) is called an excision map if
Kl - Ll = K2 - L2. The exactness of the Mayer-Vietoris sequence is closely
188
HOMOLOGY
CHAP.
4
related (in fact, equivalent) to the following excision property.
I
THEOREM
Any excision map between simplicial pairs induces an isomorphism on homology.
PROOF If (KI,LI) C (K 2 ,L 2 ) is an eXClSlon map, then K2
LI = KI n L 2 • By the Noether isomorphism theorem,
C(KI)/C(L I) :::::: [C(KI)
+ C(L2)l!C(L2) =
= KI
C(K2)/C(L 2 )
U L2 and
•
For the ordered chain complex it is still true that if KI and K2 are subcomplexes of some simplicial complex, then ~(KI U K 2) = ~(KI) + ~(K2)'
Therefore all the above results remain valid if the oriented homology is
replaced throughout by the ordered homology.
An inclusion map (XI,A I ) C (X 2 ,A 2 ) between topological pairs is called
an excision map if Xl - Al = X2 - A 2 . It is not true that every excision
map induces an isomorphism of the singular homology groups. Neither is it
true that there is an exact Mayer-Vietoris sequence of any two subsets Xl and
X 2 of a topological space.
2
Let f: R
EXAMPLE
~
R be defined by
.
f(x)
=
1
Slll-
X
o
x S; 0
and let Xl = ((x,y) E R2 I y 2: f(x) or x = 0, Iyl ::; 1) and X2 = ((x,y) E R21
y ::; f(x) or x = 0, Iyl ::; 1). Then Xl and X2 are closed path-connected subsets of R2 such that Xl U X 2 = R2 and Xl n X2 consists of two path components. Therefore there b no homomorphism HI(XI U X2 ) -> HO(XI n X2 )
which will make the sequence
fII(X I U X2 ) ~ fIO(XI n X 2 ) ~ HO(X I ) EEl fIO(X 2 )
exact at HO(XI
n
X 2 ) [the ends are both trivial, but HO(XI
n X 2 ) =1= 0].
We can, however, develop a Mayer-Vietoris sequence in singular homology for certain subsets Xl and X2 of a topological space. Let Xl and X2 be
subsets of some space. {Xl,X 2 } is said to be an excisive couple of subsets if
the inclusion chain map ~(XI) + ~(X2) C ~(XI U X 2) induces an isomorphism of homology. Our next result follows from theorem 4.4.14.
THEOREM
If Xl U Xz
3
excisive couple. •
= intxlux2 Xl
U intxlux2 X 2 , then {XI,X2 } is an
In particular, if A C X, then {X,A} is always an excisive couple. The
relation between an excisive couple {X I ,X2} and excision maps is expressed
as follows.
4
{X I ,X2 } is an excisive couple if and only if the excision map
X 2 ) C (Xl U X 2 ,X2 ) induces an isomorphism of singular homology.
THEOREM
(XI,X I
n
SEc.6
PROOF
189
MAYER·VIETORIS SEQUENCES
We have a commutative diagram of chain maps induced by inclusions
il(XI)/il(XI
n
\
<l( ")
X z) ~ il(XI U X z)/il(Xz)
[il(XI)
;;
+ il(Xz)]/ il(Xz)
where i is the excision map i: (Xl. Xl n X z ) C (Xl U Xz, Xz). By the Noether
isomorphism theorem, i is an isomorphism; therefore i*
i* i* is an isomorphism if and only if i~ is an isomorphism. Using the exactness of the homology
sequence of a pair and the five lemma, i* is an isomorphism if and only if the
inclusion map il(X I ) + il(Xz) C il(XI U X z ) induces an isomorphism of homology, which is by definition equivalent to the condition that {XI,Xz} be an
excisive couple. •
=
This yields the following excision property for singular theory.
it
COROLLARY
Let U CAe X be such that (j C int A. Then the excision
map (X - U, A - U) C (X,A) induces an isomorphism of singular homology.
PROOF The hypothesis (j C int A implies int (X - U) :J X - (j :J X - int A.
By theorem 3, {A, X - U} is an excisive couple, and the result follows from
this and from theorem 4. •
=
For any subsets Xl and Xz of a space, il(XI n X z) il(XI ) n il(Xz), and
there is a short exact sequence of singular chain complexes
o~
il(XI
n X z) ~
il(X I) EB il(Xz) ~ il(X I)
+ il(Xz) ~ 0
This yields an exact sequence
... ~ Hq(XI n X z ) ~ Hq(X I ) EB Hq(X z) ~ Hq(il(XI)
+ il(Xz)) ~
Hq-I(XI n X z)
~
If {X1,X z } is an excisive couple, the group Hq(il(XI) + il(Xz)) can be replaced
by the group Hq(XI U Xz), and the resulting exact sequence is
. .. ~ Hq(XI n X z) ~ Hq(XI) EB Hq(X2) ~ Hq(XI U X2) ~
Hq_I(X I n X2 )
=
~
=
where i*(z)
(h*z,-iz*z) and i*(ZI,Z2)
il*ZI + i2*Z2 for Z C H(XI n Xz),
ZI E H(XI)' and Zz E H(X 2 ). This is the Mayer- Vietoris sequence of singular
theory of an excisive couple {XI,Xz}. Similarly, if Xl n Xz =1= 0, there is a
reduced Mayer-Vietoris sequence of {XI'Xz}.
If (XI,A I) and (X 2,A z) are pairs in a space X, we say that {(XI,A I ), (X Z,A 2 )}
is an excisive couple of pairs if {Xh X2 } and {A I ,A 2 } are both excisive couples
of subsets. In this case it follows from the five lemma that the map induced
by inclusion
[6.(Xl) + 6.(X2)] / [6.(Al) + 6.(A2)] -+ [6.(Xl U X2)] / [6.(Al U A 2)]
induces an isomorphism of homology. Hence, if {(XhAl)' (X 2 ,A 2 )} is an
190
HOMOLOGY
CHAP.
4
excisive couple of pairs, there is an exact sequence
... ~ Hq(XI n X2, Al n A 2) ~ Hq(XI,A I ) EB Hq(X2,A2) ~
Hq(XI U X2, Al U A 2) ~
called the relative Mayer-Vietoris sequence of {(XI,A I), (X2,A2)}.
The relative Mayer-Vietoris sequence specializes to the exact sequence
of a triple (or a pair). In fact, given a triple (X,A,B), {(X,B), (A,A)} is always
an excisive couple of pairs, and the relative Mayer-Vietoris sequence of
{(X,B), (A,A)} is the homology sequence of the triple (X,A,B).
We use the Mayer-Vietoris sequence to compute the singular homology
of a sphere.
6
THEOREM
For n ~ 0
q=/=n
q=n
PROOF Let p and p' be distinct points of Sn. Because Sn - p and Sn - p' are
contractible (each being homeomorphic to Rn), fi(sn - p) = 0 = H(sn - p').
Since Sn - p and Sn - p' are open subsets of Sn, it follows from theorem 3
that {sn - p, Sn - p'} is an excisive couple. From the exactness of the corresponding Mayer-Vietoris sequence, it follows that
0*: Hq(sn) ;:::; Hq_l(sn - (p U p'))
Because Sn - (p U p') has the same homotopy type as Sn-I, there is an
isomorphism Hq_l(sn - (p U p')) ;:::; Hq_l(sn-I), and the result follows by
induction and the trivial verification that for n
0 the theorem is valid. -
=
We now show that a couple consisting of polyhedral subsets of a polyhedron is excisive.
LEMMA
Let KI and K2 be subcomplexes of a simplicial complex K.
Then {IK I I,IK21} is an excisive couple.
7
Let V be a neighborhood of IKI n K21 in IKII having IKI n K21 as a
strong deformation retract (such a V exists, by corollary 3.3.11). There is a
commutative diagram
PROOF
--,) Hq( IKI
n
i·t
... --,)
K 21) --,) Hq( IKII) --,) Hq( IKII, IKI
It
n
K21) --,)
j·t
Hq(V)
--,)
...
Because i: IKI n K21 C Vis a homotopy equivalence, i*: H(IK I n K 21) ;:::; H(V).
By the five lemma, i*: H(IKII, IKI n K21) ;:::; H(IKII, V).
Also, V U IK21 is a neighborhood of IK21 in IKI U K21 having IK21 as a
strong deformation retract. Therefore a similar proof shows that
i*: H(IK I U K21, IK21) ;:::; H(IK I U K21, V U IK21)
By theorem 4, {IKII, IK 2J} is an excisive couple if and only if the excision
SEC.
6
191
MAYER-VIETORIS SEQUENCES
map (lKll, IKl n K 21) C (IK l U K21, IK 21) induces an isomorphism of homology. In view of the isomorphisms i* and i~, this will be so if and only if the
excision map (lKll, V) c (IK l U K21, V U IK 21) induces an isomorphism of
homology. Again by theorem 4, this is equivalent to the condition that
{IKll, V U IKzl} be an excisive couple. This is so by theorem 3, since
IK21 C int (V U IK21) and IKll - IK21 C int IKll· •
8
THEOREM
For any simplicial pair (K,L) the natural transformation P
induces an isomorphism of the ordered homology sequence of (K,L) onto the
singular homology sequence of (IKI,ILI).
It suffices to prove that for any simplicial complex K, P* : H(!::..(K)) ;::::;
H(IKI), because the theorem will follow from this and the five lemma. We
prove this first for finite simplicial complexes by induction on the number of
simplexes. If K contains one simplex, then K = 8, where s is a O-simplex, and
the result follows from corollary 4.4.2.
Assume the result inductively for simplicial complexes with fewer than
m simplexes, where m
1, and let K contain exactly m simplexes. Let s be a
simplex of K of maximum dimension and let L be the subcomplex of K consisting of all simplexes other than s. Then K = L U s and oS = L n s. Because
L has exactly m - 1 simplexes, P* is an isomorphism H(!::..(L));::::; H(ILI)
and an isomorphism H(!::..(s)) ;::::; H(ISI). By corollary 4.4.2, P*: H(!::..(s)) ;::::; H(lsl).
By the exactness of the ordered Mayer-Vietoris sequence of Land s and the
Mayer- Vietoris sequence of Singular theory for ILl and lsi (which exists, by
lemma 7), it follows from the five lemma that P*: H(!::..(K)) ;::::; H(IKI).
For infinite simplicial complexes K let {K,,} be the family of finite subcomplexes of K directed by inclusion. It follows from theorem 4.3.11 that
H(!::..(K)) ;::::; lim~ H(!::..(K,,)) and from theorem 4.4.6 that H(IKI) ;::::; lim~ H(IK"I).
The theorem now holds for K because 1'* is natural. •
PROOF
>
We show next that for free chain complexes a chain map is a chain
equivalence if and only if it induces an isomorphism in homology. First we
establish an exact sequence containing the homomorphism induced by a
chain map.
9
LEMMA
Let T: C ~ C' be a chain map and let C be the mapping cone
of T. There is an exact sequence
... ~ Hq+l(C) ~ Hq(C) ~ Hq(C') ~ Hq(C) -~ ...
Let 0': C' ~ C be the chain map defined by O'(c) = (O,c). Then
0' imbeds C' as a subcomplex of G and the quotient complex GIC' is such that
(GIC')q;::::; Cq _ l ; the boundary operator of GIC' corresponds to the negative
of the boundary operator of C under this isomorphism. The desired exact
sequence is then obtained from the exact homology sequence of the short exact
sequence of chain complexes
PROOF
o~
C' ~
G ~ GIC' ~ 0
192
HOMOLOGY
CHAP.
4
by replacing Hq(G/C') by Hq-I(C) and verifying that the connecting homomorphism 0*: Hq+I(G/C') ---'> Hq(C') corresponds to T*: Hq(C) ---'> Hq(C'). •
10 THEOREM If C and C' are free chain complexes, a chain map T: C
is a chain equivalence if and only ifT*: H(C) "'" H(C').
---'>
C'
By corollary 4.2.11, T is a chain equivalence if and only if C is acyclic.
By lemma 9 and corollary 4.5.5, Gis acyclic if and only if T * : H( C) ::::: H( C'). •
PROOF
BeCltllSe b.(K)/ b.(L) and b.(IKI) / b.(ILI) are free chain complexes, we have
the following result.
I I COROLLARY For any simplicial pair (K,L),
b.(K)/ b.(L) with b.(IKI)/ b.(ILI). •
If rp: KI
---'>
/J
is a chain equivalence of
K2 is a simplicial map, there is a commutative diagram
H(K 1) ~ H( b.(KI)) ~ H( IKII )
It.(cp)·
H(K2) ~ H( b.(K2)) ~ H( IK21 )
In particular, if K' is a subdivision of K and rp: K'
imation to the identity IK'I C IKI, then
---'>
K is a simplicial approx-
and
From the commutativity of the above diagram we obtain our next result.
12 THEOREM Let K' be a subdivision of K and let rp: K' ---'> K be a simplicial
approximation to the identity map IK'I C IKI. Then
rp*: H(K') ::::: H(K)
•
By theorem 10, C(rp): C(K') ---'> C(K) is a chain equivalence. It will
be useful to construct a chain map C(K) ---'> C(K') which is a chain homotopy
inverse of C(rp). If K' is a subdivision of K, an augmentation-preserving chain
map T: C(K) ---'> C(K') is called a subdivision chain map if T: C(L) C C(K' I L)
for every subcomplex L C K [ that is, if T is a natural chain map from C to
C(K' I • ) on e(K)].
13 THEOREM If K' is a subdivision of K, there exist subdivision chain
maps T: C(K) ---'> C(K'). If rp: K' ---'> K is a simplicial approximation to
the identity IK'I C IKI, then T* = rp*-l: H(K) ::::: H(K').
If s is any simplex of K, then C(K'I s) is acyclic [because H(K' 1 s) :::::
Hence, on the category e(K) of subcomplexes of K with models
GJR.{K) = {s 1s E K}, the functor C is free and C(K' I' ) is acyclic. It follows
from theorem 4.3.3 that there exist natural chain maps T from C to C(K' I . )
preserving augmentation.
If T is any subdivision chain map and rp: K' ---'> K is a simplicial approximation to the identity map IK'I C K, the composite
PROOF
H(lsl)
= 0].
SEc.7
193
SOME APPLICATIONS OF HOMOLOGY
C(CfJ)T: C(K)
~
C(K)·
is a natural chain map over 0,K) from C to C preserving augmentation. Since
C is free and acyclic with models 0TL(K), it follows from theorem 4.3.3 that
C( CfJ)T ~ 1c(1()' Therefore CfJ* T* = 1H(K). Since, by theorem 12, CfJ* is an isomorphism, T* = CfJ*-l. •
7
SOME APPLICATIONS OF HOMOLOGY
In this section we use homology for some of the applications mentioned
earlier. We shall show that euclidean spaces of different dimensions are not
homeomorphic, and also that Sn is not a retract of En+l (which is easily seen
to be equivalent to the Brouwer fixed-point theorem). This leads to the general consideration of fixed points of maps, and we prove the Lefschetz fixedpoint theorem. Finally, we shall consider separation properties of the sphere.
Proofs are given of Brouwer's generalization of the Jordan curve theorem and
of the invariance of domain.
I
THEOREM
By theorem 4.6.6, Hn(sn) =f= 0 and Rn(sm)
PROOF
2
If n =f= m, Sn and Sm are not of the same homotopy type.
COROLLARY
= o. •
If n =f= m, Rn and Rm are not homeomorphic.
If Rn and Rm were homeomorphic, their one-point compactifications
Sn and Sm would also be homeomorphic, in contradiction to theorem 1. •
PROOF
In corollary 2 both Rn and Rm are contractible. Therefore they have the
same homotopy type and cannot be distinguished by their homology groups.
To distinguish them it was necessary to consider associated spaces having
nonisomorphic homology. We chose to consider their one-point compactifications, but another proof could have been based on the fact that Rn minus a
point has the same homotopy type as Sn-l.
These two results are applications of homology to the problem of classifying spaces up to topological equivalence. Our next application is to an
extension problem.
a
LEMMA
Let (X,A) be a pair such that A is a retract of X. Then
H(X) ;:::: H(A) EB H(X,A)
Given i: A C X and i: (X, 0) C (X,A) and a retraction r: X ~ A, then
Therefore r* i* = 1H (A) and i* is a monomorphism of H(A) onto a
direct summand of H(X). The other summand is the kernel of r*. From the
exactness of the homology sequence of (X,A)
a.
;..
j.
a.
....... Hq(X,A) .... Hq- 1(A) .... Hq- 1(X) .... Hq- 1(X,A) .... .. ·
PROOF
ri
=1
A•
194
HOMOLOGY
CHAP.
4
because ker i* = 0, 0* is the trivial map. Therefore i* is an epimorphism.
Since ker i* = im i*, i* induces an isomorphism of ker r* onto H(X,A). •
Note that lemma 3 is still valid if A is a weak retract of X.
4
COROLLARY
For n
~
0, Sn is not a retract of En+l.
*
By theorem 4.6.6, Hn(sn)
0, but because En+1 is contractible,
Hn(En+l) = O. Therefore H(Sn) is not isomorphic to a direct summand of
H(En+l). •
PROOF
This implies the following Brouwer fixed-point theorem.
S
THEOREM
For n
~
0 every continuous map from En to itself has a
fixed point.
>
PROOF
For n = 0 there is nothing to prove. For n
0 let f: En ~ En be
continuous. If f has no fixed point, define a map g: En ~ Sn-l by g(x) equal
to the unique point of Sn-l on the ray from f(x) to x, as shown in the figure.
/" f(x)
/
/ox
/
g(x)
/
/
o
I
f(x)
I
lx
I
I
g(x)
Then g is a retraction from En to Sn-l, in contradiction to corollary 4.
•
We have, in fact, proved that corollary 4 implies theorem 5. The
converse is also true, for if r: En+l ~ Sn were a retraction, the map f: En+l ~ En+ 1
defined by f(x) = - r(x) would have no fixed points.
There is an interesting generalization of theorem 5 which contains a
criterion for showing that a certain map from X to itself has a fixed point even if
not every map of X to itself has fixed points. This generalization also illustrates
another type of application of homology in that it is based on an algebraic
count of the number of fixed points, the algebraic count being formulated in
homological terms. This type of application of homology occurs frequently.
Generally it involves a set of singularities of X of a certain type (for example,
the set of fixed points of a map X ~ X, the set of discontinuties of a function
X ~ Y, the set of self-intersections of a local homeomorphism X ~ Rn, etc.)
and measures the singular set by means of a homology class associated to it.
Let C be a finitely generated graded group and let h: C ~ C be an endomorphism of C of degree o. The Lefschetz number A(h) is defined by the
formula
SEc.7
195
SOME APPLICATIONS OF H~'!OLOGY
where hq: Cq -7 Cq is the endomorphism defined by h in degree q. The following Hopf trace formula equates the Lefschetz numbers of a chain map and
its induced homology homomorphism.
6
THEOREM
Let C be a finitely generated chain complex and let
be a chain map. Then
T:
C -7 C
PROOF
The proof is similar to the proof of the corresponding statement
about the Euler characteristic (theorem 4.3.14), the Euler characteristic being
the Lefschetz number of the identity map, with theorem 4.13 of the Introduction used in place of theorem 4.12. Details are left to the reader. -
Let f: X -7 X be a map, where X has finitely generated homology. The
Lefschetz number off, denoted by "AU), is defined to be the Lefschetz number
of the homomorphism f*: H(X) -7 H(X) induced by f. It counts the algebraic
number of fixed homology classes of f*. The following Lefschetz fixed-point
theorem shows that "A( f) 0:/= 0 is a sufficient condition for f to have a fixed point.
THEOREM
Let X be a compact polyhedron and let f: X
7
If "A(f) 0:/= 0, then f has a fixed point.
-7
X be a map.
PROOF
We assume that f has no fixed point and prove "A(f) = O. Without
loss of generality, we may assume X
ILl for some finite simplicial complex L.
Because ILl is a compact metric space, if f has no fixed point, there is a
0
such that d(a,f(a» 2 a for all a E ILl. Let K be a subdivision of L with mesh
K
a/3 and let K' be a subdivision of K for which there exists a simplicial
map q;: K' -7 K which is a simplicial approximation to f: IKI -7 IKI. Since
1q;I(a) and f(a) belong to some simplex of K, d(Iq;I(a),f(a»
a/3 for a E IKI.
If s is any simplex of K, lsi is disjoint from 1q;1(lsl), for if a E lsi is equal
to 1q;1(,8) for ,8 E lsi, then
=
>
<
<
d(,8,f(,8»
s d(,8,a) + d(Iq;I(,8),f(,8» < 2a/3
in contradiction to the choice of a.
Let T: C(K) -7 C(K') be a subdivision chain map (which exists, by
theorem 4.6.13). Then C(q;)T: C(K) -7 C(K) is a chain map. If 0 is an oriented
q-simplex on a q-simplex s of K, then C(q;)T(O) is a q-chain on the largest subcomplex of K disjoint from s. Therefore C(q;)T(O) is a q-chain having coefficient 0
on o. Since this is so for every 0, all the coefficients summed in forming
Tr(C(q;)T)q are zero and Tr((C(q;)T)q)
0 for all q, which implies "A(C(q;)T)
O.
By theorem 6, "A((C(q;)T)*) = O. Let q;': K' -7 K be a simplicial approximation
to the identity map IK'I C IKI. There is a commutative diagram
=
=
H(K)
~ H(K')
~ H(K)
i~
i~
i~
H(b.(K») ~ H(b.(K'») ~ H(b.(K»)
t~
t~
t~
H(IKI) tp'I.=l H(IKI) ~ H(IKI)
196
HOMOLOGY
CHAP.
4
from which it follows that
A(f*) = A(I<pI*(I<p'I*)-I) =
By theorem 4.6.13, (<p~)-l
Therefore A(f) = O. •
= 'T*
A(<p*(<p~)-l)
and A(<p*(<p~tl)
= A(<p*'T*) = A([C(<p)'T]*).
This yields the following generalization of the Brouwer fixed-point
theorem.
8
COROLLARY
Every continuous map from a compact contractible polyhydron to itself has a fixed point.
If X is contractible, {[(X) = 0, and for any
[because f* is the identity map on Ho(X) ::::: Z]. •
PROOF
f:
X ~ X, A(f) = 1
This result is false for noncompact polyhedra. In fact, R is a contractible
polyhedron and any translation different from 1R fails to have a fixed point.
Given a continuous map f: Sn ~ Sn, the degree off is the unique integer
deg f such that
f*(z) = (degf)z
The following fact is obvious.
9
For any map f: Sn
~
Sn, A(f) = 1
+ (-l)n
deg f
•
Since the antipodal map Sn ~ Sn has no fixed points, the next result
follows from theorem 7 and statement 9.
10
COROLLARY
The antipodal map of Sn has degree ( _l)n+l.
•
I I COROLLARY If n is even, there is no continuous map f: Sn ~ Sn such
that x and f(x) are orthogonal for all x E Sn.
PROOF
Assume that such a map exists. Then a homotopy F:
fined by
f
~
l s n is de-
(1 - t)f(x) + tx
F(x,t) = -,--'-------'-"--:'-'---II (1 - t)f(x) + txll
This is well-defined, because the condition that x and f(x) be orthogonal
implies 11(1 - t)f(x) + txl1 2 = (1 - t)2 + t 2 =1= 0 for 0 ~ t ~ 1. Since f ~ lsn,
A(f) = A(lsn) = 1 + ( -l)n =1= o. Hence, by theorem 7, f must have a fixed
point, in contradiction to the orthogonality of x and f(x) for all x. •
This last result is equivalent to the statement that an even-dimensional
sphere Sn has no continuous tangent vector field which is nonzero everywhere
on Sn. For odd n such vector fields do exist because the map f: S2m-1 ~ S2m-1
defined by
f(XI, . . . ,X2m)
=
(-X2' Xl, . . . , -X2m, X2m-l)
is continuous and has the property that x and f(x) are orthogonal for all x.
SEC.
7
197
SOME APPLICATIONS OF HOMOLOGY
Instead of considering vector fields, we consider one-parameter groups
of homeomorphisms. A flow on X is a continuous map
1/;: R X
X~
X
such that
(a) 1/;(t1 + t2, x) = 1/;(t1' 1/;(t2'X))
(b) 1/;(O,x) = x
t1, t2 E R; x E X
x EX
For t E R let 1/;t: X ~ X be defined by 1/;t(x) = 1/;(t,x). Then (a) and (b) imply
1/;-t = (1/;t)-1, and so 1/;t is a homeomorphism of X for all t E R. A fixed point
of the flow is a point Xo E X such that 1/;(t,xo) = Xo for all t E R.
12 THEOREM If X is a compact polyhedron with x(X) ::/= 0, then any flow
on X has a fixed point.
Each 1/;t is homotopic to Ix [by the homotopy F: X X I
by F(x,t') = 1/;((1 - t')t, x)]. Therefore
PROOF
>-'(1/;t)
~
X defined
= >-'(lx) = x(X) ::/= 0
Hence, by theorem 7, each 1/;t has fixed points. For n ~ 1 let An be the
closed subset of X consisting of the fixed points of 1/;1I2n • Then An+1 cAn,
and {An} is a decreasing sequence of nonempty closed subsets of the compact space X. Let F = n An. Then F is nonempty, and any point of F is
fixed under 1/;t for all t of the form 1/2n for n 2: 1. This implies that each
point of F is fixed under 1/;t for all dyadic rationals t = m/2n. Since the dyadic
rationals are dense in R, each point of F is fixed under 1/;t for all t. •
We now turn our attention to separation properties of the sphere.
13
LEMMA
H(Sn - A) =
If A
o.
c Sn is homeomorphic to Ik for
0
~
k
~
n, then
We prove this by induction on k. If k = 0, then A is a point and
Sn - A is homeomorphic to Rn. Therefore H(sn - A) = o.
Assume the result for k
m, where m ~ 1, and let A be homeomorphic
to 1m. Regard A as being homeomorphic to B X I, where B is homeomorphic
to 1m -I, by a homeomorphism h: B X I ~ A. Let A' = h(B X [O,J,~]) and
A" = h(B X [%,1]). Then A = A' U A" and A' n A" is homeomorphic to
B X %. By the inductive assumption, H(sn - (A' nAil)) = O.Because
Sn - A' and Sn - A" are open sets, they are excisive and from the exactness
of the corresponding reduced Mayer-Vietoris sequence
PROOF
<
i*: Hq(sn - A) ;:::::: Hq(sn - A') EB Hq(Sn - A")
If z E Hq(sn - A) is nonzero, then either i~z::/= 0 in Hq(sn - A') or i~z::/= 0
in Hq(Sn - A"), where i': Sn - A c Sn - A' and i": Sn - A c Sn - AI/.
Assume
of sets
i* z ::/= O. We repeat the argument for A' and thus obtain a sequence
198
HOMOLOGY
CHAP.
4
such that
(a) The inclusion Sn - A c Sn - Aj maps z into a nonzero element of
fJq(sn - A j ).
(b) nA i is homeomorphic to Im-l
Because every compact subset of Sn - nA i is contained in Sn - Aj for
some f, it follows from theorem 4.4.6 that fJq(Sn - n Ai) ::::; lim~ {fJq(sn - Aj)}.
This is a contradiction because, by condition (a), the element z determines a
nonzero element oflim~ {Hq(sn - Aj)}, but by condition (b) and the inductive
assumption, fJq(sn - n Ai) = 0. -
°S; k S; n 14
COROLLARY
Let B be a subset of Sn which is homeomorphic to Sk for
1. Then
-
{O
Hq(Sn - B)::::; Z
q=/=n-k-I
q=n-k-I
We use induction on k. If k = 0, then B consists of two points and
Sn - B has the same homotopy type as Sn-l. Therefore
PROOF
-
{O
Hq(Sn - B)::::; Z
q=/=n-I
q=n-I
If k :;::: 1, set B = Al U Az, where Al and Az are closed hemispheres of Sk
and assume the result valid for k - 1. Then Al and Az are homeomorphic to
Ik and Ai n Az is homeomorphic to Sk-l. Because Sn - Al and Sn - A z are
open, {Sn - A 1 , Sn - Az} is an excisive couple, and there is an exact reduced
Mayer-Vietoris sequence
---7
fJq+1(sn - Ai) (f) fJq+l(sn - Az) ---7 fJq+l(sn - (Ai n Az)) ---7
fJq(sn - B) ---7 fJq(sn - Ai) (f) fJq(sn - Az)
---7
By lemma 13, the groups at the ends vanish. The result then follows from the
inductive assumption. For the special case of an (n - I)-sphere imbedded in Sn, we obtain the
following Jordan-Brouwer separation theorem.
15 THEOREM An (n - I)-sphere imbedded in Sn separates Sn into two components of which it is their common boundary.
PROOF
If B C Sn is homeomorphic to Sn-l, then fJo(sn - B) ::::; Z, by corollary 14. Therefore Sn - B consists of two path components. Since Sn - B is
an open subset of Sn, it is locally path connected and its path components U
and V, say, are its components.
Clearly, B contains the boundary of U and of V. To prove B C (; n \1,
let x E B and let N be a neighborhood of x in Sn. Let A C B n N be a subset
such that B - A, is homeomorphic to In-l. Then fJ(Sn - (B - A)) = 0, by
lemma 13, so Sn - (B - A) is path connected. If p E U and q E V, there is a
SEC.
8
AXIOMATIC CHARACTERIZATION OF HOMOLOGY
199
path win Sn - (B - A) from p to q. Because p and q are in different path
components of Sn - B, w meets A. Therefore A contains a point of 0 and a
point of V. Hence N meets 0 and V, and x EOn V. A related result is the following Brouwer theorem on the invariance of
domain.
16 THEOREM If U and V are homeomorphic subsets of Sn and U is open in
Sn, then V is open in Sn.
PROOF
Let h: U ---,) V be a homeomorphism and let h(x) = y. Let A be a
neighborhood of x in U that is homeomorphic to In and with boundary B
homeomorphic to Sn-l. Let A' = h(A) C Vand let B' = h(B). By lemma 13,
Sn - A' is connected, and by theorem 15, Sn - B' has two components.
Because
Sn - B'
= (sn
- A') U (A' - B')
and Sn - A' and A' - B' are connected, they are the components of Sn - B'.
Therefore A' - B' is an open subset of Sn. Since y E A' - B' C V and y was
arbitrary, V is open in Sn. -
8
AXIOMATIC CHARACTERIZATION OF HOMOLOGY
A simple set of axioms characterizing homology on the class of compact
polyhedral pairs has been given by Eilenberg and Steenrod1 . This section
describes the axiom system and related concepts. For compact polyhedral
pairs, the axioms are categorical (that is, two theories satisfying them are
isomorphic). Thus the axioms are basic theorems from which other properties
of homology theories can be deduced. In many cases, proofs based on the axioms
are simpler and more elegant than proofs which refer back to the original definition of the homology theory.
To formulate the axioms it is usual to start with a suitable category of
topological pairs and maps (called "admissible categories" by Eilenberg and
Steenrod). We shall not define these categories. The category of all topological pairs is such a category, and so are its full subcategories defined by the
polyhedral pairs and defined by the compact polyhedral pairs. For our purposes we shall always regard a homology theory as defined on the category of
all topological pairs, and we identify a space X with the pair (X, 0).
A homology theory H and a consists of
(a) A covariant functor H from the category of topological pairs and
maps to the category of graded abelian groups and homomorphisms of
degree 0 [that is, H(X,A) = {Hq(X,A)}]
1 See S. Eilenberg and N. E. Steenrod, "Foundations of Algebraic Topology," Princeton University Press, Princeton, N.J., 1952.
200
HOMOLOGY
CHAP.
4
a of degree - I from the functor H on
(X,A) to the functor H on (A, 0) [that is, a(X,A) = {aq(X,A): Hq(X,A) -)
(b) A natural transformation
H q _ 1 (A)} ].
These satisfy the following axioms.
I
HOMOTOPY AXIOM
If fa,
H(fo)
2
EXACTNESS AXIOM
h:
(X,A) -) (Y,B) are homotopic, then
= H(h):
H(X,A) -) H(Y,B)
For any pair (X,A) with inclusion maps i: A C X and
j: X C (X,A) there is an exact sequence
... ~ Hq(A)
Hii)
Hq(X) ~ Hq(X,A)
rq(X,A)
H q_ 1 (A) HQ~l(i)
.•.
3
EXCISION AXIOM
For any pair (X,A), if U is an open subset of X such
that 0 C int A, then the excision map j: (X - U, A - U) C (X,A) induces
an isomorphism
H(j): H(X - U, A - U) ::::::: H(X,A)
4
DIMENSION AXIOM
On the full subcategory of one-point spaces, there is a
natural equivalence of H with the constant functor; that is, if P is a one-point
space, then
q=l=O
q=O
Obviously, the homotopy axiom is equivalent to the condition that the
homology theory can be factored through the homotopy category of topological pairs.
Singular homology theory is an example of a homology theory. In fact,
the homotopy axiom is a consequence of theorem 4.4,9, the exactness axiom
is a consequence of theorem 4.5.4, the excision axiom is a consequence
of corollary 4.6.5, and the dimension axiom is a consequence of lemmas 4.4.1
and 4,3.1. Therefore, there exist homology theories.
Corresponding to any homology theory there are reduced groups defined
as follows. If X is a nonempty space, let c: X -) P be the unique map from
X to some one-point space P. The reduced group FI(X) is defined to be the
kernel of the homomorphism
H(c): H(X) -) H(P)
Because c has a right inverse, so does H(c). Therefore
H(X) ::::::: H(X) EEl H(P)
and the reduced groups have properties similar to those of the reduced
singular groups.
Given a triple B c A e X, let k: A C (A,B) and define a(X,A,B):
H(X,A) -) H(A,B) to be the composite
SEC.
8
201
AXIOMATIC CHARACTERIZATION OF HOMOLOGY
= H(k)o(X,A):
o(X,A,B)
H(X,A)
H(A)
~
~
H(A,B)
:;
THEOREM
For any triple (X,A,B), with inclusion maps i: (A,B) C (X,B)
and i: (X,B) C (X,A), there is an exact sequence
Hq,
(X B) -H(j)
H q(X,A) Oq(X,A,B)) Hq-l (A , B) ~ . . .
. .. ~ Hq(A ,B) -Hq(i)
--7
--7
The proof involves diagram chasing based on the exactness axiom 2.
We prove exactness at Hq(A,B) and leave the other parts of the proof to the
reader.
(a) im oq+l(X,A,B) C ker Hq(i). Hq(i)oq+l(X,A,B) is the composite
PROOF
Hq+l(X,A)
Hq(A)
Oq+l(X,A)
Hq(A,B)
Hq(k\
Hq(i\
Hq(X,B)
which also equals the composite
Hq+l(X,A)
Oq+l(X,A)
Hq(A)
Hq(X)
Hq(i')
Hq(i")
Hq(X,B)
where i': A C X and iff: X C (X,B). By axiom 2, Hq(i')Oq+l(X,A) = O. Therefore Hq(i)oq+l(X,A,B) = o.
(b) ker Hq(i) C im Oq+l(X,A,B). Let z E Hq(A,B) be such that Hq(i)z = o.
Then oq(X,B)Hq(i)z
0, and because oq(A,B)
oq(X,B)Hq(i), oq(A,B)z O.
By axiom 2, there is Z E Hq(A) such that Hq(k)z'
z. Because the composite
=
Hq(A)
=
=
Hq(i')
=
Hq(X) Hq(i"~ Hq(X,B)
equals the composite Hq(i)Hq(k), it follows that
Hq(i")Hq(i')z
= Hq(i)Hq(k)z = Hq(i)z = 0
By axiom 2, there is z" E Hq(B) such that if 1': B C X, then Hq(i')z = Hq(f')z".
Given iff: B C A, then Hq(f') = Hq(i')Hq(f"). Therefore Hq(i')(z - Hq(j")z") = O.
Again by axiom 2, there is z E Hq+l(X,A) such that Oq+l(X,A)z = z' - Hq(f")z".
Then, because Hq(k)Hq(f") = 0,
oq+l(X,A,B)z
= Hq(k)oq+l(X,A)z = Hq(k)z'
which shows that z is in im Oq+l(X,A,B).
- Hq(k)Hq(f")z"
=z
•
The exact sequence of theorem 5 is called the homology sequence of the triple
(X,A,B). If B = 0, it reduces to the homology sequence of the pair (X,A).
Let H and a and H' and a' be homology theories. A homomorphism from
H and a to H' and a' is a natural transformation h from H to H' commuting
with a and 0'. That is, for every (X,A) there is a commutative diagram
H(X,A) ~ H(A)
H'(X,A) ~ H'(A)
in which the vertical maps are homomorphisms of degree o. In view of the
dimension axiom, a homomorphism h induces a homomorphism ho: Z ~ Z
202
HOMOLOGY
CHAP.
4
that characterizes h on one-point spaces. The main result proved by Eilenberg
and Steenrod is that corresponding to any homomorphism ho: Z ~ Z there
exists a unique homomorphism h from H and a to H' and a', on the category
of compact polyhedral pairs, which induces h o. We shall not prove this, but
shall content ourselves with proving that a homomorphism h which is an isomorphism for one-point spaces is an isomorphism for any compact polyhedral
pair. This will illustrate how the axioms-1::an be used and will suffice for our
later applications.
The following is an easy consequence of the exactness axiom and the
five lemma (or of theorem 5 and axiom 2).
6
LEMMA
Let A' cAe X. Then H(A') :::::: H(A) if and only if H(X,A') ::::::
H(X,A) (both maps induced by inclusion). -
We now prove a stronger excision property. A map f: (X,A) ~ (Y,B) is
called a relative homeomorphism if f maps X - A homeomorphically onto
Y - B. Following are some examples.
7
An excision map (X - U, A - U) c (X,A), where U C A, is a relative
homeomorphism.
8
If X is obtained from A by adjoining an n-cell e and f: (En,Sn-l)
is a characteristic map for e, then f is a relative homeomorphism.
~
(e,e)
9
THEOREM
Let X be a compact Hausdorff space and let A be a closed
subset of X which is a strong deformation retract of one of its closed neighborhoods in X. Let f: (X,A) ~ (Y,B) be a relative homeomorphism, where Y
is a Hausdorff space and B is closed in Y. Then, for any homology theory
H(f): H(X,A) :::::: H(Y,B).
PROOF
Let N be a closed neighborhood of A in X such that A is a strong
deformation retract of N and let U be an open subset of X such that
A cUe (j C N (U exists because X is a normal space). Let F: N X I ~ N
be a strong deformation retraction of N to A.
Define N' = f(N) U B, U' = f(U) U B, and F': N' X I ~ N by
F'(y,t)
F'(y,t)
=Y
= fF(f-l(y),t)
Y E B, tEl
Y E f(N), tEl
Then F' is well-defined and continuous on each of the closed sets B X I and
f(N) X 1. Therefore F' is continuous and is easily verified to be a strong
deformation retraction of N' to B. Because X - 0 is open in X - A,
Y - (f( 0) U B) is open in Y - B, and because B is closed, it is open in Y.
Therefore f(O) U B is closed in Y, and 0' C f(O) U BeN. Because X - U
is a closed, and hence compact, subset of X, f(X - U) = Y - U' is a compact
subset of Y. Because Y is a Hausdorff space, Y - U' is closed in Y, and U' is
open in Y. We have B C U' c a' c N' and a commutative diagram
SEC.
8
203
AXIOMATIC CHARACTERIZATION OF HOMOLOGY
H(X,A)
~
H(X,N)
t
H(X - U, N - U)
1::::
H(Y,B)
-;::7
H(Y,N')
~
H(Y - U', N' - U')
where the vertical maps are induced by f and the horizontal maps are induced
by inclusion maps. Because A and B are deformation retracts of Nand N',
respectively, H(A) ;:::; H(N) and H(B) ;:::; H(N'). It follows from lemma 6 that
the left-hand horizontal maps are isomorphisms. The right-hand horizontal
maps are isomorphisms by the excision axiom. The right-hand vertical map is
an isomorphism because it is induced by a homeomorphism. From the commutativity of the diagram, it follows that H(f) is an isomorphism. •
10 THEOREM Let h be a homomorphism from H and a to H' and a' which
is an isomorphism for one-point spaces. Then, for any compact polyhedral pair
(X,A), h(X,A): H(X,A) ;:::; H'(X,A).
PROOF
By the five lemma, it suffices to prove h(X): H(X) ;::::; H'(X) for any
compact polyhedron X. Hence, let K be a finite simplicial complex. We need
only prove that h(IKI): H(IKI) ;:::; H'(IKI). We prove this by induction on the
number of simplexes of K. If K has just one simplex, IKI is a one-point space,
and h(IKI) is an isomorphism by hypothesis.
Assume that K has m simplexes, where m
0, and that h is an isomorphism for the space of any simplicial complex with fewer than m simplexes.
Assume dim K = n and let s be an n-simplex of K. Let L be the subcomplex
consisting of all simplexes of K different from s. By the five lemma and the
exactness axiom, h(IKI) is an isomorphism if and only if h(IKI,ILI) is an
isomorphism. If ;: (lsl,181) C (IKI,ILI), it follows from theorem 9 that H(i) and
H'(i) are isomorphisms. Hence we need only prove that h(lsl,181) is an
isomotphism.
If n = 0, (lsl,181) is a one-point space, and h(lsl,lSl) is an isomorphism by
hypothesis. If n
0, because lsi has the same homotopy type as a one-point
space, h(lsl) is an isomorphism. By the five lemma and the exactness axiom,
h(lsl,181) is an isomorphism if and only if h(181) is an isomorphism. Because 8 is a
proper subcomplex of K, h(181) is an isomorphism by the inductive hypothesis .•
>
>
To extend this result to arbitrary polyhedral pairs (not merely compact
ones), we add an additional axiom. A pair (X,A) with X compact and A closed
in X is called a compact pair.
I I AXIOM OF COMPACT SUPPORTS Given any pair (X,A) and given z E Hq(X,A),
there is a compact pair (X',A') C (X,A) such that z is in the image of
H(X',A') ~ H(X,A).
A homology theory H and a satisfying axiom 11 is called a homology
theory with compact supports (Eilenberg and Steenrod use the term "homology
theory with compact carriers"). It is clear that singular homology theory is a
204
HOMOLOGY
CHAP.
4
homology theory with compact supports. We shall see that any homology
theory with compact supports satisfies the analogue of theorem 4.4.6. The
following lemma is the main point in proving this.
12 LEMMA Let H be a homology theory with compact supports and let
(X',A') be a compact pair in (X,A). Given z E Hq(X',A') in the kernel of
Hq(X',A') ---) Hq(X,A), there is a compact pair (X",A"), with (X',A') C
(X",A") C (X,A), such that z is in the kernel of H(X',A') ---) H(X",A").
PROOF
In the proof all unlabeled maps are induced by inclusion. z is in the
kernel of the composite
Hq(X',A')
Hq(i\
Hq(X,A') ---) Hq(X,A)
By theorem 5, Hq(i)z is in the image of Hq(A,A') ---) Hq(X,A'). By axiom 11,
there is a compact space A" such that A' C A" C A and such that Hq(i)z is
in the image of the composite Hq(A",A') ---) Hq(A,A') ---) Hq(X,A'). By theorem 5,
the composite Hq(A",A') ---) Hq(X,A') ---) Hq(X,A") is trivial. Therefore z is in
the kernel of Hq(X',A') ---) Hq(X,A") for some compact A" containing A'.
Because z is in the kernel of the composite
Hq(X',A')
Hq(j)
Hq(X' U A", A") ---) Hq(X,A")
it follows from theorem 5, that Hq(;)z is in the image of
aq+l: Hq+l(X, X' U A") ---) Hq(X' U A", A")
By axiom 11, there is a compact X" containing X' U A" such that Hq(f)z is in
the image of the composite
Hq+l(X", X' U A") ---) Hq+l(X, X' U A")
~ Hq(X'
U A", A")
This composite is also equal to the map aq+l: Hq+l(X", X' U A") ---)
Hq(X' U A", A"). By theorem 5, the composite
Hq+l(X", X' U A") ~ Hq(X' U A", A") ---) Hq(X",A")
is trivial. Therefore, z is in the kernel of Hq(X',A') ---) Hq(X",A").
•
For any pair (X,A) the family of compact pairs (X',A') contained in (X,A)
is directed by inclusion. For any homology theory H and a the groups
{H(X',A') I (X',A') compact C (X,A)} constitute a direct system, and the maps
H(X',A') ---) H(X,A) define a homomorphism i: lim~ {H(X',A')} ---) H(X,A).
13 THEOREM A homology theory H and a has compact supports if and only
if for any pair (X,A), i: lim~ {H(X',A')} ;::::: H(X,A), where (X',A') varies over
the family of compact pairs contained in (X,A).
It is clear that axiom 11 is equivalent to the condition that i be an
epimorphism. Hence, if i is an isomorphism, H and a has compact supports.
Conversely, if H has compact supports, i is an epimorphism, and lemma 12
implies that i is also a monomorphism. •
PROOF
205
EXERCISES
14 THEOREM Let h be a homomorphism from H and a to H' and a' that is
an isomorphism for one-point spaces. If H and a and H' and a' have compact
supports, h is an isomorphism for any polyhedral pair.
This follows from theorems lO and 13 and from the fact that for any
polyhedral pair (X,A) the compact polyhedral pairs (X',A') contained in it are
cofinal in the family of all compact pairs in (X,A). •
PROOF
EXERCISES
A CHAIN HOMOTOPY CLASSES
1 For chain complexes C and C' show that [C;C'] is an abelian group (with group
operation [Tl] + [T2] = [Tl + T2]) and that there is a homomorphism
cP: [C;C']_ Hom (H(C),H(C'))
such that cp[T]
= T*.
2
If C is a free chain complex, prove that the homomorphism cP is an epimorphism.
3
If C is a free chain complex and H(C) is also free, prove that cp is an isomorphism.
B EULER CHARACTERISTICS
1 Let (X,A) be a pair and assume that two of the three graded groups H(A), H(X), and
H(X,A) are finitely generated. Prove that the third is also finitely generated and that
x(X) = X(A) + X(X,A).
2 Let {Xl ,X 2 } be an excisive couple of subsets of X such that H(Xl) and H(X2) are
finitely generated. Prove that H(Xl U X 2 ) is finitely generated if and only if H(Xl n X 2 )
is finitely generated, in which case
X(Xl)
+ X(X 2 ) = X(Xl
U X2 )
+ X(Xl n
X2)
3 Let y be an integer-valued function defined on the class of compact polyhedra with
base points such that
=
(a) If (X,Xo) is homeomorphic to (Y,Yo), then y(X,xo)
y(Y,Yo).
(b) If (X,A) is a compact polyhedral pair and Xo E A, then y(X,xo)
y(A,Xo) +
y(XIA,xo), where XI A denotes the space obtained by collapsing A to a Single point x&.
=
Prove that for any X
y(X,xo)
= y(SO,Po)X(X,Xo)
Prove first that if Zo is a base point of En in Sn-l, then y(En,zo) = O. Show that
the result is true for X = Sn, and then use induction on the number of simplexes in a
triangulation of X.]
[Hint: l
4
If X and Yare compact polyhedra, prove that
X(X X Y)
= X(X)X(Y)
See C. E. Watts, On the Euler characteristic of polyhedra, Proceedings of the American
Mathematical Society, vol. 13, pp. 304-306, 1962.
1
206
HOMOLOGY
C
EXAMPLES
I
Let
2
Compute the homology group of an arbitrary surface.
3
Compute the homology group of the lens space L(p,q).
S
CHAP.
4
be an n-simplex and let (s)m be its m-dimensional skeleton. Compute H((s)m).
4 Let A be a subspace of Sn which is homeomorphic to the one-point union Sp v Sq.
Compute H(sn - A).
5 Let X be the space obtained from a closed triangle with vertices Vo, Vl, and Vz by
identifying the edges VOVl, VlVZ, and VZVo linearly with the edges VlVZ, VZVo, and VOVl,
respectively. Compute H(X).
6 Given an integer n
polyhedron X such that
> 0 and an integer m > 1, prove that there exists a compact
q=/=n
q=n
7 Let H be a finitely generated nonnegative graded abelian group such that H o is a
free abelian group. Prove that there exists a compact polyhedron X such that Fl(X) :::::; H.
D
JOINS AND PRODUCTS
I
Prove that for any space X there are isomorphisms
FIq(X) :::::; FIQ+l(X
(Hint: If Y is contractible, so is X * Y.)
2
* SO)
Prove that for any space X there are isomorphisms
Hq(X X Sn, X X po) :::::;
Hq~n(X)
[Hint: Use induction on n and the fact that if Y is contractible, H(X X Y, X X yo)
3
= 0.]
Compute the homology group of the n-dimensional torus (Sl)n.
4 If a space is homeomorphic to a finite product of spheres, prove that the set of
spheres which are the factors is unique.
E ORIENTATION
I Let K be an n-dimensional pseudo manifold. Prove that it is possible to enumerate
the n-simplexes of K in a (finite or infinite) sequence So, SI, . . . ,Sq, • . . and to find a
sequence S1, S2, ... , s~, ... of (n - I)-simplexes of K such that for q :::: 1, s~ is a face
q.
of Sq and also a face of Si for some i
<
2 If K is a finite n-dimensional pseudomanifold, prove that exactly one of the following
holds:
(a) Hn(K,K):::::; Z and Hn~l(K,K) has no torsion.
(b) Hn(K,K) = 0 and Hn~l(K,K) has torsion subgroup isomorphic to Zz.
3 Let K be a finite simplicial complex which is homogeneously n-dimensional and such
that every (n - I)-simplex of K is the face of at most two n-simplexes of K. Let K be the
subcomplex of K generated by the (n - I)-simplexes of K which are faces of exactly one
n-simplex of K. Prove that if (K,K) satisfies either (a) or (b) of exercise 2 above, then K
is an n-dimensional pseudomanifold.
A finite n-dimensional pseudomanifold is said to be orientable (or nonorientable) if it
207
EXERCISES
satisfies (a) (or (b)) of exercise 2. An orientation of an orientable n-dimensional pseudomanifold K is a generator of Hn(K,K), and an oriented n-dimensional pseudomanifold is
an n-dimensional pseudo manifold together with an orientation of it.
4 Let z E Hn(K,K) be an orientation of a finite n-dimensional pseudo manifold. If 8 is
any n-simplex of K, prove that there is a unique orientation of 8, denoted by z I 8 E Hn(8,S)
and called the induced orientation of 8, characterized by the property that z and z I 8
correspond under the homomorphisms
Hn(K,K) -? Hn(K, K - 8) ~ Hn(s,s)
A collection of orientations {a(s) E Hn(s,s)} for each n-simplex s of an n-dimensional
pseudo manifold is called compatible if for any (n - I)-simplex s' of K - K which is a
face of the two n-simplexes S1 and S2 of K, a(s1) and - a(s2) correspond under the
homomorphisms
5 If z is an orientation of a finite n-dimensional pseudo manifold, prove that the collection {z I 8} is compatible. Conversely, given a compatible collection {a(s)} of orientations of the n-simplexes s of a finite n-dimensional pseudomanifold K, prove that there
is a unique orientation z of K such that z I s
a(s) for each n-simplex s of K. Use this to
define orientability for arbitrary (nonfinite) n-dimensional pseudomanifolds. [Hint: Identify Hn(K,Kn-1) with indexed collections {a(s) E Hn(s,s)}, where s varies over the
n-simplexes of K, and show that the image of the homomorphism Hn(K,K) -? Hn(K,Kn-1)
consists of the compatible collections.]
=
F
DEGREES OF MAPS
Let K1 and K2 be finite n-dimensional pseudomanifolds with orientations Z1 and Z2,
respectively. Given a continuous map f: (IK11,IK 1 1) -? (IK21,IK21), its degree, denoted by
deg f, is the unique integer such that f* (Z1) = (deg f)Z2 [where we regard
Z1 E Hn(IK 11,IK11)) and Z2 E Hn(IK21,IK21)]'
I Let cp: (K 1 ,K 1 ) -? (K 2 ,K 2 ) be a simplicial approximation to f, let S2 be a fixed
n-simplex of K2, and let m+(cp) (or m_(cp)) be the number of n-simplexes 81 of K1 such
that cp maps the induced orientation Z1 I S1 into the induced orientation Z2 I S2 (or into
-z21 S2). Prove that degf = m+(cp) - m_(cp).
2 In case K is a finite orientable n-dimensional pseudo manifold and f: (IKI,IKI)-?
(IKI,IKI), there is a unique integer degfsuch thatf* (z) = (degf)z for any z E Hn(IKI,IKI).
Prove that iff, g: (IKI,IKI) -? (IKI,IKI), then deg (g f) = (deg g) (degf).
0
3 Let f: Sn -? Sn be a map such that f(E~) C E~, f(E"-)
be the map defined by f. Prove that deg f = deg f'.
C
E"- and let 1': Sn-1 -? Sn-1
4 Show that for any n ;::: 1 and any integer m there is a map
degf = m.
G
I
f:
Sn -? Sn such that
TOPOLOGICAL INVARIANCE OF PSEUDO MANIFOLDS
Let K be a simplicial complex and let x E <s), where s is a simplex of K. Prove that
208
HOMOLOGY
CHAP.
4
there is an isomorphism
H(lKI,
IKI -
st 8) ::::::
H(IKI, IKI - x)
2 Let K be a simplicial complex and let x E (8), where 8 is a principal n-simplex of K
(that is, 8 is not a proper face of any simplex of K). Prove that
Hq(IKI,
IKI -
x) ::::::
G
q=l=n
q=n
3 Prove that a locally compact polyhedron X has dimension n if and only if n is the
largest integer such that there exist points x E X, with Hn(X, X - x) =1= O.
4 Let X be a finite dimensional polyhedron and for each n let Xn be the closure of the
set of all x E X having a neighborhood U such that Hn(X, X - '1) :::::: Z for all '1 E U. If
K is any simplicial complex triangulating X and Kn is the subcomplex of K generated by
the principal n-simplexes of K, prove that Kn triangulates Xn.
:. Prove that the property of being homogeneously n-dimensional is a topologically
invariant property of simplicial complexes (and so we can speak of a homogeneously
n-dimensional polyhedron).
6 Let K be art arbitrary simplicial complex triangulating a homogeneously n-dimensional
polyhedron X. Prove that every (n - I)-simplex of K is the face of at most two n-simplexes
of K if and only if Hq(A, A - x) = 0 for all x E A and all q 2 n - 1, where A is the
closure in X of the set {x E X I Hn(X, X - x) is noncyclic}.
7
Let X be a homogeneously n-dimensional polyhedron satisfying exercise 6 and let
x)
O} and where
Bn - l is defined in terms of B, as in exercise 4. If K is any simplicial complex triangulating X,
prove that the subcomplex of K generated by the (n - I)-simplexes of K which are faces
of exactly one n-simplex of K triangulates X.
j(
= Bn- l , where B is the closure in X of the set {x E X I Hn(X, X -
=
8 Prove that the property of being a finite n-dimensional pseudo manifold is a topologically invariant property of simplicial complexes.
D
I
EDGE-PATH GROUPS
If,
Let K be a connected simplicial complex with a base vertex Vo E K. Given an edge
e = (vo,v,,), of K, let [e] be the oriented I-simplex [vo,v,,].
= el~ ... er is a closed
edge path of K at vo, let 1/1(') = [ell + [~] + ... + [e r] E CI(K). Prove that 1/1(') is a cycle
and that if, and " are equivalent edge paths, then I/IW and 1/1(") are homologous.
2 Prove that there is a natural transformation 1ft: E(K,vo) __ HI(K) (on the category of
connected simplicial complexes with a base vertex) defined by 1ft[~l
{1ftW}.
=
3 Prove that the homomorphism 1ft is an epimorphism and has kernel equal to the commutator subgroup of E(K,vo).
I AXIOMATIC HOMOLOGY THEORY
In this group of exercises H will denote an arbitrary homology theory.
I
Let Xl and X2 be subs paces of a space X. Prove that the following are equivalent:
(a) The excision map (Xl, Xl n X 2) C (Xl U X2, X 2) induces an isomorphism of
homology.
(b) The excision map (X2' Xl n X2) C (Xl U X2, Xl) induces an isomorphism of
homology.
(c) The inclusion maps
209
EXERCISES
i l : (Xl, Xl
n X2) C (Xl U X2, Xl n X2)
and
i 2: (X 2, Xl n X 2) C (Xl U X2, Xl n X2)
induce monomorphisms on homology and
H(XI U X 2, Xl
n
X 2)::::: il*H(XI' Xl
n X2) if) i2.H(X2, Xl n
X2)
(d) The inclusion maps
il: (Xl U X2, Xl n X 2) C (Xl U X2, Xl)
and
i2: (Xl U X2, Xl n X2) C (Xl U X2, X2)
induce epimorphisms on homology and
H(Xl U X 2, Xl
n
it>
and
i2>
induce an isomorphism
X 2) ::::: H(Xl U X 2, Xl) if) H(Xl U X 2, X2)
(e) For any A C Xl n X2 there is an exact Mayer-Vietoris sequence
•.. ---'>
Hq(XI n X 2, A)
---'>
Hq(XI,A) if) Hq(X2,A)
---'> Hq(Xl U X 2, A)
---'>
Hq-l(X l
n X 2, A) ---'>
.•.
(f) For any Y :J Xl U X2 there is an exact Mayer-Vietoris sequence
... ---'>
Hq(Y, Xl
n X 2) ---'> Hq(Y,Xl)
if)
Hq(Y,X2)
Hq(Y, Xl U X 2) ---'> Hq_l(Y, Xl n X 2) ---'>
---'>
2
..•
Let Xl, . . . , Xm and A be closed subspaces of a space X such that
(a) X = U Xi.
(b) Xi n Xj = A if i =1= i(c) Xi - A is disjOint from Xj - A if i =1= iProve that the homomorphisms H(Xi,A) ---'> H(X,A) are monomorphisms and H(X,A) is
isomorphic to the direct sum of the images.
3 Let {Xj} iE J (with J possibly infinite) be a collection of closed subsets of a space X
and let A be a subspace of X such that (a), (b), and (c) of exercise 2 above are satisfied.
Assume also that every compact subset of X is contained in a finite union of {Xj} and
that H is a homology theory with compact supports. Prove that H(X,A) ::::: EBjEJ H(Xj,A).
4 Let (X,A) be a topological pair and let {Xs} be a family of subspaces of X indexed
by the integers such that
(a)
(b)
(c)
(d)
Let C
A = X_ l .
Xs C Xs+l for all s.
X = U Xs and every compact subset of X is contained in Xs for some s.
Hq(Xs,X s_l ) = 0 if q =1= s and s ::::: o.
= {Cq,a q} be the nonnegative chain complex with Cq = Hq(Xq,Xq_l ) for q ::::: 0 and
3q the connecting homomorphism of the triple (Xq, Xq- 1 ,Xq- 2 ) for q ;::: 1. If H has compact
supports, prove that H(X,A)
= H(C).
[Hint: Prove that there are exact sequences
Hq+l(Xq+l,Xq) ~ Hq(Xq,A)
---'>
Hq(X,A)
---'>
0
and
0---,> Hq(Xq,A)
---'>
Hq(Xq,Xq_l ) ---'> Hq_I(Xq_I,A)]
5 Let H be a homology theory defined on the category of compact pairs. Prove that
there is an extension of H to a homology theory H with compact supports such that
H(X,A)
lim_ {H(X',A') I (X',A') a compact pair in (X,A)}.
=
CHAPTER FIVE
PRODUCTS
WE ARE NOW READY TO EXTEND THE DEFINITION OF HOMOLOGY TO MORE GENERAL
coefficients. In this framework the homology considered in the last chapter
appears as the special case of integral coefficients. The extension is done in a
purely algebraic way. Given a chain complex C and an abelian group G, their
tensor product is the chain complex C ® G
{Cq ® G, Oq ® I}, and the
homology of C ® G is defined to be the homology of C, with coefficients G.
We shall also introduce the concepts of co chain complex and cohomology.
These are dual to the concepts of chain complex and homology and arise on
replacing the tensor-product functor by the functor Hom.
We shall establish universal-coefficient formulas expressing the homology
and cohomology of a space with arbitrary coefficients as functors of the
integral homology of the space. Although these new functors do not distinguish
between spaces not already distinguished by the integral homology functor,
it is nonetheless important to consider them, as it frequently happens that the
most natural functor to apply in a given geometrical problem is determined
by the problem itself and need not be the integral homology functor. For
example, in the obstruction theory developed in Chapter Eight we shall be
=
211
212
PRODUCTS
CHAP.
5
led to the cohomology of a space with coefficients in the homotopy groups of
another space.
A further consideration is that the cohomology of a space has a multiplicative structure in addition to its additive structure, which makes cohomology
a more powerful tool than homology. We shall present some applications of
this added multiplication structure, the most important of which is the study
of the homology properties of fiber bundles, where we establish the exactness
of the Thom-Gysin sequence of a sphere bundle.
At the end of the chapter is a brief discussion of cohomology operations.
These are natural transformations between two cohomology functors and
strengthen even further the applicability of cohomology as a tool. We shall
define the particular set of cohomology operations known as the Steenrod
squares and establish their basic properties.
Sections 5.1 and 5.2 are devoted to homology with general coefficients
and to the universal-coefficient formula for homology. Section 5.3 deals with
the tensor product of two chain complexes and contains a proof of the
Kiinneth formula expressing the homology of the tensor product as a functor
of the homology of the factor complexes. This is applied geometrically to express the homology of a product space in terms of the homology of its factors.
Sections 5.4 and 5.5 contain the dual concepts of cochain complex and
cohomology and the appropriate universal-coefficient formulas for them. In
Sec. 5.6 the cup and cap products are defined, the cup product being the
multiplicative structure in cohomology mentioned previously, and the cap
product being a dual involving cohomology and homology together. These
products are used in Sec. 5.7 to study the homology and cohomology of fiber
bundles. We establish the Leray-Hirsch theorem, which asserts that certain
fiber bundles have homology and cohomology which are additively isomorphic
to the homology and cohomology of the corresponding product of the base
and the fiber.
Section 5.8 is devoted to a study of the cohomology algebra. The exactness of the Thom-Gysin sequence is used to compute the cohomology algebra
of projective spaces, and this, in turn, is used to prove the Borsuk-Ulam
theorem. There is also a discussion of the structure of Hopf algebras, which
arise in considering the cohomology of an H space. In Sec. 5.9 the Steenrod
squares are defined and their elementary properties are proved. They will be
applied later.
I
HOMOLOGY WITH COEFFICIENTS
In this section we shall extend the concepts dealing with chain complexes to
the case where the chain groups are modules over a ring. The tensor product
of such a chain complex with a fixed module is another chain complex, and
its graded homology module is a functor of the original chain complex and
SEC.
1
213
HOMOLOGY WITH COEFFICIENTS
the fixed module. These homology modules have properties analogous to those
established in the last chapter for complexes of abelian groups. The section
closes with the definition of a homology theory with an arbitrary coefficient
module. This is analogous to the concept of homology theory (which has
integral coefficients) introduced in the last chapter.
Throughout this section R will denote a commutative ring with a unit.
We consider R modules and homomorphisms between them. A chain complex
over R, C = {Cq,a q} consists of a sequence of R modules Cq and homomorphisms aq: Cq ~ Cq_ 1 such that aqaQ+l = 0 for all q. There is then a graded
homology module
H(C)
= {Hq(C) = ker aq/im aq+1}
The concepts of chain maps and chain homotopies can be defined for chain
complexes over R, and the results about chain complexes of abelian groups
generalize in a straightforward fashion to chain complexes over R. In particular, on the category of short exact sequences of chain complexes over R,
O~C'~C~C"~O
there is a functorial connecting homomorphism
a*: Hq(C")
~
Hq_1(C')
and a functorial exact sequence
... ~ Hq(C') ~ Hq(C) ~ Hq(C") ~ Hq_1(C') ~
If C is a chain complex over Rand G' is an R module, an augmentation
of Cover G' is an epimorphism e: Co ~ G' such that e 0 a1 = O. An
augmented chain complex over G' consists of a nonnegative chain complex C
and an augmentation of Cover G'.
If C = {Cq,a q} is a chain complex over Rand G is an R module, then
C ® G = {Cq ® G, aq ® I} is also a chain complex over R, and if C is
augmented over G', then C ® G is augmented over G' ® G. The graded
homology module H(C ® G) is called the homology module of C with coefficients G and is denoted by H(C;G). If T: C ~ C' is a chain map,
T ® 1: C ® G ~ C' ® G is also a chain map, and T*: H(C;G) ~ H(C';G) denotes tlte homomorphism induced by T ® l. Given a homomorphism
<p: G ~ G', there is a chain map 1 ® <p: C ® G ~ C ® G' inducing a
homomorphism
<p*: H(C;G)
~
H(C;G')
These remarks are summarized in the follOWing statement.
I
THEOREM
There is a covariant functor of two arguments from the
category of chain complexes over R and the category of R modules to the category of graded R modules which assigns to a chain complex C and module G
the homology module of C with coefficients G. •
214
PRODUCTS
CHAP.
5
Note that if c E Cq is a cycle of C and g E C, then c ® g E Cq ® C is a
cycle of C ® C, and if c is a boundary, so is c ® g. Therefore there is
a bilinear map
Hq(C) X C
~
Hq(C;C)
which assigns to ({ c },g) the homology class {c ® g}. This corresponds to a
homomorphism
JL: H(C) ® C ~ H(C;C)
such that JL( {c} ® g) = {c ® g} for c E Z( C). The homomorphism JL is easily
verified to be a natural transformation on the product of the category of chain
complexes with the category of modules.
If C is a chain complex over Z and C is an R module, then C ® C is a
z
chain complex over R. It follows from theorem 4.5 in the Introduction that
the homology module over Z of C with coefficients C is isomorphic, as a
graded R module, to the homology module over R of C ® R with coefficients C.
z
2 EXAMPLE Let C(K) denote the oriented chain complex of the simplicial
complex K. Given an abelian group C and a simplicial pair (K,L), the oriented
homology group of (K,L) with coefficients C, denoted by H(K,L; C), is
defined to be the graded homology group of [C(K)/C(L)] ® C (which is
augmented over Z ® C ;::::; C). Then H(K,L; C) is a covariant functor of two
arguments from the category of simplicial pairs and the category of abelian
groups to the category of graded abelian groups. If C is also an R module,
H(K,L; C) is a graded R module. Similar remarks apply to the ordered chain
complex i1(K)/ i1(L).
3
EXAMPLE
If (X,A) is a topological pair and C is an abelian group, the
singular homology group of (X,A) with coefficients C, denoted by H(X,A; C),
is defined to be the graded homology group of [i1(X)/ i1(A)] ® C (which
is augmented over C). It is a covariant functor of two arguments from the
category of topological pairs and the category of abelian groups to the category of graded abelian groups. If C is an R module, H(X,A; C) is a graded
R module.
Because the ring R is commutative, there is a canonical isomorphism
C ® C' ;::::; C' ® C for R modules C and C'. Therefore, if C is a chain complex over R, C ® C is canonically isomorphic to C ® C. Hence no new
homology modules are obtained from C ® C.
We recall some general properties of tensor products which will be
important in the next section.
4
LEMMA
The tensor product of two epimorphisms is an epimorphism.
Let ex: A ~ A" and {3: B ~ B" be epimorphisms. A" ® B" is generated by elements of the form a" ® b", where a" E A" and b" E B". Since
ex and {3 are epimorphisms, A" ® B" is generated by elements of the form
PROOF
SEC.
1
215
HOMOLOGY WITH COEFFICIENTS
a(a) ® f3(b), where a E A and b E B. Since (a ® f3)(a ® b) = a(a) ® f3(b),
A" ® B" is generated by (a ® f3)(A ® B), showing that a ® 13 is an
epimorphism. •
In general, it is not true that the tensor product of two monomorphisms
is a monomorphism (see example 7 below). The following lemma shows that
something can be said about the kernel of a ® 13 when a and 13 are epimorphisms.
:;
LEMMA
If a and 13 are epimorphisms, the kernel of a ®
by elements of the form a ® b, where a E ker a or b E ker 13.
13 is
generated
PROOF
Let a: A ~ A" and 13: B ~ B" be epimorphisms and let D be the
sub module of A ® B generated by elements of the form a ® b, where
a E ker a or b E ker 13. Let p: A ® B ~ (A ® B)/D be the projection. There
is a well-defined bilinear map
A" X B"
~
(A ® B)/D
sending (a",b") to p(a ® b), where a E A and b E B are chosen so that
a(a) = a" and f3(b) = b". This bilinear map corresponds to a homomorphism
1j;: A" ® B"
~
(A ® B)/D
such that 1j;(a" ® b") = p(a ® b), where a(a)
obvious that p equals the composite
A ®B
This shows that ker (a ®
that ker (a ® 13) = D. •
6
COROLLARY
a ® (3)
13)
= a" and f3(b) = b". It is then
A" ® B" ~ (A ® B)/D
C D. The reverse inclusion is evident, showing
Given an exact sequence
A'~A~A"~O
and given a module B, there is an exact sequence
A' ® B
~
A ® B ~ A" ® B
~
0
PROOF
It follows from lemma 4 that A ® B ~ A" ® B is an epimorphism,
so the sequence is exact at A" ® B. If A. c A is the image of A' ~ A, then,
by lemma 4, A' ® B ~ A. ® B is an epimorphism. Because A. is also the
kernel of A ~ A", it follows from lemma 5 that the kernel of A ® B ~ A" ® B
is the image of A. ® B ~ A ® B. Therefore the sequence is exact at
A ® B. •
If the original sequence is assumed to be a short exact sequence, it need
not be true that the tensor-product sequence is a short exact sequence. We
present an example to illustrate this.
7
EXAMPLE
Over Z, consider the short exact sequence
O~ Z~
zL Z2~O
216
PRODUCTS
CHAP.
5
where 0:(1) = 2 and f3(1) is a generator I of Z2. The tensor product of this sequence with Z2 is not a short exact sequence because 0: ® 1; Z ® Z2 ~ Z ® Z2
is not a monomorphism [Z ® Z2 ;::::: Z2 =!= 0, but (0: ® 1)(1 ® I) = 2 ® I =
1 ® 2· i
0].
=
8
THEOREM
The tensor-product functor commutes with direct sums.
Assume A = EEl Aj and consider the bilinear map A X B ~ EEl (Aj ® B)
sending (~ aj, b) to ~ (aj ® b) and the homomorphisms Aj ® B ~ A ® B
for all i. By the characteristic properties of tensor product and direct sum,
there are commutative triangles
PROOF
AXB
L
Aj ® B
'\;
A ® B~
lL"
EEl (Aj ® B)
L
A ® B ~ EEl (Aj ® B)
Clearly, the maps <p and 1/1 are inverses, showing that A ® B ;:::::
If, also, B = EEl B k , then similarly,
A ® B;::::: EElAj ® Bk
EEl (Aj
® B).
•
j,k
9
THEOREM
The tensor-product functor commutes with direct limits.
=
Let A
lim~ {A"'} and consider the bilinear map A X B ~ lim~
{A'" ® B} sending ({a},b) to {a ® b} for a E A'" and the homomorphisms
A'" ® B ~ A ® B for all fX. By the characteristic properties of tensor product
PROOF
and direct limit, there are commutative triangles
A'" ® B
A X B
L
'\;
i/
L
A ® B.!4lim~ {A'" ® B} A ® B ~lim~ {A'" ® B}
Clearly, <p and 1/1 are inverses, showing that A ® B ;::::: lim~ {A'" ® B}. If, also,
B = lim~ {BIl}, then similarly, A ® B;::::: lim~ {A'" ® BIl}. •
We now consider a special class of short exact sequences. These sequences
have the property that their tensor product with any module is again exact.
A short exact sequence
o ~ A' ~ A 4
A" ~ 0
is said to be split if f3 has a right inverse (that is, if there exists a homomorphism f3'; A" ~ A such that f3 f3' = l A ,,). We also say that the sequence
splits.
0
10 EXAMPLE Any short exact sequence 0 ~ A' ~ A 14 A" ~ 0 with A"
free is split. To see this, let {an be a basis for A" and for each i choose aj E A
so that f3(aj) = aj'. Let f3'; A" ~ A be the homomorphism such that f3'(aj') = aj
for all i. Then f3' is a right inverse of f3.
SEC.
I
11
LEMMA
217
HOMOLOGY WITH COEFFICIENTS
Given a short exact sequence
o ---? A' ~ A ~ A" ---? 0
define A' -4 A' E8 A" 4 A" by i(a')
the following are equivalent:
(a) The sequence is split.
(b) There is a commutative diagram
A
~
A'
= (a',O)
y't
J4
i'" A' E8 A"
-:
and p(a',a")
= a".
Then
A"
(c) There is a commutative diagram
A
A'
>0
l
(d)
0'
Y~
A' E8 A"
.?I
A"
P
has a left inverse.
PROOF If [3'; A" ---? A is a right inverse of [3, let y'; A' E8 A" ---? A be defined
by y'(a',a") = O'(a') + [3'(a"). Then y' has the desired properties. Conversely,
given y', define [3'; A" ---? A by [3'(a") = y'(O,a"). Then [3' is a right inverse
of [3, so the sequence is split. Therefore (a) is equivalent to (b). A similar argument shows that (c) is equivalent to (d). It follows from the five lemma that
in the diagram of (b) [or (c)], y' [or y] is necessarily an isomorphism. Therefore (b) is equivalent to (c) with y' equal to y-1. •
12
COROLLARY
Given a split short exact sequence
o ---? A' ~ A ---? A" ---? 0
and given a module B, the sequence
o ---? A' ® B ~ A
® B ---? A" ® B ---? 0
is a split short exact sequence.
PROOF
By corollary 6 and lemma 11 we need only show that 0' Q9 1 has a
left inverse. By lemma 11, 0' has a left inverse a'. Then a' ® 1 is a left
inverse of 0' ® 1. •
In case 0 ---? C' ---? G ---? Gil ---? 0 is a split short exact sequence of chain
complexes, it follows from corollary 12 that for any module G the sequence
o ---?
C' ® G ---? G ® G ---? Gil ® G ---? 0
is a short exact sequence of chain complexes. This short exact sequence gives
rise to an exact homology sequence, and we obtain the next result.
218
13
PRODUCTS
THEOREM
CHAP.
5
Given a split short exact sequence of chain complexes
o~C'~c~c"~o
and given a module G, there is a functorial exact homology sequence
This implies the exactness of the singular homology sequence (and
reduced homology sequence) of a pair with arbitrary coefficients. Similarly,
there is an exact sequence of a triple with arbitrary coefficients. All these
sequences (except the reduced sequence of a pair) are consequences of the
exactness of the relative Mayer-Vietoris sequence, which we now establish. If
{(Xl,Al), (X 2,A2)} is an excisive couple of pairs in a topological space, the
short exact sequence of singular chain complexes
o ~ .:l(Xl n
X2 )/.:l(Al n Az) ~
.:l(Xl)/.:l(A l) EB .:l(Xz)/.:l(Az) ~ [.:l(Xl)
+ .:l(Xz)]/[.:l(Al) + .:l(Az)]
is split [by example 10, because [.:l(Xl) + .:l(Xz)]/[.:l(Al)
abelian group]. Therefore we obtain the following result.
+ .:l(A2)]
~
0
is a free
14 COROLLARY If {(XbAl)' (X2,A 2)} is an excisive couple of pairs in a space
and G is an R module, there is an exact relative Mayer- Vietoris sequence of
{(XbAl)' (X2,A2)} with coefficients G. •
If G is fixed, the singular homology of (X,A) with coefficients G satisfies
all the axioms of homology theory except the dimension axiom (all of them
are easily seen to hold except exactness, which follows from corollary 14). If
P is a one-point space, there is a functorial isomorphism Ho(P;G) ;::::; G. This
leads to the following definition.
Let G be an R module. A homology theory with coefficients G consists
of a covariant functor H from the category of topological pairs to graded
R modules and a natural transformation a: H(X,A) ~ H(A) of degree -1 satisfying the homotopy, exactness, and excision axioms, and satisfying the
following form of the dimension axiom: On the category of one-point spaces
there is a natural equivalence of H with the constant functor which assigns
to every one-point space the graded module which is trivial for degrees other
than 0 and equal to G in degree O. A homology theory with coefficients Z is
called an integral homology theory. An integral homology theory is the same
as a homology theory as defined in Sec. 4.8.
Singular homology with coefficients G is an example of a homology
theory with coefficients G. The uniqueness theorem 4.8.10 is valid for
homology theories with coefficients.
In the next section we shall show how the singular homology modules
with coefficients are determined by the integral singular homology groups.
SEC.
2
2
THE UNIVERSAL-COEFFICIENT THEOREM FOR HOMOLOGY
219
THE UNIVERSAL-COEFFICIENT THEOREM
FOR HOMOLOGY
In order to express H( C; G) in terms of H( C) and G, it is necessary to introduce certain functors of modules that are associated to the tensor-product
functor. This section contains a definition of these functors, and a study of
them in the special case of a principal ideal domain. This leads to the
universal-coefficient theorem. In the next section these new functors will
enter in a description of the homology of a product space.
Let A be an R module. A resolution of A (over R) is an exact sequence
... ~ Cn ~
...
~ C1 ~ Co -4 A ~ 0
If, in addition, each Cq is a free R module, the resolution is said to be free.
Thus a resolution of A consists of a chain complex C = {Cq,Oq} over R which
is augmented over A and is such that C is acyclic. The resolution is free
if and only if the chain complex C is free.
Any R module A has free resolutions. In fact, given an R module B, let
F(B) be the free R module generated by the elements of B and let F(B) ~ B
be the canonical map. The canonical free resolution of A is the following resolution (defined inductively):
... ~ F(ker Oq) ~ F(ker Oq-l) ~ ... ~ F(ker e) ~ F(A) -4 A ~ 0
The method of acyclic models applies to chain complexes over Rand,
when applied to a category consisting of a single object and single morphism,
implies the following result.
I
THEOREM
Let C be a free nonnegative chain complex augmented over
A and let C' be a resolution of A'. Any homomorphism q;: A ~ A' extends to
a chain map
... ~ C~+l ~ C~ ~ ... ~ Co
-4 A' ~ 0
preserving augmentations, and two such chain maps are chain homotopic.
Specializing to the case q;
= lA: A
•
C A, we obtain the next result.
2 COROLLARY If C and C' are free resolutions of A, then C and C' are
canonically chain-equivalent chain complexes. •
For modules A and B and a free resolution C of A, it follows from corollary 2 that the graded module H(C;B) depends only on A and B. Let C be the
canonical free resolution of A. For q 2:: 0 we define the qth torsion product
Torq (A,B) = Hq(C;B). It is a covariant functor of A and of B. From the short
220
PRODUCTS
CHAP.
5
exact sequence
o ----.
31C1 ----. Co
~
A ----. 0
it follows from corollary 5.1.6 that there is an exact sequence
31 C1 ® B ----. Co ® B ~ A ® B ----.0
By definition, Toro (A,B) is the zeroth homology module of the chain complex
... ----. C2 ® B ----. C1 ® B ~ Co ® B ----. 0
= (Co ® B)/im (3 1 ® 1). By the above exact sequence,
im (3 1 ® 1) = im (3 1 e l ® B ----. Co ® B) = ker (e ® 1)
Hence Toro (A,B)
Therefore
Toro (A,B)
= (Co
® B)/ker (e ® 1) ;::::; A ® B
and so Toro (A,B) is naturally equivalent to A ® B.
All the previous remarks are valid for any commutative ring with a unit.
For the remainder of this section we specialize to the case where R is a principal ideal domain. Over a principal ideal domain any submodule of a free
module is free. Therefore any module A has a short free resolution of the form
o ----. Cl ----. Co ----. A ----. 0
and C1 = ker [F(A) ----. AJ).
Such a short free resolu(simply let Co = F(A)
tion of A is the same as a free presentation of A. Because there exist short
free resolutions, Tor q (A,B) = 0 if q
1. We define the torsion product
A * B to equal Tor1 (A,B). It is characterized by the property that, given any
free presentation of A,
>
there is an exact sequence
o ----. A * B ----.
C1 ® B ----. Co ® B ----. A ® B ----. 0
In fact, A * B ;::::; H 1 (C ® B) = ker (C 1 ® B ----. Co ® B), since C 2 ® B = O.
The torsion product is a covariant functor of each of its arguments.
Because the tensor product commutes with direct sums and direct limits (by
theorems 5.1.8 and 5.1.9) and the direct limit of exact sequences is exact (by
theorem 4.5.7), the torsion product also commutes with direct sums and
direct limits. Its name derives from the fact that it depends only on the
torsion submodules of A and B (see corollary 11 below).
3
EXAMPLE
If A is free, it has the free presentation
0----. 0----. A----.A----. 0
from which we see that A
*B =
0 for any B.
SEC.
2
221
THE UNIVERSAL· COEFFICIENT THEOREM FOR HOMOLOGY
4
EXAMPLE If A is the cyclic R module whose annihilating ideal is generated
by an element v E R, then A :::::: R/vR and there is a free presentation of A
O~R~R~A~O
in which a( v') = VV ' for Vi E R. For any module B there is an isomorphism
R ® B :::::: B sending 1 ® b to b. Under this isomorphism, the map
a ® 1: R ® B ~ R ® B corresponds to a' : B ~ B, where a'(b) = vb for
b E B. Therefore ker a ' is the submodule of B annihilated by v, and so
(R/vR)
* B:::::: {b
= O}
A * B for
E B I vb
The above examples suffice to compute
a finitely generated
module A (because of the structure theorem 4.14 in the Introduction). This
theoretically determines A * B for arbitrary A, because any A is the direct
limit of its finitely generated submodules (see theorem 4.2 in the Introduction)
and the torsion product commutes with direct limits.
:.
LEMMA
* B = O.
If A or B is torsion free, then A
Because the torsion product commutes with direct limits, it suffices
to consider the case where A and B are finitely generated, in which case being
torsion free is equivalent to being free. If A is free, the result follows from
example 3. If B is free and finitely generated, it is isomorphic to a direct sum
of n copies of R. If
PROOF
o ~ C1 ~
Co
~
A
~
0
is a free presentation of A, then C 1 ® B ~ Co ® B ~ A ® B ~ 0 is isomorphic
to a direct sum of n copies of the sequence C 1 ® R ~ Co ® R ~ A ® R --7 O.
Since C 1 ® R ~ Co ® R is a monomorphism, so is C1 ® B --7 Co ® B, and
A * B = O. •
It follows that if R is a field, then A * B = 0 for all modules A and B.
The following result is proved similarly by proving it first for finitely generated
modules (where being torsion free is equivalent to being free) and taking
direct limits to obtain the result for arbitrary modules.
6
LEMMA
Given a short exact sequence of modules
o ~ A' ~ A ~ A" ~ 0
and given a module B, if A" or B is torsion free, there is a short exact
sequence
o ~ A' ® B ~ A
®B
~
A" ® B
~
0
PROOF
As remarked above, it suffices to prove the result if A" or B is free and
finitely generated. If A" is free, the original sequence splits, by example 5.1.lO,
and the result follows from corollary 5.1.12. If B is free and finitely generated,
the map A' ® B ~ A ® B is a finite direct sum of copies of A' ® R ~ A ® R,
222
PRODUCTS
CHAP.
and hence a monomorphism. The result follows from this and corollary 5.1.6.
5
•
We use this result to obtain an exact sequence of homology corresponding to a short exact sequence of coefficient modules.
7
THEOREM
On the product category of torsion-free chain complexes C
and short exact sequences of modules
o~
G' ~ G ~ Gil ~ 0
there is a natural connecting homomorphism
/3:
H(C;G")
~
H(C;G')
of degree - 1 and a functorial exact sequence
... ~ Hq(C;G') ~ Hq(C;G) ~ Hq(C;G") ~ Hq_l(C;G') ~
PROOF
By lemma 6, there is a short exact sequence of chain complexes
o~ C® G'~C® G ~ C® Gil ~ 0
Since this is functorial in C and in the exact coefficient sequence, the result
follows from theorem 4 ..'5.4. •
The connecting homomorphism /3 occurring in theorem 7 is called the
Bockstein homology homomorphism corresponding to the coefficient sequence
o~
G' ~ G ~ G' ~ O. Theorem 7 remains valid over an arbitrary commutative ring R with a unit if C is assumed to be a free chain complex over R.
Let C be a chain complex over R and let G be an R module. Recall the
homomorphism p,: H(C) ® G ~ H(C;G) defined in the last section. This
homomorphism enters in the following universal-coefficient theorem for
homology.
8
THEOREM
Let C be a free chain complex and let G be a module. There
is a functorial short exact sequence
o ~ Hq(C)
®G
~ Hq(C
® G)
~ Hq_ 1 (C)
*G~ 0
and this sequence is split.
Let Z be the subcomplex of C defined by Zq = Zq( C) with trivial
boundary operator and let B be the complex defined by Bq = Bq- 1 (C) with
trivial boundary operator. Both Band Z are free chain complexes and there
is a short exact sequence
PROOF
O~Z~CJ4B~O
where lXq(Z) = Z for Z E Zq and .8q(c) = OqC for c E Cq. Since B is a free complex; this short exact sequence is split. By theorem 5.1.13, there is an exact
sequence
SEC.
2
223
THE UNIVERSAL-COEFFICIENT THEOREM FOR HOMOLOGY
where a* {b} = {D'q-.!la qaq- 1 b} = {D'qll(b)} for b E Bq~l' Since Z and B have
trivial boundary operators, so do Z ® G and B ® G. Therefore Hq(Z;G) =
Zq ® G and Hq(B;G) = Bq ® G = Bq~l(C) ® G, and the above exact sequence becomes
... ~ Bq(C) ® G
Yq
® 1)
Zq(C) ® G ~ Hq(C;G) ~
Bq~l(C) ® G
Yq-l
® 1)
Zq~l(C) ® G ~
where Yq: Bq( C) C Zq( C). From the exactness of this sequence we obtain a
short exact sequence
o ~ coker (Yq
® 1)
HiC;G)
~
~
ker (Yq~l ® 1) ~ 0
and it only remains to interpret the modules on either side of Hq(C;G).
Since Zq( C) is free, the short exact sequence
o ~ Bq(C) ~ Zq(C) ~ Hq(C) ~ 0
is a free presentation of Hq(C). By the characteristic property of the torsion
product, there is an exact sequence
o~
Hq(C)
* G ~ Bq(C) ® G
Yq
®
1)
Zq(C) ® G ~ Hq(C) ® G ~ 0
Therefore coker (Yq ® 1) ::::: Hq(C) ® G and ker (Yq ® 1) ::::: Hq(C) * G. Substituting these into the short exact sequence above yields the short exact
sequence
o ~ Hq(C)
®G
~
Hq(C;G)
~ Hq~l(C)
*G~0
It is easily verified by checking the definitions that the homomorphism
Hq(C) ® G ~ Hq(C;G) is equal to fL.
If T: C ~ C' is a chain map, T defines a commutative diagram
O~Z~C-4B~O
o~
Z' ~ C' 4 B' ~ 0
from which we obtain the commutative diagram
Therefore the short exact sequence for Hq(C;G) is functorial.
We now prove that the short exact sequence is split (but is not functorially
split). Because Bq~l(C) is free and aqCq = Bq~l(C), there exist homomorphisms
hq: Bq~l(C) ~ Cq such that aqhq = 1. Then
hq ® 1:
Bq~l(C)
® G
~
Cq ® G
224
PRODUCTS
CHAP.
5
maps the kernel of Yq-l ® 1 into cycles of Cq ® G and induces a homomorphism Hq-1(C) * G ~ Hq(C;G) which is a right inverse of the homomorphism
Hq(C;G) ~ Hq-1(C) * G of the short exact sequence in the theorem. •
We can use this result to establish some properties of the torsion product,
beginning with the following six-term exact sequence connecting the tensor
and torsion products.
COROLLARY
Let 0 ~ B' -4 B -4 B" ~ 0 be a short exact sequence of
modules and let A be a module. There is an exact sequence
9
o ~ A * B' ~
A * B ~ A ... B" ~
A ® B'
1 ® a')
A ®B
1 ® {J')
A ® B"
~ 0
PROOF Let 0 ~ C 1 ~ Co ~ A ~ 0 be a free presentation of A and let C be
the corresponding free chain complex obtained by adding trivial groups on
both sides. Since C is free, it follows from lemma 6 that there is a short exact
sequence of chain complexes
o ~ C ® B'
1 ® a')
C® B
=
1 ® {J')
C ® B" ~ 0
=
Because Hq(C)
0 if q =t= 0 and Ho(C)
A, the homology sequence of the
above short exact sequence of chain complexes (interpreted by means of
theorem 8) gives the desired exact sequence. •
This yields the commutativity of the torsion product.
10
COROLLARY
There is a functorial isomorphism
A*B;::::;B*A
PROOF
Let 0 ~ C 1 ~ Co ~ B ~ 0 be a free presentation of B. By corollary 9, there is an exact sequence
o ~ A ... C1 ~ A * Co ~ A * B ~ A ® C1 ~ A ® Co ~ A ® B ~ 0
Since Co is free, it follows from lemma 5 that A * Co = 0, and there is
an
exact sequence
o ~ A * B ~ A ® C1 ~ A
® Co ~ A ® B ~ 0
By the characteristic property of B ... A, there is an exact sequence
o ~ B * A ~ C1 ® A ~ Co ® A ~ B ® A ~ 0
functorial isomorphism A * B ;::::; B * A then results by chasing
The
commutative diagram
*B
o~
A
o~
B ... A
~
A ® C1
~ C1
®A
~
A ® Co
~
A ®B
~ 0
®A
~
B®A
~ 0
~ Co
in the
in which the vertical maps are the functorial isomorphisms expressing the
SEC.
2
225
THE UNIVERSAL-COEFFICIENT THEOREM FOR HOMOLOGY
commutativity of the tensor product.
-
We can now show that the torsion product of A and B depends only on
the torsion submodules of A and B.
II
COROLLARY
i: Tor B C
Let A and B be modules and let i: Tor A C A and
i: Tor A * Tor B ;::::; A * B.
B. Then i *
There is a short exact sequence
PROOF
o ~ Tor B -4 B ~ BITor B ~ 0
where BITor B is without torsion. By lemma 5, A * (BITor B) = 0, and, by
corollary 9, 1 * i: A * Tor B ;::::; A * B. By a similar argument, there is an iso-
morphism i * 1: Tor A * Tor B;::::; A * Tor B, and the composite of these gives
the result. -
We use these results to extend the universal-coefficient theorem. Given a
chain complex Cover R, a free approximation of C is a chain map 1": G ~ C
such that
(a) G is a free chain complex over R.
(b) 1" is an epimorphism.
(c) 1" induces an isomorphism 1"*: H(G) ;::::; H(C).
Any chain complex C has a free approximation, uniquely determined up to homotopy equivalence.
12
LEMMA
PROOF
For each q ~ 0 choose a homomorphism aq: Fq ~ Zq(C) such that Fq
is a free R module and aq is an epimorphism. Let F~ = aq- 1(Bq(C)) and
choose a homomorphism /3q: F~ ~ Cq+1 such that Oq+1/3q = aq I F~ [such a
homomorphism exists because F~ is free and Oq+1: Cq+1 ~ Bq(C) is an epimorphism]. Define Gq Fq EEl F~_1 and define homomorphisms
=
1"q: Gq ~ Cq
by
(}q(a,b) = (b,O)
by
1"q(a,b) = aq(a)
+ /3q-1(b)
=
Then G = {Gq,() q} is a free chain complex and 1"
{1"q} is a chain map from
C to C. 1" is epimorphic because 1"q(Gq) ::::l ker Oq and Oq1"q(Gq) ::::l im Oq. Since
Zq(G)
Fq, Bq(C) = F~, and 1"q(Zq(C))
aq(Fq), it follows that
=
=
1"* : Zq(C)IBq(G) ;::::; Zq(C)IBq(C)
Therefore 1": G ~ C is a free approximation of C. The uniqueness will follow
from lemma 13 below. If 1":
G ~ C is a free approximation of C, there is a subcomplex
C = {Cq = ker 1"q:
Gq ~ Cq} of G and a short exact sequence of chain
complexes
O~C~G~C~O
Because 1"*: H( C)
;::::; H( C), it follows from the exactness of the homology
226
PRODUCTS
CHAP.
5
sequence of the above short exact sequence that C is acyclic (see corollary
4.5.5a). Since C is a free chain complex (because it is a subcomplex of a free
chain complex), it follows from theorem 4.2.5 that C is contractible. We use
this in the following lemma.
13 LEMMA Given a free approximation 'T: G ~ C of C and given a free
chain complex C' and a chain map 'T': C' ~ C, there exist chain maps
i: C' ~ G such that 'T i = 'T', and any two are chain homotopic.
0
PROOF
As above, there is a short exact sequence of chain complexes
o~c-4G~C~O
wh<=:re C is chain contractible. Let D = {Dq: Cq ~ CHI} be a contraction
of t. Because C~ is free and 'Tq: Gq ~ Cq is an epimorphism, there is a homomorphism <pq: C~ ~ Gq such that 'T q<pq = 'T q. Then
hq = aq<pq - <pq-1a~: C~ ~ Cq_1
and
'Tq_1hq = 'Tq_1aq<Pq - 'Tq-1<pq_1a~
= aq'Tq -
'T~_la~
= aq'Tq<pq -
'Tq_1aq
= 0
Therefore h q is a homomorphism of C~ into i(Cq _ 1 ). It follows immediately
that i = {iq = <pq - iDq_1i- 1hq} is a chain map i: C' ~ C such that 'Ti = 'T'.
If i, i': C' ~ C are chain maps such that 'Ti = 'Ti', then i - i' = il/; for
some chain map 1/;: C' ~ C. It follows immediately that
D
= {Dq = iDql/;q: C~ ~ Gq+1}
is a chain homotopy from i to i'.
•
If C is a chain complex over Rand G is an R module, let C * G be the
chain complex C * G = {Cq * G, aq * I}. We use this in the general universalcoefficient theorem.
14 THEOREM On the subcategory of the product category of chain complexes
C and modules G such that C * G is acyclic there is a functorial short exact
sequence
o ~ Hq(C)
® G -4 Hq(C;G) ~ Hq- 1(C)
*G~ 0
and this sequence is split.
PROOF
Let 'T: C ~ C be a free approximation to C (which exists, by lemma 12),
and consider the short exact sequence
O~C-4C-4C~O
in which Cis acyclic. By the characteristic property of the torsion product,
there is an exact sequence of chain complexes
o~ C * G ~ C ® G
i ® 1)
C®
G ~ C® G~0
SEC.
3
227
THE KUNNETH FORMULA
from which we get two short exact sequences
o ~ C * G ~ C ® G ~ im (i ® 1) ~ 0
o ~ im (i ® 1) C C ® G ~ C ® G ~ 0
In the first of these C * G is acyclic by hypothesis, and C ® G is also acyclic
(by theorem 8, because C is free and acyclic). From corollary 4.5.5c it follows
that im (i ® 1) is also acyclic. In the second exact homology sequence this
implies that
(7' ® 1)*: H(C ® G) :::::; H(C ® G)
The desired short exact sequence is now defined, so that the following diagram
is commutative
T.®lt
o~
t(T®l).
tT.*l
Hq(C) ® G ~ Hq(C ® G) ~ Hq- 1(C)
* G~ 0
where the upper row is the short exact sequence of theorem 8 (it is possible
to define the unlabeled homomorphism in the bottom sequence to make the
diagram commutative because all the vertical homomorphisms are isomorphisms). Then the bottom sequence splits because the top one does.
The functorial property of the resulting short exact sequence (and the
fact that it is independent of the particular free approximation of C) follows
from lemma 13. It should be emphasized again that the sequence of theorem 14 does not
split functorially.
l:t COROLLARY Let 7': C ~ C' be a chain map between torsion-free chain
complexes such that 7'*: H(C) :::::; H(C'). For any R module G, 7' induces an
isomorphism
7'*: H(C;G) :::::; H(C';G)
This follows from the functorial exact sequence of theorem 14 and
the five lemma. -
PROOF
In corollary 15, if C and C' are free, then 7' is a chain equivalence (by
theorem 4.6.10), and so is 7' ® 1: C ® G ~ C' ® G. Therefore 7'*: H(C;G) :::::;
H(C';G). Corollary 15 shows that the latter fact remains true (even though 7'
need not be a chain equivalence) for chain complexes without torsion.
3
THE KUNNETH FORMULA
In this section we extend the universal-coefficient theorem to obtain the
Kiinneth formula expressing the homology of the tensor product of two chain
228
PRODUCTS
CHAP.
5
complexes in terms of the homology of the factors. This is given geometric
content by the Eilenberg-Zilber theorem asserting that the singular complex
of a product space is chain equivalent to the tensor product of the singular
complexes of the factor spaces.
If C and C' are graded R modules, their tensor product C ® C' is the
graded module {(C ® C')q}, where (C ® C')q = (Bi+j=q Ci ® c;.. Similarly,
their torsion product C * C' is the graded module {( C * C')q = (B;+j=q Ci * C;}.
If C and C' are chain complexes, their tensor product [and torsion product]
are chain complexes {( C ® C')q, G~} [and {( C * C')q, Gq}], where if c E Ci
and c' E Cj with i + i = q, then
a~'(c
® c')
= GiC ®
C'
+ (-l)ic
® ajc'
[and all I Ci * Cj = ai * I + (-I) i1 * aJ]. It is easy to verify that C @ C' [and
C * C'] really are chain complexes. We shall see later that the tensor product
arises naturally in studying product spaces.
If C' is a chain complex such that C~ = 0 for q =1= 0, then C ® C' is the
same as the tensor product of C with the module Co. Therefore the tensor
product of two chain complexes is a natural generalization of the tensor
product of a chain complex with a module. It is reasonable to expect that
there is a generalization of the universal-coefficient theorem to express the
homology of C ® C' in terms of the homology of C and of C'.
We define a functorial homomorphism of degree 0
W H(C) ® H(C')
---7
H(C ® C')
If c E Zi( C) and c' E Zj( C'), then c ® c' E Zi+j( C ® C'), and if c or c' is
a boundary, so is c ® c'. Therefore there is a well-defined homomorphism fJ,
such that
fJ,( {c}
® {c'})
= {c ® c'}
This homomorphism enters in the following Kiinneth formula.
I
LEMMA
Let C and C' be chain complexes, with C' free. Then there is a
functorial short exact sequence
0---7 [H(C) ® H(C')]q ~ Hq(C ® C')
---7
[H(C)
* H(C')]q~l ---7 0
If C is also free, this short exact sequence is split.
As in the proof of theorem 5.2.8, let Z' and B' be the complexes
(with trivial boundary operators) defined by Z~ = Zq(C') and B~ = Bq~l(C').
There is a short exact sequence of chain complexes
PROOF
o ---7 Z' ---7
C'
---7
B'
---7
0
Since C' is free, so is B', and there is a short exact sequence
o ---7
C®
z' ---7
C ® C'
---7
C ® B'
from which we obtain an exact homology sequence
---7
0
SEC.
3
229
THE KUNNETH FORMULA
... ~ Hq(C ® Z') ~ Hq(C ® C') ~ Hq(C ® B') ~ Hq_1(C ® Z') ~
Note that C ® z' = ffi Ci, where (C i)q = Cq- i ® Zi(C') and C ® B' = ffi Ci,
where (C j)q = Cq_j ® Bi-l(C')' Since Zi(C') and Bj(C') are free, it follows from
theorem 5.2.14 that
Hq(C ® Z')
= ffi Hq(Ci) = ffi
i+j=q
j
Hq(C ® B') =
ffi Hq(Ci)
CB
=
i
Hi(C) ® Zj(C')
Hi(C) ® Bj(C')
i+i=q-l
a* corresponds under these isomorphisms to the homomorphism
(_l)i ® Yj, where Yi is the inclusion map yr Bj(C') C Zi(C')' Therefore there
is a short exact sequence
The map
[coker (-l)i ® Yi] ~ Hq(C ® C') ~ CB [ker (-l)i ® Yi] ~ 0
i+i=q
i+i=q-l
To compute the two sides of this sequence, consider the short exact
sequence
o ~ ffi
Because Zi(C') is free, it follows from corollary 5.2.9 that there is an exact
sequence
o ~ Hi(C) * Hi(C') ~
Hi(C) ® Bj(C')
(-I)l ®
yj)
Hi(C) ® Zi(C')
~ Hi(C) ® Hi(C')
~ 0
Hence
ffi
[coker (-l)i ® Yi]
= ffi
ffi
[ker (-l)i ® Yi]
= ffi
i+i=q
i+i=q
Hi(C) ® Hi(C')
and
i+i=q-l
* HlC')
Hi(C)
i+i=q-l
Substituting these into the short exact sequence above gives a short exact
sequence
o~
[H(C) ® H(C')]q
~
Hq(C ® C')
~
[H(C)
* H(C')]q_l ~ 0
We now verify that v is the map /-t. Given {c} E H( C) and {c'} E H( C'),
then {c} ® c' E H(C) ® Z(C') and {c} ® c'
{c ® C'}CC8Z(C')' Therefore
v( {c} ® {c'}) = {c ® C'}C0C' = /-t( {c} ® {c'}). Thus we have the desired
short exact sequence, and it is clearly functorial.
Assuming that C is also free, we can show that the sequence splits. By
lemma 5.1.11, it suffices to find a left inverse for /-t. Because C and C' are free,
so are B( C) and B( C'), and there are homomorphisms p: C ~ Z( C) and
p': C' ~ Z(C') such that p(c) = c for c E Z(C) and p'(c') = c' for c' E Z(C').
Then
=
p ® p': C ® C'
~
Z(C) ® Z(C')
230
PRODUCTS
CHAP.
5
maps B(C ® C')(which is contained in the union of im [B(C) ® C' --? C ® C']
and im [C ® B( C') --? C ® C']) into the union of im [B( C) ® Z( C') --?
Z(C) ® Z(C')] and im [Z(C) ® B(C') --? Z(C) ® Z(C')]. Therefore the composite
Z(C ® C')
c
C ® C'
p
® p')
Z(C) ® Z(C')
H(C) ® H(C')
--?
maps B(C ® C') into 0 and induces a homomorphism
H(C ® C')
which is a left inverse of fL.
--?
H(C) ® H(C')
-
A similar functorial short exact sequence can be defined if C (instead of C')
is assumed free. The two short exact sequences are identical when C and C'
are both free. 1
2
COROLLARY
If C' is a free chain complex and either C or C' is acyclic,
then C ® C' is acyclic. -
We now extend lemma 1 to obtain the following general Kiinneth
formula.
3
THEOREM
On the subcategory of the product category of chain complexes
C and C' such that C * C' is acyclic there is a functorial short exact sequence
o
--?
[H(C) ® H(C')]q -4 Hq(C ® C')
--?
[H(C)
* H(C')]q-l
--?
0
and this sequence is split.
e
PROOF
Let 'T:
--? C and 'T': C'
a short exact sequence
--?
C' be free approximations. Then there is
o --? C' ~ C' 2.,.
C'
--?
0
where C' is acyclic. Since C' is free, the six-term exact sequence becomes the
exact sequence
o --? C * C' --?
C®
C' --?
C ® G' ~ C ® C'
--?
0
Since C * C' is acyclic by hypothesis and C ® C' is acyclic by corollary 2, it
follows (as in the proof of theorem 5.2.14) that there is an isomorphism
(1 ® 'T')*: H(C ® G') :::::: H(C ® C')
There is also a short exact sequence
o--?C~e~c--?O
where C is acyclic. Since C' is free, there is a short exact sequence
o --?
By corollary 2,
C ® C'
C ® G'
--?
e ® G' $
C ® G'
--?
0
is acyclic, and we have an isomorphism
('T ® 1)*: H(G ® e') :::::: H(C ® G')
This is proved in G. M. Kelley, Observations on the Kiinneth theorem, Proceedings of the
Cambridge Philosophical Society, vol. 59, pp. 575-587, 1963.
1
SEC.
3
231
THE KUNNETH FORMULA
Hence the composite ('7' ® '7")* = (1 ® '7")* ('7' ® 1)* is an isomorphism of
H(G ® C') onto H(C ® C'). The desired short exact sequence is now defined
so that the following diagram is commutative
o~
H(G) ® H(G') -4 H(G ® G') ~ H(C)
o~
H(C) ® H(C') -4 H(C ® C') ~ H(C)
* H(C')
~ 0
* H(C')
~ 0
1T* * T*
where the top row is the short exact sequence of lemma 1 (it is possible to
define the homomorphisms in the bottom row to make the diagram commutative because the vertical homomorphisms are isomorphisms). The bottom
sequence splits because the top one does.
The functorial property of the sequence (and the fact that it is independent of the free approximations G and C') follow from the functorial property
of the sequence in lemma 1 and from lemma 5.2.13. •
If C and C' are chain complexes over Rand G and G' are R modules,
the composite
H(C ® G) ® H(C' ® G') -4 H[(C ® G) ® (C' ® G')] ~
H[(C ® C') ® (G ® G')]
[where the right-hand homomorphism is induced by the canonical isomorphism
(C ® G) ® (C' ® G') ::::: (C ® C') ® (G ® G')] is a functorial homomorphism
tt': H(C;G) ® H(C';G') ~ H(C ® C'; G ® G')
called the cross product. If z E H(C;G) and z' E H(C';G'), then
z X z' E H(C ® C'; G ® G')
denotes the image of z ® z' under this homomorphism [that is, z X z'
tt'(z ® z')].
=
4
COROLLARY
Given torsion-free chain complexes C and C' and modules
G and G' such that G G' = 0, there is a functorial short exact sequence
*
o~
[H(C;G) ® H(C';G')]q.4 Hq(C ® C'; G ® G') ~
[H(C;G)
* H(C',G')]q_l ~ 0
and this sequence is split.
PROOF This follows from theorem 3 once we verify that (C ® G) * (C' ® G')
is trivial. To show that (C ® G) * (C' ® G') = 0, let 0 ~ F' ~ F ~ G be a
free presentation of G. Because G * G' = 0, there is an exact sequence
o~ F
® G'
~
F ® G'
~
G ® G'
~
0
and since C and C' are without torsion, there is an exact sequence
o ~ (C ® F') ® (C' ® G') ~ (C ® F) ® (C' ® G')
~
(C ® G) ® (C' ® G') ~ 0
232
PRODUCTS
CHAP.
5
Because there is also a short exact sequence
O~C®F/~C®F~C®G~O
where C ® F is without torsion, it follows that (C ® G)
phic to the kernel of the homomorphism
* (C' ® G /) is isomor-
(C ® F) ® (C' ® G /) ~ (C ® F) ® (C' ® G /)
and hence is O.
-
The cross product has the following commutativity with connecting
homomorphisms.
:;
THEOREM
Let 0 ~ C ~ C ~ C ~ 0 be a split short exact sequence of
chain complexes and let z E H(C;G) and z' E H(C';G /). Then
a* (z
X Zl)
a*(z' X z)
PROOF
o~
o~
= a* z X z'
= (_l)degz'z'
X a*z
We have a commutative diagram of chain maps
C®G
~O
(C ® G) ® (C' ® G /) ~ (C ® G) ® (C' ® G /) ~ (C ® G) ® (C' ® G /) ~ 0
with exact rows, with the vertical maps defined by forming the tensor product
on the right with c' E Z(C' ® G /), where z' = {c / } [that is, r(c) = c ® c' for
c E C ® G]. Because c' is a cycle, each vertical map is a chain map. Because
the connecting homomorphism is functorial, we obtain a commutative diagram
H(C ® G) ~ H((C ® G) ® (C' ® G /))
a.t
a. t
-;;t H((C ®
C') ® (G ® G /))
tao
H(C ® G) ~ H((C ® G) ® (C' ® G /)) -;;! H((C ® C') ® (G ® G /))
in which each vertical map is a suitable connecting homomorphism. The top
row sends z into z X Zl, and the bottom row sends a* z into a* z X Z'. This
gives half the result. The second half follows by a similar argument, the only
difference being that the tensor product formed on the left with c' is not a
chain map but either commutes or anticommutes with the boundary operator,
depending on the degree of c' . This accounts for the presence of the factor
( _l)de g z' in the second equation. The following Eilenberg-Zilber theorem! is the link between the algebra
of tensor products and the geometry of product spaces.
6
THEOREM
On the category of ordered pairs of topological spaces X and
Y there is a natural chain equivalence of the functor ~(X X Y) with the
functor ~(X) ® ~(Y).
1 The theorem appears in S. Eilenberg and J. A. Zilber, On products of complexes, American
Journal of Mathematics, vol. 75, pp. 200-204, 1953.
SEC.
3
233
THE KiiNNETH FORMULA
We show that both functors are free and acyclic with models
Let dn E ~n(~n X ~n) be the singular simplex which is the
diagonal map ~n ~ ~n X ~n. If a: ~n ~ X X Y is any singular n-simplex,
then a is the composite
PROOF
{~p,M}p,q",o.
X ~n
~n ~ ~n
X X Y
a' X an)
where a' = PI a and a" = P2 a, and PI and P2 are the projections of
X X Y to X and Y, respectively. Conversely, given a': ~n ~ X and a": ~n ~ Y,
there is a corresponding a = (a' X a")dn : ~n ~ X X Y. Therefore the
singleton {d n} is a basis for ~n(X X Y), so ~(X X Y) is free with models
{~n,~n}, and hence also free with models {~p,M}. Since ~P and ~q are each
contractible, so is ~P X ~q. Therefore Li(~p X ~q) is acyclic, and we have
proved that ~(X X Y) is a free acyclic functor with models {~p,~q}.
Since ~p(X) is free with a basis ~p E ~p(~p) and ~q(Y) is free with basis
~q E ~q(M), it follows that ~p(X) ® ~q(Y) is free with the basis
0
0
~p
®
~q
E ~p(~p) ®
~q(M).
Then [~(X) ® ~(Y )]n is free with the basis {~p ® ~q}p+Q=n. Hence ~(X) ® ~(Y)
is free with models {M,~q}. Since e: ~(~p) ~ Z and e: ~(~q) ~ Z are chain
equivalences, it follows that
e ® e: ~(~p) ® ~(~q) ~ Z ® Z
=Z
is also a chain equivalence. Hence, by lemma 4.3.2, the reduced complex of
® ~(~q) is acyclic, and we have shown that ~(X) ® ~(Y) is also free
and acyclic with models {~p,M}. The theorem now follows by the method of
acyclic models. -
~(~p)
The same technique based on the method of acyclic models can be used
to prove the following results.
7
THEOREM
diagram
Given X, Y, and Z, there is a chain homotopy commutative
~((X
X Y) X Z) ::::; ~(X X (Y X Z))
::::t
[~(X)
®
~(Y)]
t::::
®
~(Z) ::::; ~(X)
®
[~(Y)
®
~(Z)]
where the vertical maps are the natural chain equivalences of theorem 6.
8
THEOREM
diagram
-
For any X and Y there is a chain homotopy commutative
~(X
X Y)
::::;
~(Y
t::::
=4
~(X)
®
X X)
~(Y)
::::;
~(Y)
®
~(X)
234
PRODUCTS
CHAP.
5
where the bottom map sends x ® y to ( - 1)de g x deg y y ® x and the vertical
maps are the natural chain equivalences of theorem 6. The sign in theorem 8 is necessary to make the map a chain map (that
is, to make it commute with the boundary operators).
Given topological pairs (X,A) and (Y,B), we define their product
(X,A) X (Y,B) to be the pair (X X Y, X X B U A X Y). Then we have the
following relative form of the Eilenberg-Zilber theorem.
9
THEOREM
On the category of ordered pairs of topological pairs (X,A)
and (Y,B) such that {X X B, A X Y} is an excisive couple in X X Y there is
a natural chain equivalence of Ll(X X Y)/Ll(X X B U A X Y) with
[Ll(X)/ Ll(A)] Q9 [Ll(Y)/ Ll(B)].
PROOF
Because {X X B, A X Y} is an excisive couple, the natural map
Ll(X X Y)/[Ll(X X B)
+ Ll(A
X Y)]
~
Ll(X X Y)/Ll(X X B U A X Y)
is a chain equivalence. By theorem 6 there is a functorial equivalence of
Ll(X) ® Ll(Y) with Ll(X X Y) taking Ll(X) ® Ll(B) and Ll(A) Q9 Ll(Y) into
Ll(X X B) and Ll(A X Y), respectively. Hence there is a functorial chain
equivalence of the quotient
Ll(X) ® Ll(Y)/[Ll(X) ® Ll(B)
+ Ll(A)
® Ll(Y)]
~
[Ll(X)/Ll(A)] ® [Ll(Y)/Ll(B)]
with the quotient
Ll(X X Y)/[Ll(X X B)
+ Ll(A
X Y)]
Combining these two chain equivalences gives the result.
-
For any two pairs (X,A) and (Y,B) we define the homology cross product
p,': Hp(X,A; G) Q9 Hq(Y,B; G') ~ Hp+q((X,A) X (Y,B); G Q9 G')
to be equal to the cross product
Hp([Ll(X)/Ll(A)] Q9 G) ® Hq([Ll(Y)/Ll(B)] Q9 G')
~
Hp+q(([Ll(X)/Ll(A)] Q9 [Ll(Y)/Ll(B)]) Q9 (G ® G'))
followed by the functorial homomorphism of the bottom module to
Hp+q(Ll(X X Y)/Ll(X X B U A X Y); G Q9 G')
If z E Hp(X,A; G) and z' E Hq(Y,B; G'), then we write
z X z'
= p,'(z ® z') E Hp+q((X,A)
X (Y,B); G Q9 G')
Because Ll(X)/Ll(A) and Ll(Y)/Ll(B) are free, we can combine theorem 9
with corollary 4 to obtain the following Kiinneth formula for singular
homology.
SEC.
3
235
THE KUNNETH FORMULA
10 THEOREM If {X X B, A X Y} is an excisive couple in X X Y and G
and G' are modules over a principal ideal domain such that G * G' = 0,
there is a functorial short exact sequence
o~
[H(X,A; G) (59 H(Y,B; G')]q ~ Hq((X,A) X (Y,B); G (59 G') ~
[H(X,A; G)
and this sequence is split.
* H(Y,B;
G')]q-l ~ 0
•
In particular, if the right-hand term vanishes (which always happens if R
is a field), then the cross product is an isomorphism
J-t': H(X,A; G) ® H(Y,B; G') ::::; H((X,A) X (Y,B); G (8) G')
The following formulas are consequences of the naturality of J-t and of
theorems 5, 7, and 8.
I I Let f: (X,A) ~ (X',A') and g: (Y,B) ~ (Y',B') be maps and let
z E Hp(X,A; G) and z' E Hq(Y,B; G'). Then, in the module
Hp+q((X',A') X (Y',B'); G (8) G')
we have
(f X g)* (z X z') = f* z X ~ z'
•
12 Let p: (X,A) X Y ~ (X,A) be the protection to the first factor and let
H(Y;G') ~ G' be the augmentation map. For z E Hq(X,A; G) and
z' E Hr(Y;G'), in Hq+r(X,A; G (8) G'),
E:
p* (z X z')
13 For
Z
= J-t(z (8) E(Z'))
•
E Hp(X,A; G), z' E Hq(Y,B; G'), and z" E Hr(Z,C; G"), in
Hp+q+r((X,A) X (Y,B) X (Z,C); G (59 G' (8) G"),
we have
Z
X (z' X z") = (z X z') X z"
•
14 Let T: (X,A) X (Y,B) ~ (Y,B) X (X,A) and qJ: G' (59 G ~ G (8) G'interchange the factors. For Z E Hp(X,A; G) and z' E Hq(Y,B; G'), in
Hp+q((Y,B) X (X,A); G ® G'), we have
T* (z X z')
= (-l)pqqJ* (z' X z)
•
15 Let {(XI,A I), (X 2 ,A 2 )} be an excisive couple of pairs in X and let
E Hp(XI U X 2 , Al U A 2 ; G) and z' E Hq(Y,B; G'). For the connecting homomorphisms of appropriate Mayer- Vietoris sequences we have
Z
a.(z x z') = a.z x z'
in Hp+q-I((X I
n X 2 , Al n A 2 ) X (Y,B); G
(8) G') and
a.(z' x z) = (-l)qz' x a.z
in Hp+q_I((Y,B) X (Xl
n X 2 , Al n A 2 ); G'
(8) G)
•
236
4
PRODUCTS
CHAP.
5
COHOMOLOGY
A chain complex has a differential of degree -1. Related to this is the concept of a cochain complex, which has a differential of degree + 1. Cochain
complexes have many of the properties of chain complexes, and this section
is devoted to a discussion of these properties. The functor Hom associates to
every chain complex a cochain complex, and vice versa. The cohomology
module of a topological pair with coefficients G is the homology module of
the cochain complex associated in this way to the singular complex of the pair.
The last part of the section is devoted to a brief discussion of axiomatic
cohomology theory.
Throughout this section R will denote a commutative ring with a unit.
A cochain complex (over R), denoted by C* = {Cq,8 q}, is a graded R module
together with a homogeneous differential 8 of degree + 1 called the coboundary operator (thus 8q: Cq ~ Cq+l and 8q+18q
0 for all q). The kernel of 8
is the module of cocycles Z( C *), and the image of 8 is the module of
co boundaries B(C*). Then B(C*) C Z(C*), and the cohomology module
H(C*) is defined to be the quotient Z(C* )/B(C*).
If C * is a co chain complex, there is a chain complex C defined by
Cq = C-q and aq: Cq ~ C q- 1 equal to 8-q: C-q ~ C-q+l. Then Hq(C) =
H-q( C *), and the cohomology module of C * corresponds to the homology
module of C. In this way results about chain complexes give results about cochain complexes. Thus we have the concepts of cochain map and cochain
homotopy, and there is a category of cochain complexes and cochain homotopy
classes of co chain maps. The cohomology module is a covariant functor from
this category to the category of graded modules. Furthermore, given a short
exact sequence of cochain complexes
=
o ~ C*
~ C* ~ C* ~ 0
there is a functorial connecting homomorphism
8*: H(C*) ~ H(C*)
of degree
+ 1 and a functorial exact cohomology sequence
... ~ Hq(C*) ~ Hq+l(C*) ~ Hq+1(C*) ~ Hq+l(C*) ~
Passing from a cochain complex to a chain complex by changing the
sign of the degree gives us the following analogues of theorems 5.2.14 and
5.3.3.
I
THEOREM
Given a cochain complex C * and a module G such that
C* * G is acyclic, there is a functorial short exact sequence
o ~ Hq(C*)
Q9 G ~ Hq(C* Q9 G) ~ Hq+1(C*)
and this sequence is split.
•
*G~ 0
SEC.
4
237
COHOMOLOGY
2
THEOREM
Given cochain complexes C * and C' * such that C * * C' *
is an acyclic cochain complex, there is a functorial short exact sequence
o~
[H*(C*) Q9 H*(C'*)]q
~
Hq(C* Q9 C'*)
~
[H*(C*)
and this sequence is split.
* H*(C'*)]q+1 ~ 0
•
There is also an analogue of corollary 5.3.4 for cochain complexes which
we shall not state as a separate theorem. If C* is a co chain complex over R
and G is an R module, an augmentation of C * over G is a monomorphism
'I): G ~ Co such that {)O
'I) = O. An augmented cochain complex over G con0) and an
sists of a nonnegative cochain complex C* (that is, Cq = 0 for q
augmentation of C * over G. Such an augmentation can be regarded as
a monomorphic chain map of the cochain complex (also denoted by G) whose
only nontrivial cochain module is G in degree 0 to C*. For this trivial
cochain complex G it is clear that Hq(G) = 0 for q 7'= 0 and HO(G) = G.
Therefore 'I) induces a monomorphism 'I) *: G ~ HO( C *). The reduced
cochain complex C* of an augmented co chain complex C * is defined to be
the quotient cochain complex Cq = Cq for q 7'= 0, Co = coker 'I), and Bq is
suitably induced by {)q. We define H( C *) = H( C*). Because there is a short
exact sequence of cochain complexes
0
<
o ~ G 4 C*
we see that Hq( C *) ~ Hq( C *) for q
o~
~
7'= 0,
C*
~ 0
and there is a short exact sequence
G ~ HO(C*) ~ [jO(C*) ~ 0
Our interest in cochain complexes is in their relation to chain complexes.
If C is a chain complex over R and G is an R module, there is a cochain
complex Hom (C,G) = {Hom (Cq,G), {)q}, where, if f E Hom (Cq,G), then
{)qf E Hom (Cq+ 1 ,G) is defined by
We also write <f,c) instead of f(c) and set <f,c) = 0 if deg f 7'= deg c.
In this notation <8qf,c) = <f,aq+1C).
If C is augmented bye: Co ~ G', then Hom (C,G) is augmented by
'I): Hom (G',G) ~ Hom (Co,G), where 'I)(f)(c) = f(e(c)) for c E Co and
f E Hom (G',G). It is easy to verify the following result.
3
THEOREM
There is a functor of two arguments contravariant in chain
complexes C and covariant in modules G which assigns to a chain complex C
and module G the cochain complex Hom (C,G). •
For a chain complex C and module G we define the cohomology module
H*(C;G) = {Hq(C;G)} ofCwithcoefficients Gby
238
PRODUCTS
CHAP.
5
Hq(C;G) = Hq(Hom (C,G))
It follows from theorem 3 that H * (C; G) is a contravariant functor of C and a
covariant functor of G to the category of graded modules. For a chain map
7": C ~ C' we use 7" * : H * (C'; G) ~ H * (C; G) to denote the homomorphism
induced by the co chain map Hom (7",1): Hom (C',G) ~ Hom (C,G), and for a
homomorphism <p: G ~ G' we use <p*: H*(C;G) ~ H*(C,G') to denote the
homomorphism induced by the cochain map Hom (1,<p): Hom (C,G) ~
Hom (C,G'). To distinguish the homology of C from its cohomology, we shall
sometimes denote H(C;G) by H* (C;G).
4 EXAMPLE Given an abelian group G and a simplicial pair (K,L), the
oriented cohomology group of (K,L) with coefficients G, denoted by
H* (K,L; G), is defined to be the graded cohomology group of the cochain
complex Hom (C(K)jC(L), G) [which is augmented over Hom (Z,G) ;:::;: G].
Then H * (K,L; G) is a contravariant functor of (K,L) and a covariant functor
of G to the category of graded abelian groups. If G is also an R module,
H* (K,L; G) is a graded R module.
S EXAMPLE If (X,A) is a topological pair and G is an abelian group, the
singular cohomology group of (X,A) with coefficients G, denoted by
H* (X,A; G), is defined to be the graded cohomology group of the cochain
complex Hom (~(X)j ~(A), G) (which is augmented over G). It is contravariant
in (X,A) and covariant in G, and if G is an R module, H* (X,A; G) is a graded
R module. If (X',A') C (X,A) and u E H* (X,A; G), we use u I (X',A') to denote
the element of H* (X',A'; G) equal to i * u, where i: (X',A') C (X,A). We also
use 1 E HO(X;R) to denote the image of 1 E R under the augmentation
1/: R ~ HO(X;R).
We recall some properties of the functor Hom. The following analogue
of corollary 5.1.6 is easily established.
6
LEMMA
Given an exact sequence of R modules
A'
~ A~A" ~
0
and an R module B, there is an exact sequence
o ~ Hom (A",B) ~ Hom (A,B) ~ Hom (A',B)
•
If A' ~ A is a monomorphism [that is, if 0 ~ A' ~ A is also exact], it
need not be true that Hom (A,B) ~ Hom (A',B) is an epimorphism, [that is,
that Hom (A,B) ~ Hom (A',B) ~ 0 is exact]. However, there is the following
analogue of corollary 5.1.12 (which follows easily by using lemma 5.1.11).
7
LEMMA
Given a split short exact sequence of R modules
o ~ A' ~ A ~ A" ~ 0
and an R module B, the sequence
SEC.
4
239
COHOMOLOGY
o~
Hom (A",B)
~
Hom (A,B)
is a split short exact sequence.
•
Hom (A',B)
~
~
0
In case 0 ~ C' ~ C ~ C" ~ 0 is a split short exact sequence of chain
complexes, it follows from lemma 7 that for any module G there is a short
exact sequence of cochain complexes
o~
Hom (C",G)
~
Hom (C,G)
~
Hom (C',G)
~
0
This gives the following result.
8
THEOREM
Given a split short exact sequence of chain complexes
o~
C'
~
C
~
C"
~
0
and a module G, there is a functorial exact cohomology sequence
... ~ Hq(C";G) ~ Hq(C;G) ~ Hq(C';G) ~ Hq+I(C";G) ~ . ..
•
This leads to an exact Mayer- Vietoris cohomology sequence analogous to
the exact sequence of corollary 5.1.14.
COROLLARY
If {(XI,A I ), (Xz,Az)} is an eXClswe couple of pairs in a
9
topological space and G is an R module, there is a functorial exact cohomology
sequence
... ~ Hq(X I U X 2 , Al U A 2 ; G) i."'o, Hq(X I , AI; G) <:B Hq(X 2 ,A 2 ; G)
i4
Hq(XI n X z , Al n A z ; G) ~ ...
where j* (u) = ut (u), j: (u)) and i*
and j2 are suitable inclusion maps. •
(Ul
+
U2)
=
it
Ul -
i: U2 and il, i2, jl,
If {Xj} is the set of path components of X, then ~(X) = ffi ~(Xj).
Therefore Hom (~(X);G) = Xj Hom (~(Xj),G), and by theorem 4.1.6, we obtain the following result.
10 THEOREM The singular cohomology module of a space is the direct
product of the singular cohomology modules of its path components. •
Because the homology functor does not commute with inverse limits, it is
not true that the singular cohomology of a space is isomorphic to the inverse
limit of the singular cohomology of its compact subsets (that is, there is no
general cohomology analogue of theorem 4.4.6).
There is an exact cohomology sequence corresponding to a short exact
sequence of coefficient modules (analogous to theorem 5.2.7).
I I THEOREM On the category of free chain complexes Cover R and short
exact sequences of R modules
o~
G' ~ G ~ G" ~ 0
240
PRODUCTS
CHAP.
5
there is a functorial connecting homomorphism
13*: H*(C;G") ~ H*(C;G')
of degree 1 and a functorial exact sequence
... ~ Hq(C;G')
PROOF
'1':4 Hq(C;G) ~ Hq(C;G") ~ Hq+l(C;G') ~
Because C is free, there is a short exact sequence of cochain complexes
o~
where <p and
Hom (C,GJ ~ Hom (C,G)
\It are induced by cp and 1/;.
-i.
Hom (C,G") ~ 0
The result follows from this.
•
The connecting homomorphism 13 * in theorem 11 is called the Bockstein
cohomology homomorphism corresponding to the coefficient sequence
o ~ G' .!4 G -4 Gil ~ O.
Let G be an R module. A cohomology theory with coefficients G consists
of a contravariant functor H* = {Hq} from the category of topological pairs
to graded R modules and a natural transformation 8 *: H* (A) ~ H* (X,A) of
degree 1 such that the following axioms hold.
12
HOMOTOPY AXIOM
If fo,
h:
(X,A)
~
(Y,B) are homotopic, then
H*(fo) = H*(h): H*(Y,B) ~ H*(X,A)
13
i: X
EXACTNESS AXIOM
For any pair (X,A) with inclusion maps i: A C X and
C (X,A), there is an exact sequence
... ~ Hq(X,A)
Hq(j\
Hq(X)
Hq(i)
Hq(A) ~ Hq+l(X,A) ~ ...
14 EXCISION AXIOM For any pair (X,A) if U is an open subset of X such
that 0 C int A, then the excision map;: (X - U, A - U) C (X,A) induces
an isomorphism
H*(;): H*(X,A):::::; H*(X - U, A - U)
15 DIMENSION AXIOM On the category of one-point spaces there is a natural
equivalence of the constant functor G with the functor H * .
Singular cohomology theory with coefficients G is an example of a cohomology theory with coefficients G (the exactness axiom following from the
application of corollary 9 to a suitable couple). The uniqueness theorem is
valid for cohomology theories (that is, a homomorphism from one cohomology
theory to another which is an isomorphism for one-point spaces is an isomorphism for compact polyhedral pairs).
The reduced cohomology modules il * of a cohomology theory are
defined as follows. If X is a nonempty space, let c: X ~ P be the unique map
from X to some one-point space P. The reduced module il* (X) is defined to
be the cokernel of the homomorphism
H*(c): H*(P)
~
H*(X)
Because c has a right inverse, H*(c) has a left inverse. Therefore
SEC.
5
THE UNIVERSAL· COEFFICIENT THEOREM FOR COHOMOLOGY
241
H* (X) ;:::; H* (X) EEl H* (P)
and the reduced modules have properties similar to those of the reduced
singular cohomology modules.
5
THE UNIVERSAL-COEFFICIENT THEOREM
FOR COHOMOLOGY
This section is devoted to relations between cohomology and homology of
chain complexes. In order to express the cohomology of a chain complex in
terms of its homology it is necessary to introduce certain functors of modules
which are associated to the module of homomorphisms of one module to
another in much the same way that the torsion products are associated to the
tensor product. Over a principal ideal domain there is one associated functor,
the module of extensions. We use this to formulate the universal-coefficient
theorems and Kiinneth formulas established in the section.
Let C be a free resolution of the module A and let B be a module. There
is a cochain complex Hom (C,B) = {[Hom (C,B)]q = Hom (Cq,B), 8q} whose
cohomology module depends only on A and B, up to canonical isomorphism
(and not on the choice of C). Let C be the canonical free resolution of A and
define Ext q (A,B) = Hq(Hom (C,B)). Then Extq (A,B) is a functor of two
arguments contravariant in A and covariant in B, and ExtO (A,B) is naturally
equivalent to Hom (A,B).
For the rest of this section we assume R is a principal ideal domain. Then,
Ext q (A, B) = 0 for q> 1, and the module of extensions Ext (A, B) is defined to
equal ExF (A, B). It is characterized by the property that given any free presentation of A
there is an exact sequence
o -? Hom (A,B) -? Hom (Co,B) Hom (a"I) Hom (C1,B) - ? Ext (A,B) -? 0
In fact, because Hom (C2 ,B) = 0,
Ext (A,B) = H1(C;B) = Hom (C 1,B)/im [Hom (hI)]
= coker [Hom (01,1)]
Clearly, Ext (A,B) is contravariant in A and covariant in B. Its name derives
from its connection with the extensions of B by A which we describe briefly
after the following examples.
I
If A is free, it has the free presentation
O-?O-?A-?A-?O
from which it follows that Ext (A,B)
= 0 for any B.
242
2
PRODUCTS
CHAP.
5
For v E R, v =;6 0 there is a short exact sequence
o~ R~ R~RlvR~
0
where a(v') = vv' for v' E R, which is a free presentation of R/vR. For any
R module B, Hom (R,B) ;::::: B and the homomorphism Hom (a, I): Hom (R,B) ~
Hom (R,B) corresponds to a*: B ~ B, where a* (b) = vb. Hence there is an
isomorphism coker Hom (a,l) ;::::: BlvB, and we have proved
Ext (RlvR,B) ;::::: BlvB ;::::: (RlvR) ® B
Since Hom commutes with finite direct sums, it follows that for any finitely
generated torsion module A there is an isomorphism (nonfunctorial)
Ext (A,B) ;::::: A ® B
because such a module A is a finite direct sum of cyclic modules (by theorem
4.14 in the Introduction).
An extension of B by A is a short exact sequence
O~B~E~A~O
With a suitable definition of equivalence of extensions (by a commutative
diagram), of the sum of two extensions, and of the product of an extension
by an element of R, there is obtained a module whose elements are equivalence classes of extensions of B by A. This module is isomorphic to Ext (A,B). In
fact, given an extension 0 ~ B ~ E ~ A ~ 0 and a free presentation of A,
o ~ C1 ~ Co ~ A ~ 0, there is, by theorem 5.2.1, a commutative diagram
o~
C1
'I'1~
o~
B
~
Co
'I'O~
~
E
uniquely determined up to chain homotopy. Then <P1 E Hom (Ct.B) is unique
up to im [Hom (Co,B) ~ Hom (C1 ,B)], and so determines an element of
Ext (A,B). This function from extensions of B by A to Ext (A,B) induces an
isomorphism of the module of equivalence classes of extensions with Ext (A,B).
Given a graded module C = {Cq}, there is a graded module Ext (C,B) =
{[Ext (C,B)]q = Ext (Cq,B)). If C is a chain complex, Ext (C,B) is a cochain
complex with
l)q
= Ext (OH1,1): Ext (Cq,B)
~
Ext (CHt.B)
A homomorphism
h: Hq(C;G) ~ Hom (Hq(C;G'), G ® G')
natural in C and G is defined by
(h{f}){~
Ci
® gi} = ~f(ci) ®
gi
for {f} E Hq(C;G) and {~Ci ® gil E Hq(C;G') [after verification that
SEC.
5
243
THE UNIVERSAL· COEFFICIENT THEOREM FOR COHOMOLOGY
~ f( Ci) ® gi is independent -of the choice of f in its cohomology class
and ~ Ci ® gi in its homology class]. For u E Hp(C;G) and z E Hq(C;G') we
define <u,z) E G ® G' to be 0 if P
q and to be h( u)(z) if P = q. In this
notation
*
<{f}, {L Ci ® gi})
= L <f,ci) ® gi
The homomorphism h enters in the following universal-coefficient
theorem for cohomology.
3
THEOREM
Given a chain complex C and module G such that Ext (C,G)
is an acyclic cochain complex, there is a functorial short exact sequence
o
-?
Ext (Hq-1(C),G)
-?
Hq(C,G) ~ Hom (Hq(C),G)
-?
0
and this sequence is split.
PROOF
We first consider the case in which C is a free chain complex. There
is then a short exact sequence of chain complexes
O-?Z-?C-?B-?O
where Zq = Zq(C) and Bq = Bq-1(C). This sequence is split because B is free,
and by theorem 5.4.8, there is an exact cohomology sequence
... -?
Hq-l(Z;G) ~ Hq(B;G)
Hq(C;G)
-?
Hq(Z;G) ~ HHl(B;G)
-?
-? ...
Since Z and B have trivial boundary operators, Hq(Z;G) = Hom (Zq(C),G)
and Hq(B;G) = Hom (Bq-1(C),G). Furthermore, the homomorphism
8*: Hq(Z;G)
-?
Hq+l(B;G)
equals Hom (Yq,l): Hom (Zq(C),G) - ? Hom (Bq(C),G), where Yq: Bq(C) C Zq(C).
Hence there is a functorial short exact sequence
o - ? coker [Hom (Yq_l,l)] - ?
Hq(C;G)
-?
ker [Hom (Yq,l)]
-?
0
To interpret the modules in the above sequence we have the short exact
sequence
o
-?
Bq(C) ~ Zq(C)
-?
Hq(C)
-?
0
which is a free presentation of Hq(C). By the characteristic property of Ext,
there is an exact sequence
o
-?
Hom (Hq(C),G)
-?
Hom (Zq(C),G)
Hom (YG,I)
Hom (Bq(C),G)
-?
Ext (Hq(C),G)
-?
0
Therefore, ker [Hom (Yq,l)] :::::: Hom (Hq(C),G) and coker [Hom (Yq,l)] ::::::
Ext (Hq(C),G). Substituting these into the short exact sequence containing
Hq(C;G) yields the desired short exact sequence
o
-?
Ext (Hq-l(C),G)
-?
with the homomorphism Hq(C;G)
Hq(C;G)
-?
-?
Hom (Hq(C),G)
-?
0
Hom (Hq(C),G) easily verified to equal h.
244
PRODUCTS
CHAP.
5
This sequence is functorial and is split (because the sequence of chain
complexes
O~Z~C~B~O
is split).
For arbitrary C such that Ext (C,G) is acyclic, the result follows by using
a free approximation to C (as in the proof of theorem 5.2.14) to reduce it to
the case of a free complex. It follows from theorem 3 that if X is a path-connected topological
space, then HO(X; R) is a free R module generated by 1 [or, in other words,
the augmentation map is an isomorphism '1/: R ::::; HO(X;R)J. From theorems 3
and 5.4.10, it follows that for any X, HO(X;G) is isomorphic to the direct product
of as many copies of G as path components of X.
COROLLARY
If (X,A) is a topological pair such that Hq(X,A;R) is finitely
generated for all q, then the free sub modules of Hq(X,A; R) and Hq(X,A;R)
are isomorphic and the torsion submodules of Hq(X,A; R) and Hq_ 1 (X,A; R)
are isomorphic.
4
Let Hq(X,A; R)
module of Hq. Then
PROOF
= Fq c:B Tq,
where Fq is free and Tq is the torsion
Hom (Hq(X,A; R), R) ::::; Hom (Fq,R) c:B Hom (Tq,R) ::::; Fq
and by example 2,
Ext (Hq(X,A; R), R) ::::; Ext (Fq,R) c:B Ext (Tq,R) ::::; Tq
The result follows from theorem 3.
-
For many purposes it would be more useful to have a formula expressing
H* (C;G) in terms of H* (C;R). Such a formula can be proved in the case of
C or G finitely generated. We begin by establishing some properties of
finitely generated modules.
Let p,: Hom (A,G) ® G' ~ Hom (A, G ® G') be the functorial homomorphism defined by ",(f ® g')(a) = f(a) ® g' for f E Hom (A,G), g' E G',
and a EA.
:;
LEMMA
If A is a free module and G' is finitely generated, then for any
module G, '" is an isomorphism.
The result is trivially true if G' = R. Because the tensor product and
Hom functors both commute with finite direct sums, it is also true if G' is a
finitely generated free module. G' is assumed to be finitely generated, so there
is a short exact sequence
PROOF
O~ G~ G~ G'~O
SEC.
5
245
THE UNIVERSAL-COEFFICIENT THEOREM FOR COHOMOLOGY
where C (hence also G) is a finitely generated free module. There is a commutative diagram
Hom (A,G) ®
C
---> Hom (A,G)
~1
®
G --->
~1
Hom (A,G) ® G' ---> 0
~1
Hom (A, G ® G) ---> Hom (A, G ® G) ---> Hom (A, G ® G') ---> 0
with exact rows (exactness follows from corollary 5.1.6 and, for the bottom
row, from the fact that A is free). Because [l and il are isomorphisms, it
follows from the five lemma that p, is also an isomorphism. There is also a functorial homomorphism
p,: Hom (A,G) ® Hom (B,G') ---> Hom (A ® B, G ® G')
=
defined by p,(f ® f')(a ® b) f(a) ® f'(b) for f E Hom (A,G),f' E Hom (B,G'),
a E A, and b E B. In case B
R, Hom (B,G') ::::::; G', and p, corresponds to
the homomorphism in lemma 5.
=
6
LEMMA
If B is a finitely generated free module, for arbitrary modules
A and G, p, is an isomorphism
p,: Hom (A,G) ® Hom (B,R)::::::; Hom (A ® B, G)
PROOF
The result is trivially true for B = R and follows for a finite sum of
copies of R because both sides commute with finite direct sums. -
7
COROLLARY
If A and B are free modules and either A and B or Band G'
are finitely generated, p, is an isomorphism
p,: Hom (A,G) ® Hom (B,G')::::::; Hom (A ® B, G ® G')
PROOF
Since A and B are free, so is A ® B. If A and B are finitely generated, so is A ® B, and there is a commutative diagram
[Hom (R,G) ® Hom (A,R)] ® [Hom (R,G') ® Hom (B,R)]
E.
Hom (R, G ® G') ® Hom (A ® B, R)
1M
M0M1
£. Hom (A ® B, G ® G')
Hom (A,G) ® Hom (B,G')
in which il((fl ® fz) ® (h ® f4)) = p,(ft ® h) ® p,(fz ® f4). By lemma 6, il
is an isomorphism and so are both vertical maps. Therefore the bottom map
is also an isomorphism.
If Band G' are finitely generated, there is a commutative diagram
Hom (A,G) ® Hom (B,R) ® G' ~ Hom (A,G) ® Hom (B,G')
/'®
11
Hom (A ® B, G) ® G'
11'
4
Hom (A ® B, G ® G')
By lemma 5, both horizontal maps are isomorphisms, and by lemma 6, the
left-hand vertical map is an isomorphism. Therefore the right-hand map
is also an isomorphism. -
246
PRODUCTS
CHAP.
5
It follows from lemma 5 that if A is free and finitely generated, /1 is an
isomorphism
/1: Hom (A,R)
® A ;:::; Hom (A,A)
The following lemma is a partial converse of this result.
8
LEMMA
If A is a module such that
/1: Hom (A,R) ® A
-0
Hom (A,A)
is an epimorphism, then A is finitely generated.
PROOF
By hypothesis, there exist fi E Hom (A,R) and ai E A for 1 :::; i :::; n
such that /1('2: fi ® ail = lAo Then, for any a E A
a
= /1('2: fi
® ai)(a) = '2: fi(a)ai
showing that A is generated by {ai}.
•
A graded module {Cq } is said to be of finite type if Cq is finitely generated for every q. Thus a graded module C of finite type is finitely generated
(as a graded module) if and only if Cq = 0, except for a finite set of integers q.
The following lemma asserts that a chain complex whose homology is of
finite type can be approximated by a chain complex of finite type.
9
LEMMA
Let C be a free chain complex such that H( C) is of finite type.
Then there is a free chain complex C' of finite type chain equivalent to C.
For each q let Fq be a finitely generated submodule of Zq(C) such
that Fq maps onto Hq(C) under the epimorphism Zq(C) -0 Hq(C). Let F~ be
the kernel of the epimorphism Fq -0 Hq(C). Define a chain complex
C' = {C~,O~} byC~ = Fq E8F~_landaq(c,c') = (c',O)forc E Fq and c' E F~-l'
Then C' is a free chain complex of finite type and Hq(C') = Fq/F~ ;:::; Hq(C).
To define a chain equivalence T: C' -0 C, choose for each q a homomorphism
<pq: F~ -0 Cq+1 such that OQ+l<Pq(C') = c' for c' E F~. Then define T by T(C,C') =
C + <Pq-l(C') for c E Fq and c' E F~-l' T is a chain map and induces an isomorphism T*: H(C') ;:::; H(C). Because C' and C are both free, it follows from
theorem 4.6.10 that T is a chain equivalence. •
PROOF
We are now ready for the universal-coefficient theorems toward which
we have been heading.
10 THEOREM Let C be a free chain complex and G be a module such that
either H( C) is of finite type or G is finitely generated. Then there is a functorial short exact sequence
0-0 Hq(C;R) ® G
~
Hq(C;G)
-0
Hq+l(C;R)
*G
-0
and this sequence is split.
PROOF
If G is finitely generated, it follows from lemma 5 that
wHom (C,R) ® G;:::; Hom (C,G)
0
SEC.
5
247
THE UNIVERSAL· COEFFICIENT THEOREM FOR COHOMOLOGY
Because Hom (C,R) is without torsion, Hom (C,R) * G = 0, and the result
follows from theorem 5.4.1.
If H(C) is of finite type, we use lemma 9 to replace C by a free chain
complex C' of finite type. By corollary 7, p.: Hom (C',R) ® G;::::: Hom (C,G),
and the result again follo~s for C' (and hence for C) from theorem 5.4.1. •
In a similar way we obtain the following Kilnneth formula for cohomology.
11 THEOREM Let C and C' be nonnegative free chain complexes and G and
G' be modules over a principal ideal domain such that G * G' = 0 and either
H(C) and H(C') are of finite type or H(C') is of finite type and G' is finitely
generated. Then there is a functional short exact sequence
0---') [H*(C;G) ® H*(C';G')]q ---') Hq(C ® C'; G ® G') ---')
[H* (C;G)
* H* (C';G')]q+1 ---') 0
and this sequence is split.
PROOF
If H(C) and H(C') are of finite type, by lemma 9, we can replace C
and C' by free chain complexes of finite type. Hence we are reduced to
proving the result for the case where C and C' have finite type or where C'
has finite type and G' is finitely generated. By corollary 7, there is an isomorphism p.: Hom (C,G) ® Hom (C',G') ;::::: Hom (C ® C', G ® G'). The result
will now follow from theorem 5.4.2 as soon as we have verified that
Hom (C, G) * Hom (C', G') is acyclic.
We show that Hom (C,G) * Hom (C',G') = O. In case C and C' are both
of finite type, Hom (Cp , G) is isomorphic to a finite direct sum of copies of G
and Hom (C~,G') is isomorphic to a finite direct sum of copies of G'. Because
0 by hypothesis, Hom (Cp,G) * Hom (C~,G')
0, and so
G * G'
Hom (C,G) * Hom (C',G') = 0 in this case.
In case C' is of finite type, Hom (C~,G') is isomorphic to a finite direct
sum of copies of G'. Hence it suffices to show that Hom (C, G) * G'
0 if G'
is finitely generated. Let
=
=
=
o ---') G ---') G ---') G' ---') 0
be a free resolution of G' with
there is a short exact sequence
G finitely generated. Because G
* G' = 0,
o ---') G ® G ---') G ® G ---') G ® G' ---') 0
and a short exact sequence of cochain complexes (because C is free)
o ---') Hom (C, G ® G) ---') Hom (C, G ® G) ---') Hom (C, G ® G') ---') 0
Using lemma 5, this implies the exactness of the sequence
0---') Hom (C,G) ®
Hence Hom (C,G)
case. •
G ---')
Hom (C,G) ®
G ---') Hom (C,G) ®
G' ---') 0
* G' = 0, and so Hom (C,G) * Hom (C',G') = 0 in either
248
PRODUCTS
CHAP.
5
If A is a free finitely generated module, then
A ;:::::: Hom (Hom (A,R), R)
Since Hom (A,R) is also free and finitely generated, it follows from corollary 7
that
A ® G ;:::::: Hom (Hom (A,R), R) ® Hom (R,G) ;:::::: Hom (Hom (A,R), G)
We use this to express homology in terms of cohomology.
12 THEOREM Let C be a free chain complex such that H( C) is of finite
type. For any module G there is a functorial short exact sequence
o ~ Ext (Hq+1(C;~), G) ~ Hq(C;G) ~ Hom (Hq(C;R), G) ~ 0
and this sequence is split.
PROOF
By lemma 9, we are reduced to the case where C is of finite type.
Then C ® G ;:::::: Hom (Hom (C,R), G), and the result follows, by theorem 3,
on changing Hom (C,R) to a chain complex by changing the sign of the
degree. •
The following result is a version of lemma 8 valid for homology that is a
partial converse to theorem 10.
13 THEOREM Let C be a free chain complex such that for every module G
the map t-t: Hom (C,R) ® G ~ Hom (C,G) induces isomorphisms of all cohomology modules. Then H* (C) is of finite type.
PROOF Because t-t: Hq(Hom (C,R) ® Hq(C)) ;:::::: Hq(Hom (C,Hq(C))), it follows
from theorem 3 that there exist fi E Hom (Cq,R) and Zi E Hq(C) such that
ht-t(~ fi ® zd·= 1Hq(C). Then, for any Z E Hq(C) we have
Z
= <t-t(~fi ® Zi}, z) = ~ <fioz)zi
showing that Hq( C) is generated by
Zi.
•
Note that if the short exact sequence of theorem 10 is valid for a given C
for all G, then the hypothesis of theorem 13 is satisfied, and so H(C) is
of finite type.
6
CUP AND CAP PRODUCTS
There is a cross product of cohomology classes from the tensor product of the
cohomology of two spaces to the cohomology of their product space. By
using the diagonal map of a space into its square, the cross product gives rise
to a product in the cohomology module of a space. This multiplicative structure provides cohomology with more structure than just the essentially additive
module structure. In this section we shall define these products and establish
some of their elementary properties.
SEC.
6
249
CUP AND CAP PRODUCTS
If {X X B, A X Y} is an excisive couple in X X Y, there is a cohomology
cross product
p,': Hp(X,A; G) Q9 Hq(Y,B; G') ~ Hp+q((X,A) X (Y,B); G ® G')
induced by the functorial homomorphism
Hom (d(X)/d(A),G)
(8l
Hom (d(Y)/d(B),G')
I't
Hom ([d(X)/d(A)] ® [d(Y)/d(B)], G ® G')
followed by an Eilenberg-Zilber cochain equivalence of the bottom module
with Hom (d(X X Y)/d(X X B U A X Y), G Q9 G'). If u E Hp(X,A; G) and
v E Hq(Y,B; G'), we define
u X v = p,'(u Q9 v) E Hp+q((X,A) X (Y,B); G Q9 G')
From theorem 5.5.11 we obtain the following Kiinneth formula for
singular cohomology.
I
THEOREM
Let {X X B, A X Y} be an excisive couple in X X Y and
let G and G' be modules such that G * G' = O. If H* (X,A; R) and H* (Y,B; R)
are of finite type or if H* (Y,B; R) is of finite type and G' is finitely generated,
there is a functorial short exact sequence
o~
[H* (X,A; G)
(8l
H* (Y,B; G')]q
4
Hq((X,A) X (Y,B); G
[H* (X,A; G)
and this sequence is split.
(8l
G') ~
* H* (Y,B;
G')]q+1 ~ 0
•
The cohomology cross product satisfies the following analogues of statements 5.3.11 to 5.3.15.
2
Let f: (X,A) ~ (X',A') and g: (Y,B) ~ (Y',B') be maps and let
u' E HP(X',A'; G) and v' E Hq(Y',B'; G'). Then, in Hp+q((X,A) X (Y,B); G ® G'),
we have
(f X g)* (u' X v') = f* u' X g * v' •
3
Let p: (X,A) X Y ~ (X,A) be the protection to the first factor and let
1/: G' ~ H*(Y;G') be the augmentation map. For u E Hq(X,A; G), in
Hq((X,A) X Y; G ® G'), we have
p* (p,(u ® g'»
=u
X 1/(g')
•
4
For u E Hp(X,A; G), v E Hq(Y,B; G'), and w E Hr(Z,C; G If ), in
HP+q+r((X,A) X (Y,B) X (Z,C); G ® G' ® G If ), we have
u X (v X w) = (u X v) X w
I:!
:. Let T: (X,A) X (Y,B) ~ (Y,B) X (X,A) and qJ: G ® G' ~ G' ® G
interchange the factors. For u E HP(X,A; G) and v E Hq(Y,B; G'), in
Hp+q((X,A) X (Y,B); G' ® G), we have
250
PRODUCTS
T*(v Xu)
= (-l)pq<p*(u
X v)
CHAP.
5
•
6
Let {(Xl,A l ), (X 2 ,A 2 )} be an excisive couple of pairs in X and let
u E Hp(X l n X 2 , Al n A 2 ; G) and v E Hq(Y,B; G'). For the connecting
homomorphisms of appropriate Mayer- Vietoris sequences we have
8*(u X v)
= 8*u
X v
in HP+q+l((X l U X 2 , Al U A 2 ) X (Y,B); G ® G') and
8*(v X u)
= (-l)qv
X 8*u
in Hp+q+l((Y,B) X (Xl U X 2 , Al U A 2 ); G' ® G).
•
Consider the twu functors il(X) and il(X) ® il(X) on the category of
topological spaces. Because il(X) is free with models {il q }Q2 0 and il(X) ® il(X)
is acyclic with models {il Q }q2 0 [that is, the reduced complex of il(M) ® il(ilq)
is acyclic for all q], it follows from the acyclic-model theorem 4.3.3 that
there exist functorial chain maps 7*: il(X) ~ il(X) ® il(X) preserving augmentation, and any two are chain homotopic. Such a functorial chain map is
called a diagonal approximation. The name stems from the fact that if
7X: il(X X X) ~ il(X) ® ~(X) is a functorial chain equivalence given by the
Eilenberg-Zilber theorem and d: X ~ X X X is the diagonal map, then the
composite
il(X)
6.(d\
il(X X X) ~ il(X) ® il(X)
is a diagonal approximation.
We construct a particular diagonal approximation called the AlexanderWhitney diagonal approximation. If a: ilq ~ X is a singular q-simplex, the
front i-face ia is defined for 0 ::; i ::; q to equal the composite a A, where
A: ili ~ ilq is the simplicial map defined by A(pj) = pj for 0 ::; i ::; i. Similarly,
the back i-face ai is defined for 0 ::; i ::; q to equal the composite a A.',
where A': ili ~ ilq is the simplicial map defined by A'(Pj)
Pj+q-i for
o ::; i ::; i. It is easy to verify that
0
=
7( a)
0
= i+j=dego
L ia ® aj
defines a functorial chain map 7: il(X) ~ il(X) ® il(X), and this chain map is
the Alexander-Whitney diagonal approximation.
Let G and G' be R modules. A pairing of G and G' to an R module G"
is a homomorphism <p: G ® G' ~ G". For example, G and G' are always
paired to G ® G'. Given such a pairing and given a diagonal approximation
7, there is a functorial cochain map
fx: Hom (il(X),G)
® Hom (il(X),G')
~
Hom (il(X),G")
defined to equal the composite
Hom (il(X),G) ® Hom (il(X),G') 4
Hom (il(X) ® il(X), G ® G')
Hom (TX'<p\
Hom (il(X),G")
SEC.
6
251
CUP AND CAP PRODUCTS
If A C X, then for f E Hom (L\(X),G) and f' E Hom (L\(X),G'), we have
Tx(f ® J') I L\(A) = TA(f I L\(A) ® J' I L\(A))
If A!, A2 C X and f vanishes on A!, J' vanishes on A 2 , it follows that
Tx(f ® f') vanishes on L\(A 1) + L\(A2). If {A 1,A 2} is an excisive couple in X,
it follows that TX induces a homomorphism
Hp(X,Al; G) ® Hq(X,A 2; G')
---?
Hp+q(X, Al U A 2; G")
which is called the cup-product homomorphism. If u E Hp(X,Al; G) and
v E Hq(X,A2; G'), their cup product is denoted by
u v v E Hp+q(X,Al U A 2; G")
This product is a bilinear function of u and v and depends on the pairing cp
but not on the particular diagonal approximation. The Alexander-Whitney
diagonal approximation yields a particular map T which defines a cup product
of cochains f v J' for f E Hom (L\p(X),G) and J' E Hom (L\q(X),G') by
(f v J')(o) = cp(f(po) ® J'(Oq))
Then {f} v {f'} = {f v J'} in Hp+q(X, Al U A 2; G").
As pOinted out above, there exist diagonal approximations which are
factored through L\(d). This implies the following relation expressing the cup
product in terms of the cross product.
THEOREM [fIX X A 2, Al X Xl is an excisive couple in X X X, if(Al,A 21 is
an excisive couple in X, and <p: G C8l G' -> Gil is a pairing, then for u E HP(X,Al;
G) and v E Hq(X,A2; G'), in Hp+q(X, Al U A 2; Gil), we have
7
uv v
= cp* (d * (u
Xv))
•
The cup product has the following properties analogous to the corresponding properties of the cross product.
8
Let f: X ---? Y map Al into Bl and A2 into B2 and let u E Hp(Y,B 1; G)
and v E Hq(Y,B 2; G'). Let fI: (X,Al) ---? (Y,Bl)' h: (X,A2) ---? (Y,B2), and
f: (X, Al U A 2) ---? (Y, Bl U B2) be maps defined by f. In Hp+q(X, Al U A 2; G"),
we have
J* (u v
9
v) =
ft u
v
nv
•
For any u E Hq(X,A; G) with the pairings R ® G :::::: G :::::: G ® R we have
10 Given a commutative diagram, where cp, cp', 1/;, and 1/;' are pairings,
G 1 ® (G 2 ® G 3 )
1®
<p't
::::::
(G 1 ® G 2) ® G 3
<p
® 1)
G 12 ® G3
Jf
252
PRODUCTS
CHAP.
5
and given UI E Hp(X,AI; GI ), U2 E Hq(X,A2; G2), and U3 E Hr(X,A3; G3),
then, in Hp+q+r(X, Al U A2 U A 3; G I23 ), we have
UI
v (U2 v U3)
= (UI v
U2)
V
U3
•
I I Given a commutative diagram of pairings
G(8) G':::::;G'(8) G
\.
.!
Gil
and given U E Hp(X,AI; G) and v E Hq(X,A2; G'), in Hp+q(X, Al U A 2; Gil),
we have
Uv v
= (- 1)pqv v
U •
12 Let {(XloAI)' (X 2,A2)} be an excisive couple of pairs in X, let A C Xl U X 2,
and let i: (Xl n X 2, A n Xl n X 2) C (Xl U X 2, A). For elements
u E Hp(X I n X 2, Al n A 2; G) and v E Hq(X I U X 2, A; G') and with the
connecting homomorphisms of the appropriate Mayer- Vietoris sequences, in
HP+q+I(X I U X 2, Al U A2 U A; Gil), we have
8*(u v i*v) = 8*u v v
8*(i*vvu)=(-1)qvv8*u.
Let T': ~(X X Y) -0 ~(X) (8) ~(Y) be a functorial chain equivalence
given by the Eilenberg-Zilber theorem and let
T:
[~(X) (8) ~(Y)] (8) [~(X) (8) ~(Y)]
-0
[~(X) (8) ~(X)] (8) [~(Y) (8) ~(Y)]
be the chain map defined by
T((c (8) d) (8) (c' (8) d')) = (_l)de g d deg c'(c (8) c') (8) (d (8) d')
If T is any diagonal approximation, it follows by the method of acyclic models
that the diagram
~(X X Y)
~
~(X X y) (8) ~(X X y)
~(X) (8) ~(Y)
TX(SSlT y )
[~(X) (8) ~(X)] (8) [~(Y) (8) ~(Y)]
is chain homotopy commutative. This implies the following additional relation
between cup products and cross products.
13 THEOREM Let cp: G I ® G 2 -0 G and G 1 (8) G 2 -0 G' be pairings and let
G I (8) G l and G 2 (8) G 2 be paired to G (8) G' by the homomorphism
(G I (8) Gl) (8) (G 2 (8) G2):::::; (G I (8) G2) (8) (G l (8) G2)
rp@rp')
G (8) G'
Given UI E HP(X,AI; G I ), U2 E Hq(X,A2; G2), VI E Hr(Y,BI: Gl ), and
V2 E HS(Y,B;; G 2) then with suitable excisiveness assumptions, we have, in
Hp+q+r+s((X, Al U A 2) X (Y, BI U B 2); G (8) G'),
(UI
X V1)
V
(U2 X V2)
= (_l)qr(UI v
U2) X
(VI
v V2)
•
SEC.
6
253
CUP AND CAP PRODUCTS
Combining theorem 13 with statements 3 and 9, we obtain the following
result expressing the cross product in terms of the cup products.
14 COROLLARY Let {X X B, A X Y} be an excisive couple in X X Y and let
Pl: (X,A) X Y ~ (X,A) and P2: X X (Y,B) ~ (Y,B) be the proiections. Given
u E Hp(X,A; G) and v E Hq(Y,B; G'), then, in Hp+q((X,A) X (Y,B); G ® G'),
we have
Uxv=pt(u)vp~(v)
•
With the last result we can give the following example of two polyhedra
having isomorphic homology and cohomology modules but not isomorphic
cup-product structures.
15 EXAMPLE Let p and q be integers ~ 1 and let X be the space which is
the union of SP, Sq, and Sp+q, all identified at one point. If i: SP C X, i: Sq C X,
and k: Sp+q C X, then i* H(SP) EEl i* H(sq) EEl k* H(Sp+q) ;::::: H(X). Computing
H(Sp X Sq) by the Kiinneth formula, we see that H(X) ;::::: H(Sp X Sq). By the
universal-coefficient theorem, X and sP X Sq have isomorphic homology and
cohomology groups for any coefficient group. Since
k*: Hp+q(X;Z) ;::::: Hp+q(sP+q;Z)
and k * commutes with the cup product, it follows that the cup product of
integral cohomology classes of degrees p and q, respectively, in X is zero.
However, it follows from corollary 14 that there are integral cohomology
classes of sP X Sq of degrees p and q, respectively, whose cup product is nonzero. Therefore H* (X;Z) and H* (Sp X Sq; Z) are not isomorphic by an isomorphism of graded modules preserving the cup product. Hence X and
SP X Sq are not homeomorphic, nor even of the same homotopy type.
There is another product closely related to the cup product that multiplies
homology and cohomology classes together. We begin with the observation
that if C and C' are chain complexes and G and G' are paired to Gil by cp,
there is a functorial homomorphism
h: Hom (C',G) ® (C ®
c' ® G') ~
C ® Gil
such that h(f® (c ® c' ® g')) = c ® cp(J,c') ® g'). A straightforward calculation shows that for J E Hom (C'q,G) and c E (C ® C')n ® G'
ah(f Q9 c) = (-1)n- q h(13J ® c)
If X is a space and
a functorial map
'7": ~(X) ~ ~(X)
®
~(X)
+ h(f Q9 ac)
is a diagonal approximation,
f: Hom (~(X),G) Q9 (~(X) ® G') ~ ~(X) ® Gil
is defined by f(f ® c)
= h(f Q9 '7"(c)). The boundary formula yields
af(f Q9 c)
= (_1)de g c-deg ff(13J ® c) + f(f ® ac)
Note that if A is a subset of X and J E Hom (~(X),G) vanishes on A, then for
O. It follows that if A1 , A2 C X,
any c E ~(A) ® G', f(f Q9 c)
=
254
PRODUCTS
CHAP.
5
f E Hom (Il(X)/ Il(Al),G) is a co cycle, and c E Il(X) ® G' is a chain such that
ac E [1l(Al) + Il(A2)] ® G', then t(f ® c) is a cha~n of Il(X) ® Gil whose
boundary is in Il(A2) ® Gil [because af(f ® c) = r(j ® ac)]. Furthermore, if
f is the coboundary of a cochain which vanishes on Il(A 1) or if c equals
a boundary modulo [1l(Al) + Il(A2)] ® G', then t(f ® c) is a boundary
modulo Il(A2) ® Gil. Hence r defines a homomorphism [sending {f} ® {c}
to {:r(f ® c)}]
Hq(X,Al; G) ®
Hn(~(X)/[~(Al)
+ ~(A2)];
G') ..... Hn-q(X,A 2; Gil)
If {A l ,A 2 } is an excisive couple in X, this yields a homomorphism
Hq(X,A 1 ; G) ® Hn(X, Al U A 2; G')
~
Hn- q(X,A2; Gil)
called the cap product. If u E Hq(X,A 1 ; G) and z E Hn(X, Al U A 2; G'), their
cap product is denoted by u r-, z E Hn_q(X,A 2; Gil). It depends on the pairing
cp but not on the particular diagonal approximation used to define :r. The
Alexander-Whitney diagonal approximation yields a map T which defines a
cap product on cochains and chains, denoted by f r-, c, by the formula
f
for fE Hom
{f_c}.
r-,
c
= f r-, (~(J
® g~) = ~n-q
(J ® cp(f,(Jq) ® g~)
a
a
(~q(X), G)
and c = :EO"
(F
®
g;" E ~n(X)
® G'. Then If} _ {c}
=
The cap product has the following properties analogous to those of the
cup product.
16 Let f: X ~ Y map Al to Bl and A2 to B2 and let u E Hq(Y,B 1 ; G) and
z E Hn(X, Al U A 2; G'). Let It: (X,Al) ~ (Y,B 1 ), fz: (X,A 2) ~ (Y,B2), and
(X, Al U A 2) ~ (Y, Bl U B 2) be maps defined by f Then, in Hn- q(Y,B2; Gil),
f
we have
fz*(fT u
r-,
z) = u
r-,
f* z •
17 For any z E Hn(X,A; G) with the pairing R ® G ;:::; G
lr-,z=z •
18 Given a commutative diagram, where cp, cp', 1/;, and 1/;' are pairings,
G 1 ® (G 2 ® G 3 );:::; (G1 ® G2 ) ® G3 ~ G 12 ® G3
1®
'I"t
~o/
for u E Hv(X,Al; Gl ), V E Hq(X,A2; G 2), and z E Hn(X, Al U A2 U A 3 ; G3),
then, in Hn- v- q(X,A 3 ; G123 ), we have
u
r-,
(v
r-,
z)
= (u v
v)
r-,
z •
19 Let u E Hq(X,A; G) and z E Hq(X,A; G') and let e: Ho(X; G ® G') ~
G ® G' be the augmentation. Then, in G ® G',
SEC.
7
255
HOMOLOGY OF FIBER BUNDLES
c(u (\ z) = <u,z)
•
20 Let {(XI,A I ), (X 2,A 2)} be an excisive couple in X and let A C Xl U X2 and
i: (Xl n X 2, A n Xl n X 2) C (Xl U X 2, A). For u E Hq(X I U X 2, A; G) and
z E Hn(XI U X 2, Al U A2 U A; G'), with the connecting homomorphisms of
the appropriate Mayer- Vietoris sequences, in Hn-q-I(X I n X 2, Al n A 2; Gil),
we have
a*(u (\ z)
= i*u (\ a*z
•
21 Let UI E HP(X,AI; G I ), U2 E Hq(Y,B I ; G 2), Zl E Hm(X, Al U A 2; Gi), and
Z2 E Hn(X, BI U B 2; G z), and let G I and Gi be paired to G'{, G 2 and G z be
paired to G 2, and (G I ® G 2) and (Gi ® G z) be compatibly paired to Gl' ® G 2.
Then, in Hm+n_p_q((X,A2) X (Y,B2); G~' ® G 2), we have
(UI
7
X U2) (\ (Zl X Z2) = (-l)p(n-q)(UI (\ Zl) X (U2 (\ Z2)
•
HOMOLOGY OF FIBER BUNDLES
Cup and cap products are used in this section to study the homology of fiber
bundles. We shall show that in case the cohomology of the total space maps
epimorphically onto the cohomology of each fiber, the homology (or cohomology) of the total space is isomorphic to the homology (or cohomology) of the
product space of the base and the fiber. For orientable sphere bundles this
leads to a proof of the exactness of the Thom-Gysin sequences, which will be
applied in the next section to compute the cohomology rings of projective
spaces.
We begin with some algebraic considerations. Let M = {Mq} be a free
finitely generated graded R module and let M * = {Mq = Hom (Mq,R)}. Let
(X,A) be a topological pair and f: X -7 Y be a continuous map. Given
a homomorphism (of degree 0) 8: M* -7 H*(X,A; R), there are homomorphisms (of degree 0) for any R module G
<1>: H(X,A; G)
<p*: H*(Y;G) ® M*
-7
-7
H(Y;G) ® M
H*(X,A; G)
defined by <1>(z) = '2.d* (8(mf) (\ z) ® mi, where {md is a basis of M and
{mT} is the dual basis of M * (<I> is uniquely defined by this formula), and
<I>*(u ® m*) = f*u v 8(m*).
1
LEMMA
With the notation above, if <I> is an isomorphism for G = R,
then <I> and <I> * are isomorphisms for all R modules G.
t
For each i let c be a co cycle of Hom (~(X)/ ~(A);R) representing
the class 8( mT ) and assume that mi (and hence also mT and cT ) have degree qi.
Let T: ~(X)/ ~(A) -7 ~(Y) ® M be the homomorphism (of degree 0) defined by
PROOF
T(C) = '2,. ~(f)(cT (\ c) ® mi
256
PRODUCTS
An easy computation shows that
homomorphisms
7'
CHAP.
5
is a chain map and that the induced
7'*: H*(X,A; G) ~ H*(i1(Y) ® M; G);:::; H*(Y;G) ® M
7'*: H*(Y;G) ® M* ;:::; H*(Hom (i1(Y) ® M, G» ~ H*(X,A; G)
equal <I> and <I>*, respectively. Since <I> is assumed to be an isomorphism for
G
R, the chain map 7' induces an isomorphism of homology. The universalcoefficient theorems for homology and cohomology then imply that <I> and <I>*
are isomorphisms for all G. •
=
A jiber-bundle pair with base space B consists of a total pair (E,E), a
jiber pair (F,F), and a protection p: E ~ B such that there exists an open
covering {V} of B and for each V E {V} a homeomorphism C]JV: V X (F,F) ~
(p-l(V), p-l(V) n E) such that the composite
V X F
.!4 p-l(V) ~ V
is the projection to the first factor. If A C B, we let EA = p-l(A) and
EA
p-l(A) n E, and if b E B, then (Eb,E b) is the fiber pair over b.
Following are some examples.
=
For a space B and pair (F,F) the product-bundle pair consists of the total
2
pair B X (F,F) with projection to the first factor.
3 Given a bundle projection p: E -> B with compact fiber P, let E be the
mapping cylinder of p and p: E -> B the canonical retraction. Then (E,E) is the
total pair of a fiber-bundle pair over B with fiber (F,F), where F is the cone
over P, and projection p.
4
If ~ is a q-sphere bundle over B, then (EE,EE) is the total pair of a fiberbundle pair over B with fiber (Eq+1,Sq) and projection Pt= E t ~ B.
Given a fiber-bundle pair with total pair (E,E) and fiber pair (F,F),
a cohomology extension of the fiber is a homomorphism 0: H*(F,F; R) ~
H* (E,E; R) of graded modules (of degree 0) such that for each b E B the
composite
is an isomorphism. The following statements are easily verified.
:;
Let p: B X (F,F) ~ (F,F) be the protection to the second factor. Then
0== p*: H*(F,F; R) ~ H*(B X (F,F); R)
is a cohomology extension of the fiber of the product-bundle pair.
•
8
Let 0: H* (F,F; R) ~ H* (E,E; R) be a cohomology extension of the fiber
of a fiber-bundle pair over B and let f: B' ~ B be a map. There is an induced
bundle pair over B', with total pair (E' ,E') and fiber (F,F), and there is a map
SEC.
f:
7
257
HOMOLOGY OF FIBER BUNDLES
(E',E') ~ (E,E) commuting with proiections. Then the composite
H*(F,P; R) ~ H*(E,E; R) ~ H*(E',E'; R)
is a cohomology extension of the fiber in the induced bundle.
•
7
Given a fiber-bundle pair over B with total pair (E,E), let the path components of B be {Bj} and let (E;,Ej) be the induced total pair over Bj.
A cohomology extension 8 of the fiber of the bundle pair over B corresponds
to a family of cohomology extensions {8j} of the induced bundle pairs
over Bj. •
We now establish the local form of the theorem toward which we are
heading. It shows that any cohomology extension of the fiber in a productbundle pair has homology properties as nice as the one given in statement 5
above.
8
LEMMA
Let (F,P) be a pair such that H * (F,P; R) is free and finitely
generated over R and let 8: H* (F,P; R) ~ H* (B X (F,P); R) be a cohomology
extension of the fiber of the product-bundle pair. Then the homomorphisms
IP: H*(B X (F,P); G) ~ H*(B;G) ® H*(F,P; R)
IP*: H*(B;G) ® H*(F,F; R) ~ H*(B X (F,P); G)
are isomorphisms for all R modules G.
PROOF
By lemma 1, it suffices to prove that IP is an isomorphism for G
If {Bj} is the set of path components of B, then
H* (B X (F,P); R) ;:::;
= R.
ffi H* (Bj X (F,P); R)
and
H* (B;R) ® H* (F,P; R) ;:::;
ffi H* (Bj;R) ® H* (F,P; R)
Therefore it suffices to prove the result for a path-connected space B. For
such a B, R ;:::; HO(B;R).
By the Kiinneth formula, H* (B X (F,P); R) ;:::; H* (B;R) ® H. (F,P; R).
We define graded submodules Ns of H* (B;R) ® H* (F,P; R) by
(Ns)q
= EB
Hi(B;R) ® Hj(F,P; R)
i+j=q.j~s
Then
H* (B;R) ® H* (F,P; R)
=
= No
::J Nl ::J ... ::J Ns ::J NS+l
=
and Ns
0 for large enough s. If u E HS(F,F; R), then 8(u)
1 X ;\(u) + ii,
where ii E ~+j=s.j<. Hi(B;R) ® Hj(F,P; R) and 8(u) I [b X (F,F)]
1 X ;\(u).
Because 8 is a cohomology extension of the fiber, ;\ is an automorphism of
H* (F,P; R). Let z' E H.(F,P; R) and consider z X z' EN•. Then
lP(z X z')
= ~ p* (8(mf)
1""\
=
(z X z')) ® mi
~
and if deg mi
< s, then 8(mt)
1""\
(z X z') E Nl and p. (Nl ) = O. Therefore
258
~(z
PRODUCTS
CHAP.
5
X z') ENs, and so ~ maps Ns into itself for all s. Because of the short
exact sequences
o ~ NS+l ~ Ns ~ Ns/NS+l ~ 0
and the five lemma, it follows by downward induction on s that ~ is an isomorphism if and only if it induces an isomorphism of NS/N.Hl onto itself for
all s. For z' E H..(F,F; R), computing ~(z X z') in Ns/NS+l' we obtain
~(z X z') =
~
degmi;o,s
p* [(1 X A(mt)
= deg ~mj=s p*[l
+ mf) '" (z
X A(mf) '" (z X z')]
X z')] ® mi
® mi
because mf '" (z X z') E Nl and p* (Nl) = O. Now, by properties 5.6.21,
5.6.19, and 5.6.17,
~
degmi=S
p* [1 X A(mf) '" (z X z')] ® mi
= deg ~
mi=S
z ® (A(mT),z')m;
= z ® A*(z')
where A* : H* (F,F; R) ~ H* (F,F; R) is the automorphism dual to A. Hence
~(z X z') = z X A* (z') in NS/N.Hl' showing that ~ induces an isomorphism
of N../Ns+l for all s. •
The following Leray-Hirsch theorem shows that fiber-bundle pairs with
cohomology extensions of the fiber have homology and cohomology modules
isomorphic to those of the product of the fiber pair and the base.
D THEOREM Let (E,E) be the total pair of a fiber-bundle pair with base B
and fiber pair (F,F). Assume that H* (F,F; R) is free and finitely generated
over R and that 0 is a cohomology extension of the fiber. Then the
homo.morphisms
~: H* (E,E; G) ~ H* (B;G) ® H* (F,F; R)
~* : H*(B;G)
®
H*(F,F; R)
-+
H*(E,E; G)
<I>*(u
®
v) = p*(u) ~ 8(v)
are isomorphisms (of graded modules) for all R modules G.
PROOF
G
By lemma 1, it suffices to prove the result for the map
C B let 0A be the composite
= R. For any subset A
~
in the case
H* (F,F; R) ~ H* (E,E; R) ~ H* (EA,EA; R)
Then OA is a cohomology extension of the fiber in the induced bundle over A.
It follows from lemma 8 that if the induced bundle over A is homeomorphic
to the product-bundle pair A X (F,P), then
~A: H* (EA,EA; R) ;::; H* (A;R) ® H* (F,F; R)
Hence ~v is an isomorphism for all sufficiently small open sets V.
If Vand V are open sets in B, then {(Ev,E v), (EV',EV')} is an excisive couple
of pairs in E, and it follows from property 5.6.20 that ~v, ~V', ~vnV',
and ~vuV' map the exact Mayer-Vietoris sequence of (Ev,Ev) and (EV',EV') into
SEC.
7
259
HOMOLOGY OF FIBER BUNDLES
the tensor product of the exact Mayer-Vietoris sequence of V and V' by
H* (F,P; R). Since H* (F,P; R) is free over R, its tensor product with any exact
sequence is exact. Therefore, if <l>v, <l>V', and <l>vnV' are isomorphisms, it follows
from the five lemma that <l>vuv' is also an isomorphism. By induction, <l>uis an
isomorphism for any U which is a finite union of sufficiently small open sets. Let
ql be the collection of these sets. Since any compact subset of B lies in some
element of "11, H* (B;R) :::::; lim~ {H* (U;R)} UE01' Also, any compact subset of E
lies in Eu for some U E Gil, so H* (E,E; R) :::::; lim~ {H* (Eu,Eu; R)}. Because
the tensor product commutes with direct limits and <I> corresponds to
lim~ {<I>U}UE"lL under these isomorphisms, <I> is also an isomorphism. The above argument proves directly that <I> is an isomorphism for any
coefficient module G. A similar argument does not appear possible for <I> * ,
because it is not true that H* (B;R) is isomorphic to the inverse limit
lim~ {H* (U;R)}UE"lL' It should be noted that in theorem 9 we have said
nothing about commutativity of <I> * with cup products, because it is not true,
in general, that <I> * preserves cup products.
We now specialize to the case of sphere bundles. Because
r=/=q+l
r=q+l
if ~ is a q-sphere bundle, a cohomology extension of the fiber in ~ is an element U E Hq+l(E~,E~; R) such that for any b E B, the restriction of U to
(p-l(b), p-l(b) n E) is a generator of Hq+l(p-l(b), p-l(b) n E; R). Such a
cohomology class is called an orientation class (over R) of the bundle. If
orientations of the bundle exist, the bundle is called orientable. An oriented
sphere bundle is a pair (~, U~) consisting of a sphere bundle ~ and an orientation
class of U~ of ~.
If U is an orientation class of ~ over Z and if 1 is the unit element of R,
then p,( U (8) 1) is an orientation class of ~ over R. Therefore a sphere bundle
orientable over Z is orientable over any R.
If (~, U~) is an oriented sphere bundle over Band f: B' ~ B, then
(f*~,f* U~) is an oriented sphere bundle over B' [wheref: (Ef*~,Ef*~) ~ (E~,E~)
is associated to fl.
From theorem 9 we get the following Thom isomorphism theorem.
10 THEOREM Let (~, U~) be an oriented q-sphere bundle over B. There are
natural isomorphisms for any R module G
<I>~: Hn(E~,E~; G) :? Hn_q_1 (B;G)
<I>~(z)
<1>(* : Hr(B;G) ;:? Hr+q+1(E~,E~; G)
<l>t(v) = p*v v U~
= p* (U~
r,
z)
Let m and m* be dual generators of Hq+1(Eq+1,Sq; R) and
Hq+1(Eq+l,Sq; R), respectively, and define a cohomology extension 8 by
8(m *) = U~. Then <I>~ is the composite
PROOF
Hn(E(,E(; G) ~ Hn_q_1 (B;G) (8) Hq+l(Eq+1,Sq;R) :::::; Hn_q_1 (B;G)
where the second map sends z ® m to z. By theorem 9, <I> is an isomorphism,
260
PRODUCTS
CHAP.
5
and so <I>~ is an isomorphism. A similar argument shows that <I>~* is an isomorphism. These isomorphisms are natural for induced bundles because of
naturality properties of the cup and cap products. This result implies the exactness of the following Thom-Gysin sequences
of a sphere bundle.
I I THEOREM Let (~, V~) be an oriented q-sphere bundle with base Band
proiection P = pIE: E ~ B. For any R module G there are natural
exact sequences
... ~ Hn(E~;G) ~ Hn(B;G) ~ Hn_q_1 (B;G) -4 Hn_l(E~;G) ~ ...
...
~
Hr(B;G)
p*
.
p*
~ Hr(E~;G) ~
'i' •
Hr-q(B;G) -4 HT+1(B;G)
~
...
in which 'I' ~ and 'I' ~* have properties
'I'~(v ~ z) = (-l)(q+1) deg v 'I'~* (v)
'1'.;* (VI v V2) = VI v 'I'~*(V2)
PROOF
~
Z
There is a commutative diagram (with any coefficient module)
. .. ~ Hn(E) ~ Hn(E) ~ Hn(E,E) ~ Hn- 1 (E) ~
the top row of which is exact. Since p is a deformation retraction of E onto B,
p* is an isomorphism. By theorem 10, <I>~ is an isomorphism. The desired sequence is obtained by defining 'I'~ = <l>d*p* -1 and p = 0<1>.;-1. Similarly, the
cohomology sequence is defined by 'I' ~* = p* -Ii * <1>.;* and p* = <I>~* -18.
We verify the formula for
'I',,(v
~
z)
'I'~.
= <l>d*p* -l(V ~ z) = <l>d* (p* (v) ~ p* -l(Z))
= <I>~(p*(v) ~ i*p* -l(Z)) = p*(V ~ [P*(v) ~ i*p* -l(Z)])
= p*(i*[Vvp*(v)] ~P* -l(Z))
= (_l)(q+1) deg v p* [;* <l>l (v)
= (_l)(q+1)de g v'l'.;*(v) ~ Z
~
p* -l(Z)]
-
Note that the isomorphisms <I> and <I> * of the Thorn isomorphism theorem
depend on the choice of the orientation class V of the bundle. Therefore the
homomorphisms p and 'I' and p* and 'I' * of the Thom-Gysin sequences also
depend on the orientation class. In case B is path connected and V and V'
are orientation classes of a sphere bundle over B, it follows from theorem 10
that there is an element r E R such that
V'
= p*(r X
1) v V
= r[p*(l) v
V]
If bo E B, then
V' I (p-l(b o), p-l(bo)
n E) = r[V I (p-l(b o), p-l(bo) n E)]
Therefore we have the next result.
SEC.
7
261
HOMOLOGY OF FIBER BUNDLES
12 LEMMA Two orientation classes U and U' of a sphere bundle over
a path-connected base space B are equal if and only if for some b o E B
U I (p-l(bfj), p-l(b o)
n E) =
U' I (p-l(b o), p-l(b o)
n E) •
If B is not path connected, let {Bj} be the set of path components of B
and let (E),E j ) be the part of (E,E) over Bi . Then
H*(E,E; R) = Xj H*(Ej,~; R)
and we also obtain the following result.
13 LEMMA Two orientation classes U and U' of a sphere bundle with base
space B are equal if and only if for all b E B
U I (p-l(b), p-l(b) n E) = U' I (p-l(b), p-l(b) n E)
•
In case R = Z2, then Hq+l(p-l(b), p-l(b) n E; Z2) ::::: Z2 for all b E B.
Therefore this module has a unique nonzero element, and we obtain the following consequence of lemma 13.
14
COROLLARY
equal.
Any two orientation classes over Z2 of a sphere bundle are
•
Thus, for R = Z2 the homomorphisms <1>, p, and 'I' and <1> * , p * , and 'I' *
are all unique.
The characteristic class Q< of an oriented q-sphere bundle (~, Uf,) is
defined to be the element
Q< = '1'<* (1) E Hq+l(B;R)
This is functorial (that is, Qr*< = f* Q<). From the multiplicative properties of
'I' < and '1'<* in theorem 11 we obtain the following equations.
15 For
Z
E Hn(B;G)
'¥«z) = Q< ,..., z
and for v E W(B;G)
We now investigate the existence of orientation classes for a sphere
bundle. Let (X,X') be a pair and let {Ai}i EJ be an indexed collection of subsets Ai C X. An indexed collection
{ui E Hn(Ai' Ai
is said to be compatible if for all
ui I (Ai
f, l'
n X'; G)}iEJ
EJ
n Ai', Ai n Ai' n X') = Ui' I (Ai n Ai" Ai n Ai' n X')
The compatible collections {Ui} constitute an R module Hn( {Ai },X'; G).
Clearly, the restriction maps
Hn(X , X'·, G)
-i>
Hn(A·J' A·J
n
X'., G)
262
PRODUCTS
define a natural homomorphism Hn(X,X'; G)
~
CHAP.
5
Hn( {Aj} ,X'; G).
16 LEMMA Let (E,E) be a fiber-bundle pair with base B, proiection p: E ~ B,
and fiber pair (F,F). Assume that for some n
0, Hi(F,F; R) = 0 for i
n.
Then
<
>
(a) For all A C B and all R modules G
Hi(P-1(A), p-1(A)
n E; G)
= 0 = Hi(p-1(A), p-1(A) n E; G)
i
<n
(b) If {V} is any open covering of B, then in degree n the natural homomorphism is an isomorphism
Hn(E,E; G) ;:::: Hn( {p-1 V},E; G)
By the universal-coefficient formula, it suffices to prove (a) for G = R.
If A C B is such that (p-1(A), p-1(A) n E) is homeomorphic to A X (F,F),
then by the Kiinneth formula,
PROOF
Hi(P-1(A), p-1(A)
n E; R) ;:::: Hi(A X (F,F); R)
=0
i
<n
From this it follows (as in the proof of theorem 9) by induction on the
number of coordinate neighborhoods of the bundle needed to cover A (using
the Mayer-Vietoris sequence and the five lemma) that (a) holds for all compact A C B. By taking direct limits, (a) holds for any A.
For (b), let {W} be the collection of finite unions of elements of {V}.
By (a) and the universal-coefficient formula for cohomology, there is a commutative diagram
::::: Hom (Hn(E,E;R), G)
Hn(E,E; G)
!::::
!
lim~{Hn(p-1(W), p-l(W)
n
E; G)} ::::: lim~{Hom (Hn(p-l(W), p-l(W)
n
E; R),G)}
Hence we need only prove that a compatible collection {uv} V E {V} extends to a
unique compatible collection {UW}WE {W}. This follows by using Mayer-Vietoris
sequences again and from the fact that Hi(p-1(W), p-1(W) n E; G) = 0 for
i
n. •
<
For sphere bundles we have the following immediate consequence.
17 COROLLARY A sphere bundle ~ with base B is orientable if and only if
there is a covering {V} of B and a compatible family {uv}, where Uv is an
orientation class of ~ I V for each V E {V}. •
Since a trivial sphere bundle is orientable, corollaries 17 and 14 imply
the following result.
18
COROLLARY
Any sphere bundle has a unique orientation class over Z2.
•
By theorem 2.8.12, there is a contravariant functor from the fundamental
groupoid of the base space B of a sphere bundle ~ to the homotopy category
which assigns to b E B the fiber pair (Eb,Eb) over b and to a path class [w] in
B a homotopy class h[w] E [E.,(o),E.,(o); E.,(1),E.,(1)]' For fixed R there is then a
SEC.
8
263
THE COHOMOLOGY ALGEBRA
covariant functor from the fundamental groupoid of B to the category of
R modules which assigns to b E B the module Hq+l(Eb,E b; R) and to a path
class [w] the homomorphism
h[w] *: Hq+l(Ew(l»Ew(l); R) ~ Hq+l(Ew(o»Ew(o); R)
19 THEOREM A sphere bundle ~ is orientable over R if and only if for
every closed path w in B, h[ w] * = 1.
If ~ is orientable with orientation class U E Hq+l(E,E; R), for any
small path w in B (and hence for any path)
PROOF
h[w] * (U I (Ew(l»Ew(l»))
= U I (Ew(o),Ew(o»)
Since U I (Eb,E b) is a generator of Hq+l(Eb,E b; R), this implies that h[w] * = 1
for any closed path w.
Conversely, if h[ w] * = 1 for every closed path w in B, there exist generators
Ub E Hq+l(Eb,E b; R) such that for any path class [w] in B, h[ w] * (Uw(l») = Uw(O)'
If V is any subset of B such that ~ I V is trivial, it is easy to see that there is an
orientation class Uv of ~ I V such that Uv I (Eb,E b) = Ub for all b E V. If {V}
is an open covering of B by sets such that ~ I V is trivial for all V, then {Uv }
is a compatible family of orientations, and by corollary 17, ~ is orientable. 20
COROLLARY
over any R.
8
A sphere bundle with a simply connected base is orientable
-
THE COHOMOLOGY ALGEBBA
The cup product in cohomology makes the cohomology (over R) of a topological pair a graded R algebra. In the first part of this section we define the
relevant algebraic concepts and compute this algebra over Z2 for a real projective space and over any R for complex and quaternionic projective space.
This is applied to prove the Borsuk-Ulam theorem.
For the case of an H space, there is even more algebraic structure that
can be introduced in the cohomology algebra. The cohomology of such a
space is a Hopf algebra, and the second part of the section is devoted to its
definition and some results about its structure. The section concludes with
a proof of the Hopf theorem about the cohomology algebra of a compact connected H space.
A graded R algebra consists of a graded R module A = {Aq} and a
homomorphism of degree 0
p,:A®A~A
called the product of the algebra (p, then maps Ap ® A q into Ap+q for all
p and q). For a, a' E A we write aa' = p,(a ® a'). The product is associative
if (aa')a" = a( a' a") for all a, a', a" E A and is commutative if
aa' = (_l)deg a deg a' a' a for all a, a' EA.
264
PRODUCTS
CHAP.
5
I
EXAMPLE
If (X,A) is a topological pair, then H * (X,A; R) is a graded
R algebra whose product is the cup product (with respect to the multiplication pairing of R with itself to R). It follows from property 5.6.10 that this
product is associative and from property 5.6.11 that it is commutative. If
A = 0, it follows from property 5.6.9 that 1 is a unit element of the algebra
H* (X;R). H* (X,A; R) is called the cohomology algebra of (X,A) over R.
2
n
EXAMPLE
The polynomial algebra over R generated by x of degree
> 0, denoted by Sn(x), is defined by
[Sn(x)]q =
{ofree
R module generated by Xp
q F 0 (n) or q
q = pn, p ::::: 0
<0
with the product (axp )(f3xq ) = (af3)xp + q for a, f3 E R. It is then clear that
Xo is a unit element and that Xp = (XI)P. If we denote Xl by x, then Xp = xp.
Thus, disregarding the graded structure, Sn(x) is simply the polynomial algebra over R in one indeterminate x. The truncated polynomial algebra over R
generated by x of degree n and height h, denoted by Tn,h(X), is defined to be
the quotient of Sn(x) by the graded ideal generated by xh. If h = 2, this
is called the exterior algebra generated by x of degree n and is denoted
by En(x).
If A and B are graded R algebras, their tensor product A ® B is also a
graded R algebra with product
(a ® b)(a' ® b' ) = (_I)de g b deg a'aa' ® bb'
If A and B have associative or commutative products, so does A ® B.
3
EXAMPLE
If R is a field and (X,A) and (Y,B) are topological pairs such
that either H* (X,A; R) or H* (Y,B; R) is of finite type, it follows from theorem
5.5.11 that
H*(X,A; R) ® H*(Y,B; R);::::; H*((X,A) X (Y,B); R)
We compute the graded Zz algebra H*(pn;Zz) for real projective space
pn. Note that the double covering p; Sn -7 pn is a O-sphere bundle. We let
Wn E HI(pn;ZZ) be the characteristic class (over Zz) of this bundle.
4
THEOREM
For n ::::: 1, H*(pn;Zz) is a truncated polynomial algebra over
Zz generated by Wn of degree 1 and height n + 1.
All coefficients in the proof will be Zz and will be omitted. By
corollary 5.7.18 and theorem 5.7.11, there is an exact Thom-Gysin sequence
PROOF
. . . -7
Hq(Sn) ~ Hq(pn) ~ Hq+l(pn) ~ Hq+I(Sn)
starting on the left with 0
-7
-7 . . .
HO(pn) E..";. HO(Sn) and terminating on the right
with Hn(Sn) ~ Hn(pn) -7 0 [note that Hq(pn)
polyhedron of dimension n]. Because Hq(Sn)
that
= 0 for q > n, because pn is a
= 0 for 0 < q < n, it follows
SEC.
8
265
THE COHOMOLOGY ALGEBRA
<
is an epimorphism for 0:::; q
n - 1 and is a monomorphism for
0< q :::; n - 1. Because pn and Sn are connected for n ~ 1, p* HO(pn) = HO(Sn),
which implies that '1'*: HO(pn) ~ Hl(pn) is also a monomorphism. Therefore
Hq(pn) -=1= 0 for 0 :::; q :::; n, and because p* Hn(Sn) = Hn(pn) and Hn(Sn) ;::::; Z2,
it follows that p* is a monomorphism and that '1'*: Hn-l(pn) ~ Hn(pn) is also
an epimorphism.
We have shown that for 0 :::; q :::; n - 1
'1'*: Hq(pn) ;::::; Hq+l(pn)
Then Wn = '1'*(1) is the nonzero element of Hl(pn), and by equation 5.7.15,
'I'*(wnq) = wnq+l. Therefore, for 1 :::; q :::; n, wnq is the nonzero element of
Hq(pn). •
By corollary 3.8.9, Pn(C) and Pn(Q) are simply connected. It follows
from corollary 5.7.20 that the Hopf bundles S2n+1 ~ Pn(C) with fiber Sl and
S4n+3 ~ Pn(Q) with fiber S3 are orientable over any R. Let Xn E H2(Pn(C);R)
and Yn E H4(Pn(Q);R) be the characteristic classses of these Hopf bundles
(based on some orientation class of each bundle). An argument analogous to
that of theorem 4, using the Thom-Gysin sequences of the Hopf bundles,
'establishes the following result.
S THEOREM For n ~ 1, H* (Pn(C);R) is a truncated polynomial algebra
over R generated by Xn of degree 2 and height n + 1, and H* (Pn(Q);R) is a
truncated polynomial algebra over R generated by Yn of degree 4 and height
n + 1. •
6
COROLLARY
Let n > m ~ 1 and let i: pm C pn be a linear imbedding.
Then for q:::; m
i *: Hq(pn;Z2) ;::::; Hq(pm,Z2)
The hypothesis that i is a linear imbedding implies that the O-sphere
bundle over pm induced by i from the double covering Sn ~ pn is the double
covering Sm ~ pm. By the naturality of the characteristic class, i * Wn = W m .
The result now follows from theorem 4 and the fact that i * (wnq) = (i * wn)q. •
PROOF
7
COROLLARY
a map f': pn
covering.
~
>
Let n
m ~ 1 and let f: pn ~ pm be a map. There exists
Sm such that p f' = f, where p: Sm ~ pm is the double
0
PROOF
By the lifting theorem 2.4.5, it suffices to prove f #( 7T(pn)) = O.
If m = 1, this follows from the fact that 7T(pn) = Z2 and 7T(Pl) = Z. Assume
that m
1 and observe that because Hl(pn) has just the two elements 0 and
W n, either f* (w m) = 0 or f* (w m) = W n· Because f* is an algebra homomorphism, the latter is impossible [since 0 -=1= w nm+1 and f* (w mm+1 ) = 0]. Therefore f* (w m ) = O.
We know that 7T(pn) = Z2, and a generator for this group is the homotopy class of the linear inclusion map i: pI C pn. Because f* (w m ) = 0, it follows that i * f* (w m) = O. If i: pI C pm is the linear inclusion map, by
>
266
PRODUCTS
CHAP.
5
corollary 6, i * (w m) =1= O. Since (f i) * (w m) =1= i * (w m), f i is not homotopic
to j. Since 'IT(pm) = Zz, f i is null homotopic. Hence f#[i] = [f i] = 0, and
so f#('lT(pn)) = 0 in this case also. •
0
0
0
0
>
8
COROLLARY
For n
m 2:: 1 there is no continuous map g: Sn _
such that g( -x) = -g(x) for all x E Sn.
Sm
PROOF
If there were such a map, it would define a map J: pn _ pm such
that the following square (where p and p' are the double coverings) is
commutative
p'l
lp
pnLpm
By corollary 7, f can be lifted to a map 1': pn _ Sm. Then
P1'p'
= fp' = pg
Therefore 1'p' and g are liftings of the same map. For any x E Sn either
g(x) = 1'P'(x) or g( -x) = 1'P'(x) = 1'P'( -x). In any event, 1'P' and g must
agree at some point of Sn. By the unique-lifting property 2.2.2, 1'P' = g. This
is a contradiction, because for any x E Sn, p' maps x and - x into the same
point, while g maps them into separate points. •
This last result is equivalent to the Borsuk-Ulam theorem, which is next.
9
THEOREM
Given a continuous map J: Sn _ Rn for n
x E Sn such that f(x) = f( -x).
PROOF
2::
1, there exists
Assume there is no such x and let g: Sn _ Sn-l be the map defined by
f(x) - f( -x)
g(x) = IIf(x) - f(-x)11
Then g( - x) = - g(x), which would contradict corollary 8.
•
Dual to the concept of graded R algebra is that of graded R coalgebra,
which is defined by dualizing the concept of product. A graded R coalgebra
consists of a graded R module A = {Aq} and a homomorphism of degree 0
d: A_A ®A
called the coproduct of the coalgebra (so d maps A q into Gji+j=q Ai ® Aj for
all q). The coproduct is said to be associative if
(d ® l)d = (1 ® d)d: A _ A ® A ® A
and is said to be commutative if Td = d, where T: A ® A _ A (8) A is the
homomorphism T(a ® a') = (_l)de g a deg a'a' ® a. A counit for the coalgebra
is a homomorphism f: A _ R (where R is regarded as a graded R module
SEC.
8
267
THE COHOMOLOGY ALGEBRA
consisting of R in degree 0) such that each of the composites
E®l~R ®A
A ~ A ®A
~
l®~A
'i.'-
A
®R?
is the identity map.
A Hopf algebra over R is a graded R algebra B which is also a coalgebra
whose coproduct
d: B----') B ® B
is a homomorphism of graded R algebras. A Hopf algebra B is said to be connected if BO is the free R module generated by a unit element 1 for the
algebra and the homomorphism e: B ----,) R defined by e(al)
a for a E R is
a counit for the coalgebra.
=
10 EXAMPLE If X is a connected H space whose homology over a field R is
of finite type, then the multiplication map }L: X X X ----,) X defines a coproduct
d =
}L *:
H* (X;R) ----,) H* (X;R) ® H* (X;R)
H* (X;R) with this coproduct is a connected Hopf algebra of finite type whose
product is associative and commutative (the fact that X has a homotopy unit
Xo implies that the map H* (X;R) ----,) H* (xo;R) ::::; R is a counit).
We shall study connected Hopf algebras having an associative and commutative product and describe the algebra structure of those which are of
finite type over a field of characteristic O. The following is the inductive step
of the structure theorem toward which we are heading.
I I LEMMA Let B be a connected Hopf algebra with an associative and
commutative product over a field R of characteristic O. Let B' be a connected
sub Hopf algebra of B such that B is generated as an algebra by B' and some
element x E B - B'. If x has odd degree n, then as a graded algebra
B::::; B' ® En(x) and if x has even degree n, then as a graded algebra
B ::::; B' ® Sn(x).
PROOF Because B' is a sub Hopf algebra of B, the unit element of B belongs
to B'. Since x E B - B', x has positive degree n. Let A be the ideal in B generated by the elements of positive degree in B', and if 1J: B ----,) B/A is the
projection, let
= (1 ® 1J)d: B ----,) B ® B ----,) B ® (B/A)
Then d' is an algebra homomorphism, d'(f3) = f3 ® 1 for f3 E B', and d'(x)
d'
=
x ® 1 + 1 ® 1J(x). Note that x ¢ A, because A consists of finite sums ~i~O fJixi,
where f3i E B' is of positive degree, so f3ixi is of degree larger than n unless
i = O. Therefore 1J(x) =1= 0 in B/A.
Assume that x is of odd degree. Because B has a commutative product
and R has characteristic different from 2, x 2 = O. We show that there is no
268
PRODUCTS
relation of the form f30
such a relation, then
o = d'(f3o +
= f31
+
f31X)
® 1I(X)
f31X
= 0 with f3o,
= f30 ®
1
CHAP.
5
f31 E B' and f31 =1= O. If there were
+ (f31 ® l)[x ®
1
+
1 ® 1I(X)]
Since 1I(X) =1= 0, this implies f31 = 0, which is a contradiction. Therefore the
homomorphism B' ® En(x) ~ B sending f3 ® 1 to f3 and f3 ® x to f3x is an
isomorphism of graded algebras.
Assume that x is of even degree. We shall show that there is no relation
of the form ~O~i~T f3ixi = 0 with f3i E B', r ::::: 1, and f3T =1= O. If there were
such a relation, consider one of minimal degree in x. Then
o=
d'(~ f3iXi) = ~ (f3i ® l)[x ® 1
+1®
1I(X)]i
= (~ if3i Xi - 1 ) ® 1I(X) + ... + f3T ® (1I(X))'
The only term on the right in B ® (B/A)n is the term (~if3iXi-l) ® 1I(X).
It must be 0, and because 1I(X) =1= 0, ~ if3ixi-1 = O. If r
1, this is a relation
of smaller degree in x (note that rf3r =1= 0 because R has characteristic 0), and
this is a contradiction. If r = 1, we get f31 = 0, which is also a contradiction.
Therefore there is no relation, and the homomorphism B' ® Sn(x) ~ B
sending f3 ® xq to f3~ for f3 E B' and q ::::: 0 is an isomorphism of graded
algebras. -
>
We use this result to establish the following Leray structure theorem for
Hopf algebras over a field of characteristic 0 1 .
12 THEOREM Let B be a connected Hopf algebra with an associative and
commutative product and of finite type over a field R of characteristic O. As
a graded R algebra either B ;::::; R or B is the tensor product of a countable
number of exterior algebras with generators of odd degree and a countable
number of polynomial algebras with generators of even degree.
Because B is of finite type, there is a countable sequence
1 = Xo, xl, X2, . . . of elements of B such that i
implies that deg Xi :S deg Xi
and such that as an algebra B is generated by the set {Xi}j:o.O. For n ::::: 0 let
Bn be the sub algebra of B generated by Xo, Xl, . . . , xn. We can also assume
that Xn+1 does not belong to Bn. Because of the condition that deg Xi is a nondecreasing function of ;, each Bn is a connected sub Hopf algebra of B (that is,
d maps Bn into Bn ® Bn). Since Bn+1 is generated as an algebra by Bn and
Xn+1, lemma 11 applies. Since Bo ;::::; R, Bl ;::::; R ® E(X1) or B1 ;::::; R ® S(X1)'
Therefore B = Bo ;::::; R or B1 is either an exterior algebra on an odd-degree
generator or a polynomial algebra on an even-degree generator. By induction
on n, using lemma 11, each Bn+1 is a tensor product of the desired form.
Since B has finite type, B ;::::; lim~ Bn, and B has the desired form. PROOF
<;
1 A structure theorem valid over a perfect field of arbitrary characteristic can be found
in A. Borel, Sur la cohomologie des espaces fibres principaux et des espaces homo genes
de groupes de Lie compacts, Annals of Mathematics, voL 57, pp. 115--207, 1953.
SEC.
9
269
THE STEENROD SQUARING OPERATIONS
For a connected H space whose homology is finitely generated over a
field F no polynomial algebra factors can occur in the above structure theorem,
and we obtain the following Hopf theorem on H spaces.
Let X be a connected H space whose homology over a field
R of characteristic 0 is finitely generated. Then the cohomology algebra of X
over R is isomorphic to the cohomology algebra over R of a product of a
finite number of odd-dimensional spheres. •
13
COROLLARY
In particular, we obtain the following result about spheres that can be
H spaces.
14
COROLLARY
H space.
9
No even-dimensional sphere of positive dimension is an
•
THE STEENROD SQUARING OPERATIONS
In the last section the cup product in cohomology was used to prove the
Borsuk-Ulam theorem, a geometric result. Any other algebraic structure which
can be introduced into cohomology (or homology) and which is functorial can
be similarly applied. A particular example of such an additional algebraic
structure is a natural transformation from one cohomology functor to another.
These natural transformations are called cohomology operations. In this section we introduce the concept of cohomology operation and define the particular set of cohomology operations called the Steenrod squares.
Let p and q be fixed integers and G and G' fixed R modules. A cohomology
operation B of type (p,q; G,G') is a natural transformation from the functor
HP( ;G) to the functor Hq( ;G') (both functors being contravariant singular
cohomology functors defined on the category of topological pairs). Thus B
assigns to a pair (X,A) a function (which is not assumed to be a homomorphism)
B(X,A):
HP(X,A; G)
~
Hq(X,A; G')
such that if f: (X,A) ~ (Y,B) is a map, there is a commutative square
Hp(Y,B; G)
B(y.B»
Hq(Y,B; G')
f* ~
Hp(X,A; G)
~f*
B(x,A»
Hq(X,A; G')
A homology operation is defined Similarly, but we shall not discuss homology
operations.
Following are some examples.
1 If C[!: G ~ G' is a homomorphism,
(q,q; G,G') for every q, where
C[!*:
Hq(X,A; G)
C[!*
~
is a cohomology operation of type
Hq(X,A; G')
270
PRODUCTS
CHAP.
5
is defined as in Sec. 5.4. <p* is called the operation induced by the coefficient
homomorphism <po
2 Given a short exact sequence of R modules 0 - G' _ G - G" - 0,
the Bockstein cohomology operation f3* of type (q, q + 1; G",G') for every q
is defined to equal the Bockstein homomorphism
f3*: Hq(X,A; G") _ Hq+1(X,A; G')
corresponding to the coefficient sequence 0 _ G' _ G _ G" _ 0 as defined
in theorem 5.4.11.
For any p and q there is an operation 8p of type (q,pq; R,R), called the
3
pth-power operation, defined by
An operation 8 is said to be additive if 8(X,A) is a homomorphism for
every (X,A). The operations in examples 1 and 2 are additive; however, the
operation 8p of example 3 is not additive, in general.
Any cohomology operation provides a necessary condition for a homomorphism between the cohomology modules of two pairs to be the induced
homomorphism of some continuous map between the pairs. For example, if 8
is of type (p,q; G,G), a necessary condition that a homomorphism
1/;: H*(Y,B;
G) _ H*(X,A; G)
be induced by some map f: (X,A) _ (Y,B) is that
1/;8(y,B)
=
8(x,A)1/;: Hp(Y,B; G) _
Hq(X,A; G)
In these terms the algebraic idea underlying corollaries 5.8.7 and 5.8.8 is that
for n
m 2': 1 there is no homomorphism
>
1/;: H* (pm;Z2) -
H* (pn;Z2)
such that 1/; sends the nonzero element of Hl(pm;Z2) to the nonzero element
of H1(pn;Z2) and commutes with the (m + l)st-power operation 8m + 1 of type
(1, m + 1; Z2,Z2).
We shall now define a sequence of operations Sqi called the Steenrod
squares, each Sqi being a cohomology operation of type (q, q + i; Z2,Z2) for
every q. These operations include the squaring operation 8 2 and are related
to it by "reducing" the value of 82 (u) in a certain way. For this reason, the
operations Sqi are also called the reduced squares.
For the remainder of this section we make the assumption that all
modules are over Z2 and all homology and cohomology modules have coefficients Z2. The Steenrod squares, or reduced squares, {Sqi h;:.o are additive
cohomology operations
SEC.
9
271
THE STEENROD SQUARING OPERATIONS
defined for all q such that
(a) SqO = 1.
(b) If deg u = q, then Sqqu = u v u.
(c) If q > deg u, then Sqqu = O.
(d) If u E H*(X,A) and v E H*(Y,B) and IX X B, A X Y} is an excisive
couple in X X Y, the following Cartan formula is valid:
Sqk(U X v) = . ~ Sqiu X Sqiv
'+J=k
The above properties characterize the cohomology operations Sqi. We
shall not prove the uniqueness 1 , but shall content ourselves with their construction. First we establish a formula equivalent to the Cartan formula.
4
LEMMA
If U, v E H* (X,A), then
v) = . ~ Sqiu v Sqiv
'+J=k
PROOF
Since u v v = d* (u X v), where d: (X,A) -7 (X,A) X (X,A) is the
diagonal map, this follows from the Cartan formula and functorial properties
of Sqi. •
Sqk(u
v
For any chain complex C let T: C ® C -7 C ® C be the chain map
interchanging the factors [T( Cl ® C2) = C2 ® Cl is a chain map over Z2J.
5 LEMMA There exists a sequence {Ddi;>O of functorial homomorphisms
Di : ~(X) -7 ~(X) ® ~(X) of degree i such that
(a) Do is a chain map commuting with augmentation.
(b) For i > 0, ODi + Dio + Di - 1 + TDj-l = O.
If {Dj} and {Dj} are two such sequences, there exists a sequence {Ej }j;>o of
functorial homomorphisms Ej: ~(X) -7 ~(X) ® ~(X) of degree i such that
(c) Eo = O.
(d) For i ~ 0, oEj+1 + Ej+1o
+ Ej + TEj + Dj + Dj
= O.
We use the method of acyclic models. Let R be the group ring of Z2
over the field Z2. We regard R as the quotient ring of the polynomial ring Z2(t)
modulo the ideal generated by the polynomial t 2 + 1 = O. Thus the elements
of R have the form a + bt, where a and b E Z2.
Let Z2 be regarded as a trivial R module (that is, the element t of R induces the identity map of Z2) and let C be the free resolution of Z2 over R in
which C q is free with one generator d q for all q 2': 0 and which has boundary
operator o(dq) = (1 + t)dq_1 for q 2': 1 and augmentation e(do) = 1. The
functor which assigns to a space X the chain complex ~(X) ® C is augmented
PROOF
zz
and free over R with models {~q} q;>O and basis Uq ® dj}. We regard
For a proof see N. Steenrod and D. Epstein, Cohomology operations, Annals of Mathenwtics
Studies No. 50, Princeton University Press, Princeton, N.J., 1962.
1
272
PRODUCTS
CHAP.
5
il(X) ® il(X) as a chain complex over R, with t acting on il(X) ® il(X) in the
Z2
same way T does. Then il(X) ® il(X) is augmented and acyclic, with models
{ilq}q",o, It follows from theorem 4.3.3 (which is valid for chain complexes
over R) that there exist natural chain maps T: Ll(X) ® C ~ il(X) ® il(X) preserving augmentation, and any two are naturally chain homotopic.
A map T: il(X) ® C ~ il(X) ® il(X) of degree 0 corresponds bijectively
to a sequence of maps
D j : il(X) ~ il(X) ® il(X)
; ~ 0
of degree; such that Die) = T(e ® dj ). Then T is a chain map preserving
augmentation if and only if {Dd satisfies (a) and (b). Thus there exist families
{Dj} satisfying (a) and (b), and any such family corresponds to some T.
Similarly, a map H: il(X) ® C ~ il(X) ® il(X) of degree 1 corresponds
bijectively to a sequence of maps
Ej: Ll(X)
~
Ll(X) ® Ll(X)
;
~
0
of degree; such that Eo = 0 and Ej(e) = H(e ® dj - 1 ) for; ~ 1. Then H is a
chain homotopy from T to T' if and only if {Ej} satisfies (e) and (d) for the
sequences {Dj} and {Dj} corresponding to T and T', respectively. Thus,
if {Dj} and {Dj} are two sequences satisfying (a) and (b), there is a sequence
{Ej} satisfying (e) and (d) . •
Given a sequence {Dj }hO as in lemma 5, we define homomorphisms
Dj :Hom (.l(X) Q9 .l(X),
Z2)
-->
Hom (.l(X), Z2)
of degree -; by (Dj*f)(a) = f(Dja) for a E Llq(X) and f E Hom (Ll(X) ®
il(X), Z2). If e* E Hom (ilq(X), Z2) is a q-cochain of il(X), then
e* ® c* E Hom (il(X) ® Ll(X), Z2),
and we define a (q
+ i )-cochain Sqic*
Sq'e. * = {ODt-i(e*
E Hom (il(X), Z2) by
®e*)
i>q
i ~ q
Let us now establish some properties of these cochain maps. It will be
convenient to understand Dj = 0 for;
O. Then lemma 5b holds for all ;.
<
6
If e* is zero on Ll(A) for some A C X, then Sqie* is zero on Ll(A).
PROOF
7
This follows from the naturality of {Dj}, and hence of {Sqi}.
•
If 8c* = 0, then 8(Sq ie*) = O.
PROOF
This is trivial if i
> q. If i
~
q, we have
8(Sq ic*)(a) = Dt-i(e* ® e*)(aa) = (c* ® e*)(Dq_iaa)
= (c* ® c* )(aDq_ia) + (e* ® c* )(Dq_i_1a
= (c* ® e*)(aDq_ia)
+
TDq_i_1a)
SEC.
9
273
THE STEENROD SQUARING OPERATIONS
the last equality because (c* ® c* )(Tc) = (c* ® c*)c for any c E .:l(X) ® .:l(X).
Then we have
(c* ® c* )(oDq_ia) = 8(c* ® c* )(Dq_ia) = 0
because 8c* = O.
•
If c* = 8c*, then SqiC* = 8[D"'_i(C* ® c*)
8
PROOF
If i
> q, both sides are zero. If i
+ Dd-i-l(C* ® c* )].
::; q, we have
(Sqic*)(a) = D~_i(8c* ® 8c*)(a) = 8(c* ® 8c*)(Dq_i(a))
= (c* ® 8C* )(Dq_ioa + Dq_i_1a + TDq_i_1a)
= D:_i(C* ® c*)(oa) + 8(c* ® c*)(Dq_i_1a)
the last equality because
(c* ® 8c*)(Dq_i _1a
+
TDq_i_1a) = (c* ® 8c*
+ 8c*
® c*)(Dq_i_1a)
We also have
8(c* ® c* )(Dq_i_1a) = (c* ® c* )(Dq_i_1oa + Dq- i_2 a
= D~_i_1(c* ® c* )(oa)
+
TDq_i_2 a)
The result follows by substituting this into the right-hand side of the other
equation. •
9
If ct and
c~
are cocycles, then
Sqi(c1
PROOF
Sqi(c1
+
If i
+ c~)
= Sq ic1
+
Sqic~
> q, both sides are zero. If i
+ 8Dd-i+l(c1
®~)
::; q, we have
[(c1 + 4) ® (c1 + ~)](Dq_ia)
= (c'j' ® c1 + ~ ® c~ )(Dq_ia) + (c1 ® ~ )(Dq_ia + TDq_ia)
= (Sq ic1 + Sqi~)(a) + (c1 ® c~)(Dq-i+lOa + ODq-i+1a)
= [Sq ic1 + Sqic~ + 8D:-i+l(c1 ® c~)](a)
~)(a) =
the last equality because 8(c1 ® ~) =
o. •
It follows that there is a well-defined functorial homomorphism
Sqi: Hq(X,A)
~
Hq+i(X,A)
defined by Sqi{ c*} = {Sqic*}. If {Dj} is another system satisfying lemma 5a
and 5b, and Sq'i is defined using this system, let {Ej} satisfy 5c and 5d.
If c* is a q-cocycle of .:l(X)j.:l(A) , then
(c* ® c* )(Dq_ia
+ D~_ia + Eq+1_ioa)
= 0
Therefore
SqiC*
+
Sq'ic*
+ 8E:+ 1_i(C*
® c*) = 0
showing that Sqi{ c*} = Sq'i{ c* }. Hence Sqi is uniquely defined independent
274
PRODUCTS
CHAP.
5
of the particular choice of {Dd. We shall now verify that these cohomology
operations {Sqi} satisfy the axioms characterizing the Steenrod squares.
10 THEOREM The additive cohomology operations {Sqi} defined above
satisfy conditions (a) to (d), inclusive, on page 271.
Let C(ll q) denote the oriented chain complex of the simplex. Over
Z2 there is a unique orientation for each simplex, and C(ll ~ is isomorphic
to the subcomplex of ~(~q) generated by the singular simplexes which are the
faces of ~q. We regard C(~q) as imbedded in ~(M) in this way. E(~q) is
acyclic, and if ,\: M -7 ~q is a p-face of M, then ~('\)(C(M)) C C(~q). It
follows that a sequence {Dj} can be found satisfying lemma 5a and 5b such
that Dj(I;q) E C(~q) ® C(~q) for all q and j. For such a sequence, Dj(I;q) = 0 if
j
q (because [C(~q) ® C(M)]s = 0 if s
2q), whence Dj(a) = 0 for any
a E ~q(X) with q < j.
We now shall prove Dq(I;q) = I;q ® I;q for all q by induction on q. If q = 0,
then Do(I;o) must have nonzero augmentation, by lemma 5a. The only element
of C(~o) ® C(~o) with nonzero augmentation is I;o ® I;o. Therefore Do(I;o) =
I;o ® I;o. Assume that q 0 and Dq-1(I;q-l) = I;q-l ® I;q-l. Either Dq(I;q) =
I;q ® I;q or Dq(I;q) = O. In the latter case, by lemma 5b, we have [because
Dq(oI;q) = 0]
PROOF
>
>
>
Dq-1(I;q)
+
TDq-1(I;q)
=0
From this it follows that Dq-1(I;q) = L ai(I;q ® I;q(i)
or ai = 1. This is a contradiction, because
+
I;/i) ® I;q), where ai
=0
= oDq-1(I;q) + Dq-1(oI;q)
and I;q(i) ® I;q(i) has a coefficient of 2ai + 1 = 1 on the right and a coefficient
Dq- 2 (I;q)
+
TDq- 2 (I;q)
of 0 on the left.
Therefore, with this choice of 1Dj } we have Dq(a-)
degree q. Then
(SqOc*)(a)
= (c*
® c*)(Dq(a))
=
a- (2) a- if a- has
= [c*(a)J2
Because a2 = a for a E Z2, we see that SqOc* = c* , and so SqO = 1, showing
that condition (q) is satisfied.
By definition, Do is a chain approximation to the diagonal. Therefore
(c* ® c*)} = {c*} v {c* } for any co cycle c* , and so Sqqu = u v u if
deg u = q. Hence condition (b) is satisfied. From the definition of Sqi condition (c) is trivially satisfied.
It merely remains to verify the Cartan formula. Let {Dj} be a system
satisfying lemma 5a and 5b and let {D/} be the collection of homomorphisms
for ~(X). On the category of pairs of topological spaces X and Y the system
{DkXXY} and the system {T Li+j=k TkDix ® DjY}, where
{m
T:
[~(X) ® ~(X)] ® [~(Y) ® ~(Y)]
-7
[~(X) ® ~(Y)] ® [~(X) ® ~(Y)]
interchanges the second and third factors, both satisfy lemma 5a and 5b.
SEC.
9
275
THE STEENROD SQUARING OPERATIONS
Then a system {EkXXY} satisfying 5c and 5d with respect to them can be
defined by the method of acyclic models. Therefore the system
{T ~ TkDix ® D/}
i+i=k
can be used to define 5qk(U X v) for u E H*(X,A) and v E H*(Y,B). Let ct
be a p-cochain of X, c~ a q-cochain of Y, (Jl a singular p'-simplex of X with
P ~ p' ~ 2p, and (J2 a singular q' -simplex of Y with q ~ q' ~ 2q, where
p' + q' = P + q + k. Then
5qk(ct ®
q )(Jl
= [(q
= [(ct
® (J2)
® c~) ® (q ® c~)](q~~_k(Jl ® (J2))
® cf) ® (c~ ® c~)K . ~
TP+q-kDix(Jl ® D/(J2)
'+J=p+q-k
= [(C{ (8) ci)(D2';,-P'O"l)] [(c: (8) c:)(D!rn'0"2)j
= (5qP'-Pq ® 5qq'-qC~)(Jl (>9 (J2)
Letting (Jl and (J2 vary, we see that 5qk(ct ® c~) = ~i+i=k 5qi ct ® 5qiq.
Passing to cohomology and using the natural homomorphism
H*(X,A) ® H*(Y,B)
~
H*([fl(X)/fl(A)] ® [fl(Y)/fl(B)]) ;::;H*((X,A) X (Y,B))
sending the tensor product to the cross product, we obtain
Sqk(U X v)
=
L
i+j=k
showing that condition (d) is satisfied.
II
EXAMPLE
Sqiu
x
Sqjv
•
Observe that, by condition (b) on page 271 and theorem 5.8.5,
5q2: H2(P2(C))
~
H4(P2(C))
is nontrivial. If u E H2(P 2(C)) is such that 5q2u -=1= 0 and v E Hl(I,i) is the
nontrivial element, it follows from condition (d) that
5q2(U X v) = 5q 2u X v
and 5q2: H3(P2 (C) X (I,i)) ~ H5(P 2(C) X (I,i)) is nontrivial. Let X be the
unreduced suspension of P2(C) obtained from P2(C) X 1 by identifying
P2(C) X 0 to one point Xo and P2(C) X 1 to another point Xl. There is then a
continuous map
f:
P2(C) X (I,i) ~ (X, Xo U Xl)
inducing an isomorphism
f*: Hq(X, Xo U Xl) ;::; Hq(P 2(C) X (1,1))
for all q. Therefore 5q2: H3(X) ~ H5(X) is nontrivial. Let Y be the one-point
union of 53 and 55. An easy computation shows that X and Y have isomorphic
homology and cohomology for any coefficient group, and even isomorphic
cup and cap products. However, because 5q2: H3(X) ~ H5(X) is nontrivial
276
PRODUCTS
and Sq2: H3(Y)
~
CHAP.
5
H5(Y) is trivial, X and Yare not of the same homotopy type.
Further applications of the Steenrod squares will be given in the next
chapter and in Chap. 8.
It is obvious that cohomology operations of the same type can be added
and that the sum is again a cohomology operation of the same type. Given
cohomology operations 8 of type (p,q; G,G') and 8' of type (q,r; G',G"), their
composite 8'8 (of natural transformations) is a cohomology operation of type
(p,r; G,G"). In this way the Steenrod squares can be added and multiplied,
and they generate an algebra of cohomology operations called the modulo 2
Steenrod algebra.
In this algebra the following Adem relations 1 hold:
0< i < 2;
where [i/2] denotes as usual the largest integer:;; i/2 and the binomial coefficient (t~;?) is reduced modulo 2. Using these relations, it is easily shown that
the algebra of cohomology operations generated by Sqi, where i is a power of 2,
contains all the Steenrod squares. This implies that the only spheres that can
be H spaces have dimension 2 n - 1 for some n. By using deeper properties
of the algebra of cohomology operations Adams 2 has shown that the only
spheres that can be H spaces are the spheres So, Sl, S3, and S7. Each of these
is, in fact, an H space, with multiplication defined to be the multiplication of
the reals, complex numbers, quatemions, or Cayley numbers, respectively,
of norm l.
EXERCISES
A DISSECTIONS
Let C be a graded module over R. A filtration (increasing) of C is a sequence {FsC} of
graded submodules of C such that FsC C FS+l C for all s. It is said to be bounded below
if for any t there is s(t) such that Fs(t)Ct = 0, and it is convergent above if U FsC = C.
I
If {FsC} is a filtration of a chain complex C by subcomplexes, there is an increasing
filtration of H* (C) defined by FsH* (C) = im [H* (FsC) -? H* (C)J. If the original filtration
on C is bounded below or convergent above, prove that the same is true of the induced
filtration on H* (C).
An increasing filtration {FsC) of a chain complex C by subcomplexes is called a dissection if it is bounded below, convergent above, and if
1 See J. Adem, The iteration of the Steenrod squares in algebraiC topology, Proceedings of the
National Academy of Sciences, USA, vol. 38, pp. 720-726, 1952, or H. Cartan, Sur !'iteration
des operations de Steenrod, Commentarii Mathematici Helvetici, vol. 29, pp. 40-58, 1955.
2 See J. F. Adams, On the non-existence of elements of Hopf invariant one, Annals of Mathematics, vol. 72, pp. 20-104, 1960.
277
EXERCISES
Given a dissection {F.C} of a chain complex C, the sequence
... ~ Hq+l(FQ+1C,FQC) ~ HQ(FQC,FQ_1C) ~ HQ-l(FQ-lC,FQ-2C) ~ ...
is a chain complex C, called the chain complex associated to the dissection.
2
If C is the chain complex associated to a dissection of C, prove that H. (C) :::::: H. (C).
3 Let {F.G} be a dissection of a free chain complex C by free subcomplexes such that
Fs+IC/F.C is free for all s. If C is the chain complex associated to the dissection, prove
that C and C have isomorphic homology and cohomology for all coefficient modules.
[Hint: The freeness hypotheses ensure that the universal-coefficient theorems hold for
both homology and cohomology. Then {F.C ® G} is a dissection of C ® G whose associated chain complex is isomorphic to C ® G. Dual considerations apply to {Hom (F.C,G)}
and Hom (C,G).]
A block dissection of a chain complex C is a collection of subcomplexes {EjQ}, called
blocks, where q varies over the set of integers and for each q, i varies over a set JQ, such
that if F.C is the subcomplex of C generated by {EjQ}q". and if E;Q = EjQ n F._ 1 C, then
EjQ n EkQ C FQ - 1 C
=0
U F.C = C
EjQ
H;(E;Q,E;Q)::::::
(~
i*k
q sufficiently small
i*q
i=q
4 If {EjQ} is a block dissection of a chain complex C, prove that the corresponding
collection {F.C} is a dissection of C whose associated chain complex C is free with
generators for CQin one-to-one correspondence with tlte set JQ.
A block dissection of a simplicial complex K is a collection of subcomplexes {K;Q},
where q varies over tlte set of integers and for each q, i varies over some indexing set JQ,
such that if F.K
U j ,;. KjQ and K;Q
F._1K n K;Q, then
=
=
KF n KkQ C FQ_1K
K;Q
U F.K
.
=0
i*k
q sufficiently small
=K
(0
H;(KjQ,KjQ):::::: Z
i*q
i =q
5 If {KjQ} is Ii block dissection of K, prove that {C(KjQ)} is a block dissection of the
chain complex C(K) by free subcomplexes. If C is the chain complex associated to the
dissection, prove that C and C(K) have isomorphic homology and cohomology with any
coefficient group.
B HOMOLOGY MANIFOLDS
A homology n-manifold is a locally compact Hausdorff space X such that for all x E X,
HQ(X, X - x) = 0 for q n and either Hn(X, X - x) = 0 or Hn(X, X - x) :::::: Z. Furthermore, if the boundary X of X is defined to be the subset
*
X = {x E X I Hn(X, X - x)
= O}
then we also assume that X - X is a nonempty connected set. If X
without boundary.
= 0, X is said to be
278
PRODUCTS
CHAP.
5
I If X is a homology n-manifold and Y is a homology m-manifold, prove that X X Y is
a homology (n + m)-manifold whose boundary equals X X Y u X X Y.
2
Prove that if a polyhedron is a homology n-manifold, its boundary is a subpolyhedron.
3 If K is a simplicial complex triangulating a homology n-manifold X, prove that K is
an n-dimensional pseudomanifold and K triangulates 1<.. (A polyhedral homology n-manifold is said to be orientable or nonorientable, according to whether any triangulation of
it is orientable or nonorientable as a pseudomanifold.)
4 Let (K,K) be a simplicial pair triangulating a polyhedral homology n-manifold (X,X)
and let L be the subcomplex of the barycentric subdivision K' consisting of all simplexes
disjoint from K'. If sq is a q-simplex of K - K, let En-q(sq) be the subcomplex of L generated by the star of the barycenter b(sq). Prove that {En-q(sq)} s'I € K-X is a block dissection of L and that if C is the chain complex associated to this block dissection, then C
has homology and cohomology isomorphic to that of X-X. (Hint: let st sq = sq * B(sq),
where B(sq) is a subcomplex of K. Then En-q(sq) = b(sq) * [B(sq)l' and En-q(sq) = [B(sq)]'.
Also note that ILl is a strong deformation retract of IKI - IKI.)
:; Lefschetz duality theorem. Let (K,K) be a simplicial pair triangulating a compact
homology n-manifold (X,X) and assume that z E Hn(K,K) is an orientation of K. For each
q-simplex sq of K - K let z(sq) E Hn(K, K - st sq) be the image of z, and assume an
orientation aq of sq chosen once and for all. Then z(sq) = aq * z(a q), where z(aq) E
H n_q_1 (B(sq)). Define z'(a q) E Hn_q(En-q(sq),En-q(sq)) to correspond to z(a q) under the
isomorphisms
Hn_q_1(B(sq)) ::::: Hn_q_1 (En- q(sq)) ::::: Hn_q(En-q(sq),En-q(sq))
Let <p: Hom (Cq(K,K), G) ~ Cn- q ® G be the homomorphism defined by
<p(u) =};q z'(a q) ® u(a q)
u E Hom (Cq(K,K), G)
cr
Prove that <p is an isomorphism and that it commutes up to sign with the respective coboundary and boundary operators. Deduce isomorphisms
X; G)
Hq(X,X; G) ::::: Hn_q(X -
and
Hq(X,X; G) ::::: Hn-q(X -
X; G)
C PROPERTIES OF THE TORSION PRODUCT AND EXT
In this group of exercises all modules will be over a principal ideal domain R.
I
Prove that the torsion product is associative.
2
If A, B, and C are modules, prove that
A ® (B
* C) EB A * (B ® C)
is symmetric in A, B, and C.
3
Given a module A and a short exact sequence of modules
o ~ B' ~ B ~ B" ~ 0
prove there is an exact sequence
o ~ Hom (A,B')
4
~
Hom (A,B)
~ Hom (A,B") ~
Ext (A,B') ~ Ext (A,B)
Given a short exact sequence of modules
O~A'~A~A"~O
and given a module B, prove there is an exact sequence
~
Ext (A,B")
~
0
279
EXERCISES
o
---?
Hom (A",B)
---?
Hom (A,B) ---? Hom (A',B)---?
Ext (A ",B) ---? Ext (A,B)
---?
Ext (A',B)
0
---?
If C = {Cd and C * = {Ci} are graded modules, there is a graded module
Hom (C,C*) = {Hom q (C,C*)}, where Hom q (C,C*) = X i+i=q Hom (C;,Ci) [thus an
element of Hom q (C,C *) is an indexed family {C:Pi: Ci ---? Cq-i };J. Similarly, there is
a graded module Ext (C,C*) = {Extq (C,C*)}, where Extq (C,C*) = Xi+i=q Ext (Ci,O).
5 If C is a chain complex and C * is a cochain complex, prove that Hom (C, C *) is a
cochain complex, with
(8c:p)i,j
= C:Pi-l,j
0
Ji
+ (_1)i8 j- 1
0
C:Pi,j-l
c:P
= {C:Pi,j}
E Hom q (C,C*)
and that Ext (C,C*) is a cochain complex with
(8l/;)i,j
= Ext (o;,l)(l/;i-u) + (_l)i Ext (1,8i- 1)(l/;i,j-l)
6 If C is a chain complex and C* is a cochain complex such that Ext (C,C*) is
acyclic, prove that there is a split short exact sequence
o ---? Extq-l (H* (C),H* (C*)) ---? Hq(Hom (C,C*)) ---?
Hom q (H* (C),H* (C*))
---?
0
7 If C and C' are chain complexes and C * is a cochain complex, prove that the exponential correspondence is an isomorphism
Hom (C, Hom (C',C*)):::::: Hom (C ® C', C*)
8 Let (X,A) and (Y,B) be topological pairs such that {X X B, A X Y} is an excisive
couple in X X Y. For any module G prove that there is a split short exact sequence
o ---?
where H*
Ext q- 1 (H* ,H*)
---?
Hq((X,A) X (Y,B); G)
---?
Hom q (H* ,H*)
---?
0
= H* (X,A; R) and H* = H* (Y,B; G).
D CATEGORY
A topological space X is said to have category::;; n, denoted as cat X ::;; n, if X is the
union of n closed sets, each deformable to a point in X.
I
If X is a connected polyhedron of dimension n, prove that cat X
2
If X is any space, prove that cat (SX) ::;; 2.
s: n + 1.
:I If cat X ::;; n, prove that all n-fold cup products of positive-dimensional cohomology
classes of X vanish.
= n + 1 and cat (pnl
4
Prove that cat pn
E
HOMOLOGY OF FIBER BUNDLES
X ... X pnk)
= nl + ... + nk + 1.
I Let p: E ---? B be a fiber-bundle pair, with total pair (E,E) and fiber pair (F,F), such
that H* (F,F) = O. Prove that H* (E,E) = o.
2 If p: E ---? B is a fiber-bundle pair over a path-connected base space B, prove that a
homomorphism B: H* (F,F; R) ---? H * (E,E; R) is a cohomology extension of the fiber if
and only if for some b E B the composite
is an isomorphism.
3 Let p: E ---? B be a fiber-bundle pair over a path-connected base space. If for some
b E B the pair (Eb,Eb) is a weak retract of (E,E), prove there exists a cohomology extension of the fiber.
280
PRODUCTS
CHAP.
5
4 Prove that a q-sphere bundle ~ with base space B is orientable over R if and only if
for every map a: Sl ~ B the bundle a* (~) is orientable over R.
5 Prove that a q-sphere bundle ~ is orientable over Z if and only if there is an element
U E Hq+1(E~"E(; Z4) whose image in Hq+1(EE,EE; Z2) is the unique orientation class of ~
over Z2. (Hint: Show that there is such an element U if and only if for every closed path
w in the base space, h[ wJ* is the identity map of Hq+l(E w(l),E w(l); Z4), and this, in turn, is
equivalent to the condition that h[wJ * is the identity map of Hq+l(E w(l),E w(l); Z).)
6 Let ~ be a q-sphere bundle with base space B and with orientation class
U, E Hq+l(E"E~; R) and let Q~ E Hq+1(B;R) be the corresponding characteristic class.
Prove that <I> (Q,) = U~ v U(.
r
7 Prove that the characteristic class
over Z has order 2.
Q~
of an even-dimensional sphere bundle ~ oriented
8 Let ~ be a sphere bundle oriented over R, with base space B. If ~ has a section in E~,
(that is, if the map Pt: E;; -+ B has a right inverse), prove that its characteristic class
Q, = O. [Hint: Any two sections B ~ E~ are homotopic in E~. Since E~ is the mapping
cylinder of p~: E~ ~ B, there is an inclusion map k: B C EE which is a section. There is
a section in EE if and only if k is homotopic to a map B ~ E" in which case the composite
Hq+l(E(,E~; R) ~ Hq+l(E(;R)
is trivial, because p* -1
F
I
pH)
Hq+l(B;R)
= k*.J
HOPF ALGEBRAS
Prove that the tensor product of connected Hopf algebras is a connected Hopf algebra.
2 If B is a connected Hopf algebra of finite type over a field R, prove that
B* = Hom (B;R) is a connected Hopf algebra over R whose product and coproduct are
dual, respectively, to the coproduct and product of B.
*
3 Let B be a connected Hopf algebra over a field of characteristic p
0 and assume
that B has an associative and commutative product and is generated as an algebra by a
2, then B = E(x),
single element x of positive degree. Prove that if deg x is odd and p
and if deg x is even or p = 2, then either B = Sdeg xix) or B = I deg x,h(X), where h = pk
for some k ~ 1.
*
4 Let B be a connected Hopf algebra of finite type over a field of finite characteristic
p 0 and assume that B has an associative and commutative product. If the pth power
of every element of positive degree of B is 0, prove that B is the tensor product of exte2) and truncated polynomial algebras
rior algebras (with generators of odd degree if p
2).
of height p (with generators of even degree if p
*
**
G THE BOCKSTEIN HOMOMORPHISM
I Show that the Bockstein homomorphism in homology (or cohomology) anticommutes
with the boundary homomorphism (or coboundary homomorphism) of a pair.
For any prime p let f3p be the Bockstein homomorphism in either homology or
cohomology for the short exact sequence of abelian groups
o ~ Zp ~ Zp2 ~ Zp ~ 0
Let j3p be the Bockstein homomorphism for the short exact sequence
O~ Z~ Z~ Zp ~ 0
281
EXERCISES
== pn and /Lp is reduction modulo p.
that f3p == (/Lp)* /lp'
that f3p f3p == O.
that f3p(u \J v) == f3p(u) \J V + (_I)deg U \J f3p(v).
that Sq2i+l == 132 Sq2i for i ?: O. [Hint: Show that there exist functorial
where Ap(n)
2
Prove
a
Prove
4
Prove
0
0
U
5 Prove
homomorphisms {Ddj2 0 , with Dj of degree i from the integral singular chain complex Ll(X)
to Ll(X) ® Ll(X), such that Do is a chain map commuting with augmentation and
0
oD 2j _ 1 + D 2j _ 1 0 == D2j - TD2j
aD2j - D2j a == D Zj_ 1 + TD 2J _1
where
T(U1
® (2) == ( _I)de g
0)
deg
02
U2
®
U1.]
6 Let ~ be a q-sphere bundle and let U, E Hq+1(E"E,; Z2) be its unique orientation
over Z2. Prove that ~ is orientable over Z if and only if f32(U,) == o.
H STIEFEL-WHITNEY CHARACTERISTIC CLASSES
Let ~ be a q-sphere bundle, with base space B, and let U, E Hq+1(E"E(; Zz) be its orientation class over Z2. The ith Stiefel-Whitney characteristic class Wi(~) E Hi(B;Z2) for
i ?: 0 is defined by
<I>r(Wi(~))
Let f: B'
2
If ~ is a product bundle, prove that wM)
a
Prove the following:
---7
== Sqi(U()
B be continuous. Prove that f* (Wi(~)) == Wi(f* ~).
I
== 0 for i
> o.
(a) wo(~) is the unit class of HO(B;Zz).
(b) f32(W2i(~)) == W2i+1W + W1W V W2i(~) for i ?: o.
(c) If ~ is a q-sphere bundle, then Wi(~) == 0 for i
characteristic class of ~ over Z2.
(d) ~ is orient able over Z if and only if W1W = O.
> q + 1,
and
Wq+l(~)
is the
If ~ is a q-sphere bundle over Band f is a q' -sphere bundle over B', their cross product
~ X f is a (q + q' + I)-sphere bundle with E(x(' = E, X E(', E(X(' == E( X E" U E( X E('
and p,x(' == p, X p(,.
4 If U, E Hq+1(E(,E,; Z2) and U(' E Hq'+l(E("E,'; Zz) are respective orientation classes,
prove that
U, XU" E Hq+q'+2(E(x",E(x('; Z2)
is the orientation class of ~ X
5
== ~i+j=k Wi(~) X Wj(f).
f are sphere bundles with the same base space B, their Whitney sum
EB f is the sphere bundle over B induced from g X f by the diagonal map B ---7 B X B.
If
~
6
f.
Prove that Wk(~ X f)
~ and
Whitney duality theorem. Prove that
Wk(~ EB f)
== . ~ Wi(g)
HJ=k
v Wj(e)
I
HOMOLOGY WITH LOCAL COEFFICIENTS
If u: Llq ---7 X is a singular q-simplex of X, with q ?: 1, let Wo be the path in X obtained
by composing the linear path in Llq from va to Vl with cr. Given a local system r of
282
PRODUCTS
CHAP.
5
R modules on X, define D.q(X;f) to be the R module of finitely nonzero formal sums ~ Ciaa
in which a varies over the set of singular q-simplexes of X and Cia E [(a(vo)) is zero
except for a finite set of a. For q
0 define a homomorphism 0: D.q(X;f) ~ D.q-l(X;f) by
>
o(Cia) =
~
O<~s.q
(-I)iCia(i)
+
f(Wa)(Ci)a(O)
I Prove that D.(X;f) = {D.q(X;f), o} is a chain complex which is free (or torsion free)
if f is a local system of free (or torsion free) R modules, and if A C X, show that
D.(A; f I A) is a subcomplex of D.(X;f).
The horrwlogy of (X,A) with local coefficients f, denoted by H* (X,A; f), is defined
to be the graded homology module of D.(X,A; f) = D.(X;f)/ D.(A; f I A).
2 For a fixed ring R let 2 be the category whose objects are topological pairs (X,A),
together with local systems f of R modules on X, and whose morphisms from (X,A) and
f to (Y,B) and f' are continuous maps f: (X,A) ~ (Y,B), together with indexed families
of homomorphisms {fx: f(x) ~ f'(f(X))}XEX such that fw(o) a [(w) = f'(f W) fw(1) for
any path W in X. Prove that H* (X,A; f) is a covariant functor from (; to the category of
graded R modules.
0
0
3 Exactness. Given A C B C X and a local system f of R modules on X, prove that
there is an exact sequence
...
~
Hq(B,A; f I B)
~
Hq(X,A; f)
~
Hq(X,B; f)
~
Hq_l(B,A; f I B)
~
...
4 Excision. Let Xl and X2 be subsets of a space X such that Xl U X2 =: int Xl U int X 2 .
For any local system f of R modules on X prove that the excision map il from
(Xl, Xl n X 2 ) and f I Xl to (Xl U X 2 , X 2 ) and f I (Xl U X2 ) induces an isomorphism
il *: H* (Xl, Xl n X 2 ; f I Xl) :::; H* (Xl U X 2 , X 2 ; f I (Xl U X 2 ))
:. Two morphisms f and g in (; from (X,A) and f to (Y,B) and f' are said to be
homotopic in (; if there is a homotopy F: (X,A) X I ~ (Y,B) from f to g and an indexed
family of homomorphisms {F(x,tj: [(x) ~ f'(F(x,t)) }(X,t)EXXI such that F(x,o) = fx and
F(x,l) = gx. Prove that homotopy is an equivalence relation in the set of morphisms from
(X,A) and f to (Y,B) and f' and that the composites of homotopic morphisms are
homotopic (so that the homotopy category of (; can be defined).
6 Homotopy. If f and g are morphisms from (X,A) and f to (Y,B) and f' and f is
homotopic to g in 2, prove that f* = g *: H* (X,A; f) ~ H* (Y,B; f').
7 If f and f' are local systems of R modules on X, there is a local system f ® f' on
X with (f ® f')(x) = [(x) ® ['(x) and (f ® f')(w) = [(w) ® f'(w). In case f' is the
constant local system equal to G, then prove that
D.(X,A; f ® G) :::; D.(X,A; f) ® G
Deduce a universal-coefficient formula for homology with local coefficients.
8 If f and f' are local systems of R modules on X and Y, respectively, let r X r' =
p* (r) ® p' * (r') be the local system on X X Y, where p* (r) and p' * (r') are induced
from rand r', respectively, by the projections p: X X Y ~ X and p': X X Y ~ Y.
Prove that there is a natural chain equivalence of D.(X;r) ® D.(Y;r') with D.(X X Y; r X r').
Deduce a Klinneth formula for homology with local coefficients.
J COHOMOLOGY WITH LOCAL COEFFICIENTS
If r is a local system of R modules on X, define D.q(X;r) to be the module of functions cp
assigning to every singular q-simplex a of X an element cp(a) E [(a(vo)). Define a homomorphism 0: M(X;r) ~ D.q+1(X;r) by
283
EXERCISES
I Prove that .1* (x;r) = {.1q(x;r), 8} is a cochain complex and that if A C X, the
restriction map .1* (X;f) ~ .1* (A; r I A) is an epimorphism.
The cohomology of (X,A) with local coefficients
defined to be the graded cohomology module of
.1* (X,A; r)
r,
r),
denoted by H* (X,A;
= ker [.1* (X;r) ~ .1* (A;
is
r I A)l
2 For a fixed ring R let e be the category whose objects are topological pairs (X,A),
together with local systems r of R modules on X, and whose morphisms from (X,A) and
r to (Y,B) and 1" are continuous maps f: (X,A) ~ (Y,B), together with indexed families
of homomorphisms {fx: r'(f(x)) ~ r(x) h,x such that f(w) f w(1) = fw(O) r'(f w) for
any path w in X. Prove that H * (X,A; f) is a contravariant functor from c:" to the
category of graded R modules.
0
0
0
3 Prove that the cohomology with local coefficients has exactness, excision, and homotopy properties analogous to those of the homology with local coefficients.
4 If r is a local system of R modules on X and e is an R module, there is a local system Hom (r,e) of R modules on X which assigns to x E X the module Hom (r(x),e).
Prove that
.1* (X,A; Hom (r,e))
~
Hom (.1(X,A; r), e)
Deduce a universal-coefficient formula for cohomology with local coefficients.
r,
Let ~ be a q-sphere bundle with base space B and let
be the local system on B
such that r«(b) = H q + 1 (E b ,Eb ). Let pt (r() be the local system on E, induced from r( by
p( E( ~ B. A Thom class of ~ is an element U( E Hq+l(E(,E(; pt (ri )) such that for every
b E B the element
U( I (Eb,E b) E Hq+l(E b,E b; pt (r() I E b) = Hq+l(E b,E b; Hq+1 (E b,Eb))
corresponds to the identity map of H q + 1 (E b,E b ) under the universal-coefficient isomorphism
Hq+1(E b,Eb; Hq+1 (E b,E b)) ~ Hom (Hq+l(Eb,Eb), Hq+1 (E b,E b))
5 Prove that every q-sphere bundle has a unique Thorn class. (Hint: Prove the result
first for a product bundle, and then use Mayer-Vietoris sequences to extend the result to
arbitrary bundles.)
6 Let ~ be a q-sphere bundle with a base space B and let U( be its Thorn class. If
any local system of abelian groups on X, prove that the homomorphism
r
is
<1\: Hn(E(,E,; p*(r)) ~ Hn_q_1 (B; r, 0 r)
such that <I>,(z) = p* (U( r-, z), where U, r-, z is an element of Hn- q_ 1 (E; p* (r, 0 r)), is
an isomorphism. If B is compact, prove that the homomorphism
<I> t: HT(B;r) ~ HT+q+l(E"E,; p* (r
such that <I> t (v) = p* (v) v U( is an isomorphism.
0 r,))
CHAPTER SIX
GENERAL COHOMOLOGY
THEORY AND DUALITY
IN THIS CHAPTER WE CONTINUE THE STUDY OF HOMOLOGY AND COHOMOLOGY
functors, with particular emphasis on the homological properties of topological
manifolds. For this important class of spaces we shall establish the duality
theorem equating the cohomology of a compact pair in an orientable manifold
with the homology, in complementary dimensions, of the complementary pair.
The cohomology which enters in the duality theorem is the direct limit
of the singular cohomology of neighborhoods of the pair, with the family of
neighborhoods directed downward by inclusion. For the case of a closed pair
in a manifold, the resulting direct limit depends only on the pair itself. In fact,
it is isomorphic to the Alexander cohomology of the pair, Alexander cohomology being another cohomology theory distinct from the singular cohomology.
Thus we are led to consider Alexander cohomology. We define it and
prove that it is a cohomology theory in the sense that it satisfies the axioms
of cohomology theory. We also establish the special properties of tautness and
continuity possessed by this theory and not generally valid for singular cohomology. For deeper properties of the Alexander theory we introduce the
cohomology of a space with coefficients in a presheaf. The definition of this
285
286
GENERAL COHOMOLOGY THEORY AND DUALITY
CHAP.
6
cohomology involves a Cech construction, using nerves of open coverings.
We use general properties of this cohomology to prove that for paracompact
spaces the Alexander and Cech cohomologies are isomorphic, and with this
result establish universal-coefficient formulas for the Alexander cohomology
of compact pairs and for the Alexander cohomology with compact supports
of locally compact pairs.
The cohomology of presheaves is also applied to compare the singular
and Alexander cohomology theories, and we prove that they are isomorphic
for manifolds. Another application of the cohomology of presheaves is in the
proof of the Vietoris-Begle mapping theorem. The final topic is a discussion of
homological properties of one manifold imbedded in another.
In Sec. 6.1 we define the slant product as a pairing from the cohomology
of a product space and the homology of one of its factors to the cohomology
of the other factor. This furnishes the map that is the isomorphism in the
duality theorem for manifolds, and the duality theorem itself is proved in
Sec. 6.2. In Sec. 6.3 we consider various formulations of orientability for
manifolds.
The Alexander cohomology theory is defined in Secs. 6.4 and 6.5, and
the axioms of cohomology theory are verified for it. Section 6.6 contains
a proof of the tautness property for Alexander cohomology, that the Alexander
cohomology of a closed pair in a paracompact space is isomorphic to the direct
limit of the Alexander cohomology of its neighborhoods. We deduce the continuity property of Alexander cohomology and show that the continuity
property characterizes Alexander cohomology on compact pairs. We also
define the Alexander cohomology with compact supports.
Sections 6.7, 6.8, and 6.9 develop the theory of the cohomology of spaces
with coefficients in a presheaf and illustrate its application to the Alexander
theory. In this way we equate the Alexander and singular cohomology for
paracompact spaces that are homologically locally connected in all dimensions.
Section 6.lO contains definitions of the characteristic classes of a manifold
and the normal characteristic classes of one manifold imbedded in another.
These are related in the Whitney duality theorem, which is a useful tool for
establishing non-imbeddability results.
I
THE SLANT PRODUCT
We are ready now to introduce a new product which pairs cohomology of a
product space and homology of one of the factors to the cohomology of the
other factor. This product will be used in the next section to prove the duality
theorem for topological manifolds. In this section we shall establish some of
its properties. We shall also introduce new cohomology modules of a pair
(A,B) in a space X which appear to depend on the imbedding of (A,B) in X.
These will be used in the proof of the duality theorem in the next section.
Later in the chapter, we shall introduce the Alexander cohomology modules
SEC.
1
287
THE SLANT PRODUCT
and prove that these are isomorphic to the abovementioned ones in all
relevant cases.
Given chain complexes C and C' over R and a cochain
c* E Hom ((C ® C')n, G)
and chain c' E C' q 09 G', their slant product c*/c' E Hom (Cn- q , G 09 G') is
the (n - q)-cochain such that if c' = ~i ci ® g; with ci E C~ and g; E G', then
(c* /c',c) = ~, (c*, c ® ci) ®
g;
c E Cn- q
It is easily verified that
o(e* / e') = [(oe*)/ e'l + (- l)n-'lc* / ae'
Therefore the slant product of a cocycle and a cycle is a cocycle, and if the
co cycle is a coboundary or the cycle is a boundary, the slant product is
a coboundary. Hence there is a slant product of Hn(C ® C'; G) and Hq(C';G')
to Hn-q(C; G ® G') such that {e* }/{c'} = {c* /c'} for {c*} E Hn(C ® C'; G)
and {c'} E Hq(C';G').
For topological pairs (X,A) and (Y,B) let
T:
[Ll(X)/Ll(A)] ® [Ll(Y)/Ll(B)]
~
[Ll(X X Y)]/[Ll(X X B U A X Y)]
be a functorial chain map given by the Eilenberg-Zilber theorem. For
u E Hn((X,A) X (Y,B); G) and z E Hq(Y,B; G'), their slant pmduct
u/z E Hn-q(X,A; G ® G')
is defined to equal the slant product (T* u) / z. The following properties of this
slant product are easy consequences of the definitions.
I
Given f: (X,A) ~ (X',A'), g: (Y,B) ~ (Y',B'), U E Hn((X',A') X (Y',B'); G),
and z E Hq(Y,B; G'), then, in Hn-q(X,A; G ® G'),
[(f X g)*u]/z = f*(u/g*z)
•
2
Given u E Hp(X,A; G), v E Hq(Y,B; G'), and z E Hq(Y,B; Gil), if {X X B,
A X Y} is an excisive couple in X X Y, then, in Hp(X,A; G ® G' ® Gil),
(u X v)/z = Jl(u ® (v,z»)
3
•
Let {(Xl,A I ), (X 2,A 2)} and {(YI,B I ), (Y 2,B2)} be excisive couples in X and
Y, respectively. Given
u E Hn((Xl U X2 )
x (Yl U Y2 ), Xl x
Bl
U X2 X
B2
U Al X Yl U A2
x Y2 ; G)
and
z E Hq(YI U Y 2, BI U B2; G')
then, in Hn-q+l(Xl U X2 , Al U A 2 ; G 09 G').
[u I (Xl U X2, Al U A2) X (YI n Y 2, BI n B2)]/a*z
= (-1)n- Q- 1 8*([u I (Xl n X 2, Al n A 2) X (YI U Y 2, BI U B2)]/Z)
•
288
GENERAL COHOMOLOGY THEORY AND DUALITY
CHAP.
6
The following formulas express relations between the slant product and
the cup and cap products. We sketch proofs in which the Alexander-Whitney
diagonal approximation a ~ ~i+j=deg a ia (8) aj is used in Ll(X) and its tensor
product with itself
a (8) a'
~ ~
',}
(-l)j(P-i)(ia (8) ja') (8) (ap_i (8)
a~_j)
deg a
= p, deg a' = q
is used in Ll(X) (8) Ll(Y).
4 Given v E Hp(X,A; G), u E Hn((X,A') X (Y,B); G'), and z E Hq(Y,B; Gil),
then, in Hp+n-q(X, A U A'; G (8) G' (8) Gil),
(u/z)
V V
= [(v
X 1) v u]!z
PROOF
Let c! be a p-cochain of Ll(X), c~ an n-cochain of Ll(X) 0 Ll(Y),
and a' E Llq(Y). It suffices to prove that
c! v (c~ fa')
= [(c!
(8) 1) v c~]!a'
If a E Llp+n_q(X), then
<cT v
(c~
fa'), a)
= <c!, pa)
= <c!, pa)
= <c!
=«cf
(8) <c~ /a',a n _ q)
(8) <c~, an _ q (8) a')
(8) 1, pa (8) oa') 0 <c~, an _ q (8)
a')
(8)l)vc~,a(8)a') =<[(cf01)vc~]!a',a)-
:; If u E Hn((X,A) X (Y,B); G), v E Hp(Y,B'; G'), and z E Hq(Y,B
then, in Hn-(q-p)(X,A; G (8) G' 0 Gil),
u/(v
r'\
z)
= [u v
/ (c ~
f"'\
a')
B'; Gil),
(1 X v)]!z
PROOF
Let c! be an n-cochain of Ll(X) (8) Ll(Y),
and a' E Llq(Y). It suffices to prove that
cf
U
= [c T v
c~
be a p-cochain of Ll(Y),
(1 (8) c ~ )]! a'
If a E Lln_(q_p)(X), then
<cT /(c~
f"'\
a'), a)
= <c!, a 0 (c! a')
= <c!, (1 0 c~) (a (8) a')
= <c! v (1 (8) c~), a 0 a')
= <[c! v (1 (8) c!)]!a', a)
f"'\
f"'\
-
6 Given u E Hn((X,A) X (Y,B); G), w E Hr(X,A; G'), and z E Hq(Y,B; Gil),
let p: X X Y ~ X be the protection to the first factor and let
T: G (8) Gil 0 G'
~
G (8) G' (8) Gil
interchange the last two factors. Then, in Hr_(n_q)(X; G 0 G' (8) Gil),
p* (u
PROOF
f"'\
(w X z))
= T* [(u/z)
f"'\
w]
Let c* be an n-cochain of Ll(X) (8) Ll(Y), a E Llr(X), and a' E Llq(Y).
SEC.
1
289
THE SLANT PRODUCT
Then
fl(p)( c* ("'"\ (a ® a')) = fl(p)[ L
i+j=n
(- 1)i(q-j)(r_ia ® q_p') ® <c*, ai ® aj)]
= r-(n-q)a ® <c* , an_q ® a')
= r-(n-q)a ® <c* / ai, an_q)
= (c* / a') ("'"\ a •
For a topological space X let SIX) be the diagonal of X defined by
SIX) = {(x,x' ) E X X X I x = x'}. Given u E Hn(X X X, X X X - SIX); R)
and a pair (A,B) in X, define
Yu: Hq(X - B, X - A; G)
~
Hn-q(A,B; G)
by Yu(z) = [u I (A,B) X (X - B, X - A)l!z (with R ® G identified with G).
If i: (A,B) C (A',B') and i: (X - B', X - A') C (X - B, X - A), it follows
from property 1 that there is a commutative diagram (all coefficients G)
Hq(X - B', X - A')
~
i*l
Hq(X - B, X - A)
Hn-q(AI,B ' )
1i '
~
Hn-q(A,B)
Thus Yu is a natural transformation from Hq(X - B, X - A) to Hn-q(A,B) on
the category of pairs of subspaces and inclusion maps in X. It follows from
property 3 that Yu commutes up to sign with the connecting homomorphisms
of relative Mayer-Vietoris sequences.
For a pair (A,B) in a topological space X we define a neighborhood
(U, V) of (A,B) to be a pair in X such that U is a neighborhood of A and V is
a neighborhood of B. The family of all neighborhoods of (A,B) in X is directed
downward by inclusion. Hence
{Hq( U, V; G) I (U, V) a neighborhood of (A,B)}
is a direct system, and we define
fIq(A,B; G) = lim~ {Hq(U,V; G)}
where (U, V) varies over neighborhoods of (A,B) [or over the cofinal family of
open neighborhoods of (A,B)]. The restriction maps Hq(U'y; G) ~ Hq(A,B; G)
define a natural homomorphism
i: fIq(A,B; G) ~ Hq(A,B; G)
The pair (A,B) is said to be tautly imbedded in X, or to be a taut pair in X
(with respect to Singular cohomology), if i is an isomorphism for all q and G.
The definition of tautness can be formulated for any cohomology theory (or
any contravariant functor). We shall see examples later of a subspace taut
with respect to one cohomology theory but not with respect to another.
Following are some examples.
7
If (A,B) is an open pair, or, more generally, if it has arbitrarily small
290
GENERAL COHOMOLOGY THEORY AND DUALITY
CHAP.
6
neighborhoods which are homotopy equivalent to (A,B), then (A,B) is a taut
pair in X.
>
8
Let A' = {(x,y) E RZI x 0, y = sin l/x}, let A" = {(x,y) E RZI x = 0,
Iyl ::::; I}, and let A = A' U A" c RZ. Then A' and A" are the path components of A, and so HO(A;Z) :::::: Z EEl Z. Since A is connected, in any open
neighborhood V of A in RZ, A' and A" must be in the same path component
of V (the path components of V are the same as the components of V because
V is locally path connected). It follows that HO(A;Z)
lim~ {HO( V;Z)}, where
V varies over the connected open neighborhoods of A in RZ. Therefore
HO(A;Z) :::::: Z and i: HO(A;Z) -7 HO(A;Z) is not an epimorphism. Thus A is not
a taut subspace of RZ with respect to singular cohomology.
=
9
LEMMA
Let (A,B) be a pair in X. Then, if two of the three pairs (B, 0),
(A, 0), and (A,B) are taut in X, so is the third.
PROOF
This follows from the exa,ctness of the cohomology sequence of a
triple, from the fact that a direct limit of exact sequences is exact, and from
the five lemma. •
Recall (exercise set l.C) that a normal space X is an absolute neighborhood retract if it has the property that whenever it is imbedded as a closed
subset of a normal space, it is a retract of some neighborhood. Also recall that
a space X is binormal if X X I (hence also X) is normal.
10 THEOREM Any imbedding of an absolute neighborhood retract as a
closed subspace of a binormal absolute neighborhood retract is taut.
PROOF
Assume A C X, where A and X are absolute neighborhood retracts
and A is closed in the binormal space X. There is a neighborhood V of A in
X such that A is a retract in V. Then H*(V) -7 H*(A) is an epimorphism,
and this implies that
i: H*(A)
-7
H*(A)
is an epimorphism.
To show that it is also a monomorphism, let V be an open neighborhood
of A in X. There is a closed neighborhood V' of A in V of which A is a retract.
Let r: V' -7 A be a retraction and define a map
F: (V' X 0) U (A X 1) U (V' X 1)
-7
V
by F(x,O) = x and F(x,l) = r(x) for x E V' and F(x,t) = x for x E A and t E I.
Because A is closed in X, (V' X 0) U (A X 1) U (V' X 1) is closed in V' X I,
the latter being a normal space because it is a closed subset of the normal
space X X I: Since V is an open subset of the absolute neighborhood retract X,
it follows (see exercise l.e.4) that V is an absolute neighborhood retract and
F can be extended to a map F: N -7 V, where N is a neighborhood of
(V' X 0) U (A X 1) U (V' X 1) in V' X I. N contains a set of the form V X I,
where V is a neighborhood of A in U', and F' I V X I is a homotopy from the
SEC.
1
291
THE SLANT PRODUCT
inclusion map j: V C U to kr'. where r'
Therefore there is a commutative triangle
= r I V:
V
~
A and k: A C U.
H*(U) ~ H*(A)
J*\
,jr'*
H*(V)
which shows that ker k* C ker j * . Thus, if an elementin H * (U) restricts to 0
in H* (A), it restricts to 0 in H* (V) for some smaller neighborhood V, hence
it represents 0 in lim~ {H*(U)} = H*(A). Therefore i: H*(A) ~ H*(A) is
a monomorphism and A is taut in X. •
II
If A, B, and X are compact polyhedra, any imbedding of
COROLLARY
(A,B) in X is taut.
PROOF
This follows from the fact (exercise 3.A.l) that a compact polyhedron
is an absolute neighborhood retract and from theorem 10 and lemma 9. •
One reason for introducing the modules Hq(A,B; G) is the following
result, which asserts that any pair (A,B) in X is taut with respect to the
functor Hq.
I2
THEOREM
As U varies over the neighborhoods of A, there is an
isomorphism
Restricting U to the cofinal family of open neighborhoods, we have
Hq( U; G) = Hq( U; G), and the limit on the left is, by definition, equal to the
module on the right. •
PROOF
If (A,B) and (A',B') are pairs in X and (U,V) and (U',V') are respective
open neighborhoods, there is a relative Mayer-Vietoris sequence of
{(U,Y), (U',Y')}. As (U,Y) and (U',V') vary over open neighborhoods of (A,B)
and (A',B'), respectively, (U U U', V U V') varies over a cofinal family of neighborhoods of (A U A', BUB'). If (A,B) and (A',B') are closed pairs in a normal space X, it is also true that (U n U', V n V') varies over a cofinal family
of neighborhoods of (A n A', B n B'). Because the direct limit of exact
sequences is exact, we obtain the following result, which is another reason for
our interest in the modules H* (A,B).
13 THEOREM If (A,B) and (A',B') are closed pairs in a normal space X,
there is an exact relative Mayer- Vietoris sequence (for any coefficient
module G)
...
~
Hq(A U A', BUB')
~
Hq(A,B) EEl Hq(A',B') ~
Hq(A n A', B
n
B') ~
•
Given u E Hn(x X X, X X X - 8(X); R), as (U, V) varies over neighborhoods of (A,B), the homomorphisms
Yu: Hq(X - V, X - U; G) ~ Hn- q( U, V; G)
292
GENERAL COHOMOLOGY THEORY AND DUALITY
CHAP.
6
define a homomorphism
lim~
{Hq(X - V, X - U; G)}
---'> lim~
{Hn-q(u,v; G)}
Because singular homology has compact supports, if X is a Hausdorff space
the limit on the left is isomorphic to Hq(X - B, X - A; G). Therefore we
obtain a natural homomorphism
Yu: Hq(X - B, X - A; G)
---'>
fln-q(A,B; G)
such that if (U,v) is a neighborhood of (A,B), there is a commutative diagram
(all coefficients G)
Hq(X - V, X - U)
Hn- q(U, V)
---'>
---'>
Hq(X - B, X - A)
fln-q(A,B)
--4
Hn-q(A,B)
If (A,B) and (A',B') are closed pairs in a normal space X, then
exact Mayer-Vietoris sequence of the couple of open pairs
Yu maps the
{(X - B, X - A), (X - B', X - A')}
into the exact Mayer-Vietoris sequence of theorem 13 in such a way that each
square is commutative up to sign.
2
DUALITY IN TOPOLOGICAL MANIFOLDS
This section is devoted to a study of homology properties of topological
manifolds. Over a connected manifold as base space there is a fiber-bundle
pair called the homology tangent bundle. An orientation class of this bundle
gives rise to a duality in the manifold asserting that the cohomology of
a compact pair in the manifold is isomorphic to the homology of its complement. This duality theorem is proved by using the orientation class and the
slant product to define a natural homomorphism from homology to cohomology.
The resulting homomorphism is shown to be an isomorphism by proving it
first in euclidean space and then in an arbitrary manifold using the piecingtogether technique based on Mayer-Vietoris sequences.
A topological n-manifold (without boundary) is a paracompact Hausdorff
space in which each point has an open neighborhood homeomorphic to Rn
(called a coordinate neighborhood in the manifold). Following are some
examples of n-manifolds.
I
Rn and Sn are n-manifolds.
2
An open subset of an n-manifold is an n-manifold.
3
The product of an n-manifold and an m-manifold is an (n
+ m)-manifold.
SEC.
2
293
DUALITY IN TOPOLOGICAL MANIFOLDS
4 pn is an n-manifold, Pn(C) a 2n-manifold, and Pn(Q) a 4n-manifold for all
n. In fact, if X denotes one of these spaces and is coordinatized by homogeneous coordinates [to,tl, . . . ,tn ], then for each 0 ::::; i ::::; n the subset Ai C X
of points having ith coordinate 0 is a projective space of dimension n - 1 and
X - Ai is homeomorphic to R, RZ, or R4, respectively. Hence, X - Ai is a
coordinate neighborhood of X, and X is covered by these n + 1 coordinate
neighborhoods.
:.
LEMMA
In an n-manifold X each point x has an open neighborhood V
such that (V X X, V X X - o(V)) is homeomorphic to V X (X, X - x) by a
homeomorphism preserving first coordinates.
PROOF
Let U be a coordinate neighborhood containing x. Without loss of
generality, we can suppose that there is a homeomorphism <p: U:::::; Rn such
that <p(x) = O. Let D' = {z E Rn Illzll ::::; 2} and V = {z E Rn Illzll
I} and
define D = <p-l(D') and V = <p-l(V'). Then V is an open neighborhood of x
contained in the compact set D. If (X',X") E V X D - o(V), there is a unique
point z'" E Rn such that Ilz'" II = 2 and <p(x") belongs to the closed segment
from <p(x') to Z"'. If <p(x")
t<p(x') + (1 - t)Zll', with t E I, let h(x',x") E D - x
be the point such that <ph(x',x") = (1 - t)Zll', as illustrated
<
=
x' •
x'
D
and define h(x',x')
D'
= x. A homeomorphism
1/;: (V X X, V X X - o(X)) :::::; V X (X, X - x)
having the desired properties is defined by
'"
I/;(x,x )
=
")
{((x',x',xh(x',x"))
x"
¢D
x"ED
•
It follows from lemma 5 that if x' E V then (X, X-x') is homeomorphic
to (X, X - x). Hence we obtain the following result.
6
COROLLARY
In a connected n-manifold X the group of homeomorphisms
acts transitively; in particular, the topological type of (X, X -' x) is independent of x. Furthermore, projection to the first factor p: X X X -~ X is the projection of a fiber-bundle pair (X X X, X X X - o(X)) with fiber pair
(X, X - x). •
If V is a coordinate neighborhood of x in an n-manifold X, the couple
{V, X - x} is excisive, and so there is an excision isomorphism
294
GENERAL COHOMOLOGY THEORY AND DUALITY
H*(V, V -
X;
G);:::::; H*(X, X -
X;
CHAP.
6
G)
Since H* (V, V - x; G) ;:::::; H* (Rn, Rn - 0; G), it follows that
Hq(X, X - x; G) ;:::::;
q=l=n
q=n
{~
and so the fiber pair (X, X - x) of the fiber-bundle pair of corollary 6 has the
same homology as (Rn, Rn - 0). For this reason the fiber-bundle pair of corollary 6 will be called the homology tangent bundle of X (the tangent bundle
itself is an n-plane bundle defined if X is a differentiable manifold and having
homology properties isomorphic to those of the homology tangent bundle).
A connected n-manifold X is said to be orientable (over R) if its
homology tangent bundle is orientable [that is, if there exists an element
V E Hn(x X X, X X X - 8(X); R) such that for all x E X, V I x X (X, X - x)
is a generator of Hn(x X (X, X - x); R)]. Such a cohomology class V is called
an orientation of X. An n-manifold X (which is not assumed to be connected)
is said to be orientable if each component is orientable, and an orientation of
X is defined to be a cohomology class V E Hn(x X X, X X X - 8(X); R)
whose restriction to each -component is an orientation of that component.
7 EXAMPLE For Rn the fiber-bundle pair (Rn X Rn, Rn X Rn - 8(Rn)) is
trivial, because the map
f(z,z') = (z,
Z' -
z)
is a homeomorphism f: (Rn X Rn, Rn X Rn - 8(Rn)) ;:::::; Rn X (Rn, Rn - 0)
preserving first coordinates. Therefore Rn is an orientable n-manifold.
The results of Sec. 5.7 dealing with the homology properties of sphere
bundles carry over to the homology tangent bundle. We list some of these
explicitly.
8
Two orientations V and U' of a connected manifold X are equal if and
only if for some Xo E X
U I Xo X (X, X - xo)
9
= V'I Xo
X (X, X - xo)
Any manifold has a unique orientation over Z2.
•
•
lOA simply connected manifold is orientable over any R.
•
I I An n-manifold X is orientable if and only if there is an open covering
{V} of X and a compatible family {U v E Hn(V X X, V X X - 8(V); R)},
where Uv corresponds to an orientation of V under the excision isomorphism
Hn(V X X, V X X - 8(V); R) ;:::::; Hn(V X V, V X V - 8(V); R)
•
The duality theorem asserts that if U E Hn(X X X, X X X - 8(X); R) is
an orientation of X, then for any compact pair (A,B) in X, Yu is an isomorphism
of Hq(X - B, X - A; G) onto Hn-q(A,B; G). We prove this first for Rn by a
sequence of lemmas.
SEC.
2
295
DUALITY IN TOPOLOGICAL MANIFOLDS
12 LEMMA Let A C Rn be homeomorphic to a simplex and let ao EA.
Then Hq(Rn - ao, Rn - A; G) = 0 for all q and G.
Regarding Rn as an open subset of Sn, there is an excision isomorphism
Hq(Rn - ao, Rn - A; G) ;:::; Hq(Sn - ao, Sn - A; G). Because Sn - ao is
homeomorphic to Rn, Hq(sn - ao; G) = O. From lemma 4.7.13 and the
universal-coefficient formula, Hq(sn - A; G) = O. The lemma now follows from
exactness of the reduced homology sequence of the pair (Sn - ao, Sn - A). •
PROOF
13 COROLLARY If A C Rn is homeomorphic to a simplex and U is an
orientation of Rn over R, then for all q and R modules G
Yu: Hq(Rn, Rn - A; G) ;:::; Hn-q(A;G)
PROOF
Let ao E A and consider the diagram (all coefficients G)
y'l
y,
1
y"
1
Hn-q+!(A,ao)
---)
...
The rows are exact, and each square either commutes or anticommutes.
Since A is contractible, H* (A,ao) = O. Using lemma 12, we see that trivially
Yu: Hq(Rn - ao, Rn - A) ;:::; Hn-q(A,ao). By the five lemma, to complete the
proof we need only verify that Yu: Hq(Rn, Rn - ao) ;:::; Hn-q(ao). Because U is
an orientation, U I lao X (Rn, Rn - ao)] = 1 X u, where u E Hn(Rn, Rn - ao; R)
is a generator. By property 6.1.2,
yu\z) =
<u,z) 1
Since u is a generator of Hn(Rn, Rn - ao; R) ;:::; Hom (Hn(Rn, Rn - ao; R), R),
it follows that the map z -c> <u,z) of Hn(Rn, Rn - ao; R) to R is an isomorphism; and hence so is Yu: Hn(Rn, Rn - ao; R) ;:::; HO(ao;R). If q =1= n, it is
trivially true that Yu: Hq(Rn, Rn - ao; R) ;:::; Hn-q(ao;R), since both modules
are trivial. •
If U is an orientation of Rn over Rand (A,B) is a compact
polyhedral pair in Rn, then for all q and all R modules G there is an
isomorphism
14
THEOREM
Because of the naturality properties of Yu, it suffices to prove this for
the case where B is empty. The theorem follows for A from corollary 13 by
induction on the number of simplexes in a triangulation of A, using MayerVietoris sequences and the five lemma. •
PROOF
15 COROLLARY If U is an orientation of Rn over Rand (A,B) is a compact
pair in Rn, then for all q and R modules G there is an isomorphism
Yu: Hq(Rn - B, Rn - A; G) ;:::; Hn-q(A,B; G)
PROOF
Since the family of compact polyhedral pairs is cofinal in the family
296
GENERAL COHOMOLOGY THEORY AND DUALITY
CHAP.
6
of all neighborhoods of a compact pair (A,B) in Rn, the corollary follows from
theorem 14 by taking direct limits. •
Because of the commutativity of the triangle
Hq(Rn - B, Rn - A; G)
'I'i!
~['
iin-q(A,B; G) ~ Hn-q(A,B; G)
it follows from theorem 14 and corollary 15 that any imbedding of a compact
polyhedral pair in Rn is taut (which is also a consequence of corollary 6.1.11).
As an immediate result of corollary 15, we obtain the following Alexander
duality theorem.
16
THEOREM
If A is a compact subset of Rn, then for all q and R modules G
Hq(Rn - A; G) ;:::: Hn- q-l(A;G)
PROOF
Because H* (Rn;G)
= 0, there is an isomorphism
0*: Hq+l(Rn, Rn - A; G) ;:::: Hq(Rn - A; G)
The result is obtained by composing the inverse of this isomorphism with the
isomorphism of corollary 15. •
For general orientable manifolds there is the following duality theorem.
17 THEOREM Let U be an orientation over R of an n-manifold X and let
(A,B) be a compact pair in X. Then for all q and R modules G there is an
isomorphism
Yu: Hq(X - B,
X - A; G) ;:::: Hn-q(A,B; G)
Because of the naturality properties of Yu, it suffices to prove the
theorem for the case where B is empty. If A is contained in some coordinate
neighborhood Vof X and U' = U I (V X V, V X V - 8(V)) is the induced
orientation of V, there is a commutative triangle (all coefficients G)
PROOF
Hq(V, V - A) :? Hq(X, X - A)
Y"\
,!?u
By corollary 15, YU' is an isomorphism, hence Yu is also an isomorphism. The
result for arbitrary compact A follows by induction on the finite number of
coordinate neighborhoods needed to cover A, using naturality of Yu, the usual
Mayer-Vietoris technique, and the five lemma. •
In case X is compact, by applying theorem 17 to the pair (X, 0) and
observing that i: Hq(X;G) ;:::: Hq(X;G), we obtain the following Poincare
duality theorem.
SEC.
2
297
DUALITY IN TOPOLOGICAL MANIFOLDS
18 THEOREM If U is an orientation over R of a compact n-manifold X,
then for all q and R modules G there is an isomorphism
Yu: Hq(X;G) ::::: Hn-q(X;G)
-
A pair (X,A) is called a relative n-manifold if X is a Hausdorff space, A is
closed in X (A may be empty), and X - A is an n-manifold. For relative
manifolds there is the following Lefschetz duality theorem.
Let (X,A) be a compact relative n-manifold such that X - A
is orientable over R. For all q and R modules G there is an isomorphism
19 THEOREM
Hq(X - A; G) ::::: iin-q(X,A; G)
Let {N} be the family of closed neighborhoods of A directed downward by inclusion. There are isomorphisms
PROOF
lim~
lim~
{Hq(X - N; G)} ::::: Hq(X - A; G)
{iin-q(X,N; G)} ::::: iin-q(X,A; G)
the first because singular homology has compact supports and the second as a
consequence of theorem 6.1.12. Let V be an open neighborhood of A with V
contained in the interior of N and let U be .an orientation of X - A over R. By
theorem 17 and standard excision propertiesl there are isomorphisms (all
coefficients G)
Hq(X - N)
;?
Hq((X - A) - (N - V), (X - A) - (X - V))
:::1 'Iv
fIn-q(X,N)
:::7
iin-q(X -
v, N - V)
which yield the result on passing to the limit.
-
An n-manifold X with boundary X is a paracompact Hausdorff space
such that (X,X) is a relative n-manifold and every point x E X has a neighborhood V such that (V, V n X) is homeomorphic to Rn-l X (1,0). Since X may
be empty, the concept of manifold with boundary encompasses that of manifold without boundary.
If X is an n-manifold with boundary X, then Xhas neighborhoods N such
that (N,X) is homeomorphic to X X (1,0).1 Such a neighborhood N is called a
collaring of X, and its interior is called an open collaring of X. (In case X is
compact, any neighborhood of X contains a collaring of x.) Because of the
existence of such collarings, X - X is a weak deformation retract of X, and
the pair ((X - X) X (X - X), (X - X) X (X - X) - 8(X -X)) is a weak
deformation retract of (X X X, X X X - 8(X)).
An n-manifold X with boundary X is said to be orientable over R if
X - X is orientable over R. An orientation over R of X is a class
1
See M. Brown, Locally flat imbeddings of topological manifolds, Annals of Mathematics,
vol. 75, pp. 331-341, 1962.
298
GENERAL COHOMOLOGY THEORY AND DUALITY
CHAP.
6
U E Hn(X X X, X X X - 8(X); R) whose restriction to ((X - X) X (X - X),
(X - X) X (X - X) - 8(X - X)) is an orientation of X - X over R. For
manifolds with boundary the Lefschetz duality theorem takes the following
form.
20 THEOREM Let X be a compact n-manifold with boundary X and orientation U over R. For all q and R modules G there are isomorphisms (where
i: X - X eX)
Hq(X;G) ~ Hq(X -
X;
G) ~ Hn-q(x,x; G)
Hq(X,X; G) ~ Hn-q(x -
X;
G) ~ Hn-q(X;G)
Because i is a homotopy equivalence, i* and i * are isomorphisms.
Let N be a collaring of X with interior IV. Let U' be the orientation of X - X
obtained by restricting U. In the following commutative diagram each horizonal map is induced by inclusion and is an isomorphism because it is an
excision (labelled e) or a homotopy equivalence (labelled h) (all coefficients G):
PROOF
Hq(X - X) ~ Hq(X - N) ~ Hq((X - X) - (N - IV), (X - X) - (X - IV))
yut
yut
LyU'
Hn-q(x,x) ~ Hn-q(X,N) ~ Hn-q(x -
IV, N
- IV))
Because (X - IV, N - IV) has arbitrarily small neighborhoods of which it is a
deformation retract i: Fln-q(x - IV, N - IV) ~ Hn-q(X - IV, N - IV), and it
follows from theorem 17 that the right-hand vertical map is an isomorphism
(because it corresponds to the isomorphism '10'). Therefore the left-hand
vertical map is also an isomorphism proving the first part of the theorem.
Similarly, there is a commutative diagram
~ Hq(X ~L
Hn-q(X - X)
Lw
~L
-tt Hn-q(X -
X, (X - X) - (X - IV))
IV)
~ Hn-q(x -
IV)
Because X - IV has arbitrarily small neighborhoods of which it is a deformation retract, it follows from theorem 17 that the right-hand vertical map is an
isomorphism. Therefore the left-hand vertical map is also an isomorphism,
proving the second part of the theorem. •
From the isomorphisms of theorem 20 and the universal-coefficient
theorem for homology, we obtain a short exact sequence
o ~ Hq(X;R) ® G ~ Hq(X;G) ~ Hq+l(X;R) * G ~ 0
and a similar short exact sequence for Hq(X,X; G). Since this is so for every
R module G, from theorem 5.5.13 we have the following result.
If X is a compact n-manifold with boundary
over R, then H* (X;R) and H* (X,X; R) are finitely generated. •
21
COROLLARY
X orientable
SEc.3
299
THE FUNDAMENTAL CLASS OF A MANIFOLD
Later in the chapter (see theorem 6.9.11) we shall prove that corollary 21
is also valid for nonorientable manifolds.
3
THE Ft:NDA.MENTAL CI.ASS OF A M."-NIFOLD
In view of the importance of the concept of orientability of manifolds, we
shall now investigate some equivalent formulations. We shall show that a
compact connected n-manifold is orientable if and only if its n-dimensional
homology module is nonzero. In fact, any orientation class of the manifold
will be shown to correspond to a generator of the n-dimensional homology
module. Moreover, if z is the element of Hn corresponding to the orientation,
then the cap product of z and a cohomology class defines a homomorphism
which equals, up to sign, the inverse of the duality isomorphism. The methods in
this section rely heavily on the technique of piecing together homology classes, 1
analogous to the piecing together of cohomology classes in lemma 5.7.16.
Let X be a space, X' a subspace of X, and If = {A} a collection of subsets of X - X'. A compatible c? family is a family {ZA E Hq(X, X - A; G)}
(for some fixed q and G) indexed by U' such that if A, A' E If, then ZA and ZA'
map to the same element of Hq(X, X - A n A'; G) under the homomorphisms
Hq(X, X - A; G)
---7
Hq(X, X - A n A'; G)
~
Hq(X, X - A'; G)
The compatible If families form a module with respect to componentwise
operations that will be denoted by H8'(X,X'; G). For the collection If of all
compact subsets of X - X' we use Hqc(X,X'; G) to denote the corresponding
module.
We are interested in the module HnC(X,X; R) for an n-manifold X with
boundary X. The following lemma is important in this connection.
I
LEMMA
Let X be an n-manifold with boundary
pact subset of X - X. For all R modules G
X and let A be a com-
Hq(X, X - A; G) = 0
Assume first that A is contained in some coordinate neighborhood V
in X - X. By excision, Hq(V, V - A) ;::::; Hq(X, X - A), and since V is homeomorphic to Rn, we can use corollary 6.2.15 to obtain
PROOF
Hq(V, V - A) ;::::; fIn-q(A)
=0
For arbitrary compact A the result follows by induction on the number of coordinate neighborhoods needed to cover A, using Mayer-Vietoris sequences. •
In an n-manifold X with boundary X a small cell in X - X is defined to
be a compact subset A having an open neighborhood V C X - X such that
This technique can be found in H. Cartan, Methodes modernes en topologie a1gebrique,
Commentarii Mathematici Helvetici, vol. 18, pp. 1-15, 1945.
1
300
GENERAL COHOMOLOGY THEORY AND DUALITY
CHAP.
6
(V,A) is homeomorphic to (Rn,En). Every point of X - X has arbitrarily small
neighborhoods which are small cells. If A and V are as above, there is an
excision isomorphism
Hq(X, X - A; G) ;:::: Hq(V, V - A; G) ;::::
q-=l=-n
q=n
{~
If Xo E A, then the inclusion map induces isomorphisms
Hq(X, X - A; G) ;:::: Hq(X, X -
Xo; G)
We use HqSC(X,X; G) to denote the module of compatible ef families, where ef
consists of the collection of small cells of X - X. Since the collection of small
cells is contained in the collection of compact subsets of X - X, there is
a natural homomorphism
HqC(X,X; G) ~ Hqsc(X,X; G)
which assigns to a compatible family {ZA} indexed by all compact A the compatible subfamily of elements indexed by small cells.
2
LEMMA
Let X be an n-manifold with boundary
X.
Then, for all G
HnC(X,X; G) ;:::: HnSC(X,X; G)
PROOF
For each positive integer i let ~ be the collection of compact subsets
of X - X contained in the union of i small cells. Then ~ C ~+1 and U ~ is
the collection of all compact subsets of X - X. There are homomorphisms
and an isomorphism Hn c ;:::: lim {H~i}.
Since every element of ef1 is contained in some small cell, it is obvious
that H~l ;:::: Hn sc . By the usual Mayer-Vietoris technique and lemma 1, it
follows that for any i;::: 1 H~i+l ;:::: H~i. Combining these isomorphisms
yields the result. •
This gives the following important result.
3
THEOREM
Let X be an n-manifold with boundary X and let
{ZA} E Hnc(X,X; G)
(a) {ZA} = 0 if and only if Zx = 0 for all x E X-X.
(b) If X is connected, {ZA}
oif and only ifzx ofor some x E
=
=
x-x.
(a) follows from lemma 2 and the observation that if A is a small cell
and x E A, then
PROOF
Hn(X, X - A; G) ;:::: Hn(X, X -
=
=
x;
G)
and so ZA
0 if and only if Zx
O.
To prove (b), assume zXo = 0 for some Xo EX - X. Because X is
connected, so is its weak deformation retract X - X. This implies that if
SEC.
3
301
THE FUNDAMENTAL CLASS OF A MANIFOLD
x E X - X, there is a finite sequence of small cells AI,
, Am in X - X
such that Xo E Al and x E Am, and Ai meets A i + l for 1 S i
m. Choose a
m. There are isomorphisms
pOint Xi E Ai n Ai+l for 1 S i
<
<
Hn(X, X - xo)
~
Hn(X, X - AI) ;;? Hn(X, X - Xl) ~ ...
~ Hn(X, X - Am) ;;? Hn(X, X - X)
from which it follows that if Z"'o = 0, then z'"
X E X - X, the result follows from (a).
•
= 0.
Since this is so for all
If X is an n-manifold with boundary X, a fundamental family of X over
R is an element {ZA} E HnC(X,X; R) such that for all X E X - X, z" is a generator of Hn(X, X - X; R). The relation between fundamental families and
orientations is made precise in the next result.
THEOREM
Let X be an n-manifold with boundary X. There is a one-toone correspondence between orientations V (over R) of X and fundamental
families {ZA} (over R) of X such that V and {ZA} correspond if and only if
Yu(ZA) = 1 E HO(A;R) for all compact A in x-x.
4
If V is an orientation of X, let V' be the induced orientation of
x-x. For any compact A C X - X we have the commutative diagram (all
coefficients R)
PROOF
Hn(X,X - A) ~ Hn(X - X, (X - X) - A)
l'iu
fjO(A)
By theorem 6.2.17, the right-hand vertical map is an isomorphism, and since
1 E HO(A) is the image of 1 E j{O(A), there is a unique ZA E Hn(X, X - A)
such that 'Yui* -1(ZA) = 1 E j{O(A). Because of the uniqueness of ZA and the
naturality of Yu and 'iu', the collection {ZA} is a compatible family. From the
commutativity of the above diagram, Yu(ZA) = 1 E HO(A) for all compact A
in X-X. Hence we need only verify that {ZA} is a fundamental family. In
case A
x, it follows from the commutativity of the above square and the
fact that i: j{O(x);::::; HO(x) that Yu: Hn(X, X - x) ;::::; HO(x). Therefore
z'" = Yu- I (I) is a generator of Hn(X, X - x). Hence {ZA} is a fundamental
family with the desired property, and the collection {Z"'}"'EX-X (and hence, by
theorem 3a, {ZA}) is uniquely characterized by the property yu(z",)
1 E HO(x).
Conversely, given a fundamental family {ZA}, let Vbe any open subset
of X - X homeomorphic to Rn. If Xo E V, then H* (V;R) ;::::; H* (xo;R), which
implies that
=
=
H*(V X X, V X X - 8(V); R);::::; H*(xo X (X, X - xo); R)
If u E Hn(V X X, V X X - 8(V); R), it follows from the Kiinneth formula
for cohomology (theorem 5.6.1) that u I Xo X (X, X - xo) = 1 X u' for a
unique u' E Hn(x, X - Xo; R) ;::::; Hom (Hn(X, X - Xo; R), R). By property 6.1.2,
302
GENERAL COHOMOLOGY THEORY AND DUALITY
[u I xo X (X, X - XO)]!Z,xo
CHAP.
6
= (u',z,xo) 1
Since z,xo is a generator of Hn(X, X - Xo; R), (u' ,z,xo> completely determines
u'. Therefore there is a unique element V E Hn(V X X, V X X - 8(V); R)
such that [V I Xo X (X, X - xo)]!z"'o = 1 E HO(xo;R).
We now show that for any x E V, [Vlx X (X, X - x)]!z", = 1 E HO(x;R).
If x and x' belong to a small cell A C V, then ZA maps to z'" and to Z",'.
Therefore [V I A X (X, X - A)]!ZA E HO(A;R) maps to [V I x X (X, X - x)]!z",
and to [V I x' X (X, X - x')]!z"" by naturality of Yu. Since HO(A;R) ~ HO(x;R)
and HO(A;R) ~ HO(x';R), it follows that both [V I x X (X, X - x)l!z", =
1 E HO(x;R) and [V I x' X (X, X - x')]!z"" = 1 E HO(x';R) or neither equation
is true. Hence the set of x E V for which [V I x X (X, X - x)]!z", = 1 E HO(x;R)
is open and its complement in V is open. Since V is connected and
[V I Xo X (X, X - xo)]!z"'o = 1, it follows that [V I x X (X, X - x)]!z", = 1
for all x E V.
This means that V is an orientation of V, and if V' is a similarly defined
orientation for another coordinate neighborhood V' in X - X, then for any
x E V n V', V I x X (X, X - x) = V' I x X (X, X - x). This implies that V
and V' induce the same orientation of V n V'. Hence the collection {Vv} for
coordinate neighborhoods V in X - X is compatible. Therefore there is an
orientation V of X such that V I (V X X, V X X - 8(V)) = V v. From the
construction of Vv we see that yu(z",) = 1 E HO(x;R) for all x E X - X. By the
first half of the proof, there is a fundamental family {ZA} such that
Yu(ZA) = 1 E HO(A;R). Then z~ = Z,x for all x E X - X, and by theorem 3a,
ZA = ZA for all compact A C X - X. Therefore Yu(ZA) = 1 E HO(A;R) for all A,
proving that every fundamental family {ZA} corresponds to ~ome orientation V.
The orientation V is uniquely characterized by the fundamental family
{ZA}' for if V and V' are two orientations of X such that yu(z",) = Yu,(z,x) for all
x E X - X, then V I x X (X, X - x) = V' I x X (X, X - x) for all x E X-X.
Therefore, by lemma 5.7.13, V = V'. •
This last result gives the following useful characterization of orientability
for connected manifolds.
it THEOREM Let X be a connected n-manifold with boundary X. If
HnC(X,X; R) =1= 0, then HnC(X,X; R) ~ R and any generator is a fundamental
family of X.
PROOF
From theorem 3b it follows that, given Xo E X -
X, the homomorphism
HnC(X,X; R) ~ Hn(X, X - Xo; R)
sending {ZA} to z"'o is a monomorphism. Since Hn(X, X - Xo; R) ~ R, either
HnC(X,X; R) = 0 or HnC(X,X; R) ~ R. Assume HnC(X,X; R) ~ R and let {ZA}
be a generator of Hnc(X,X; R). Assume that for some x E X - X, z'" is not a
generator of Hn(X, X - x; R). There is then a noninvertible element r E R
such that z'" = rz~ for some z~ E Hn(X, X - x; R). It follows that for any small
cell A containing x, ZA
rZA for some ZA E Hn(X, X - A; R). Because X
=
SEC.
3
303
THE FUNDAMENTAL CLASS OF A MANIFOLD
is connected, it follows, as in the proof of theorem 3b, that for any small cell
A in X - X, ZA = rZA for some ZA E Hn(X, X - A; R). If A' is a small cell in A,
then rZA maps to rzA' in Hn(X, X - A';.R). Because Hn(X, X - A'; R) is
torsion free, by lemma 1, ZA maps to ZA" Therefore {ZA} E HnSC(X,X; R).
By lemma 2, it follows that the original element {ZA} E HnC(X,X; R) is divisible
by the element r E R. Since r is not invertible, this contradicts the hypothesis
that {ZA} is a generator of HnC(X,X; R). •
6
COROLLARY
If X is a connected n-manifold with boundary X, then X is
orientable over R if and only if HnC(X,X; R) -=1= O.
PROOF
This is immediate from theorems 4 and 5.
•
We now specialize to the case of a compact manifold.
7
LEMMA
If X is a compact n-manifold with boundary X, there is an
isomorphism
Hn(X,X; G) ;::::; Hnc(X,X; G)
= image of Z in Hn(X, X - A; G)}.
Let V be an open collaring of X and let B = X-V. Then B is com-
sending Z E Hn(X,X; G) to {ZA
PROOF
pact and there is a homomorphism
Hnc(X,X; G) ~ Hn(X, X - B; G)
sending {ZA} to ZB. Since X - B
equivalence, the composite
=V
and (X,X) C (X, V) is a homotopy
Hn(X,X; G) ~ HnC(X,X; G) ~ Hn(X, X - B; G)
is an isomorphism. To complete the proof we need only show that the righthand map is a monomorphism. Assume that {ZA} is a compatible family such
that ZB = 0 and let A be any compact set in X-X. There is then an open
collaring V' of X such that V' C Vand V' is disjoint from A. Let B' = X - V'.
Then A, B C B', and we have homomorphisms (all coefficients G)
Hn(X, X - A)
~
Hn(X, X - B')
~
Hn(X, X - B)
the second map being an isomorphism because (X, V') C (X, V) is a homotopy
equivalence. Since ZB
0, ZB'
0 and ZA
O. Therefore {ZA}
0 in
Hnc(X,X; G).
•
=
=
=
=
8 COROLLARY A compact connected n-manifold X with boundary X is
orientable over R if and only if Hn(X,X; R) -=1= O.
PROOF
This is immediate from corollary 6 and lemma 7.
•
If X is a compact n-manifold with boundary X, a fundamental class over
R of X is an element Z E Hn(X,X; R) whose image in HnC(X,X; R) under the
isomorphism of lemma 7 is a fundamental family [that is, for every x E X - X
the image of Z in Hn(X, X - x; R) is a generator of the latter].
304
GENERAL COHOMOLOGY THEORY AND DUALITY
CHAP.
6
9
THEOREM
If X is a compact n-manifold with boundary X, there is a
one-to-one correspondence between orientations U over R and fundamental
classes z over R such that U corresponds to z if and only if yu{z) = 1 E HO(X;R).
PROOF
This follows from theorem 4 and lemma 7 on observing that an
element v E HO(X;R) equals 1 if and only if v I x = 1 E HO(x;R) for all
x E X-X. •
10 COROLLARY If X is a compact n-manifold with boundary X, then if X
is orientable, so is X, and any fundamental class of X maps to a fundamental
class of X under the connecting homomorphism
0*: Hn(X,X; R) ~ Hn_1(X;R)
Let N be a collaring of X with interior N. Then N is an n-manifold
with boundary X U (N - N), and there is a commutative diagram (all
coefficients R)
PROOF
Hn(X,XU (X-
N»)
j* 1':::::
.
Hn_1(X)
~
~
.
0
0
Hn_1(X U (N - N), N - N)
Hn(N, X U (N - N))
It is clear from the definition of fundamental class that if z E Hn(X,X) is a
fundamental class of X, then i* -li*z = z' is a fundamental class of N. Because
N is homeomorphic to X X I in such a way that X and N - N correspond to
X X 0 and X X 1, respectively, the Kiinneth formula implies
Hn(N, X U (N -
N)) ,:::; Hn-1(X)
(8) H1(I,i)
Let wE H1(I,i) be a generator and let {Xj} be the components of X.
Then z' corresponds to 2: zj X w for some zj E Hn-1(Xj ), and "4 -lO*Z =
-+- 2: zj. Hence 0* z = -+- 2: zj, and since z is a fundamental class of X,
zj X w corresponds to a fundamental class of Xj X 1. Therefore zj is nonzero and is a generator of Hn-l(X j ). Then zj is a fundamental class of Xj,
whence -+- 2: zj = 0* z is a fundamental class of X. •
We are now heading toward a proof that cap product with a fundamental
class is an isomorphism which, up to sign, is inverse to the duality isomorphism
in a compact manifold. First we need a lemma.
I I LEMMA Let X be a compact orientable n-manifold with boundary X
and let Pl, P2: X X X ~ X be the proiections. Given
u E Hq(X X X, X X X - 8(X); R), z E Hm(X X X, X X X - 8(X); G),
and v E Hr(X;G), then
Pl*(U 1""\ z) = P2*(U 1""\ z)
u v pT v = u v p! v
PROOF
in
in
Hm_q(X;G)
Hq+r(x X X, X X X - 8(X); G)
Let T: (X X X, X X X - 8(X)) ~ (X X X, X X X - 8(X)) be the
SEC.
3
305
THE FUNDAMENTAL CLASS OF A MANIFOLD
map interchanging the factors. If w E Hn(X,X; R) is a fundamental class of X,
then w X w E H2n((X,X) X (X,X); R) is a fundamental class of X X X (whence
(-I)nw X w. By theorem 9, T maps
X X X is orientable), and T* (w X w)
the orientation of X X X corresponding to w X w into ( _1)n times itself. Let
=
y: Hm(X X X, X X X - ~(X); G) ;:::; j[2n-m(~(x),~(X);G)
be the duality map associated to this orientation. Then we have a commutative diagram (all coefficients G)
Hm(X X X, X X X -
~(X);
G)
T
~
Hm(X X X, X X X -
~
~(X);
G)
y<-l)ny
i!2n-m( ~(X),~(X);G)
Therefore T* (z) = ( -I)nz for any z E H* (X X X, X X X implies T*(u) = (-I)nu for any u E H*(X X X, X X X -
~(X);
~(X);
= P1* T* (u "z) = P1* (T* u " T*z) = P1* (u "
and uv p~v = (-I)nT*(u vp~v) = u v T*p~v = uv p!v •
P2* (u "z)
G) (which
G)). Then
z)
Let z be a fundamental class over R of a compact n-manifold
X. For all q and R modules G the homomorphism
K z(v) = v " z defines isomorphisms
12 THEOREM
X with boundary
K
K
z: Hq(X;G) ;:::; Hn_q(X,X; G)
z: Hq(X,X; G) ;:::; Hn_q(X;G)
which are, up to sign, the inverse of the duality isomorphisms of theorem 6.2.20
defined by the orientation corresponding to z.
PROOF
Let U be the orientation of X corresponding to z as in theorem 9,
and let j: X - X c X. We prove commutativity up to sign in the triangle (all
coefficients G)
i* \.
Hq(X)
/L"'Kz
For w E Hq(X - X), by property 6.1.6,
kzyu(w)
= {[U I (X,X) X (X - X)]!w} "
= P1*{[U I (X,X) X (X - X)] "
z
(z X w)}
By lemma 11, this equals
P2*{[U I (X,X) X (X - X)] " (z X w)}
= P1*T* {[U I (X,X) X (X - X)] " (z X w)}
= +j*]11*{[U I (X - X) X (X,X)] " (w X z)}
where ]11: (X - X) X X ~ X - X is projection to the first factor. Again by
property 6.1.6,
306
GENERAL COHOMOLOGY THEORY AND DUALITY
(h* {[U I (X - X) X (X,X)] r-. (w X z)}
CHAP.
6
= yu(z) r-. w = w
Therefore
Kzyu(W)
= +i* (w)
Similarly, we prove commutativity up to sign in the triangle
Hq(X) ~ Hn-q(X,X)
j*\
bO'
Hq(X - X)
For v E Hq(X), by property 6.1.5,
YUKz(V) = [U I (X - X) X (X,X)]/(v r-. z)
= {[U v p~(v)]1 (X - X) X (X,X)}/z
By lemma 11 and property 6.1.4, this equals
+{[p!i*(v) v
Therefore
UJ I (X - X) X (X,X)}/z
= +i*(v) v yu(z)
YUKz(V) = +i*v
4
= +i*(v)
•
THE ALEXANDER {;OHOMOLOGY THEORY
We shall now describe a cohomology theory particularly suited for applications
in which a space is mapped into polyhedra (the singular theory is more suitable
for applications where polyhedra are mapped into a space). One approach to
the theory, called the Cech construction, is based on approximating a space
by nerves of open coverings; another approach, called the Alexander-Kolmogoroff construction, is based on complexes built of "small" simplexes consisting
of finite sets of points. We shall begin with the Alexander construction, and
show later in the chapter (see corollary 6.9.9 and the follOWing paragraph)
that if (A,B) is a closed pair in a manifold X, then [[q(A,B; G) as defined in
Sec. 6.1 is the Alexander cohomology of (A,B) with coefficients G.
Let G be an R module and let X be a topological space. For q ~ 0 let
Cq(X;G) be the module of all functions cP from Xq+l to G with addition and
scalar multiplication defined pointwise. Thus, if xo, Xl, . . . , Xq E X, then
CP(XO,Xl, . . . ,Xq) E G, and if CPl, qJ2 E Cq(X;G) and r E R, then
(CPl
rcpl(xO, ... ,xq) = r(CPl(xO, ... ,xq))
. . . ,xq) = CPl(XO, ... ,xq) + CP2(XO, ... ,Xq)
+ CP2)(XO,
We shall omit the symbol G from Cq(X;G) where its absence will not cause
confusion.
A coboundary homomorphism 8: cq(X) ~ Cq+l(X) is defined by the
formula
SEC.
4
307
THE ALEXANDER COHOMOLOGY THEORY
(8<p) (xo, ... ,Xq+l)
= OS'Sq+l
.~
(_I)i<p(XO,
... ,Xi, ... ,Xq+l)
Then 88 = 0 and C*(X) = {Cq(X),8} is a cochain complex over R. If X is
nonempty, it is augmented over G by 1/: G ~ CO(X), where (1/(g))(x) = g for
g E G and all x E X. So far the topology of X has played no role, and the following result shows that C* (X) has uninteresting cohomology.
I
LEMMA
If X is a nonempty space, 1/*: G:::::: H*(C*(X;G)).
PROOF
Let x be a fixed point of X and define a cochain homotopy
D: C* (X) ~ C* (X) by
(D<p) (xo, ... ,Xq)
Then
8D<p
= <p(x,xo,
..
+ D8<p = {: _ 1/(<p(x))
Therefore, if '7": C(X;G)
~
,Xq)
deg <p
deg <p
q
~
0
>0
=0
G is the cochain map defined by
deg <p
deg <p
=
>0
=0
then '7"1/
IG and D is a cochain homotopy from Ic*(X) to 1/'7". Therefore 1/ is a
cochain equivalence, whence the result. We now use the topology of X to pass to a more interesting quotient complex. An element <p E Cq(X) is said to be locally zero if there is a covering 'Yi of
X by open sets such that <p vanishes on any (q + I)-tuple of X which lies in
some element of GIl. Thus, if we define 'Yiq+l = U U E'1l Uq+l C Xq+l, then <p
vanishes on 'Yiq+1. The subset of Cq(X) consisting of locally zero functions is a
submodule, denoted by Coq(X), and if <p vanishes on 'Yiq+1, then 8<p vanishes
on Gllq+2 , whence ct(X)
{Coq(X),8} is a co chain subcomplex of C*(X). We
define C * (X) to be the quotient cochain complex of C * (X) by C t (X). If X is
nonempty, the composite
=
G 4 C*(X) ~ C*(X)
is an augmentation of G*(X), also denoted by 1/. The cohomology module of
C*(X) of degree q is denoted by jiq(X;G).
Given a function f: X ~ Y (not necessarily continuous), there is an induced cochain map
f#: C*(Y;G)
~
C*(X;G)
defined by the formula
(f#<p)(xo, ... ,Xq)
= <P(f(xo),
... ,f(Xq))
<p E Cq(y); Xo, . . . ,Xq E X
If <p vanishes on 'Y q+1, where 'Y is an open covering of Y, and if there is
an open covering GIL of X such that f maps each element of 'Yi into some element of CV; then f#<p vanishes on 'Yiq+l. In particular, if f is continuous, f-I'Y
is an open covering of X which can be taken as Gil., and therefore f#
308
GENERAL COHOMOLOGY THEORY AND DUALITY
maps C~(Y) into
cochain map
C~(X).
CHAP.
6
It follows that if fis continuous, there is an induced
f#: C*(Y;G) ~ C*(X;G)
Let A be a subspace of X and let i: A c X. Then .i#: C * (X; G) ~ C * (A; G)
is an epimorphism. Therefore the kernel of i# is a co chain subcomplex
of C*(X;G), denoted by C*(X,A; G). The relative module fIq(X,A; G) is defined to be the cohomology module of C*(X,A; G) of degree q.
Since there is a short exact sequence of cochain complexes
o~ C*(X,A; G)
L
C*(X;G) ~ C*(A;G)~ 0
it follows that there is an exact sequence
2
... ~ fIq(x,A; G)
4
fIq(X;G) ~ fIq(A;G) ~ fIq+l(X,A; G) ~
The graded module fI*(X,A) = {fIq(X,A; G)} is the module function of
the cohomology theory we are constructing, and the homomorphism
8 *: fIq(A;G) ~ fIq+l(X,A; G) is the connecting homomorphism of the theory.
Given a continuous map f: (X,A) ~ (Y,B), there is induced by f a commutative diagram of cochain maps
o~
C* (Y,B; G) ~ C* (Y;G) ~ C* (B;G) ~ 0
f#
o~
1
(fI
X )#
1
l(fI A )#
C* (X,A; G) ~ C* (X;G) ~ C*(A;G) ~ 0
The homomorphism f*: fI* (Y,B; G) ~ fi* (X,A; G) is defined to be the
homomorphism induced by the cochain map f# in the above diagram. It is
then clear that for fixed G, fi * (X,A; G) and f* constitute a contravariant
functor from the category of topological pairs to the category of graded
R modules. Furthermore, the connecting homomorphism 8 * is a natural
transformation of degree 1 from fI* (A;G) to fI* (X,A; G). Therefore we have
the constituents of a cohomology theory, and we shall verify that the axioms
are satisfied. The resulting cohomology theory is called the Alexander (or
Alexander-Spanier l ) cohomology theory, and fiq(X,A; G) is called the Alexander cohomology module of (X,A) of degree q with coefficients G.
The exactness axiom is a consequence of the exactness of the sequence 2.
The dimension axiom will follow from the next result.
:I
LEMMA
If X is a one-point space,
1)
*: G ::::; fI* (X;G).
Because X is a one-point space, a locally zero function on X is zero.
Therefore C*(X;G) = C*(X;G) and the result follows from lemma 1. •
PROOF
Before proving the excision axiom it will be useful to introduce another
eo chain complex for the relative theory. If A C X, let C* (X,A) be the sub1 See E. Spanier, Cohomology theory for general spaces, Annals of Mathematics, voL 49
pp. 407-427, 1948.
SEC.
4
309
THE ALEXANDER COHOMOLOGY THEORY
complex of C* (X) of functions
short exact sequence
qJ
which are locally zero on A. Thus there is a
o ~ C*(X,A) ~ C*(X) ~ C*(A) ~ 0
C*(X,A). It follows that C*(X,A) = C*(X,A)/C~(X).
and C~(X) C
excision axiom follows from the next result.
The
4
LEMMA
Let U be a subset of A C X such that U has an open neighborhood W with W C int A. Then the inclusion map;: (X - U, A - U) C (X,A)
induces an isomorphism
;#: C*(X,A);::::: C*(X -
There is a commutative diagram with exact rows
PROOF
O~C~(X)
~
C*(X,A)
~
o~
u, A - U)
~
C*(X,A)
~O
~k#
C~(X - U) ~ C*(X - U, A - U) ~ C*(X - U, A - U) ~ 0
It suffices to prove that Ak# is an epimorphism and that (k#t1(C~ (X - U)) =
C~(X). If Cf' EO Gq(X - U, A - U), let cp EO Gq(X) be defined by
_
qJ(xo, . . . ,Xq) =
(O (Xo,
qJ
... ,Xq
)
E W for some 0 :::; i :::; q
Xo, . . . ,Xq E X - W
Xi
If C\,( is an open covering of A - U such that qJ vanishes on "q+1, then
~ = {V U W I V E C\,(} is an open covering of A such that cp vanishes on
~q+1. Therefore cp E O(X,A), and from the definition of cp, k#cp - qJ vanishes
on Yrq+! where Yr = I V n int A I V EO Y) U IX - W), which is an open
covering of X - U. Therefore Ak#cp = AqJ, and because A is an epimorphism,
so is Ak#.
Assume that qJ E Cq(X,A) is such that k#qJ E Coq(X - U). Because qJ is
locally zero on A, there is an open covering 0(L1 of A such that qJ vanishes on
"Il1 q + 1 . Because k#qJ E Coq(X - U), there is an open covering 0~ of X - U
such that qJ vanishes on 0J2q+1. Let
Then 'Y = ''Vi U '\2 is an open covering of X such that
,\q+1. Therefore qJ E Coq(X) and sc
(k#)-1(C~(X
- U)) =
C~(X)
qJ
vanishes on
•
The homotopy axiom will be proved in the next section. We conclude
this section with a study of Bo. A function qJ from a topological space X to a
set is said to be locally constant if there is an open covering ql of X such that
qJ is constant on each element of "It.
5
THEOREM
If A C X, then BO(X,A; G) is isomorphic to the module of
locally constant functions from X to G which vanish on A.
310
GENERAL COHOMOLOGY THEORY AND DUALITY
PROOF
CHAP.
A locally zero function from X to G is zero. Therefore CoO(X)
6
= 0,
and so
CO(X,A)
= CO(X,A)jCoO(X) = CO(X,A)
Therefore fIO(X,A; G) is the kernel of the composite
CO(X,A) ~ O(X,A) ~ Cl(X,A)
CO(X,A) is the module of functions from X to G which vanish on A. If
cp E CO(X,A), then cp is in the kernel of the above composite if and only
if there is some open covering Gil of X such that Scp vanishes on Gl12.
Since (Scp)(x,y)
cp(y) - cp(x), this is equivalent to the condition that there is
an open covering GIl such that cp is constant on each element of ,,?t. Hence the
kernel of the above composite equals the module of functions vanishing on A
that are locally constant on X. -
=
6
COROLLARY
Let X be a topological space in which every quasi-component
is open and let A C X. Then flO(X,A; G) is isomorphic to the module of functions from the set of those quasi-components of X which do not intersect
A to G.
PROOF
This follows from theorem 5 and the fact that a locally constant
function on X is constant on every quasi-component of X. -
7
COROLLARY
A nonempty space X is connected if and only if
1/ *: G ;:::; flO(X;G)
PROOF
This follows from theorem 5 and the trivial observation that every
locally constant function on X is constant if and only if X is connected. -
It follows that there exist spaces for which the singular cohomology and
Alexander cohomology differ. In fact, for any connected space which is not
path connected, corollary 7 and theorem 5.4.lO show that they differ in
degree O.
We now present a version of theorem 5.4.lO valid for the Alexander
theory.
THEOREM
Let {Uj } be an open covering of X by pairwise disioint sets.
Then there is a canonical isomorphism
8
fIq(X;G) ;:::; X fIq(Uj;G)
PROOF
Because {Uj } consists of pairwise disjoint sets, the map induced by
restriction
,i#:
C*(X)
~
X C*(Uj)
is an epimorphism. Because {Uj } is an open covering of X, it follows that
(i#)-l(X
C~(Uj))
Therefore i# induces an isomorphism
= C~(X)
C* (X) ;:::; X C* (Uj ).
-
SEC.
5
THE HOMOTOPY AXIOM FOR THE ALEXANDER THEORY
311
9
COROLLARY
Let {Cj } be the collection of components of a locally connected space X. Then there is a canonical isomorphism
Jiq(X;G) ;::::; X jiq(cj;G)
PROOF
Because X is locally connected, its components are open, and the result follows from theorem 8. •
THE HOMOTOPY AXIOM FOR THE ALEXANDER THEORY
In this section we shall prove the homotopy axiom for the Alexander cohomology
theory. The proof will be based on a description of the Alexander cochain
complex as the limit of cochain complexes of abstract simplicial complexes.
We shall also use this description to construct a homomorphism of the
Alexander cohomology theory into the singular cohomology theory. Because
the Alexander theory satisfies all the axioms, this homomorphism is an isomorphism from the Alexander theory to the singular cohomology theory on the
category of compact polyhedral pairs.
We shall be considering a fixed R module G as coefficient module for
cohomology and will usually not mention G explicitly. Let "11 be a collection
of subsets covering a set X. Let X("It) be the abstract simplicial complex
whose vertices are the points of X and whose simplexes are finite subsets F of
X such that there is some U E "II containing F. Let C('YL) be the ordered chain
complex of X(ql) over R. Given a subset A C X and a subcollection "11' C "11
which covers A, we let A(ql') be the subcomplex of X("Il) whose vertices are
the points of A and whose simplexes are finite subsets of A lying in some element of "11'. Then C'('YL') will denote the chain subcomplex of C("It) corresponding to A('YL').
Let ('Y,'Y') be another pair consisting of a covering 'Yof X and a subset
"If' C 'Ywhich is II covering of A. Assume that ('Y,'Y') is a refinement of ("11,"11')
in the sense that every element of 'Y is contained in some element of "It and
every element of "V' is contained in some element of "11'. Then the pair
(C('Y),C'('Y')) is mapped injectively into the pair (C(GIl),C'(ql')) by the identity
map of (X,A) to itself.
Let X be a topological space and A a subspace of X. Consider pairs ("11,"11'),
where GIl is an open covering of X and "11' is a subset of "It which covers A.
Such a pair is called an open covering of (X,A). Let C*("I1,"It') be the
cochain complex of the pair (C(Gll),C'("II')) (with coefficients in G). An element
u of Cq(Gll,ql') is a function defined on (q + I)-tuples of X which lie in some
element of "11, taking values in G, and vanishing on (q + I)-tuples of A which
lie in some element of Gll'. If (CV~'Y') is a refinement of ("11,"11'), the restriction
map is a cochain map
312
GENERAL COHOMOLOGY THEORY AND DUALITY
CHAP.
6
If (GIl, "11') and ('Y,'Y') are two open coverings of (X,A) as above, let
VI V E GIl, V E 'Y} and let G)Jl' = {V' n V'I V' E GLL', V E 'Y'}.
Then (G)Jl, G)Jl ') is another open covering of (X,A) and (G)Jl, GlJ) ') is a refinement of
( Ok, ~') and of (r,Y'). Therefore the cochain complexes (C*( Ok, ~')} form a
direct system, and we have a limit cochain complex
G)Jl
= {V n
lim~
{C * ("11, ql')}
We shall show that this limit cochain complex is canonically isomorphic
to G* (X,A). If cP E Cq(X,A), let ql' be a collection of open subsets of X covering A such that cP vanishes on (GIl')q+1 n A q+1 (such a GIl' exists because cP is
locally zero) and let GIl = GIl' U {X}. Then (qL,''11') is an open covering of (X,A)
and cP determines by restriction an element cP I (GIl, GLL') E Cq(GIl, GIl'). Passing to
the limit, we obtain a homomorphism (by restriction)
A: C*(X,A)
---7lim~
{C*(GIl,"I1')}
which is a canonical cochain map. Th~ folloWing result explains our interest
in the cochain complexes C * ("11, "11').
I
THEOREM
The canonical cochain map
A: C*(X,A)
---7lim~
is an epimorphism and has kernel equal to
{C*(GIl,GIl')}
C~ (X).
PROOF
To prove that A is an epimorphism, let u E Cq(0il,"Il'). Define
CPu E Cq(X) by
CPu (Xo,
• • • ,Xq
)
= ( U(xo,...
0
,Xq)
if xo, . . .
otherwise
,Xq
E V, where V E GLL
Then CPu vanishes on (0il')q+1 n Aq+1, and therefore CPu E O(X,A). By definition, CPu I (GIl,GIl') = u, and A is an epimorphism.
An element cp E Cq(X,A) is in the kernel of A if and only if there is some
(0il,GLL') such that cp I (G11,GIl') = O. Thus A(cp) = 0 if and only if there is some
open covering ql such that cp vanishes on ql q+1. By the definition of C ~ (X),
A(cp)
0 if and only if cp E C~ (X). •
=
From theorem 1 and the analogue of theorem 4.1. 7 for cochain complexes, we have the following corollary.
2
COROLLARY
For the Alexander cohomology theory there is a canonical
isomorphism
iIq(X,A; G);:::::::
lim~ {Hq(C*("Il,'~l';
G))}
•
We are now ready for the proof of the homotopy axiom for the Alexander
cohomology theory. In the presence of the other axioms, it suffices to prove it
for the case of the two mappings
ho, h 1: (X,A) ---7 (X X I, A X 1)
SEC.
5
313
THE HOMOTOPY AXIOM FOR THE ALEXANDER THEORY
=
=
where ho(x)
(x,O), hl(x)
(x,I). The proof consists in showing that if ("ll,"ll')
is any open covering of (X X I, A X I), there is an open covering (<'\;'Y') of
(X,A) such that ho and hI induce chain-homotopic chain maps from
(C(T),Cn-')) to (C("l1),C("l1')). This is a result about free chain complexes, and
the technique of acyclic models is available for obtaining the desired chain
homotopy.
Let Y be an arbitrary set and n a nonnegative integer. Let C(Y,n) be the
chain complex over R of the abstract simplicial complex (Y X I)("l1(Y,n)),
where ''It(Y,n) is the covering of Y X I defined by
I
~(y,n)={ YX [;, m2~1] O~ m<2n}
3
LEMMA
If Y is nonempty, the chain complex C(Y,n) is acyclic.
<
For 0 ::::; m
2 n let Km be the subcomplex of (Y X I)("ll(Y,n)) consisting of all the finite subsets of Y X [m/2n, (m + I)/2n]. For 0 ::::; m ::::; 2 n let
Lm be the subcomplex of (Y X I)("ll(Y,n)) consisting of all the finite subsets of
Y X (m/2 n ). Then (Y X I)("ll(Y,n)) = U m Km and Ki n Kj = 0 if Ii - il
1
and Ki n Ki+1 = Li+l. Because Y is nonempty, each Km (and Lm) is nonempty and is the join of Km (or Lm) with any vertex in it. Therefore,
by theorem 4.3.6, C(Km) and C(Lm) are acyclic. Let Nq = Um,;q Km. Then
Nq+1 = Nq U Kq+1 and Nq n Kq+1 = Lq+I. By induction on q, using the
exactness of the reduced Mayer-Vietoris sequence, it follows that C(Nq ) is
acyclic for all q. Therefore C(Y,n) = C(N2 n_l) is acyclic. •
PROOF
>
From this we have our next result, which will provide the acyclic model
for the homotopy axiom.
LEMMA
Let YI , . . . , Yq be subsets of a nonempty set Y, where
4
Y = YI , and for each i let ni be a nonnegative integer. Let K be the simplicial
complex defined by
K
=V
, (Yi
X I)("It(Yi,ni))
Then C(K) is acycliC.
PROOF
We prove the lemma by induction on q. If q = 1, it follows from
lemma 3. Assume that q
1, and the result is valid for fewer than q sets Yi .
Let K
Ui,;q-I (Yi X I)(0.l(Y;,ni)). Then K U (Yq X I)(G1.l(Yq,nq))
K. If Yq
is empty, C(K) = C(K) is acyclic, by the inductive assumption. If Yq is nonempty, C(Yq,nq) is acyclic, by lemma 3, and C(K) is acyclic, by the inductive
assumption. To prove that C(K) is acyclic, from the exactness of the reduced
Mayer-Vietoris sequence it suffices to prove that C(K n (Yq X I)(G1.l(Yq,nq)))
is acyclic. However, K n (Yq X I)(0.l(Yq,nq)) = UI,;i<q (Yi X I)(G1.l(Yi,ni)),
where Yi = Yi n Yq are subsets of Yq (and Y1 = Yq) and ni = max (n;,nq).
Therefore, by the inductive assumption, C(K n (Yq X I)(0.L(Yq,nq))) is
acyclic. •
=
>
=
314
GENERAL COHOMOLOGY THEORY AND DUALITY
CHAP.
6
We now come to the following main step in the proof of the homotopy
axiom.
:» LEMMA Let (GiL, "ll') be any open covering of (X X I, A X I). There is an
open covering (cv,'Y') of (X,A) such that ho and h1 induce chain-homotopic
chain maps from (C('Y),C'('Y')) to (C(G(l),C'("ll')).
PROOF
For each x E X it follows from the compactness of x X I that there
is an open set Vx about x and an integer n ~ 0 such that for 0 :::;; m
2n the
set Vx X [m/2n, (m + 1)/2n] is contained in some element of G(1. Furthermore,
if x E A, we can choose Vx and n so that Vx X [m/2n, (m + 1)/2n] is contained in some element of GLl'. Let 'Ybe the collection {VX}XEX and 'Y' the subcollection {VX}XEA. To show that ('Y,'Y') has the desired property, let e be the
category consisting of the subcomplexes of X('Y) partially ordered by inclusion. For each subcomplex K of X('Y) let G(K) be the ordered chain complex
of K. For each Simplex s of X('Y) [or A('Y')] define n(s) to be the smallest nonnegative integer such that for 0 :s; m < n(s) each set s X [m/2 n(sl, (m + 1)/2 n(S)]
is contained in some element of ''It [or 01']. Such an integer exists because of
the way ('Y,'Y') was chosen. For a sub complex K of X(V) let K be the
subcomplex of (X X 1)(01) defined by
<
R=u
{(s X 1)("l[(s,n(s)) Is E K}
and let G'(K) be the ordered chain complex of K. Then G and G' are covariant functors from e to the category of augmented chain complexes.
Let GJTL be the set of subcomplexes {s C X(CY) I s E X('Y)}. Then G is free
on e with models 0TL, and by lemma 4, G' is acyclic on e with models 0lL If
a = (XO,X1, . . . ,Xq) is an ordered q-simplex of X('Y), then
~(O')
= ((:ro,O), . . .
,(xq,O))
and
hl(O')
= ((:ro,1), . . .
,(xq,l))
are natural chain maps preserving augmentation from G to G'. It follows from
theorem 4.3.3 that there is a natural chain homotopy from ho to h 1. •
If u E Hq(C*(Gi1,GLl')), where (Cll,GLl') is an open covering of (X X I, A X 1),
it follows from lemma 5 that there is an open covering ('Y,'Y') of (X,A) such
that ho('Y,'Y') C (Gi1,01') , h1('Y,'Y') C (Cll,GLl'), and h~u = hTu in Hq(C*('Y,'Y')).
Passing to the limit and using corollary 2 gives us the final result.
S
THEOREM
axiom.
The Alexander cohomology theory satisfies the homotopy
•
We have now verified all the axioms of cohomology theory for the
Alexander cohomology theory. We construct a homomorphism fl from the
Alexander cohomology theory to the singular cohomology theory. Let (Cll,GLl')
be an open covering of (X,A). There is a canonical chain transformation
(~(Cll),~(GLl'
n A))
~
(C(Cll),C'(Cll'))
which assigns to a singular q-simplex a: ~q ~ X the ordered simplex
(a(vO),a(v1), . . . ,a(vq)) of C(Cll). This induces a homomorphism
SEC.
6
315
TAUTNESS AND CONTINUITY
C*("Il,"Il'; G)
~
C* (~("Il),
~(ql'
n A);
G)
Passing to the limit and using corollary 2, we obtain a canonical homomorphism
Jl': Hq(X,A; G) ~ lim~ {Hq(~("Il), ~(G2L'
n A); G)}
By theorem 4.4.14, there is a canonical isomorphism
and the homomorphism
J.t: Hq(X,A; G) -+ Hq(A(X), A(A); G)
is defined to equal the composite Jl"-IJl'. It can be verified that this homomorphism has the commutativity properties necessary to be a natural transformation of cohomology theories.
We now introduce a cup product in the Alexander theory, which will
have the usual properties of a cup product (as in Sec. 5.6) and will be compatible with the singular cup product by the homomorphism Jl.
Let G and G' be R modules paired to an R module G". Given
C(JI E Cp(X;G) and C(J2 E Cq(X;G'), we define C(JI v C(J2 E Cp+q(X;G") by
(C(JI
v C(J2)(XO, . . . ,xp+q)
= C(JI(XO,
. . . ,Xp)C(J2(Xp, . . . ,xp+q)
If C(JI is locally zero on AI. so is C(JI V C(J2, and if C(J2 is locally zero on A 2, so is
C(JI V C(J2. Therefore C(JI v C(J2 induces a cup product from CP(X;G) and
Cq(X;G') to Cp+q(X;G"). An easy verification shows that
S(C(JI
v C(J2)
= SC(JI v
C(J2
+ (-l)PC(JI V
SC(J2
Therefore the cup product induces a cup product on cohomology classes, and
this cup product is clearly mapped by Jl to the singular cup product.
In order to get a cup product from Cp(X,A I; G) and Cq(X,A 2; G') to
Cp+q(X, Al U A 2; G"), we need to ensure that an element of Cp+q(X;G")
which is locally zero on Al and locally zero on A2 will be locally zero on
Al U A 2. If Al U A2 = intA, uA2AI U intA, UA2A2, this is so. With this modification properties 5.6.8 to 5.6.12 are all valid for the resulting cohomology
product.
6
TAUTNESS AND CONTINUITY
In this section we shall consider tautness for the Alexander theory and establish the strong result that any paracompact space imbedded as a closed subspace of a paracompact space is tautly imbedded. This implies a strong excision
property for paracompact pairs (X,A) with A closed in X. It also implies the
continuity property (that the Alexander cohomology theory commutes with
limits of compact Hausdorff spaces directed by inclusion). This continuity
property, together with the other axioms of cohomology theory, characterizes
316
GENERAL COHOMOLOGY THEORY AND DUALITY
CHAP.
6
the Alexander theory on the category of compact Hausdorff pairs (that is,
pairs with X compact Hausdorff and A closed in X). The section closes with
a brief discussion of the Alexander cohomology with compact supports. Our
proof of the special tautness properties of the Alexander cohomology theory is
based on techniques of Wallace. 1
Let "11 be a collection of subsets of a set X. Let "Il* = {U * }U E 'l!, where
u* = u {U'
E
OU IU'
n u ¥- <1>1
A collection 'Y is said to be a star refinement of "Il if 'Y* is a refinement of "11.
A topological space X is said to be fully normal if every open covering of X
has an open star refinement. It is known that for Hausdorff spaces paracompactness is equivalent to full normality.
I
LEMMA
Let A be a subset of a topological space X and let'\:- be an open
covering of X. There exist a neighborhood N of A and a function f: N -7 A
(not necessarily continuous) such that
(a) f(x) = x for x E A.
(b) If V E 'If, then f(V
n N)
C V*.
PROOF
If A is empty, let N = A and fbe the identity map. If A is nonempty,
let N = U (V E YI V n A¥- <1>1 and define f: N -+ A by f(x) = x for x E A,
or if x ¢ A, choose f(x) E A so that there is V E 'Y with' x, f(x) E V. Such a
choice of f(x) is always possible because of the way N was defined. Clearly, if
x E V n N, there is V' E'Y with x, f(x) E V'. Therefore x E V n V' and
V' C V*. Hence, f(V n N) C V* and (a) and (b) are satisfied. •
This last result may be interpreted as asserting that A is a discontinuous
neighborhood retract of X with a retraction that is not too discontinuous.
If A is a closed subset of a paracompact space, it is similar enough to an absolute neighborhood retract so that we have the following generalization of
theorem 6.1.10 for the Alexander theory.
2 THEOREM A closed subspace of a paracompact Hausdorff space is a
taut subspace relative to the Alexander cohomology theory.
Let A be a closed subspace of a paracompact space X and let
cp E Cq(A) be a cochain such that 8cp vanishes on "'Jlq+2, where G1JI- is an open
covering of A. Let "11 = {W U (X - A) I W E "2l1} and observe that "II is an
open covering of X because A is closed in X. Let 'Y be an open star refinement
of "II and let N be a neighborhood of A and f: N -7 A a function (not necessarily continuous) satisfying lemma 1 relative to '\~ Then f#cp E Cq(N), and we
show that 8f#cp = f#l3cp vanishes on 'Yq+2 n Nq+2. By lemma lb, for any
V E 'Ythere is U E %"such thatf(V n N) C U. Thenf(V n N) C UnA C W
for some WE G)J). Therefore 8f#cp vanishes on (V n N)q+2. This means that
f#cp represents a co cycle of Eq(N) and, by lemma la, (f#cp) I A = cpo Hence
PROOF
1 See A. D. Wallace, The map excision theorem, Duke Mathematical Journal, vol. 19,
pp. 177-182, 1952.
SEc.6
317
TAUTNESS AND CONTINUITY
the cohomology class {<p} E flq(A) is the image under restriction of the cohomology class {f#<p} E flq(N), showing that lim_ {fiq(N)} ~ fiq(A) is an
epimorphism.
To prove that it is a monomorphism, let N' be a paracompact neighborhood of A and assume that <p E Cq(N') is such that o<p vanishes on L'11lq+2 and
<p I A = o<p' on (61LI-,)q+1, where q[) is an open covering of N' and lUI' is an open
covering of A. Let GIL = {W' U (N' - A) I w' E "lll"} and observe that "It is an
open covering of N' (because A is closed.) Let 'Ybe an open star refinement
of both ('ll\' and G/J.,('Y is a covering of N') and let N be a neighborhood of A in
N' and f: N ~ A a function (not necessarily continuous) defined with respect
to 'Y to satisfy lemma 1. If V E 'Y, then f(V n N) C W' for some W' E q[)'.
Therefore f#(<p I A) = of#<p' on Vq+1 n Nq+l.
To show thatf#(<p I A) is cohomologous in Cq(N) to <p I N, for l/; E Cp(N)
define Dl/; E Cp-l(N) by
(Dl/;) (xo, . . . ,Xp-l) =
.~
O:':J:,:p-l
(-l)jl/;(xo, . . . ,Xj, f(Xj), ... ,f(Xp-l))
An easy computation establishes the formula
oDl/;
+ Dol/; = f#(l/; I A)
- l/;
For every V E 'Y, (V n N) U f(V n N) C W for some WE "llI- (by lemma Ib),
and because o<p vanishes on q[) q+2, oD( <p IN) = f #( <p IA) - <p IN on
'Yq+l n Nq+l. Therefore the cohomology class {<p} E fiq(N) maps to zero in
fiq(N). This suffices to show that lim_ {fiq(N)} ~ flq(A) is a monomorphism,
and so A is a taut subspace of X. •
3
COROLLARY
Let X ~ A ~ B, where X is a paracompact Hausdorff
space and A and B are closed subspaces of X. Then, relative to the Alexander
cohomology theory, (A,B) is a taut pair in X.
PROOF
This is an immediate consequence of theorem 2 and lemma 6.1.9.
•
EXAMPLE
Let X be the subspace of R2 C S2 defined in example 2.4.8.
4
The space X obtained by retopologizing X by the topology generated by the
path components of open sets in X is a half-open interval. Since X has the
same singular homology as X, Hl(X;Z) = O. Since S2 - X has two components,
it follows from the Alexander duality theorem that lim_ {Hl( U;Z)} = Z as U
varies over neighborhoods of X. Therefore lim_ {Hl( U;Z)} ~ Hl(X;Z) is not
a monomorphism, and so X is not a taut subspace of R2 with respect to singular cohomology. Since X is closed in R2, it is taut with respect to Alexander
cohomology.
Note that the above example is one in which lim_ {Hl( U;Z)} ~ Hl(X;Z) is
not a monomorphism, whereas in example 6.1.8 a subspace A C R2 was given
such that lim_ {HO(U;Z)} ~ HO(A;Z) was not an epimorphism.
The tautness property 3 implies that the Alexander cohomology theory
satisfies the following strong excision property.
318
GENERAL COHOMOLOGY THEORY AND DUALITY
CHAP.
6
5
THEOREM
Let (X,A) and (Y,B) be pairs, with X and Y paracompact
Hausdorff and A and B closed. Let f: (X,A) ~ (Y,B) be a closed continuous
map such that f induces a one-to-one map of X - A onto Y - B. Then, for
all q and all G
f*: Hq(Y,B; G) :::::: Hq(X,A; G)
Because f is closed, continuous, and one-to-one from X - A onto
Y - B, it follows that f is a homeomorphism of X - A onto Y - B. Let {Ua }
be the family of open neighborhoods of B in Y and let Va
f-l( Ua). Then
Va is an open neighborhood of A in X, and because f is a closed map, the collection {Va} is coRnal in the family of all neighborhoods of A in X. We have
a commutative diagram
PROOF
=
Hq(Y,B)
~
lim~ {Hq(y,Ua)} ~ lim~ {Hq(y - B, Ua - B)}
f!~
f*~
Hq(X,A)
~
lim~
{Hq(X, Va)}
~n
~
lim~ {Hq(X -
A, Va - A)}
in which the vertical m~ps are induced by f and the horizontal maps are
induced by inclusions. By corollary 3 and lemma 6.4.4, the horizontal maps
are isomorphisms. Because f I X - A is a homeomorphism of X - A onto
Y - B, it follows that for each a, f I (X - A, Va - A) is a homeomorphism of
(X - A, Va - A) onto (Y - B, Ua - B). Therefore f~ is an isomorphism,
and by commutativity of the diagram, f* is also an isomorphism. -
The following weak continuity property of the Alexander cohomology
theory is another consequence of its tautness properties.
6
THEOREM
Let {(Xa,Aa)}a be a family of compact Hausdorff pairs in
some space, directed downward by inclusion, and let (X,A) = (n X a, n Aa).
The inclusion maps ia: (X,A) C (Xa,Aa) induce an isomorphism
{i: }: lim~ Hq(Xa,A a; M) :::::: Hq(X.A; M)
If F is a closed subset of Xf3 for some /3, the collection {Xa n F} a
consists of compact sets directed downward by inclusion, and X n F =
n (Xa n F). It follows that if X n F = 0, there is some a such that
Xa n F
0. Therefore, if U is any neighborhood of X in X f3 , there exists a
such that Xa C U. Similarly, if (U, V) is any neighborhood of (X,A) in Xf3 ,
there is a such that (Xa,Aa) C (U,V).
To show that {i: } is an epimorphism, let u E Hq(X,A) be arbitrary. For
any /3, (X,A) is a taut pair in Xf3 , by corollary 3. Therefore there is a neighborhood (U, V) of (X,A) in Xf3 and an element v E Hq(U, V) such that v I (X,A) = u.
Let a be such that (Xa,Aa) C (U, V) and Va = V I (X",Aa). Then Va E Hq(X",A a)
and i:va = u, which proves that {i:} is an epimorphism.
To prove that {iii'} is a monomorphism, let u E Hq(Xf3 ,Af3 ) be such that
qu O. By corollary 3, (X,A) is a taut pair in Xf3. Therefore there is a
neighborhood (U, V) of (X,A) in Xf3 such that u I (U, V n Af3) = O. Choose a
PROOF
=
=
SEc.6
319
TAUfNESS AND CONTINUITY
so that (X",A,,) C (U, V
isomorphism. -
n AfJ). Then u I (X",A,,)
= 0,
and {i;;-} is an
The continuity property involves an assertion analogous to that of
theorem 6 for an arbitrary inverse system {(X",A,,)} of compact Hausdorff
pairs, where (X,A) = lim_ {(X",Aa)}. It is not hard to prove that the continuity
property is equivalent to the weak continuity property. I A cohomology theory
having the weak continuity property is called weakly continuous. Such
theories are characterized on the category of compact Hausdorff spaces in
view of the following result.
7
LEMMA
Any compact Hausdorff pair can be imbedded in a space in
which it is the intersection of a family of pairs directed downward by inclusion, each pair of the family being a compact Hausdorff space of the same
homotopy type as a compact polyhedral pair.
I
It is a standard fact that any compact Hausdorff space can be imbedded
in a cube [J; hence we assume (X,A) imbedded in fJ. For each finite subset a C J let PiX: [J ~ I" be the projection map and let (U, V) be a compact
polyhedral neighborhood of (p,,(X),p,,(A)) in la. It can be verified that the collection of pairs {(p" -1( U ),p" -l(V))} corresponding to all finite a C J and compact polyhedral neighborhoods of (p,,(X),p,,(A)) in I" is directed downward by
inclusion and has (X,A) as intersection. Furthermore, (p" -l(U),p" -leV)) is a
compact pair in [J homeomorphic to (U, V) X [J-a, and the projection map
PROOF
PiX: (p" -l(U),p" -1(V))
~
(U,v)
is a homotopy equivalence. Therefore the family {(pa -1( U ),Pa -1(V))} has the
desired properties. This yields the following extension of the uniqueness theorem for weakly
continuous cohomology theories.
8
THEOREM
Given two weakly continuous cohomology theories, any
homomorphism between them which is an isomorphism for some one-point
space is an isomorphism for all compact Hausdorff pairs. -
We now describe the Alexander cohomology with compact supports.
This is a cohomology theory on a suitable category of topological pairs and
maps, and we shall discuss the category first.
A subset A of a topological space X is said to be bounded if A is compact.
A subset B C X is said to be cobounded if X - B is bounded. A function f
from a space X to a space Y is said to be proper if it is continuous and if for
every bounded set A of Y, f-I(A) is a bounded set of X (or, equivalently, for
every cobounded set B of Y, f-l(B) is a cobounded set of X). Clearly, the
composite of proper maps is proper, and there is a category of topological
spaces and proper maps. There is also a category of topological pairs and
1 See S. Eilenberg and N. E. Steenrod, "Foundations of Algebraic Topology," Princeton University Press, Princeton, N.J., 1952, or exercise 6.C.2 at the end of this chapter.
320
GENERAL COHOMOLOGY THEORY AND DUALITY
CHAP.
6
proper maps, a proper map from (X,A) to (Y,B) being a proper map from X
to Y which maps A to B. This is the category on which the Alexander cohomology theory with compact supports will be defined.
Given a topological pair (X,A), let Ccq(X,A; G) be the submodule of
Cq(X,A; G) consisting of all cp E Cq(X,A; G) such that cp is locally zero on
some cobounded subset of X. If cp is locally zero on B, so is 8cp, and therefore
there is a cochain complex C~(X,A; G) = {Ccq(X,A; G), 8} which is a subcomplex of C*(X,A; G). Clearly, C~(X;G) C C~(X,A; G), and we define
C~ (X,A; G) = C~ (X,A; G)/C~ (X;G)
The Alexander cohomology of (X,A) with compact supports, denoted by
B~ (X,A; G), is the cohomology module of C~ (X,A; G). If f: (X,A) ---7 (Y,B)
is a proper map, f# maps C~(Y,B; G) to C~(X,A; G) and induces a
homomorphism
f*: B~ (Y,B; G) ---7 B~ (X,A; G)
The Alexander cohomology with compact supports satisfies suitable modifications of all the axioms of cohomology theory.
The homotopy axiom holds for proper homotopies, a proper homotopy
being a proper map (X,A) X 1---7 (Y,B). In general, an inclusion map
(X',A') C (X,A) is not a proper map. It is a proper map, however, if X' is
closed in X. Because of this, the coboundary homomorphism
8*: Bcq(A;G) ---7 Bc q+ 1(X,A; G)
is defined only when A is a closed subset of X. When A is a closed subset of
X, there are proper inclusion maps i: A C X and i: X C (X,A) and there is a
short exact sequence of cochain complexes (for any coefficient module G)
o ---7 C~ (X,A) 4
c~ (X) ~ C~ (A) ---70
The connecting homomorphism of this short exact sequence is a natural transformation from B~ (A) to B~ (X,A), of degree 1 on the category of pairs
(X,A), with A closed in X and proper maps between such pairs. The exactness axiom then holds for pairs (X,A) with A closed in X.
The excision axiom holds for proper excisions, a proper excision map
being an inclusion map i: (X - U, A - U) C (X,A) such that U is an open
subset of X with (j C int A, in which case it can be shown (analogous to the
proof of lemma 6.4.4) that
W:
C~(X,A);:::: C~(X -
U, A - U)
The dimension axiom is obviously satisfied.
We now consider relations between the Alexander cohomology with
compact supports and the Alexander cohomology theory previously defined.
The following is one case in which they agree.
9
LEMMA
If A is a co bounded subset of X, then
B~(X,A; G)
= B*(X,A;
G)
SEc.6
PROOF
321
TAUTNESS AND CONTINUITY
Because A is cobounded in X,
C~(X,A)
and so C~ (X,A)
= C* (X,A).
= C*(X,A)
•
10 LEMMA Let B be a closed subset of a Hausdorff space A. Then a subset
U of A - B is cobounded in A - B if and only if U U B is a neighborhood
of B cobounded in A.
PROOF
If U' is a neighborhood of B in A, then the closure of A - U' in A
equals the closure of (A - B) - (U' - B) in A - B. Hence one is compact if
and only if the other is. Therefore the result will follow once we have verified
that if U is a cobounded subset of A - B, then U U B is a neighborhood of
B in A. However, if C is the compact set which equals the closure of
(A - B) - U in A - B, then C is closed in A (because A is Hausdorff).
Therefore A - C is an open subset of A containing B. Since (A - B) - C c U,
it follows that (A - C) c U U B, and U U B is a neighborhood of B in A. •
Let B be a closed subset of a normal space A. If U is a neighborhood of
B in A which is a cobounded subset of A, then C (A, U) C C ~ (A,B). Therefore lim~ {C (A, U)}
U C (A, U) is imbedded as a subcomplex of
C~ (A,B). By the excision property 6.4.4,
*
=
*
*
U C*(A,U);:::; U C*(A - B, U - B)
As U varies over cobounded neighborhoods of B in A, it follows from lemma
lO that U - B varies over cobounded subsets of A - B. Therefore
U C* (A - B, U - B)
= C~ (A
- B)
and we have defined a functorial imbedding
i: C~ (A - B) C C~ (A,B)
=
such that i(C~(A - B))
lim~ {C*(A,U)}, where Uvaries over cobounded
neighborhoods of B in A. Hence i induces an isomorphism of cohomology if
and only if
lim~ {H*(A,U)} ;:::; H~(A,B)
We shall now consider cases in which i induces an isomorphism of cohomology.
I I LEMMA If A is a compact Hausdorff space and B is closed in A, for all
q and all G there is an isomorphism
Hcq(A - B; G) ;:::; Hq(A,B; G)
PROOF
By lemma 9 and the above remarks, it suffices to prove that as
U varies over neighborhoods of B in A (any such neighborhood being
cobounded because A is compact), there is an isomorphism
lim~ {Hq(A, U; G)} ;:::; Hq(A,B;
G)
322
GENERAL COHOMOLOGY THEORY AND DUALITY
CHAP.
6
Since A is paracompact, this is a consequence of the tautness property 3 of
Alexander cohomology. •
This result allows the following interpretation of the cohomology with
compact supports of a locally compact space.
12 COROLLARY If X is a locally compact Hausdorff space and X+ is the
one-point compactification of X, there is an isomorphism
iicq(X;G) ::::::: Hq(X+;G)
~ROOF
By lemma 11, iicq(X;G)::::::: iiq(x+, X+ - X; G) and because
H* (X+ - X; G)
0, there is an isomorphism
=
iiq(x+, X+ - X; G) ::::::: Hq(x+;G)
13
EXAMPLE
•
It follows from corollary 12 that
q-=l=n
q=n
because (Rn)+ is homeomorphic to Sn. Hence, if n -=1= m, Rn and Rm are not of
the same proper homotopy type.
14
EXAMPLE
Regarding Rl as a linear subspace of R2, then
q-=l=2
q=2
15
THEOREM
Let B be a closed subset of a locally compact Hausdorff space
A. For all q and all G there is an isomorphism
lim~
{iiq(A, U; G)} ::::::: iicq(A,B; G)
where U varies over cobounded neighborhoods of B in A.
If A is compact, this follows from lemmas 9 and 11. If A is not compact, let A + be the one-point compactification of A. Set p+ = A + - A and
B+
B U p+ c A +. Then B+ is closed in the compact space A +. There is a
commutative diagram of chain maps
PROOF
=
C~(A-B)~
C~(A)
~ C~(B)
~
o ~ C*(A+,B+) ~ C*(A+,p+) ~ C*(B+,p+) ~ 0
~
~
and, by corollary 12 and lemma 11, each vertical map induces an isomorphism
on cohomology. Since the bottom row is exact and C~ (A - B) C C~ (A), it
follows that C~ (A)/C~ (A - B) ~ C~ (B) induces isomorphisms of cohomology. Since there is a short exact sequence of cochain complexes
o ~ C~(A,B)/C~(A
- B) ~ C;C(A)/Cc*(A - B) ~ Gc*(B) ~ 0
it follows that C~(A,B)/C~(A - B) has trivial cohomology. Therefore
SEC.
7
323
PRESHEAVES
if ~ (A - B) ;:::; if ~ (A,B), and this is equivalent to the statement of the
theorem. •
The last result is a form of tautness for Alexander cohomology with
compact supports. This and the five lemma easily imply the next result.
16 THEOREM Let (A,B) be a pair of closed subsets of a locally compact
Hausdorff space X. For all q and all G there is an isomorphism
lim~
{ifq( U,v; G)} ;:::; ifcq(A,B; G)
where (U, V) varies over neighborhoods of (A,B) in X, both U and V being
cobounded subsets of X. •
In a similar fashion, we may consider the singular cohomology with
compact supports. A singular cochain c* E Hom (Llq(X)j Llq(A),G) is said to
have compact support if there is some cobounded set U C X such that
for every x E U there is a neighborhood Vof x such that c*(O') = 0 for every
singular q-simplex 0' in V. The singular cochains with compact support form a
subcomplex of the singular cochain complex, whose cohomology module is denoted by Ht (X, A; G).
7
PRESHEAVES
In this section the Cech construction will be introduced. Because of the
ultimate applications, we define the Cech cohomology of a space not merely
for coefficients in a module, but, more generally, for coefficient modules
which may vary from one point of X to another. This leads to the concepts of
presheaf and sheaf. We shall introduce these and give the definition of
the Cech cohomology of a space with coefficients in a presheaf. Applications
will be given in the next two sections.
A presheaf f of R modules on a topological space X is a contravariant
functor from the category of open subsets U of X and inclusion maps U C V
to the category of R modules such that f( 0) = O. Thus f assigns to every
open subset U C X an R module f(U) and to every inclusion map U C Va
homomorphism Puv: f(V) -'> f( U), called the restriction map, such that
Puu
Puw
=
= Puv
If(U}
0
pvw:
f(W)
-'>
f(U)
UC Vc
w
Given y E f(V) and U C V, we use y I U to denote the image puv(y) E f(U).
In a similar manner, we define presheaves on X with values in any category. We are interested primarily in the case of a presheaf of modules or of
cochain complexes. Following are some examples.
Given an R module G, the constant presheaf G on X assigns to every
I
nonemptyopen U C X, the module G (and to 0 the trivial module).
2
Given a subset A C X, the relative Alexander presheaf of (X,A) with
324
GENERAL COHOMOLOGY THEORY AND DUALITY
CHAP.
6
coefficients G, denoted by C * ( ., . n A; G), assigns to an open U C X the
co chain complex C* (U, UnA; G).
3
The relative singular presheaf of (X,A) with coefficients G, denoted by
Ll * ( ., . n A; G), assigns to an open U C X the cochain complex
a *( U, unA; G) equal to the subcomplex of Hom (a*( U), G) of cochains
locally zero on UnA (i.e. cochains that are zero on Ll * (qL) for some open
covering GIL of UnA).
Given two presheaves f and f' on X taking values in the same category,
a homomorphism lX: f _ f' is defined to be a natural transformation from
f to f'. It is then clear that there is a category of presheaves on X with values
in any fixed category and homomorphisms between them. In particular, there
is a category of presheaves of modules and a category of pres heaves of
cochain complexes. If lX: f _ f' is a homomorphism of presheaves of
modules (or cochain complexes), it is clear how to define ker lX, im lX, and
coker lX so as to be presheaves of modules (or cochain complexes) on X.
Therefore it is meaningful to consider exact sequences of presheaves of
modules (or cochain complexes) on X.
If f and f' are presheaves of modules (or cochain complexes) on X, their
tensor product f ® f' is the presheaf of modules (or cochain complexes) on X
such that for open U C X
(f ® f/)(U)
= f(U)
® f/(U)
Consider two examples.
4
There is a homomorphism
7:
C* ( ., . n A; G) _ Ll * ( ., • n A; G)
such that if cp E Cq( u, UnA; G) and a: Llq _ U, then 7(cp)(a) =
cp(a(po), . . . ,a(pq)), where po, . . . , pq are the vertices of M.
:;
There is a homomorphism
7: C*(·, •
n A; R) ® G_ C*(·, . n A; G)
such that if cp E Cq( u, UnA; R) and g E G, then
7(cp ® g)(xo, . . . ,Xq) = cp(xo, . . . ,Xq)g
Similar to the concept of presheaf on X with values in a category is the
concept of sheaf on X with values in a category. We are interested only in
sheaves of modules, and for this case the following formulation will do.
Let f be a presheaf of modules on X. If GIL. = {U} is a collection of open
subsets of X, a compatible qlJamily of f is an indexed family {yu E f( U)} U€"ll
such that
Yu I U
n u' = YU' I U n u'
U, U' E GIL
The presheaf f is said to be a sheaf if both the following conditions hold:
SEC.
7
325
PRESHEAVES
(a) Given a collection ''It of open subsets of X with V = U Udl U and
given y E f(V) such that y I U = 0 for all U E 01, then"y = O.
(b) Given a collection ''It of open subsets of X with V = U UE"l1 U and
given a compatible 0l family {YU}UE"lb there is an element y E f(V)
such that y I U = Yu for all U E 0l.
It follows from (a) that the element y in (b) is unique.
We now associate to every presheaf f of modules another presheaf f,
called its completion, whose elements are compatible families of f. Given a
collection of open sets 0l = {U}, let f(0l) be the module of compatible
0l families of f. If 'Y is another collection of open sets which refines 01, there
is a homomorp-hism f(0l) ---7 f('Y) which assigns to a compatible 0l family
{yu} the compatible '\'family {yv} such that if V E 'Yis contained in U E ' 11,
then Yv = Yu I V (yv is uniquely defined by this condition because of the compatibility of {Yu}). As 'Yl varies over the family of open coverings of a fixed
open set W C X, the collection {f(0l)} is a direct system of modules, and we
define
f(W)
= lim~
{f(0t)}
If W' C Wand 0t is an open covering of W, then 0l' = {U n W' I U E 0l}
is an open covering of W' which refines 0l. Hence there is a homomorphism
f(0l) ---7 f(0l') which defines (by passage to the limit) a homomorphism
f(W) ~ f(W'). A trivial verification shows that f is a presheaf [if GIL = {0},
then trivially f(GIl) = 0, and so f( 0) = 0]. There is a natural homomorphism a: f ---7 f such that a assigns to y E f(V) the element of f(V)
represented by the compatible 'Y family {y}, where 'Y consists solely of V. The
presheaf f is called the completion of f. It depends only on the values f( U)
for small open sets U C X.
6
LEMMA
A presheaf f is a sheaf if and only if
a: f ;::::; f
PROOF
In fact, condition (a) above is satisfied if and only if a is a monomorphism. If condition (b) is satisfied, a is an epimorphism. If a is an isomorphism,
then (b) is satisfied. •
7
EXAMPLE
The constant presheaf G defined by a module G is not generally a sheaf [if U is a disconnected open set, G(U) ;j:: 6(U)).
8
EXAMPLE
If C * is the relative Alexander presheaf of (X,A) (with some
coefficient module G), the kernel of a: C* ---7 C* is C~ (the locally
zero functions). To show that a satisfies condition (b) (and hence induces an
isomorphism C* ;::::; C*), let q/ E cq(V, V n A) and assume q/ represented by
a compatible 0t family {<PU}UE'Il, where GIL is an open covering of V. Then
<Pu: Uq+l ---7 G for U E GIl is locally zero on UnA and
<pul (U
n U')q+1 =
<PU' I (U
n U' )q+l
U, U' E 01
326
GENERAL COHOMOLOGY THEORY AND DUALITY
CHAP.
6
Therefore there is a function cp: Vq+1 ~ e such that cp I Uq+l = cpu for U E "II
and cp(xo, . . . ,xq) = 0 if Xo, . . . , Xq do not all lie in some element of L1l.
Then cp is locally zero on A, whence cp E Cq(V, V n A) and a(cp) = cp'.
This example shows that, in general, H * (C *) =1= H * (C *), so it is not
generally true that a presheaf of cochain complexes and its completion have
isomorphic cohomology.
EXAMPLE
If /1 * is the relative singular presheaf of (X,A) (with some
9
coefficient module e), the kernel of a: /1 * ~ Li * is the subcomplex of
locally zero cochains [that is, c* E Hom (/1 q(V),e) is in the kernel of a if and
only if there is some open covering "II of V such that c* is zero on
/1 q (G)l) C /1 q (V)]. Also a satisfies condition (b) (as can be shown by an argument similar to that of example 8). If G).L is an open covering of X, it is clear
that /1 * ("11) = U Hom (/1 * ("11)//1* ("1['), e) where the union is over all open
coverings ql' of A that refine "11 n A. As "It and "It' vary over open coverings,
respectively, of X and A such that "II' refines "It n A, there is an inverse
system of chain complexes {/1 * (G)L) 1/1* (L1I')} and a direct system of cochain
complexes
Therefore there is an isomorphism
lim.... {Hom (Il*( ~ ) !Il*( ~ '),
e)l = J. *(.
,.
n A; e)(X)
It follows from theorem 4.4.14 that
induces isomorphisms of the cohomology modules. Therefore a induces an
isomorphism
H* (/1 * ( .,
.n
A; e)(X)) ;:::; H* (Li * ( .,
. n A;
e)(X))
I 0 EXAMPLE Let ~ be an n-sphere bundle with base space B and let R be
fixed. A presheaf r on B is defined by f(V) = Hn+l(p€-l(V), P€-l(V) n E€; R)
for an open V C B. r is called the orientation presheaf of ~ over R.
can be
verified that if B is connected, ~ is orientable over R if and only if f(B) =1= O.
It
I I EXAMPLE Let X be an n-manifold with boundary j( and let R be fixed.
Define a pre sheaf r on X - X such that f(V) = Hn(X, X - V; R) for open
V C X - X. r is called the fundamental presheaf of X over R. It can
be verified (using lemma 6.3.2) that f(X) ;:::; HnC(X,X; R). By theorem 6.3.5, it
follows that if X is connected, it is orientable over R if and only if f(X) =1= O.
There are cohomology modules of X with coefficients in sheaves, 1 and
cohomology modules with coefficients in presheaves. For paracompact spaces
1
See R. Godement, "Theorie des fllisceaux," Hennann et Cie, Paris, 1958.
SEC.
7
327
PRESHEAVES
these theories are equivalent. We now define the Cech cohomology with
coefficients in a presheaf of modules.
Let f be a presheaf of modules on a space X and let "11 be an open covering of X. For q ;::: 0 define Cq("I1;f) to be the module of functions 1f; which
assign to an ordered (q + I)-tuple Uo, U1 , . . . , Uq of elements of "11 an element 1f;( Uo, . . . , Uq) E f( Uo n ... n Uq). A coboundary operator
0: Cq("Il;f)
~
Cq+l("Il;f)
is defined by
(o1f;)(U o, ..
,Uq+1 )
=
~ (-I)i1f;(UO, . . . ,(li, . . . , Uq+1) I (Uo n ... n Uq+l)
O"i<:q+l
Then 00 = 0 and C*("I1;f) = {Cq("IL;f),o} is a cochain complex. Its cohomology module is denoted by H* ("I1;f).
12 EXAMPLE It is an immediate consequence of the definition that
HO("IL;f) = [(Gil) (the module of compatible GiL families).
Let 'Y be a refinement of 0(1 and let A: 'I;' ~ "2l be a function such that
V C A(V) for all V E 'Y There is a cochain map A*: C* (Gi1;f) ~ C* ('Y;f)
defined by
(A* 1f;)(Vo, . . . ,Vq) = 1f;(A(VO), . . . ,A(Vq)) I (Vo n ... n Vq)
If W 'Y ~ "11 is another function such that V C ,u(V) for all V E '\ a cochain
homotopy D: Cq(0(l;f) ~ Cq-l('\;f) from A* to ,u * is defined by
(D1f)(Vo, . . . ,vq-l)
=
.~
O<:J"q-l
(-I)j1f;(A(Vo), . . . ,A(Vj), ,u(Vj), . . . ,,u(Vq- 1 )) I (Vo n ... n Vq_1 )
It follows that there is a well defined homomorphism
A* : H* (GiL; f)
~
H* ('Y;f)
such that A* {rp} = {A* rp} that is independent of the particular choice of A.
As "11 varies over open coverings of X, the collection {H*(Ulj;f)} is a direct
system, and the Cech cohomology of X with coefficients f is defined by
H*(X;f) = lim~ {H*(0(I;f)}
13
EXAMPLE
For any presheaf f, iIO(X;f) = f(X).
14 EXAMPLE The Cech cohomology of X with coefficients G, denoted by
H * (X;G), is- defined to be the cohomology of X with coefficients the constant
presheaf G.
We now establish some basic properties of the cohomology with coefficients in a presheaf.
328
GENERAL COHOMOLOGY THEORY AND DUALITY
CHAP.
6
15 THEOREM There is a covariant functor from the category of short exact
sequences of pres heaves on X to the category of exact sequences which
assigns to a short exact sequence 0 ~ r' ~ r ~ r" ~ 0 of presheaves on X
an exact sequence
... ~ Hq(X;r') ~ Hq(X;f) ~ Hq(X;r") ~ Hq+1(X;f') ~ ...
For any open covering "Il there is a short exact sequence of co chain
complexes
PROOF
o
~
C*(cYl;r')
~
C*(cYl;f)
~
C*(cYl;r")
~
0
This yields an exact cohomology sequence, and the result follows from this on
passing to the direct limit. Given a short exact sequence of modules
o~
G'
~
G
~
G"
~
0
the corresponding constant presheaves on X constitute a short exact sequence
of presheaves. The corresponding exact cohomology sequence of Cech cohomology modules given by theorem 15 is an analogue for Cech theory of the
exact sequence of theorem 5.4.1I.
Given a presheaf r on X and given a subspace A C X, define a presheaf
r A on X by
UnA=;F0
unA = 0
r A on X by
Also define a presheaf
rA(U)
= {~(U)
UnA = 0
UnA=;F0
Then r A is a sub-pre sheaf of r, and there is a short exact sequence of presheaves
o~rA~r~rA~O
The corresponding exact cohomology sequence given by theorem 15 is an
exact Cech cohomology sequence of the pair (X,A) with coefficients r when
we define ih(A;f)
Hq(X,r A) and Hq(X,A; f)
Hq(X;rA). Thus the exact
sequence of theorem 15 gives rise to exact sequences corresponding to a
change of coefficients or to a change of space.
A presheaf r of modules on X is said to be locally zero if, given y E f(V),
there is an open covering "Il of V such that y I U = 0 for all U E qt. This is so
if and only if the completion t of r is the zero presheaf and is equivalent to
the condition that for all x E X, lim~ {f( U)} = 0 as U varies over open
neighborhoods of x.
=
=
16 THEOREM If X is a paracompact Hausdorff space and
presheaf on X, then H* (X;f) = O.
PROOF
r
is a locally zero
Let ql be a locally finite open covering of X and cp a q-cochain of
SEC.
8
329
FINE PRESHEAYES
C*( %';r). Let Ybe a locally finite open star refinement of Ok. For x E X, because
r is locally zero, there is an open neighborhood Vx contained in some element
of Y such that x E Uo n ... n Uq with 00, . . . , Uq E %' implies that Vx C
Uo n ... n Uq and cp(Oo, . . . , Uq ) I Vx = 0 (only a finite number of conditions
because %' is locally finite). Let r = {Vx}XEX and define t..: r --+ %' so that for
each x E X there is Wx E Y with Vx C Wx C W: C t.. (Vx). Then if Vxo n ...
n VXq ¥ 0, Vxo C WXj for eachjso that Vxo C t.. (VXj ) for eachj: Therefore, cf>(t..(Vxo )'
... ,t.. (Vx q)) I Vxo = 0, so t..*cf> = 0 in C*(Y; r). Therefore, Hq(X; r) = 0 for all q.
A homomorphism a: f -7 f' between presheaves on X is called a local
isomorphism if ker a and coker a are both locally zero. This is equivalent to
the condition that for all x E X, a induces an isomorphism
lim~
{f(V)} ;::::;
lim~
{f'(V)}
where V varies over open neighborhoods of x. There are short exact sequences
of presheaves
o -7 ker a -7 f ~ im a -7 0
o -7 im a 4 f' -7 coker a -7 0
with a = alia'. Combining theorems 15 and 16, we obtain the following result.
17 COROLLARY If a: f -7 f' is a local isomorphism of pres heaves on a
paracompact Hausdorff space X, then
a.: H*(X;r) "'" H*(X;r') •
18
COROLLARY
phism a: f
-7
If X is a paracompact Hausdorff space, the natural homomor-
t induces isomorphisms
a* : I!* (X;f) ;::::; I!* (X;f)
It suffices to prove that a: f
PROOF
-7
t
is a local isomorphism. Let
y E (ker a)(V). Then y E f(V), and there is an open covering Gllof V such that
y IV
0 for all V E GU. Hence ker a is locally zero.
=
If y' E (coker a)(V), there is an open covering Gll of V and a compatible
Gllfamily {Yu} which represents y'. For each V E GIl, y'l Vis represented by
Yu E a(f(V)). Therefore y' I V = 0, and coker a is locally zero. •
8
FINE PRESHEAVES
In this section we shall introduce the concept of fine presheaf and show that
the positive dimensional cohomology of a paracompact space with coefficients
in a fine presheaf is zero. This leads to uniqueness theorems for cohomology
of co chain complexes of fine presheaves on a paracompact space, which we
apply to compare the Alexander and Cech cohomology. Further applications
will be given in the next section.
330
GENERAL COHOMOLOGY THEORY AND DUALITY
CHAP.
6
A presheaf r on X is said to be fine if, given any locally finite open
covering 6)l of X, there exists an indexed family {eu} u E qj of endomorphisms of
r such that (for every open set Y in X)
(a) For y E f(V), eu(y) I (V - 0) = O.
(b) If V meets only finitely many elements of {O}, then for y E f(V),
y = ~UE'It eu(y).
Note that the sum in condition (b) is finite because, by (a), eu(y) = 0 if
On V = 0.
EXAMPLE
The relative Alexander pre sheaf of (X,A) of degree q with
coefficients G is fine. In fact, if GLl is a locally finite open covering of X, for
each x E X choose an element Ux E 6)l containing x and for cp E O( V, V n A; G)
define eucp E Cq(V, V n A; G) by
I
(eucp )( xo, . . . ,Xq)
If V'
= {CP(xo,
0
. . . ,Xq)
c V, there is a commutative square
Cq(V,v
n A;
G)
~
Cq(V, V
1
Cq(V',
n A;
G)
1
v' n
A; G) ~ Cq(V', V' n A; G)
showing that eu is an endomorphism of Cq. If
(xo, ... ,
Xq) E
(yq+1_
Oq+l) C
(yq+l - U q+1 ),
then Uxo i=- U and (eucp)(xo, . . . ,Xq) = O. Hence eucp I V - 0 = 0, and condition (a) is satisfied. To show that (b) is also satisfied, observe that, given
xo, ... , xq, there is a unique U, namely Uxo ' such that (eucp)(xo, ., ,Xq) i=- O.
Then
(~eucp)(xo,
. . . ,Xq)
= (euxo cp)(xo,
. . . ,Xq)
= cp(xo,
. . . ,Xq)
It should be noted that eu does not commute with the coboundary operator
in C * (V, V n A; G). Therefore eu is not an endomorphism of the Alexander
presheaf C * ( " . n A; G) of cochain complexes.
EXAMPLE
The relative singular presheaf of (X,A) of degree q with coeffi2
cients G is also fine. If G2l is a locally finite open covering of X and Ux is
chosen so that x E Ux E "it, then
eu: Hom (Llq(V)/Llq(V
n A),
G)
~
Hom (Llq(V)/Llq(V
n
A), G)
is defined by
(eue*)(a)
= {~*(a)
U
U
= Ua(po)
i=-
Ua(Po)
Then the family {eu} UE 'It satisfies conditions (a) and (b) of the definition of fine-
SEc.8
331
FINE PRESHEAVES
ness [but eu is not an endomorphism of t::.* ( " • n A; G) so t::.* ( " • n A; G)
has not been shown to be a fine presheaf of co chain complexes].
Given a pre sheaf f on X and a continuous map f: X ~ Y, there is a presheaf f* f on Y defined by (f* f)(V) = f(f-l V) for an open V C Y. Clearly,
f defines a covariant functor from the category of presheaves of any type on
X to the category of presheaves of the same type on Y. Some of the nice
properties of fine presheaves are made explicit in the following result.
3
THEOREM
Let f be a fine presheaf of modules on X.
(a) For any presheaf f' of modules on X, f ® f' is fine.
(b) Iff: X ~ Y is continuous, f* f is fine on Y.
(c) t is a fine presheaf on X.
PROOF
For (a), observe that if "II is a locally finite open covering of X and
{eu} u E 'It are the corresponding endomorphisms of f, then {eu ® I} u E 'It is a
family of endomorphisms of f ® f', showing that f ® f' is fine.
For (b), observe that if c~l is a locally finite open covering of Y,
then f- 1"11 = {f-l U I U E ~} is a locally finite open covering of X. If {eu} u E 'It
is a family of endomorphisms of f corresponding to the covering f- 1"11, they
induce endomorphisms of f* f, showing that f* f is fine.
(c) follows easily on observing that any endomorphism of f induces an
endomorphism of t. •
Given an open covering "I[ of a space X, a ,shrinking of "II is an open covering T of X in one-to-one correspondence with "II such that if U E Gil. corresponds to Vu E 'Y, then Vu C U. Any locally finite open covering of a normal
Hausdorff space has shrinkings. Any shrinking of a locally finite open covering
is clearly locally finite.
The following theorem is the main result on fine presheaves.
4
THEOREM
If f is a fine presheaf on a paracompact Hausdorff space X,
then j[q(X;f) = 0 for q
O.
>
PROOF
Let "II = {U} be a locally finite open covering of X and let
"II' = {U'} be a shrinking of "It. Let {eu} u E 'It be fineness endomorphisms of f
corresponding to the covering "II' (but indexed by the covering "11). Let
'I - = {V} be an open refinement of "II covering X such that each V E 'V
meets only a finite number of elements of "II and for any U E 01 either V C U
or V C X - (j'. Let ,\: T ~ "II be a function such that V C '\(V) for
all V E 'I:
Since each eu is an endomorphism of f, eu induces a co chain map,
denoted by eu: C * ("Il;f) ~ C * (c~l;f) such that for 1[; E Cq("Il;f) and
Uo, . . . , Uq E C~l
(eul/;)(Uo, ... ,Uq) = eu(1[;(Uo, ... ,Uq ))
Then eu acts similarly as a cochain map on C* ('IT) and commutes with the
cochain map ,\*: C*("il;f) ~ C*('\f).
Let q
0 and 1[; E Cq("II;f) be a cocycle. Define 1[;u E Cq('Y; f) by
>
332
GENERAL COHOMOLOGY THEORY AND DUALITY
CHAP.
6
t/;u = eu(A* t/;). Then t/;u is a cocycle for each U E 611, and if Yo, ... , Vq E 'Y,
then t/;U(Vo, ... , Vq) = 0, except for a finite number of U E UL1. Therefore
~ t/;u exists, and ~ t/;u = A* t/;.
Define t/;u E Cq-I('Y;f) by
l/;'u(Vo, ... ,va-I)
= {eou(\jJ(u,A(Vo), ... ,A(Vq - 1)) I (Vo
n ... n V
q - 1))
Vo
Vo
n ... n V
n ... n
q- 1 C U
Vq- 1 C X - 0'
Then ot/;'u = t/;u for all U, and because ~ t/;u can be formed [for given
Yo, ... , Vq_l , t/;f:r(Vo,
,vq-I) = 0, except for a finite number of U E ql],
we see that
A*t/; = ~ t/;u = o(~ t/;'u)
Therefore A*t/; is a coboundary, and fIq(X;f)
= O.
•
Our next results are technical lemmas about cochain complexes of
presheaves. If f * is a cochain complex of presheaves of modules on X, we
use Zq and Bq+1 to denote the kernel and image, respectively, of 0: fq ~ fq+1
and Hq to denote Zq/Bq, all of these being presheaves of modules on X. (Note
that a fine presheaf of cochain complexes is a cochain complex of fine presheaves, but the converse is not generally true.)
5
LEMMA
Let f * be a cochain complex of presheaves of modules on X.
For every q there is an exact sequence, functorial in f * ,
o ~ ker (fIO(X;Bq) ~
fII(X;Zq-I)) ~ fIo(X;zq) ~ Hq(f * (X)) ~ 0
PROOF
By example 6.7.13, rq(X) = iIO(X;f q). From the short exact sequence
of presheaves
there follows, by theorem 6.7.15, an exact sequence
o~
fIo(X;zq) ~ fIO(X;f q) ~ fI O(X;Bq+1) ~ fII(X;Zq)
Because Bq+l C f q +1, it follows from a similar exactness property that
fIO(X;Bq+l) C fIO(X;f q +1). Combining these, we see that
fIo(X;zq) ::::::: ker [fIO(X;f q) ~ fI O(X;Bq+1)]
::::::: ker [fIO(X;f q) ~ fIO(X;fq+1)]
and also that
im [fIO(X;f q) ~ fIO(X;f q +1)] ::::::: ker [fIO(X;Bq+l) ~ Hl(X;Zq)]
Since
Hq(r* (X))
= ker [HO(X;fq) ~ HO(X;fq+1))/im [HO(X;f q- 1) ~ HO(X;f q)]
the result follows.
•
SEC.
8
333
FINE PRESHEAVES
6
COROLLARY
Let f * be a cochain complex of pres heaves of modules on
a paracompact Hausdorff space X. For any q there is a short exact sequence,
functorial in f * ,
0----,> im [HO(X;Bq)
----'>
Hl(X;Zq-l)]
----'>
Hq([*(X))----,>
ker [HO(X;Hq)
----'>
Hl(X;Bq)]
----'>
0
If [q-l is fine, this becomes
0----,> Hl(X;Zq-l)
PROOF
----'>
Hq([ * (X))
ker [HO(X;Hq)
----'>
----'>
Hl(X;Bq)]
----'>
0
From the short exact sequence of presheaves
o ----'> Bq ----'>
Zq
----'>
Hq
----'>
0
it follows, by theorem 6.7.15, that there is an isomorphism
HO(X;Zq)/HO(X;Bq) ;::::: ker [HO(X;Hq)
----'>
Hl(X;Bq)]
From lemma 5, there is an isomorphism
HO(X;Zq)/ker [HO(X;Bq)
----'>
Hl(X;Zq-l)];::::: Hq([*(X))
It follows that Hq([ * (X)) maps epimorphically to ker [HO(X;Hq)
with kernel isomorphic to
HO(X;Bq)/ker [HO(X;Bq)
----'>
Hl(X;Zq-l)] ;::::: im [HO(X;Bq)
----'>
----'>
Hl(X;Bq)]
Hl(X;Zq-l)]
This gives the first short exact sequence. For the second, there is a short exact
sequence of presheaves
and if fq-l is fine, it follows from theorems 6.7.15 and 4 that
im [HO(X;Bq)
----'>
Hl(X;Zq-l)] = Hl(X;Zq-l)
•
7
THEOREM
Let f * be a nonnegative cochain complex of fine pres heaves
of modules on a paracompact Hausdorff space X. Assume that for some
integers 0 :c::: m < n, Hq(f *) is locally zero for q < m and m < q < n. Then
there are functorial isomorphisms
and a functorial monomorphism
Hn-m(X;Hm(f*))
PROOF
----'>
Hn(f*(X))
For each q there is a short exact sequence of presheaves
o ----'> Zq ----'> fq ----'>
Bq+l
----'>
0
Because fq is fine, it follows from theorems 6.7.15 and 4 that
(a)
p '2 1
334
GENERAL COHOMOLOGY THEORY AND DUALITY
CHAP.
6
For each q there is also a short exact sequence of presheaves
o ~ Bq ~
Because Hq is locally zero for q
theorems 6.7.15 and 6.7.16 that
(b)
Zq
~
<m
q
Hq
0
~
and m
< q < n,
it follows from
< m or m < q < n, all p
Since BO is the zero presheaf, it follows by induction on q from equations
(b) and (a) that for q < m
(c)
p ?:. 1
<
From this and corollary 6, it follows that Hi(f' * (X)) = 0 for i
m. Hence
the theorem holds for q < m (both modules being trivial). For q = m we
have [by corollary 6 and equation (c)]
Hm(I'* (X)) ::::::; HO(X;Hm)
and the theorem holds in this case too.
To obtain the result for m < q ~ n, note that, by equation (c),
HP(X;Bm) = 0, if P ?:. 1. From the short exact sequence of presheaves
o ~ Bm ~
Zm
~
Hm
0
~
it follows that
HP(x;zm) ::::::; HP(X;Hm)
For m
p ?:. 1
< i < n it follows from corollary 6 that
Hl(X;Zi-l) ::::::; Hi(I' * (X))
and for i
= n there is a monomorphism
Hl(x;zn-l) ~ Hn(I' * (X))
Using equations (b) and (a), we see that for m
<i
~
n
Hl(X;Zi-l) ::::::; Hl(X;Bi-l) ::::::; H2(X;Zi-2) ::::::; ... ::::::; Hi-m(x;zm) ::::::; Hi-m(X;Hm)
and this gives the result for m
< q ~ n.
•
This last result has as an immediate consequence the following isomorphism between the Cech and Alexander cohomologies with coefficients G.
8
COROLLARY
For any paracompact Hausdorff space and module G there
is a functorial isomorphism
H*(X;G)::::::; H*(X;G)
of the Cech and Alexander cohomology modules.
Let C * be the Alexander presheaf of X with coefficients G. Since Cq
is fine for all q (by example 1), this is a nonnegative cochain complex of fine
sheaves. Furthermore, for any nonempty U, by lemma 6.4.1,
PROOF
SEC.
8
335
FINE PRESHEAYES
q¥=O
q=O
>
Therefore Hq( C *) is locally zero for q
0 and HO( C *) is isomorphic to the
constant pre sheaf G. The hypotheses of theorem 7 are satisfied with m = 0
and any n, and there is a functorial isomorphism
fIq(X;G) ;:::; Hq(C*)
for all q. As pointed out in example 6.7.8, there is a canonical isomorphism
C* ;:::; C*, and so j{q(X;G) ;:::; Hq(C*). Combining these isomorphisms yields
the result. The last result is also true without the assumption of paracompactness
(see exercise 6.D.3). The next result is the main uniqueness theorem of the
cohomology of presheaves.
9
THEOREM
Let X be a paracompact Hausdorff space and let T: f * ~ f' *
be a cochain map between nonnegative cochain complexes of fine pres heaves
of modules on X. Assume that for some n ?: 0, T *: Hq(f *) ~ Hq(f' *) is a
n and a local monomorphism for q = n. Then the
local isomorphism for q
induced map
<
f*: Hq(t * (X)) ~ Hq(t' * (X))
is an isomorphism for q
< n and a monomorphism for q =
n.
PROOF
Let f ~ be the mapping cone of T (defined for cochain complexes analogous to the definition in Sec. 4.2 for chain complexes). Then f T q = fq+l EEl f'q,
and for "1 E rq+l(U) and "1' E r'q(U), 8("1,"1') = ( - 8("1), T('Y) + 8("1')). r1 is
a nonnegative cochain complex of fine presheaves on X, and for any open
U C X there is an exact sequence
...
~
Hq(f'*(U))
~ Hq(f~(U)) ~
Hq+1(f*(U))
~
Hq+1(f'*(U))
~
...
Taking the direct limit as U varies over open neighborhoods of x E X, we see
that T*: Hq(f *) ~ Hq(f' *) is a local isomorphism for q
n and a local
n. By
monomorphism for q = n if and only if Hq(f~) is locally zero for q
theorem 7, it follows that Hq(t ~ (X)) = 0 for q
n (if n = 0 this is trivially
0 it follows from theorem 7 with m = 0).
true, and if n
It is obvious that t ~ is the mapping cone f ~ of the induced map
f: f' * ~ f" * between the completions. Therefore
<
<
>
<
... ~ Hq(f" * (X)) ~ Hq(f'~(X)) ~ Hq+l(f' * (X)) ~ Hq+1(t' * (X)) ~ ...
<
Since Hq(t ~ (X)) was shown to be zero for q n in the first paragraph above,
the result follows from the exactness of this sequence. For compact spaces there is the following universal-coefficient formula
for tech cohomology.
10 THEOREM
Let X be a compact Hausdorff space. On the product category
336
GENERAL COHOMOLOGY THEORY AND DUALITY
of pres heaves
r
CHAP.
6
on X consisting of torsion free R modules and the category of
R modules G there is a functorial short exact sequence
o ----,) fIq(X;f)
PROOF
C8l G ----,) fIq(X;
r
C8l G) ----,) fIq+1(x;r)
* G ----,) 0
Let G(l be a finite open covering of X. The cochain map
7:
C* (ql;r) C8l G ----,) C* (GLl; r C8l G)
defined by 7(l/; C8l g)(Uo, . . . ,Uq ) = l/;(Uo, . . . ,Uq ) ® g is an isomorphism
(this is a consequence of the finiteness of G(l analogous to lemma 5.5.6). From
the universal-coefficient formula for cochain complexes (theorem 5.4.1), there
is a functorial short exact sequence
0----') Hq(G(l;r) C8l G ----,) Hq(G(l; r ® G) ----,) Hq+l(ql;f)
* G ----,) 0
The result follows by taking direct limits over the cofinal family of finite open
coverings of X (because the tensor product and the torsion product both
commute with direct limits). From corollary 8, this gives a universal-coefficient formula for Alexander
cohomology of compact spaces. The following theorem generalizes this result
to compact pairs and includes the statement that the short exact sequence in
question is split.
I I THEOREM On the product category of pairs (X,A), where A is a closed
subset of a compact Hausdorff space X, and the category of R modules G, there
is a functorial short exact sequence
o ----,)
fIq(X,A; R) ® G ----,) fIq(X,A; G) ----,) fIq+l(X,A; R)
* G ----,) 0
and this sequence is split.
Let 7: C* ( " . n A; R) ® G ----,) C* ( " . n A; G) be the homomorphism of presheaves defined as in example 6.7.5 [that is, 7(cp C8l g)
(xo, . . . ,xq) = cp(xo, . . . ,xq)g]. Both C* (., . n A; R) C8l G and
C* (', . n A; G) are nonnegative co chain complexes of fine presheaves.
First we prove that
PROOF
7*: H*(C*(','
n A; R) C8l G) ----,) H*(C*(',' n A; G))
is a local isomorphism. If U eX - A, C*(U, UnA; R) = C*(U;R), and
C * (U, UnA; G) = C * (U; G), it follows from lemma 6.4.1 and theorem 5.4.1
that
7*: H*(C*(U,
un
A; R) Q9 G) = H*(C*(U,
un
A; G))
Since A is closed in X, for any x E X - A, 7* is an isomorphism of
{H*(C*(U, unA; R) ® G)} onto lim~ {H*(C*(U, UnA; G))},
both limits as U varies over open neighborhoods of x in X.
For any U intersecting A there is a commutative diagram with exact
rows
lim~
SEC.
8
o~
337
FINE PRESHEAYES
C*(U, UnA; R) ®
o ~ C * (U,
G~
C*(U;R) ®
G~
C*(U
n A; R) ®
J
~
C*(U
n A;
J
J
unA; G)
~
C*(U;G)
G)
G~ 0
~O
By lemma 6.4.1, the middle co chain complexes have trivial reduced modules.
Therefore there is a commutative square
fjq(G* (U n A; R) ® G)
~
J
fjq(G* (U
J
n A;
G))
To complete the proof that
prove that for x E A
lim~
Hq+l(C* (U, UnA; R) ® G)
~
Hq+1(C* (U, unA; G))
* is a local isomorphism, therefore, we need only
'T
{fjq(G*(U n A; R) ® G)} ::::: lim~ {fjq(G*(U n A; G))}
as U varies over neighborhoods of x in X. This is equivalent to the condition
that
fjq(lim~
{G*(U n A; R)}) ® G::::: fjq(lim~ {G*(U n A; G)})
where UnA varies over neighborhoods of x in A. This is trivially true
because both sides are zero for all q (this follows from the tautness property
of x in the paracompact space A but can be proved without assuming the
paracompactness of A, because anyone-point subspace is taut in any space
with respect to Alexander cohomology).
We have verified that 'T satisfies the hypotheses of theorem 9 for all n.
Therefore 'T induces an isomorphism
n A; G)(X))
By example 6.7.8, the right-hand side is isomorphic to H* (G * (X,A; G)). By
•
f*: H*([G*("
n A; R)
® G](X))::::: H*(G*("
•
example 6.7.13, the left hand side is the qth cohomology module of
fIo(X; C * ( " . n A; R) ® G). By theorem 10 and the fineness of
C*(', . n A; R) ® G, this is isomorphic to
fIo(X;c*("
• :l A; R)) ® G::::: (G*(" • n A; R)(X)) ® G
;::- C*(X,A; R) ® G
It follows that the map
f: C*(X,A; R) ® G ~ C*(X,A; G)
induced by 'T induces an isomorphism of cohomology. The result now follows
from the universal-coefficient formula for cochain complexes (theorem 5.4.1). •
This implies the following universal-coefficient formula for Alexander
cohomology with compact supports.
338
GENERAL COHOMOLOGY THEORY AND DUALITY
CHAP.
6
12 COROLLARY On the product category of pairs (X,A), where A is a closed
subset of a locally compact Hausdorff space X, and the category of R modules
G, there is a functorial short exact sequence
o ~ Hcq(X,A;
R) ® G ~ Hcq(X,A; G) ~ Hcq+l(X,A; R)
*G~ 0
and this sequence is split.
Let N be a closed cobounded neighborhood of A in X. There is
a commutative square of cochain maps
PROOF
C* (X,N; R) ® G
C*(X,N; G)
l
l
C*(X - N, X - N n N; R) ® G ~ C*(X - N, X - N n N; G)
in which, by theorem 6.6.5, each vertical map induces an isomorphism of
cohomology. By theorem 11, the bottom horizontal map induces an isomorphism of cohomology. Therefore the top horizontal map also induces an isomorphism of cohomology.
There is also a commutative square (in which the limit is over closed
cobounded neighborhoods N of A in X)
lim~ {C*(X,N; R)}
® G ---7lim~ {C*(X,N; G)}
l
C~(X,A; R)
l
®G
C~(X,A; G)
It follows from the first paragraph above that the top horizontal map induces an isomorphism of cohcmology. Since the closed cobounded neighborhoods of A in X are cofinal in the family of all co bounded neighborhoods of A
in X, it follows from theorem 6.6.15 that each vertical map induces an isomorphism in cohomology. Therefore the bottom horizontal map induces an
isomorphism in cohomology. The result follows from this and theorem 5.4.1. •
9
APPLICATIONS OF THE COHOMOLOGY OF PRESHEAVES
This section is devoted to two main applications of the theory developed in
the last two sections. One is the study of the relation between Alexander and
singular cohomology. We shall prove that in a homologically locally connected
space (for example, a manifold) the two are isomorphic. The other application
is to a study of the relation between the Alexander cohomology of two spaces
connected by a continuous map. We conclude with a proof of the VietorisBegle mapping theorem.
Let (X,A) be a pair and let G be an R module. Recall the homomorphism
T:
C* (. , .
n A;
G)
~
Ll * (., .
n A;
G)
SEc.9
339
APPLICATIONS OF THE COHOMOLOGY OF PRES HEAVES
defined in example 6.7.4. This induces a homomorphism
f:
C* ( " . n
A; ~e) ~
i *(" . n
A;
e)
such that the following square is commutative
C*(', . n A; e)
~
ll*(·, . n A; e)
C*(', . n A; e).i,. &*(" . n A; e)
By examples 6.7.8 and 6.7.9, there are isomorphisms
IX
C*(', . n A; e);:::; C*(', . n A; e)
A; e)) ;:::; H* (& * ( " . n A; e))
* : H* (ll * ( " . n
In Sec. 6.5 a natural homomorphism
p,: fi*(X,A; e) ~ H*(X,A; G)
was defined, and it is a simple matter to check that commutativity holds in
the diagram
H* (C* (X,A; e))
~
H* (ll * (X,A; e))
::oJ
H*(C*(', . n A; e)(x)) ~ H*(&*(" . n A; G)(X))
Therefore p, is an isomorphism if and only if f
* is.
I
THEOREM
Let X be a paracompact Hausdorff space and suppose there
is n ~ 0 such that each x E X is taut with respect to singular cohomology
with coefficients
e in degrees < n.
Then
p,: fiq(X;G) ~ Hq(X;e)
is an isomorphism for q
< n and a monomorphism for q = n.
Both C * ( . ; e) and II * ( . ;e) are nonnegative cochain complexes of
fine presheaves. The tautness assumption of the points of X with respect to
singular cohomology implies that 'T *: Hq(C* ( . ;e)) ~ Hq(ll * (. ;G)) is a local
isomorphism for q < n and a local monomorphism for q = n (in fact, it
is always a local monomorphism for all q). By theorem 6.8.9,
PROOF
T*: Hq(C*(X;G)) ~ Hq(&*(X;e))
is an isomorphism for q
< n and a monomorphism for q = n.
-
There is a partial converse of theorem 1 which asserts that if
< n and every open U C X,
then each point x E X is taut with respect to singular cohomology in degrees
n. This follows from commutativity of the following diagram (where
U varies over open neighborhoods of x EX):
p,: fiq(u;e) ~ Hq(U;e) is an isomorphism for q
<
340
GENERAL COHOMOLOGY THEORY AND DUALITY
lim~
6
{Hq(U;G)} ~ Hq(x; G)
~l
lim~
CHAP.
~l~
{Hq(U;G)}
~
Hq(x;G)
In case X is a Hausdorff space in which every open subset is paracompact
(for example, X is metrizable), we see that each point x E X is taut with respect
n if and only if J.L: Hq( U; G) ~ Hq( U; G)
to singular cohomology in degrees
is an isomorphism for all q
n and all open U C X.
A space X is said to be homologically locally connected in dimension n
if for every x E X and neighborhood U of x there exists a neighborhood V of
x in U such that Hq(V) ~ Hq( U) is trivial for q ::; n. It is said to be
homologically locally connected if it is homologically locally connected in
dimension n for all n.
<
<
2
EXAMPLE
Any locally contractible space, in particular any polyhedron
or any manifold, is homologically locally connected in dimension n for all n.
3
EXAMPLE
Let Xq = Sq for q :2: 1 and let Xq be a base point of Xq- The
subspace of X Xq consisting of all points having at most one coordinate different from the corresponding base point is homologically locally connected
in dimension n for all n but is not locally contractible.
4
Hq(~
LEMMA
If X is homologically locally connected in dimension n, then
* (. ;G)) is locally zero for q ::; n and all G.
Let c* E Hom (t.q(U),G) be a co cycle (0::; q ::; n) and let x E U. If
q = 0, let V be a neighborhood of x in U such that Ho(V) ~ Ho(U) is trivial.
If c E t.o(V), there is c' E ~l(U) such that c = ac'. Then c* (c) = c* (ac') =
(8c* )(c') = O. Therefore c* I Lio(V) = 0, proving that HO(~* (. ;G)) is locally
trivial.
If 0
q, let V and V be neighborhoods of x in U, with V C V' and such
that Hq_1 (V) ~ Hq_ 1 (V') and Hq(V') ~ Hq(U) are both trivial. If c is a reduced singular (q - I)-cycle of V, let c' be a q-chain of V' such that ac' = c.
Then c* (c') EGis independent of the choice of c'; if c" is another q-chain in
V' such that ac" = c, then c' - c" = ad for some (q + I)-chain din U and
PROOF
<
c*(c' - c")
= c*(ad) = (8c*)(d) = 0
Hence there is a homomorphism c*: Zq_l(V) ~ G such that c* (c) = c* (c') if
ac' = c. Because ~q_l(V)/Zq_l(V) is free (since it is isomorphic to a subgroup
of ~q-2(V) if q lor to Z if q = 1), there is a homomorphism d*: ~q_l(V) ~ G
which is an extension of c* . Then c* I ~q( V) = 8d * , proving that Hq( ~ * ( . ;G))
is locally trivial. -
>
:.
COROLLARY
If X is a paracompact Hausdorff space homologically locally
connected in dimension n, then J.L: fIq(X;G) ~ Hq(X;G) is an isomorphism for
q ::; n and a monomorphism for q = n + 1. -
SEc.9
341
APPLICATIONS OF THE COHOMOLOGY OF PRESHEAVES
6
COROLLARY
Let A be a closed subset, homologically locally connected in
dimension n, of a Hausdorff space X, homologically locally connected in
dimension n. If X has the property that every open subset is paracompact,
}-t: fIcq(X,A; G) ~ Hcq(X,A; G) is an isomorphism for q ::;; n and a monomorphism for q = n + 1.
PROOF
From the definitions, there is a commutative square (where U varies
over open cobounded subsets of X)
lim~
{Hq(X,U; G)}
~
til
Ilt
lim~
{Hq(X, U; G)}
Hcq(X;G)
~
Hcq(X;G)
Since an open subset of a space homologically locally connected in dimension
n is again a space homologically locally connected in dimension n corollary 5
applies to X and to every open U C X. By the five lemma,
}-t:
fIq(X, U; G) ~ Hq(X, U; G)
is an isomorphism for q ::;; n and a monomorphism for q = n + 1. Passing to
the limit, }-t: fIcq(X;G) ~ Hcq(X;G) is an isomorphism for q ::;; n and a monomorphism for q = n + 1. Since A has the same properties as X,
}-t:
fIcq(A;G) ~ Hcq(A;G)
is an isomorphism for q ::;; n and a monomorphism for q = n
now follows from the five lemma. -
+ 1. The result
Since a manifold is homologically locally connected in dimension n for
all n, and every open subset is paracompact, this implies the next result.
7
COROLLARY
If X is a manifold, }-t: fI*(X;G)::::::: H*(X;G). If A is a
closed homologically locally connected subset of X,
}-t:
fI~ (X,A; G) ::::::: H~ (X,A; G).
-
8
COROLLARY
If X is a homologically locally connected space imbedded as
a closed subset of a manifold Y, then X is taut in Y with respect to singular
cohomology.
PROOF
By corollary 5, fI* (X;G) ::::::: H* (X; G), and for an open set U in Y, by
corollary 7, fI * (U; G) ::::::: H * (U; G). Since X is taut in Y with respect to
Alexander cohomology, these isomorphisms imply that it is also taut with
respect to singular cohomology. -
9
COROLLARY
If A is any closed subset of a manifold X, then as U varies
over neighborhoods of A in X,
lim~ {H*(U;G)} ::::::: fI*(A;G)
where the right-hand side is Alexander cohomology.
PROOF
By corollary 7, lim_ {fI * (U; G)} ::::::: lim~ {H * (U; G)}, so the result
342
GENERAL COHOMOLOGY THEORY AND DUALITY
CHAP.
6
follows from the tautness of A with respect to the Alexander cohomology
theory. This shows that the modules H * (A; G) and H * (A,B; G) introduced in
Sec. 6.1 are the Alexander cohomology modules if A [or (A,B)] is a closed subset
[or pair] of a manifold. The next result generalizes the duality theorem 6.2.17
to arbitrary closed pairs.
10 THEOREM Let X be an n-manifold orientable over R. For any closed
pair (A,B) in X and any R module G there is an isomorphism
Hq(X - B, X - A; G) ::::: Hcn-q(A,B; G)
PROOF
Let N be a closed cobounded neighborhood of B in A. By theorem
6.6.5, there is an isomorphism
Hn-q(A,N; G) ::::: Hn-q(A - N, A - N
n
N; G)
Since (A - N, A - N n N) is a compact pair in X, by theorem 6.2.17,
Hq(X - (A - N
n N),
X - (A - N); G) ::::: Hn-q(A - N, A - N
n N;
G)
Since X - (A - N) and X - N are open, there is an excision isomorphism
Hq(X - N, X - A; G) ::::: Hq(X - (A - N
n N), X - (A - N); G)
Combining these gives an isomorphism
Hq(X - N, X - A; G) ::::: Hn-q(A,N; G)
As N varies over closed cobounded neighborhoods of B in A, the limit of the
modules on the left is Hq(X - B, X - A; G) and the limit of the modules on
the right is Hcn-q(A,B; G), whence the result. I I THEOREM If X is a compact Hausdorff space which is homologically
locally connected in dimension n, then Hq(X) is finitely generated for q S n.
PROOF
This follows from corollary 5, theorem 6.8.11, and theorem 5.5.13.
-
The last result gives a generalization of corollary 6.2.21 to arbitrary compact manifolds (orientable or not). We now work toward a proof of the
Vietoris-Begle mapping theorem.
12 LEMMA Let (X,A) be a pair and let f be the presheaf on X defined by
f(V) = Cq(V, V n A; G) for open y C X (q and G being fixed).
(a) For any open covering ql of X the map f(X) ~ f(ql) sending y E f(X)
to the compatible '11 family {y I U} U E -'I is a monomorphism.
(b) If GIl· is a locally finite open covering of X and 'Y is a shrinking ofql,
the image of f(GIl) ~ f('Y) equals the image of the composite
f(X)
PROOF
~
f(q1)
~
f('\")
For (a), assume that y E Cq(X,A; G) is in the kernel of f(X) ~ f(ql)
SEc.9
343
APPLICATIONS OF THE COHOMOLOGY OF PRES HEAVES
(that is, Y I V = 0 for all V E 621). Let!p E O(X,A; G) be a representative of y.
Then y I V
0 implies that !p I V is locally zero on V. Since this is so for all
V E "It, !p is locally zero on X and y = 0, proving (a).
To prove (b), let {YU}UE'lL be a compatible 621 family and suppose that
!Pu E Cq( V, V n A; G) is a representative of Yu for V E 621. Then, for V, V' E G21,
!Pu I V n V' - !PU' I V n V'is locally zero on V n V'. If x E X, some neighborhood of x meets only finitely many elements of Gil, and there is a smaller
neighborhood W", of x such that
=
W", intersects Vu <=> x E
xEV
W",
x E Vu= W",
x E Vu n Vu =!Pul
(i)
(ii)
(iii)
(iv)
=
Vu
c V
C Vu
W", =!Pu I W",
The first three conditions are clearly satisfied by taking W", small enough
(because there are only a finite number of conditions to be satisfied) and (iv)
can also be satisfied, because for x E Vu n Vu" cpu I V n U' - CPU' I V n U'
is locally zero.
For x E X choose V so that x E Vu and set !p", = !Pu I W", E Cq(W""
W", n A; G). By (iv), this is independent of the choice of V. If x" E W'" n W"",
then x" E Vu for some V E Gil. Then W'" and W"" meet V u , and by (i), x,
x' E Vu. Therefore!p",
!Pu I W'" and !p",'
!Pu I W"", whence!p", I w'" n W"" =
!p",' I w'" n W"'" Hence the collection {!p", E Cq( w"" w'" n A; G)} is a compatible {W",} family [of Cq(·, • n A; G)J. By example 6.7.8, there is an element!p E O(X,A; G) such that!p I W'" = !p", for all x E X. To complete the proof
of (b) it suffices to prove that for each V E GiL, !P I Vu - !Pu I Vu is locally zero
on Vu. However, if x E V u, then, by (iii), w'" C Vuand!p I W'" = !p", = !Pu I W"'.
Hence {W"'}"'EVU is an open covering of Vu on which !P I Vu and !Pul Vu
agree. •
=
=
13 THEOREM Let f: X' - ? X be a closed continuous map between paracompact Hausdorff spaces. Let A' be a closed subset of x' and suppose there
n such that fiq(f-lx, f-1x n A'; G)
0 for all x E X
are integers 0 :::;: m
and for q
m or m
q
n. Let f be the presheaf on X defined by
f(V) = fim(f-l(V), f-l(V) n A'; G). Then there are isomorphisms
<
<
=
< <
Hq-m(X;f) ;::::; fiq(XI,A ' ; G)
q
<n
and a monomorphism
fIn-m(X;f)
-?
fin(XI,A'; G)
PROOF
Let f * be the nonnegative cochain complex of presheaves on X defined by f * (V) = C* (f-l( V), f-l( V) n A'; G). Thus fq is the image under
f* of the fine presheaf on X' which assigns Cq(V' , V' n A'; G) to V' C X'.
By theorem 6.8.3c, the latter is a fine presheaf on X' [being the completion of
the fine presheaf 0(', . n A'; G); see example 6.8.1J, and by theorem 6.8.3b,
fq is fine on X. As V varies over neighborhoods of x in X, (f-l( V), f-l( V) n A')
344
GENERAL COHOMOLOGY THEORY AND DUALITY
CHAP.
6
varies over a cofinal family of neighborhoods of (f-IX, f-IX n A') in (X',A')
(because f is closed and continuous). From the standard tautness properties
and the hypothesis about fj * (f-IX, f-IX n A'; G), it follows that Hq(f *) is
locally zero for q
m and m
q
n. By theorem 6.8.7, there are functorial
isomorphisms
<
< <
and a monomorphism
fIn-m(X;Hm(f *)) ~ Hn(f * (X))
Since f
= Hm(f *), it merely remains to verify that
HP(I' * (X)) ;:::::: fjp(X',A'; G)
all P
As ~ varies over the cofinal family of locally finite open coverings of X it follows from lemma 12 that
f*(X)
= lim~
{r*(~)}
= lim~
and this yields the result.
{C*(·,·
n A'; G)(f-101)};:::::: C*(X',A'; G)
-
If ~ is an m-sphere bundle over a paracompact Hausdorff base space B,
then fjq(pl; -1(X), PI; -l(x) n E) = 0 if q T m + 1. Therefore the hypotheses of
theorem 13 are satisfied for all n. Since the presheaf r that occurs in theorem
13 is the tensor product of the orientation presheaf of ~ and G, we obtain the
folloWing generalization of the Thorn isomorphism theorem to nonorientable
sphere bundles.
14 THEOREM Let ~ be an m-sphere bundle over a paracompact Hausdorff
base space B and let r be the orientation presheaf of ~ over R. For all
R modules G and all q there is an isomorphism
fIq(B;
r
0 G) ;:::::: fjq+m+1(EI;,EI;; G)
-
Another interesting consequence of theorem 13 is the following VietorisBegle mapping theorem.
15 THEOREM Let f: X' ~·X be a closed continuous surjective map between
'f!!1racompact Hausdorff spaces. Assume that there is n ~ 0 such that
[jq(f-Ix;G)
0 for all x E X and for q
n. Then
=
<
f*: fjq(X;G) ~ fjq(X';G)
is an isomorphism for q
< n and a monomorphism for q =
n.
Let Z be the mapping cylinder of f and regard X' as imbedded in Z.
Then Z is a paracompact Hausdorff space, X' is closed in Z, and the retraction r: Z ~ X is a closed continuous map. For x E X, rl(x)Js contractible
[since it is homeomorphic to the join of x withf-l(x)], and so H*(r-l(x)) = O.
Because r-l(x) n X' = f-I(X) is nonempty, we have
PROOF
SEc.9
345
APPLICATIONS OF THE COHOMOLOGY OF PRESHEAVES
It follows from theorem 13 that fiq(Z,X'; G) = 0 for q ~ n. Since there is a
commutative diagram with an exact row
...
~
Hq(Z,X')
Hq(z)
~
~
Hq(X')
~
Hq+1(Z,x')
~
/lr·
the result follows.
•
There is a partial converse of theorem 15 asserting that if f: X' ~ X is a
closed continuous surjective map between paracompact Hausdorff spaces and
there is n 2': 0 such that for every open Q C X,f*: Hq(U;G) ~ Hq(f-1(U);G)
is an isomorphism for q
n, then Hq(f-1(X);G) = 0 for all x E X and
for q
n. This follows from commutativity of the following diagram (where
U varies over open neighborhoods of x EX):
<
<
lim_ {Hq( U; G) }
r·l
In particular, if X and X' are metrizable (or have the property that every
open subset is paracompact), then for n 2': 0, f*: fiq(U;G) ~ fiq(f-1(U);G)
i~ an isomorphism for all open U C X and all q
n if and only if
Hq(f-1(X);G) = 0 for all x E X and all q
n.
We present an example to show that the condition that fbe a closed map
is necessary in theorem 15.
<
<
16 EXAMPLE Let X' = {(x,y) E R21 x2 + y2 = 1 or x2 + y2
and let X = [0,1]. Define f: X' ~ X by
f(x,y)
= {~
< 1, x> O}
x~O
x2':O
Then f is a continuous surjective map but not a closed map. Furthermore,
closed semicircle
f-1(t)
= 1closed interval
single point
t =0
0
t
t =1
< <1
Because the unit circle 51 is a strong deformation retract of X',
fi1(X';G) ;::::: fi1(5 1;G) ;::::: G.
Since fi1(X;G)
isomorphism.
17
EXAMPLE
= 0, the homomorphism f*:
W(X;G) ~ fi1(X';G) is not an
Let X C R2 be the space of example 2.4.8, illustrated below:
346
GENERAL COHOMOLOGY THEORY AND DUALITY
(0,-2)
CHAP.
6
(1,-2)
A,
There is a closed continuous surjective map f of X onto the space Y consisting
of the four sides of the rectangle
(0,0)
(0,-2)
0
(1,0)
(1,-2)
such that
f
~1
_
(y) -
{single point
closed interval
y -=/= (0,0)
y = (0,0)
It follows from theorem 15 thatf*: H*(Y;G) ::::::: H*(X;G) for any G, and
therefore the map f is not null homotopic.
18 THEOREM Let f: X' -> X be a proper surjective map jJetween locally
compact Hausdorff spaces and assume that for some n > 0, Hq(f~l(X); G) = 0
for all x E X and all q < n. Then
f*: Hcq(X;G) -> Hcq(X';G)
is an isomorphism for q
< n and a monomorphism for q =
n.
If either X or X' is compact, the other one is also compact, and the
result follows from lemma 6.6.9 and theorem 15. If neither X nor X' is
compact, let X+ and X'+ be their one-point compactifications and extend fto a
map f+: X'+ -+ X+ mapping the point at infinity of X'+ to the point at infinity
of X+. Then f+ satisfies the hypotheses of theorem 15, and the result follows
from corollary 6.6.12 and theorem 15. •
PROOF
I 0
CHARACTERISTIC CLASSES
This section is a culmination of our general work on homology theory. We
use the cup product and Steenrod squares to define characteristic classes of a
manifold and of one manifold imbedded in another. These characteristic
classes are important invariants of the manifold and have interesting applications to nonimbedding problems.
Let X be an n-manifold with boundary Xand U E Hn(x X X, X X X - 8(X))
SEC.
10
347
CHARACTERISTIC CLASSES
be an orientation (over R) of X. Let
Then the maps
i:
X - X C X be the inclusion map.
i
X 1: (X - X) X (X,X) C X X (X,X)
1 X i: (X,X) X (X - X) C (X,X) X X
are both homotopy equivalences. Therefore there are elements
such that
(f X 1) * UI
= U I (X -
(1 X i) * U2
X) X (X,X)
= U I (X,X)
X (X - X)
If X is compact, let z E Hn(X,X) be the fundamental class of X corresponding
to U, as in theorem 6.3.9. The Euler class of a compact oriented manifold X,
denoted by X E Hn(X,X), is defined by
= (UI v
X
U2 )/z
The reason for the name is furnished by theorem 2 below.
Assume that R is a field and that X is a compact n-manifold with boundary X. By theorem 6.9.11, H* (X) and H* (X,X) are finitely generated. If {u;)
is a basis of H*(X) and {Vj} is a basis of H*(X,X), then by the Kiinneth formula for cohomology, {Ui X Vj} is a basis of H* (X X (X,X)). Hence
= ~ aijUi X
UI
Vj
i,j
for some scalars aij. Let b jk = (Vj V Uk, z), where z is the fundamental class
corresponding~ to U. Then we have matrices A = (au) and B = (bjk ), and the
following expresses their relation to each other.
I
LEMMA
With the above notation,
(AB)ik
= (-l)n deg Uk 8ik
The proof is essentially the same as that for theorem 6.3.12. Because
z is the fundamental class corresponding to U, it follows that
PROOF
Udz
= 1 E HO(X)
By property 6.1.4, for any k
Uk
= Uk V
1
= Uk v
UI/z
= [(Uk
X 1) v
UIJlz
From lemma 6.3.11 it follows readily that
(Uk
X 1)
V
UI
= (1
=
X
Uk)
v
UI
= (_l)ndeg UkUI
~
(-l)n deg Uka;jUi X
Uk
=
'.1
(Vj
v
Hence by property 6.1.2
~
( -l)n deg UkaijbjkUi
'.1
Since {u;) is a basis, this implies the result.
-
Uk)
V
(1 X
Uk)
348
GENERAL COHOMOLOGY THEORY AND DUALITY
CHAP.
6
2
THEOREM
If X is the Euler class of a compact n-manifold X oriented
over a field, then (X,z) is the Euler characteristic of X.
PROOF
We first compute V2. Let T: X X X ~ X X X be the map interchanging the factors. There is a commutative diagram, with all vertical maps
induced by maps defined by T and all horizontal maps induced by inclusions,
Hn(X X X, X X X - 8(X)) ~ Hn((X - X) X (X,X))
TTl
«(j
~ 1)* Hn(X X (X,X))
Tn
IT!
Hn(X X X, X X X - 8(X)) ~ Hn((x,x) X (X - X))
«(1
~j)* Hn((x,x) X X)
In the proof of lemma 6.3.11 it was shown that T! V = (-I)nu. Therefore
1": Vj = (-I)nV2, and so
= (-I)nT~ ( k,l~ aklUk
V2
(-I)n ~
=
k,l
X
VI)
(_I)de g Uk deg
VI
aklVI
X
Uk
Therefore
VI
V
V2
=
(_I)n ~
(_I)de g
=
(-I)n
( -1 )deg VI
~
VI
deg
where the summation is over all i,
deg
Uk)
Vi
+ deg Uk deg VI + deg Ui deg VI aijakl( VI v
Ui)
f, k,
VI
aijakl(Ui
v
X
v
deg Vi + deg Uk deg
VI)
(Vj
and 1 such that
Ui
+ deg Vj = n = deg Uk + deg VI
=
~ (_I)de g Ukaijakl(VI
It follows that
VI v V2
V
Ui)
X
(Vj
v
Uk)
Using lemma 1,
(X,z)
= (VI V
=
.~
t,J,k,1
V2,
Z
X z)
(_I)de g Ukaijbjkaklbli
= t,k
~ (_1)de g Uk(AB)ik (ABhi
=~(_I)degUk
k
and the last sum is the Euler characteristic of X.
•
Classically, the Euler class is usually taken to be the Euler class (in our
sense) over Z. For any pair (Y,B) whose homology is of finite type, it follows
from the universal-coefficient formula for cohomology (theorem 5.5.10) that
Hq(Y,B; R) :::::: Hq(Y,B; Z) ® R
Therefore the monomorphism Z ~ R induces a monomorphism
SEC.
10
349
CHARACTERISTIC CLASSES
In particular, the monomorphism Hn(x,x; Z) ~ Hn(x,x; R) maps Euler class
to Euler class, and therefore theorem 2 remains valid for the integral Euler
class of X.
We now specialize to the case where the coefficient field is Z2, in which
case U, hence also Ub and (if X is compact) z, are all unique. There is the
Thorn isomorphism
<1>*: Hq(X - X) ;:::::, Hq+n((x - X) X (X - X), (X - X) X (X - X) - 8(X - X))
defined by <1>* (v) = (v X 1) v U', where
U' = UI ((X - X) X (X - X), (X - X) X (X - X) -8(X - X))
<1>* can be extended to
<1>*: Hq(X)
~
Hq+n(X X X, X X X -8(X))
by <1>* (v) = (v X 1) v U. There is a commutative diagram whose vertical
maps are isomorphisms
Hq+n(X X X, X X X - 8(X))
:::::1
Hq(X - X) ~ Hq+n((X - X) X (X - X), (X - X) X (X - X) -8(X - X))
from which it follows that <1>* is also an isomorphism on Hq(X). For i ;:::: 0 the
ith Stiefel- Whitney class of X, Wi E Hi(X;Z2)' is defined by the formula
<1>* (Wi) = Sqi U
[that is, SqiU = (Wi X 1) v U]. Following are some examples.
3
By condition (a) on page 271, Wo = 1.
4
By condition (b) on page 271, if X is a compact n-manifold without
boundary, Wn is the Euler class of X over Z2.
5
By condition (c) on page 271, Wi = 0 for i
> dim X.
6
A manifold X is orientable over Z if and only if
5.H.3d).
Wl
= 0 (see exercise
If X is compact and z E Hn(X,X) is the fundamental class of X over Z2,
then, by property 6.1.4,
Wi = [(Wi X 1)
v
U1l!z = SqiUdz
where Ul E Hn(X X (X,X)) corresponds to U. We use this to determine the
Stiefel-Whitney classes of a compact X in terms of cohomology operations
in X. For i ;:::: 0 the homomorphism Sqi: Hn-i(X,x) ~ Hn(x,x) has a transpose
homomorphism Sqi: Hn(X,X) ~ Hn_i(X,X) such that
<Sqiu,z) = <u,SqiZ)
where z is the fundamental class of X. By the isomorphism of theorem 6.3.12,
350
GENERAL COHOMOLOGY THEORY AND DUALITY
CHAP.
6
Kz: Hi(X) ::::::; Hn_i(X,X)
and there is a unique Vi E Hi(X) such that Kz(Vi ) = Sqi(Z). Then for
U E Hn-i(X,X; Z2)
= (U,SqiZ) = (U,Kz(Vi )
= <u, Vi z) = (u V Vi, z)
(Sqiu,z)
r'\
This equation holds trivially if deg U =1= n - i. The following Wu formula
shows that the classes Vi and the Stiefel-Whitney classes Wi determine each
other.
7
THEOREM
In a compact n-manifold, for q 2':
Wq =
~
O<;t<;q
°
Sqq-i Vi
We have U 1 = L aijUi X vj, where {u;} is a basis of H* (X,Z2) and
{Vj} is a basis of H* (X,X; Z2)' By the Cartan formula, condition (d) on page 271
PROOF
Sqqu1 =
avSq"u; X Sqlvj
~
k+l=q
Let VI
= ~ ClmU m. Then we have
Wq = (Sqqu1 )! z = ~ aij(Sqlvj,z)Sq"u;
k+l=q
= k+l=q
~ ai/ Vj v VI, z) SqkUi
= k+l=q
~ aijClm( Vj V Um, z) SqkUi
= k+l=q
~ aijbjmClmSqkui
Using lemma 1, we find that
Wq =
~
k+l=q
cliSqkUi =
~
k+l=q
Sqk VI
•
Let pn be the real projective n-space and let W be a generator of Hl(pn)
for any n 2': 1. We use lemma 5.9.4 to compute Sqi(wj) in the following
examples.
For the real projective plane p2, Sql(W) = w 2; therefore Vl(f2)
8
Wl(f2) = w, and W2(f2) = w 2.
9
For P3, Sq2(W)
Wi(P3) = for i > 0.
°
= ° and
Sql(WZ)
= 0,
so Vi(P3)
= w,
= ° for i > ° and
= w 4 and Sql(W 3) = w4, so V 1(P4) = W, V Z(P4) = w Z,
Wl(F4) = W, WZ(P4) = 0, W3(F4) = 0, and W4(P4) = w 4.
I I For P5, SqZ(w 3) = w 5 and Sql(W4) = 0, and V 2(P5) = WZ is the only nonzero V i(P5) , where i > 0. Hence Wl(P5) = 0, WZ(P5) = w Z, W3(P5) = 0,
W4(P5) = w 4, and W5(P5) = 0.
10 For P4, SqZ(WZ)
The Euler class and Stiefel-Whitney classes of a manifold X are topological invariants associated to X. We shall now define characteristic classes for a
SEC.
10
351
CHARACTERISTIC CLASSES
manifold X imbedded in a manifold Y. These will be topological invariants of
the imbedding. First, however, we need an algebraic digression.
In our consideration of the slant product we limited ourselves to one of
the two possible slant products. We now introduce the other one. Given chain
complexes C and C', a cochain c* E Hom ((C ® C')n, G), and chain
c E Cq ® G', there is a slant product c\c* E Hom (C~_q, G ® G') which is
the cochain such that if c = ~ Ci ® gi, with Ci E Cq and gi E G', then
(c\c* ,c')
Then
= ~ (c*, Ci ® c')
® gi
c' E C~_q
c5(c\c*) = (-I)q(c\c5c* - oc\c*)
from which it follows that there is an induced slant product of Hn( C ® C'; G)
and Hq(C;G') to Hn-q(C'; G ® G'). This gives rise to a topological slant
product of Hn((X,A) X (Y,B); G) and Hq(X,A; G') to Hn-q(Y,B; G ® G') having
properties analogous to 6.1.1 to 6.1.6. We list without proof two of these, to
which we shall have occasion to refer.
12 Given u E Hn((X,A) X (Y,B); G), z E Hq(X,A; G"), and v E HP(Y,B; G'),
let T: G ® Gil ® G' ~ G ® G' ® Gil interchange the last two factors.
In Hn-q+p(Y,B; G ® G' ® G") we have
T* ((z\u) v v)
= z\[u v
(1 X v)]
•
13 Given u E Hn((X,A) X (Y,B); G), v E Hp(X,A; G'), and z E Hq(X,A; Gil),
then, in Hn+p-q(Y,B; G ® G' ® Gil),
(v
I".
z)\u
= z\[u v
(v Xl)]
•
Let Y be an m-manifold without boundary and
U E Hm(y X Y, Y X Y - c5(Y); R)
an orientation of Y over R. Given a pair (A,B) in Y, we define
yo: Hq(A,B; G)
~
Hn-q(Y - B, Y - A; G)
by
yU(z) = z\[U I (A,B) X (Y - B, Y - A)]
z E Hq(A,B; G)
Then we have the following complement to the duality theorem.
14 LEMMA Let X be a compact homologically locally connected space in
an m-manifold Y with orientation class U. Then we have an isomorphism for
all q and all G
yo: Hq(X;G) :::::; Hm-q(Y, Y - X; G)
PROOF
Since X is compact and homologically locally connected, it follows
from theorem 6.9.11 that H(Ll(X)) is of finite type. By lemma 5.5.9, there is a
free chain complex C of finite type which is chain equivalent to Ll(X).
Let i\: C ~ Ll(X) be a chain equivalence. Let Ll' and C' be the chain complexes
obtained by reindexing the cochain complexes Hom (Ll(X),R) and Hom (C,R),
respectively, so that Ll~
Hom (Llm_q(X),R) and C~
Hom (Cm_q,R). The
=
=
352
GENERAL COHOMOLOGY THEORY AND DUALITY
CHAP.
6
chain equivalence ,\ defines a chain equivalence ,\': tl' ----') C. Because C is
free and of finite type, so is C [tl' will not be free, in general, because tl(X)
need not be of finite type].
Let c* E Hom ([tl(X) ® (tl(Y)/tl(Y - X))]m, R) be an m-cocycle corresponding to U I X X (Y, Y - X) under the Eilenberg-Zilber isomorphism and
define a map
T:
by r(c)
=
tl(Y)/tl(Y - X) ----') tl'
c* / cfor c E Ii( Y)/ Ii( Y- X). If deg c= q,
a(r(c))
=
o(c* / c) = (-1) m- qc* / ac = (-1) m- qr(ac)
so T either commutes or anticommutes with G, depending on degree. Hence T
induces homomorphisms T* on homology and T* on cohomology for any
coefficient module. Clearly,
T* = Yu: Hq(Y, Y - X; R) ----') Hm-q(X;R)
Because X is homologically locally connected, by corollary 6.9.8, X is taut in Y,
and by the duality theorem, Yu, and hence T*, is an isomorphism. Therefore
the composite ,\' T induces an isomorphism ,\'* T* of Hq(Y, Y - X; R)
with Hq(C) = Hm-q(C;R). Since tl(Y)/tl(Y - X) and C are both free, it
follows from the universal-coefficient formula for cohomology (theorem 5.5.3)
that for any G
0
(,\10 T)* = T*
0
0
,\'*: H*(Hom (C,G))::::: H*(Hom (tl(Y)/tl(Y - X), G))
There is also a commutative diagram
Hq(Ii(X) Q$l G) «("~l)' Hq(CQ$l G)
1
1=
,,'* Hm-q(Hom (C', G))
Hm-q(Hom (Ii', G)) <---;;;;where the vertical maps are induced by the canonical map
A ® G ----') Hom (Hom (A,R), G)
for any module A (the right-hand vertical map being an isomorphism because
C is of finite type). Hence there are isomorphisms
Hq(X;G) :? Hm-q(Hom (tl',G))
~
Hm-q(Y, Y - X; G)
It only remains to verify that this composite is
g E G, the composite
tlq(X) ® G ----') Hom (tl' m_q,G)
Hom (T,l)
yu.
If a E tlq(X) and
Hom (tlm_q(Y)/ tlm_q(Y - X),G)
maps a ® g to the homomorphism h such that if a' E tlm-q(Y),
h(a') = T(a')(a ® g) = (c* /a')(a ® g)
= (c*, a ® a')g = [(a ® g)\c*](a')
SEC.
10
353
CHARACTERISTIC CLASSES
Therefore h
= (a ® g) \c*, and this gives the result on passing to homology.
-
Let X be a closed subset of a space Y tautly imbedded with respect to
singular cohomology and let A C X. Assuming X - A taut in Y - A, we define
Hp(y, Y - X; G) v Hq(X, X - A; G')
~
Hp+q(Y, Y - A; Gil)
where G and G' are paired to Gil. If V is any neighborhood of X in Yand V'
is a neighborhood of X - A in V - A, there is a cup product
Hp(V, V - X; G)
v
Hq(V,V'; G')
~
Hp+q(V, (V - X) U V'; Gil)
There are excision isomorphisms (for all coefficients)
Hp(y, Y - X) ;:::::: Hp(V, V - X)
Hp+q(Y, Y - A) ;:::::: Hp+q(V, V - A)
Since V - A = (V - X) U V' we have a cup product
Hp(y, Y - X; G) v Hq(V,V'; G')
~
Hp+q(Y, Y - A; Gil)
As V varies over neighborhoods of X in Y and V' varies over neighborhoods
of X - A in V - A, it follows from the tautness assumptions and the five
lemma that lim~ {H*(V,V'; G')} ;:::::: H*(X, X - A; G'). The desired cup
product is thus obtained by passing to the direct limit with the above cup
product.
Let X be a compact n-manifold without boundary imbedded in an
m-manifold Y without boundary. Assume that U and U' are orientations of
X and Y, respectively, over R. There is then an isomorphism (for any R
module G)
(): Hq(X;G) ;:::::: Hm-n+q(Y, Y - X; G)
characterized by commutativity in the triangle of isomorphisms (note that X
is homologically locally connected, and so lemma 14 applies to X C Y)
Hn_q(X;G)
Yu.(
'1u'
Hq(X;G) J4 Hm-n+q(y, Y - X; G)
This map () is similar to a Thorn isomorphism and has the following
multiplicative property.
15 LEMMA The isomorphism (): Hq(X,G) ;:::::: Hm-n+q(Y, Y - X; G) has the
property that for v E Hq(X;G)
()(v) = +()(1) v v
where ()(1) E Hm-n(y, Y - X; R)
PROOF
Let z E Hn(X;R) be the fundamental class of X corresponding to U
and suppose v
i * v' for v' E Hq(V;G) and i: X C V, where Vis a neighborhood of X in Y. By theorem 6.3.12, Yu- 1 (v)
+v" z
+i*v' "z. Then,
using properties 12 and 13 (with all equations holding up to sign),
=
=
=
354
GENERAL COHOMOLOGY THEORY AND DUALITY
O(v) I (V, V - X)
= -+-(i*v'
CHAP.
6
z)\[U'1 X X (V, V - X)]
(V, V - X)]
= -+-(v' f'I i*z)\[U' I V X (V, V - X)]
= -+-i*z\{[U'1 V X (V, V - X)] v (v' X Iv)}
= -+-i*z\{[U'1 V X (V, V - X)] v (Iv Xv')}
= -+-z\{[U'1 X X (V, V - X)] v (Ix X v')}
= -+-[0(1) I (V, V - X)] v v'
= -+-[0(1) v v] I (V, V - X)
f'I
= -+-i*(i*v' f'lz)\[U'1 Vx
Since H* (Y, Y - X) ;:::; H* (V, V-X), this gives the result.
-
Our next result, a consequence of lemma 15, follows immediately from
the definition of the cup product, Hp(y, Y - X) v Hq(X) ~ Hp+q(Y, Y - X).
16 COROLLARY Let X be a compact oriented n-manifold imbedded in an
oriented m-manifold Y, both without boundary. For any element v E Hq(Y;G)
we have
O(v I X)
= -+-0(1) v
v
-
The normal Euler class of X in Y, denoted by XX,Y E Hm-n(X;R), is
defined by the equation
O(XX,Y)
= 0(1) v
0(1) E }{2(m-n)(Y, Y - X; R)
=
Since 0(1) v 0(1)
0(1) v [0(1) I Y], we obtain from corollary 16 the following characterization of the normal Euler class.
17 THEOREM If a compact n-manifold X is imbedded in an m-manifold Y,
both without boundary and oriented over R, the normal Euler class
XX,Y = 0(1) I X. -
In particular, if Hm-n(Y;R) ~ Hm-n(X;R) is trivial, it follows that the
normal Euler class is zero. Thus, if Y is Euclidean space, the normal Euler
class of any compact X imbedded in Y is zero.
For i ~ 0 the ith normal Stiefel- Whitney class of X in Y, wiE Hi(X;ZZ),
is defined by
Here are some examples.
18 By condition (a) on page 271, Wo
= l.
19 By condition (b) on page 271, if k
normal Euler class of X in Y over Zz.
20 By condition (c) on page 271, Wi
= dim Y -
dim X then Wk is the
= 0 for i > dim Y -
dim X.
There is an important relation between the Stiefel-Whitney classes of X
and Y and the normal Stiefel-Whitney classes of X in Y toward which we are
heading.
SEC.
10
355
CHARACTERISTIC CLASSES
21 LEMMA Let X be a compact n-manifold imbedded in an m-manifold Y,
both without boundary. Let U and U' be the orientation classes of X and Y,
respectively, over Z2 and let 0(1) E Hm-n(Y, Y - X; Z2). Then
U' I (X X Y, X X Y - 8(X))
= [1
X 0(1)] v U
PROOF
If X' is a component of X, it suffices to prove that
u' I (X'
X Y, X' X Y - 8(X/))
= ([1
X 0(1)] v U) I (X' X Y, X' X Y - 8(X/))
Hence we may assume X connected, in which case (X X Y, X X Y - 8(X)) is
a fiber-bundle pair over X with fiber pair (Y, Y - xo), where Xo E X. Since
U' I (X X Y, X X Y - 8(X)) is an orientation over Z2 of this bundle pair, and
there is a unique orientation over Z2, it suffices to prove that [1 X 0(1)] v U
is also an orientation over Z2 of this bundle pair. That is, we need only show
that for x E X, ([1 X 0(1)] v U) I x X (Y, Y - x) is nonzero. This will be so
if its image in x X (Y, Y - X), which equals ([1 X 0(1)] v U) I x X (Y, Y - X),
is nonzero. Because U E Hn(x X X, X X X - 8(X)) is an orientation,
U I x X (X, X - x) = Ix X u, where u E Hn(X, X - x) is nonzero. Because
Hn(x, X - x) ~ Hn(X) is a monomorphism [dual to the monomorphism
Ho(x) ~ Ho(X)], u I X is nonzero. We have
([1 X 0(1)] v U) I x X (Y, Y - X) = [Ix X 0(1)] v (Ix X u I X)
Ix X [0(1) v u I Xl
Ix X O(u I X)
=
=
Since 0 is an isomorphism, this implies that ([1 X 0(1)] v U) I x X (Y, Y - X)
is nonzero. •
From this result we have the following Whitney duality theorem.
22 THEOREM Let X be a compact n-manifold imbedded in an m-manifold
Y, both without boundary. For k :?: 0
Wk(Y) I X =, ~ Wi V Wj(X)
'+J=k
where Wk(Y), Wj(X), and Wi denote the Stiefel- Whitney classes of Y,X, and
X in Y, respectively.
The result follows easily from lemma 21 and the Cartan formula
(rather, the equivalent form of lemma 5.9.4):
PROOF
([Wk(Y)
I Xl
X Iv) v U' I (X X Y, X X Y - 8(X))
X Iv] v U' ) I (X X Y, X X Y - 8(X))
= ([Wk(Y)
= SqkU' I (X X
Y, X X Y - 8(X))
= Sqk(U' I (X X
= Sqk([Ix X 0(1)] v U) =, ~ [Ix X SqiO(I)] v
'+J=k
= ,~ (Ix X [0(1) v Wi]) v [Wj(X) X Ix] v U
Y, X X Y - 8(X)))
SqjU
'+J=k
= i+j=k
~ (Wi X Ix) v [Wj(X) X Ix]"-' [Ix X 0(1)] v U
= (([,'+J=k
~ Wi v Wj(X)] X Iy) v U') I (X X Y, X X Y -
8(X))
356
GENERAL COHOMOLOGY THEORY AND DUALITY
By the Thorn isomorphism theorem, this implies the result.
CHAP.
6
-
>
=
In case Y is Euclidean space, Wk( Y)
0 for k
0, and theorem 22 shows
that Wi and Wj(X) determine each other recursively. In particular, the classes
Wi are independent of the imbedding of X in the Euclidean space. If X is a
compact n-manifold imbedded in Rn+d, it follows from example 19 and 20
and from the fact that the Euler class of X in Rn+d is zero that Wi
0 for
i ;;::: d. This gives the following necessary condition for imbeddability of X in
Rn+d.
=
23 COROLLARY Let X be a compact n-manifold imbedded in Rn+d and let
Wi E Hi(X;Z2) be defined by
. ~ Wi
'+J=k
Then Wi
= 0 for i
;;::: d.
V
w;(X) =
k=O
k>O
{01
-
We present some examples.
= wand W2(P2) = 0, so p2 cannot be imbedded in R3.
For P3, Wi(P3) = 0 for i > O.
For P4, WI(J'4) = w, W2(J'4) = w 2, W3(J'4) = w 3, and W4(J'4) = O. There-
24 For PZ, WI(PZ)
25
26
fore J'4 cannot be imbedded in R7.
27 For P5, WI(P5) = 0, W2(F5) = w 2, W3(P5) = 0, W4(P5) = 0, and W5(P5) = O.
Hence p5 cannot be imbedded in R 7 (which is also a consequence of example 26).
The last examples show the importance of calculating Wi(pn), which we
now do.
28
THEOREM
Let (?)z be the binomial coefficient (?) = n!/i!(n - i)! reduced
modulo 2. Then
Wi(pn)
= (ntl)zWi
Since (ntl)2 == n + 1 = X(pn), the result is true for i = n. For
< n, where n > 1, we suppose pn-Ilinearly imbedded in pn. Then pn _ pn-I
PROOF
i
is an affine space, hence fJ*(pn - pn-l) = 0 and Hq(pn, pn - pn-l) = fJq(pn).
Then the normal Thorn class 8(1) E HI(pn, pn - pn-I) maps to W in Hq(pn) ,
so WI
W. By theorem 22, Wi(pn) I pn-I
Wi(pn-I) + W v Wi_l(pn-I).
Since Hq(pn) :::::: Hq(pn-l) for q
n, it follows by induction on n that
=
<
Wi(pn)
=
= [C~l)z + (?)z]wi = (ntl)zwi
-
EXER(;ISES
A MANIFOLDS
I If X is an n-manifold with boundary X, prove that X is a homology n-manifold whose
boundary, as a homology manifold, equals X.
357
EXERCISES
In the rest of the exercises of this group, X will be an n-manifold without boundary and
R will be a fixed principal ideal domain.
If f is a local system of R modules on X, prove that for any A
2
Hq(A X X, A X X - 8(A); R X f)
C
X
=0
(Hint: Prove this first for A contained in a coordinate neighborhood of X. Prove it next
for compact A by using the Mayer-Vietoris technique. Then prove it for arbitrary A by
taking direct limits over the family of compact subsets of A.)
Prove that there is a local system f x of R modules on X such that
= Hn(x, X - x; R) for x EX.
3
fx(x)
For x E X let
isomorphism
Zx
E Hn(X, X - x; f x) be the generator corresponding under the
Hn(X, X - x; f x) :::::: Hom (Hn(x, X - x; R), Hn(x, X - x; R))
to the identity homomorphism of Hn(x, X - x; R). A Thorn class of X is an element
V E Hn(x X X, X X X - 8(X); R X Hom (fx,R))
such that (V I [x X (X, X - x)])/zx
4
VI
= 1 E HO(x;R) for all x E X.
If V is an open subset of X and V is a Thorn class of X, prove that
(V X V, V X V - 8(V)) is a Thorn class of V.
:;
Prove that Rn has a unique Thorn class.
6
Prove that X has a unique Thorn class. [Hint: Use exercise 2 to show that
Hn(x X X, X X X - 8(X); R X Hom (fx,R)) ::::::
lim~ tHn(V X X, V X X - 8(V); R X Hom (fx,R))}
where V varies over finite unions of coordinate neighborhoods. Then the result follows
from exercises 4 and 5 by Mayer-Vietoris techniques.]
If (A,B) is a pair in X and
e is an R module, define
y: Hq(X - B, X - A; f x ® e)
--'>
Hn-q(A,B; e)
=
by y(z)
[V I (A,B) X (X - B, X - A)l/z, where V is the Thorn class of X. As (V, W)
varies over neighborhoods of a closed pair (A,B) in X, there are isomorphisms
lim_ {Hq(X - W, X - V; fx ® e)} ::::::Hq(X - B, X - A; fx ® e)
lim_ {Hn-q(V, W;
and
en :::::: Hn-q(A,B; e)
and a homomorphism
y: Hq(X - B, X - A; r x ® e) --'> Hn-q(A,B; e)
is defined by passing to the limit with y.
7
Duality theorem. Prove that for a compact pair (A,B) in X, y is an isomorphism.
B THE INDEX OF A MANIFOLD
I Let X be a compact n-manifold, with boundary X oriented over a field R, and let
[X] E Hn(X,X; R) be the corresponding fundamental class. For u E Hq(X,X; R) and
v E Hn-q(X;R) prove that qJx(u,v) = <u v v, [X]) E R is a nonsinguiar bilinear form from
Hq(X,X) X Hn-q(X) to R [that is, u = 0 if and only if qJx(u,v) = 0 for all v].
=
2 With the same hypotheses as above, let [X]
a[X] E Hn_1 (X;R) and let qJx be the
corresponding bilinear form from Hq-l(X;R) X Hn-q(X;R) to R. Let i: X C X, and
if u E Hq-l(X;R) and v E Hn-q(X;R), prove that
358
GENERAL COHOMOLOGY THEORY AND DUALITY
cpx(u,i*(v))
CHAP.
6
= cpx(8(u),v)
3 Prove that the Euler characteristic of any odd-dimensional compact manifold is 0
and the Euler characteristic of an even-dimensional compact manifold which is a boundary
is even. (Hint: If X is the boundary of a (2n + I)-manifold X, then, with Z2 coefficients,
dim im
U*:
Hn(x) ~ Hn(x)])
= dim im [8:
Hn(x) ~ Hn+1(X,x)]
and their sum equals dim Hn(X).)
Let Y be a compact 4m-manifold, without boundary oriented over R, and define the
index of Y to be the index of the non singular bilinear form cPy from H2m(Y;R) X H2m(Y;R)
to R (when cPy is represented as a sum of k squares minus a sum of i squares, the index
of cpy is k - i).
4 If Y is oppositely oriented, prove that its index changes sign. Show that the index of
the product of oriented manifolds is the product of their indices.
5 If X is a compact (4m + I)-manifold, with boundary X oriented over R, prove that
The index of Xis O. [Hint: Prove that j*(H2 m(X;R)) is a subspace of H2m(X;R) whose dimension equals one-half the dimension of H2m(X;R) and on which 'Pi is identically zero. This
implies the result.]
(;
CONTINUITY
I Let {(Xj,Aj), 'I1'l}jEJ be an inverse system of compact Hausdorff pairs and let
(X,A) = lim_ {(Xj,Aj)}. Prove that (X,A) can be imbedded in a space in which it is
a directed intersection of compact Hausdorff pairs {(Xj,Aj) }jEJ, where (X;,Aj) has the same
homotopy type as (Xj,A j). [Hint: For each i E J imbed Xj in a contractible compact
Hausdorff space Yj, ~r example, a cube, and let (X",A,,) C XjEJ Yj be defined as the pair
of all points (Yj) with Yk in Xk or in A k, respectively, such that if i ~ k, then Yj = 'I1'l(Yk),
and if i $ k, then Yj is arbitrary.]
2 Prove that a cohomology theory has the continuity property if and only if it has the
weak continuity property.
3
The p-adie solenoid is defined to be the inverse limit of the sequence
51
J- 51
~ . .. ~ 51
J- 51
~ ...
where J(z) = zP. Compute the Alexander cohomology groups of the p-adic solenoid for
coefficients Z, Zp, and R.
4 Generalize the solenoid of the preceding example to the case where there is a
sequence of integers n1, n2, ... such that the mth map of 51 to 51 sends z to znm. Compute the integral Alexander cohomology groups of the resulting space.
5
Find a compact Hausdorff space X such that jjq(X;Z)
= 0 if q =1= 1 and W(X;Z) ;::: R.
D CECH COHOMOLOGY THEORY
I Let (GiJ.,G(j;') be an open covering of (X,A) ("11 is an open covering of X and "ll' C "ll is
a covering of A) and let K(02l) be the nerve of ql and K'("ll') the subcomplex of
K(Gll) which is the nerve of GIl' n A = {U' n A I u' E qr}. Prove that the chain complexes (C(K(02l)),C(K'(GIl'))) and (C(X(ql)),C(A(GIl'))) are canonically chain equivalent. (Hint:
If s
{Uo, . . . ,Uq } is a simplex of K(GIl) [or of K'(q1')], let .\(s) be the subcomplex of
X(GIl) [or of A("ll')] generated by all simplexes of X("l1) [or of A (GlL')] in n Ui. If
S' = {Xo, ... ,Xq} is a simplex of X('OIl) [or of A'(ql')], let !lis') be the subcomplex
=
359
EXERCISES
of K(01) [or of K'(01')] generated by all simplexes {Uo, . . . , Ur } of K(01) [or of
K'(''21')] such that Ui contains s' for 0 ::;; i ::;; r. Then C(.\(s)) and C(}.t(s')) are acyclic, and
the method of acyclic models can be applied to prove the existence of chain maps
T: (C(K(ql)),C(K'(ql'))) __ (C(X(cYl)),C(A(ql')))
T': (C(X(ql)),C(A(qJ))) __ (C(K(ql)),C(K'(cYl')))
such that T(C(S)) C C(.\(s)) and T'(C(S')) C C(p.(s')). Similarly, the method of acyclic
models shows that T and T' are chain homotopy inverses of each other.!)
2 Let ('Y,"V') be a refinement of ("11,"11'), let 7T: (K("V),K'("V')) __ (K(0J),K'(GiL')) be a projection map, and let i: (X('Y),A('Y')) C (X(cYl),A(01')). For any abelian group G prove that
there is a commutative diagram
H* (K(Gil),K'(01'); G)
H* (X(Gil),A(0J'); G)
1j·
"·1
H*(K("V),K'(T); G) :::::: H*(X('J),A('Y'); G)
where the horizontal maps are induced by the canonical chain equivalences of exercise 1
above.
3
lim~
The Cech cohomology group of (X,A) with coefficients G is defined by fl* (X,A; G) =
{H*(K(01),K'(01'); G)}. Prove that there is a natural isomorphism
fl* (X,A; G) :::::: H* (X,Ai G).
= 0 for all q > n and all G.
4
If dim (X - A) ::;; n, prove that HQ(X,A; G)
E
THE KUNNETH FORMULA FOR CECH COHOMOLOGY
If Kl and Kz are simplicial complexes, their simplicial product Kl ~ K z is the simplicial
complex whose vertex set is the cartesian product of the vertex sets of Kl and of Kz and
whose simplexes are sets {(vo,wo), . . . ,(vq,Wq)}, where Vo, . . . , Vq are vertices of
some simplex of Kl and Wo, . . . , Wq are vertices of some simplex of K z.
I Prove that Kl D. Kz is a simplicial complex, and if Ll C Kl and Lz C K z, then
Ll D. L z C Kl D. K z.
2
For simplicial pairs (K1,L1) and Kz,L z) define
(K1,L1) ~ (Kz,Lz)
= (Kl ~
Kz, Kl D. Lz U Ll
~
K z)
Prove that C((Kl,L 1) D. (Kz,L z)) is canonically chain equivalent to C(K1,L 1) ® C(K 2 ,L2 ).
(Hint: Use the method of acyclic models.)
3 Call an open covering (uk, JtI') of (X,A) special if '2;' = (U E '2; I unA op 4>1. If
(~ '2;') is a special open covering of (X,A) and (~Y,Y') is a special open covering of (Y, B),
let (U~ uti') X (~Y') = ('if", ~') be the special open covering of (X,A) X (Y,B) where
''If = ( U X V I u E ~ V E ~r 1and YI"' = ( U x V I U E '2;' or V E Y'I. Prove that
(K(Yf/), KTYf/')) = (K( '2;), K'( uti')) tl(K(Y), K' (Y')).
4 If A is closed in X, prove that the family of special open coverings of (X,A) is colinal
in the family of all open coverings of (X, A). If (X, A) and (Y, B) are compact Hausdorff
pairs, prove that the family of coverings of (X,A) X (Y,B) of the form (~'2;') X Cf(Y')
where (UZ; uti') is a special open covering of (X,A) and (~Y') is a special open covering
of (Y,B) is colinal in the family of all open coverings of (X,A) X (Y,B).
1 For details see C. H. Dowker, Homology groups of relations, Annals of Mathematics, vol. 56,
pp. 84-95, 1952.
360
5
G
GENERAL COHOMOLOGY THEORY AND DUALITY
CHAP.
6
If (X,A) and (Y,B) are compact Hausdorff pairs and G and G' are modules such that
* G' = 0, prove that there is a short exact sequence
0-> (H! ® iI~)q -> flq((X,A) X (Y,B); G ® G') -> (fI!
* iI~)q+1 -> 0
where H! = H*(X,A; G) and H~ = H*(Y,B; G').
6 Let (X,A) and (Y,B) be locally compact Hausdorff pairs with A and B closed in X
and Y, respectively. If G and G' are modules such that G * G'
0, prove that there is a
short exact sequence
=
0-> (fH,1
® H~ ,2)q -> Hcq((X,A) X (Y,B); G
* G') -> (FI~,I * H~ ,2)q+1 -> 0
where H~,I = H~ (X,A; G) and H~,2 = H~ (Y,B; G').
F
LOCAL SYSTEMS AND SHEAVES
Throughout this group of exercises we assume X to be a paracompact Hausdorff space.
I If f is a local system on X, let f be the presheaf on X such that for an open
set V C X, f(V) is the set of all functions f assigning to each x E X an element
f(x) E f(x) with the property that for any path w in V, f(w(I)) = f(w)(f(w(O))). Prove that
f is a sheaf on X and the association of f to f is a natural transformation from local systems to sheaves.
2 A presheaf f on X is said to be locally constant if there is an open covering 0t = {U}
of X such that if U E '71 and x E U, then f( U) ;::: lim_ {f( V)}, where V varies
over open neighborhoods of x. If U E 0t and U' IS a connected open subset of U, prove
that the composite
f(U) -> f(U') -> f(U')
is an isomorphism. Deduce that if f is a locally constant sheaf and U' is a connected
open subset of U E 0t, then f( U) ;::: f( U').
3 If X is locally path connected and f' is a locally constant sheaf on X, prove that
there is a local system f on X such that f ;::: f'.
4 If X is locally path connected and semilocally I-connected, prove that there is a oneto-one correspondence between equivalence classes of local systems on X and equivalence classes of locally constant sheaves on X.
5 If f is a local system of R modules on X, let tl q ( • ;f) be the presheaf on X such that
tl q(· ;f)(V) = tlq(V;f I V) for V open in X. Prove that tl q ( • ;f) is fine.
6 If f is a local system of R modules on X, let tI* ( . ;f) be the cochain complex of presheaves tl q ( • ;f) on X and let 6.* ( . ;f) be the cochain complex of completions 6. q ( • ;f).
Prove that there is an isomorphism
H* (tI* ( . ;f)(X)) ;::: H* (3.* ( . ;f)(X))
7 Let f be a local system of R modules on X and assume that Hq(tI* ( . ;f)) is locally
zero on X for all q
O. Prove that there is an isomorphism
>
fl*(X;f);::: H*(X;f)
(Hint: Note that f
G
I
= HO(tI* (. ;f)) and apply theorem 6.8.7.)
SOME PROPERTIES OF EUCLIDEAN SPACE
Find a compact subset X of H2 that is n-connected for all n and such that HI(X;Z) ;::: Z.
361
EXERCISES
If X is a compact subset of Rn and dim X
2
<n -
1, prove that Rn - X is connected.
Let Al and A2 be disjoint closed subsets of Rn and let Zl E Hp(Al;R) and
Z2 E Hq(A 2;R), with P + q = n - 1. If Zl E Hp(Al;R), let zl E Hp+l(Rn,Rn - A 2;R) be
the image of Zl under the composite
Hp(AI)
-->
Hp(Rn - A 2) ~ Hp+l(Rn, Rn - A 2)
The linking number Lk (Zl,zz) E R is defined by
Lk (Zl,Z2)
= (Yu(Zl),Z2)
where U is an orientation class of R n over R fixed once and for all.
3
Prove that Lk (ZI,Z2)
= (U, i*(z2
X zl), where
i: A2 X (Rn, Rn - A 2) C (Rn X Rn, Rn X Rn - 8(Rn))
4 Assume that Lk (Z2,Zl) is also defined [that is, Z2 E Hq(A2)]. Prove that Lk (Zl,Z2) =
(_l)pq+1 Lk (Z2,ZI).
5
p
+
Let Al be a p-sphere and A2 a q-sphere imbedded as disjoint subsets of Rn, where
q = n - l. Prove that Hp (AI) -> Hp (R n - A 2) is trivial if and only if
Hq(A2)
->
Hq(Rn - AI) is trivial.
H IMBEDDINGS OF MANIFOLDS IN EUCLIDEAN SPACE
I Prove that a compact n-manifold which is nonorientable over Z cannot be imbedded
in Rn+l.
2 Let X be a compact connected n-manifold imbedded in Rn+l and let U and V be
the components of Rn+l - X. Let i: X C Rn+l - U and i: X C Rn+1 - V and prove
that over any R, i*(H*(Rn+1 - U)) and j*(H*(Rn+l H*(X)
3
V)) are sub algebras of
Prove that for n 2': 2 the real projective n-space pn cannot be imbedded in Rn+1.
~HAPTER
SEVEN
HOMOTOPY THEORY
WITH THIS CHAPTER WE RETURN TO THE CONSIDERATION OF GENERAL HOMOTOPY
theory. Now that we have homology theory available as a tool, we are able to
obtain deeper results about homotopy than we could without it. We shall consider the higher homotopy groups in some detail and prove they satisfy
analogues of all the axioms of homology theory except the excision axiom. We
introduce the Hurewicz homomorphism as a natural transformation from the
homotopy g£Oups to the integral singular homology groups. It leads us to the
Hurewicz isomorphism theorem, which states roughly that the lowest-dimensional nontrivial homotopy group is isomorphic to the corresponding integral
homology group.
We discuss next the concept of CW complex. The class of CW complexes
is particularly suited for homotopy theory because it is the smallest class of
spaces containing the empty space and, up to homotopy type, is closed with
respect to the operation of attaching cells (even an infinite number).
The last main result is the Brown representability theorem. It characterizes by means of simple properties those contravariant functors from the
homotopy category of path-connected pOinted CW complexes to the category
363
364
HOMOTOPY THEORY
CHAP.
7
of pointed sets that are naturally equivalent to the functor assigning to a
CW complex the set of homotopy classes of maps from it to some fixed
pointed space.
Section 7.1 contains a general exactness property for sets of homotopy
classes. Section 7.2 contains definitions of the absolute and relative homotopy
groups and proofs of the exactness of the homotopy sequences of a pair, a
triple, and a fibration. In Sec. 7.3 we consider the extent to which the homotopy groups depend on the choice of the base point used in their definition and
prove analogues for the higher homotopy groups of properties established in
Chapter One for the fundamental group.
The Hurewicz homomorphism is defined in Sec. 7.4 and the Hurewicz
isomorphism theorem is proved in Sec. 7.5. The proof establishes the absolute
and relative Hurewicz theorems, as well as a homotopy addition theorem, by
simultaneous induction. The Hurewicz theorem implies the Whitehead
theorem, which asserts that a continuous map between simply connected
spaces induces isomorphisms of all homotopy groups if and only if it induces
isomorphisms of all integral singular homology groups.
Section 7.6 introduces the concept of CW complex. Among the elementary
properties established is the cellular-approximation theorem, which is an
analogue for CW complexes of the simplicial-approximation theorem. Section
7.7 deals with contravariant functors on the homotopy category of pathconnected pOinted spaces. We prove the representability theorem cited above,
and apply it in Sec. 7.8 to obtain CWapproximations to a space or a pair and
to discuss the related concept of weak homotopy type. The representability
theorem will be used again in Chapter Eight.
I
EXACT SEqUENCES OF SETS OF HOMOTOPY CLASSES
One of the most important properties of the homology functor is the exactness
property relating the homology of the pair and the homology of each of the
spaces in the pair. A similar exactness property is valid for functors defined
by homotopy classes. This section is devoted to preliminaries about homotopy
classes and a proof of this exactness property. Throughout the section we
shall work in the category of pointed spaces, and unless stated to the contrary,
(X,A) will be understood as a pair of pointed spaces (that is, A has the same
base point as X) in which the subspace A and the base point are closed in X.
Homotopies in this category are understood to preserve base points. If A C X,
we use X/A to denote the space obtained from X by collapsing A to a single
point (this point serving as the base point of X/A). If X' and A are closed
subsets of X, then A/(A n X') is a closed subset of X/X'. Hence, if (X,A) is a
pair and X' is closed in X, there is a pair (X/X', A/(A n X')), which will also
be denoted by (X,A)/X'.
SEC.
I
365
EXACT SEQUENCES OF SETS OF HOMOTOPY CLASSES
The unit interval 1 will be a pointed space with 0 as base point. The
reduced cone CX over X is defined to be the space obtained from X X 1 by
collapsing X X 0 U Xo X 1 to a point (so CX = X X l/(X X 0 U Xo X 1)).
We shall use [x,t) to denote the point of CX corresponding to the point
(x,t) E X X I under the collapsing map X X 1 -7 CX. X is imbedded as a
closed subset of CX by the map x -7 [x,l). If (X,A) is a pair, then CA is a
subspace of CX and C(X,A) is defined to be the pair (CX,CA).
I
LEMMA
A map f: (X,A) -7 (Y,B) is null homotopic if and only if there
-7 (Y,B) such that F[x,l) = f(x) for all x E X.
is a map F: C(X,A)
There is a one-to-one correspondence between null homotopies
H: (X,A) X 1-7 (Y,B) of f and maps F:C(X,A) -7 (Y,B) such that F[x,l) = f(x),
given by the formula
PROOF
F[x,t) = H(x, 1 - t)
•
The following relative homotopy extension property can also be deduced
from the relative form of theorem 1.4.12.
2
LEMMA
Given f: C(X,A) -7 (Y,B) and a homotopy G: (X,A) X 1 -7 (Y,B)
of f I (X,A), there is a homotopy F: C(X,A) X 1-7 (Y,B) of f such that
F I (X,A) X 1 = G.
PROOF
An explicit formula for F is
f[x, t(l + t'))
F([x,t), t') = { G(x, t(l + t') - 1)
+ t') ::::: 1
1 ::::: t(l + t')
t(l
•
The homotopy class of the unique constant map (X,A) -7 (Y,B) is denoted
by 0 E [X,A; Y,B) [it consists of the null-homotopic maps (X,A) -7 (Y,B)).
Because the composite, on either side, of a null-homotopic map and an arbitrary map is null homotopic, the element 0 is a distinguished element of
[X,A; Y,B), and we regard [X,A; Y,B) as a pointed set with this distinguished
element. Given a map f: (X',A') -7 (X,A), the kernel of the induced map
f#: [X,A; Y,B)
-7
[X',A'; Y,B)
is defined to be the pointed set f#-l(O) and is denoted by ker f#.
We now show how to map another set of homotopy classes into
[X,A; Y,B) so that its image equals ker f#. This will be the basis for the exactness property we seek. The mapping cone Cr of a map f: X' -7 X is defined
to be the quotient space of CX' v X by the identifications [x',l) = f(x') for all
x' EX'. Given a map f: (X',A') -7 (X,A), let f': X' -7 X and f": A' -7 A be
maps defined by f. Then C{" is a closed subspace of Cr and there is a pair
(Cr,Cd. There is a functorial imbedding i of (X,A) as a closed subpair of
(Cr,Cd·
A three-term sequence of pairs and maps
(X',A') ~ (X,A) --f4 (X",A")
366
HOMOTOPY THEORY
CHAP.
7
is said to be exact if for any pair (Y,B) (where B is not necessarily closed in Y)
the associated sequence of pOinted sets
[Y,B; X',A'] ~ [Y,B; X,A]
!!!4 [Y,B; X",A"]
is exact (that is, ker g# = im f#). Similarly, it is sajrl to be coexact if the
sequence of pointed sets
[X",A"; Y,B] ~ [X,A; Y,B]
4
[X',A'; Y,B]
is exact (that is, ker f# = im g#). A sequence of pairs and maps (which may
terminate at either or both ends)
... ~ (Xn+I,A n+1 ) ~ (Xn,An)
fn-l)
(Xn-I,A n- 1 ) ~
..•
is said to be an exact sequence (or a coexact sequence) if every three-term
sequence of consecutive pairs is exact (or coexact).
3
THEOREM
For any map f: (X',A')
(X',A')
~
(X,A) the sequence
-4 (X,A) ~
(C("C",)
is coexact.
Let (Y,B) be arbitrary (with B not necessarily closed in Y) and consider the sequence
PROOF
[C("C(,,; Y,B] ~ [X,A; Y,B]
L
[X',A'; Y,B]
We now show that im ,i# C ker f#. The composite i
equals the composite
(X',A') C C(X',A') C C(X',A') v (X,A)
0
f:
(X',A')
~
(C("Cd
.!4 (C("C",)
where k is the canonical map to the quotient. However, the inclusion map
(X',A') C C(X',A') is null homotopic [by lemma 1, because this inclusion map
can be extended to the identity map of C(X',A')]. Therefore i f is null
homotopic, and so im (f# i#) = 0, proving that im i# C ker f#.
Assume that g: (X,A) ~ (Y,B) is such that f#[g] = (that is, g f is null
homotopic). By lemma 1, there is a map G: C(X',A') ~ (Y,B) which extends
go f. Then G and g define a map G': C(X',A') v (X,A) ~ (Y,B) such that
G' I C(X',A') = G and G' I (X,A) = g. Since
0
°
0
G'[x',l]
= G[x',l] = g(f(x')) = G'(f(x'))
0
x' EX'
there is a map h: (C("C{,,) ~ (Y,B) such that G' = h k. Then hi (X,A) = g,
showing that hoi = g or [g] = i#[h]. Therefore ker f# C im i#. •
0
For a map f: (X',A')
4
~
(X,A) we have a sequence
(X',A').4 (X,A) ~ (C("Cd
-4 (Ci"Cd ~
and by theorem 3, it follows that this sequence is coexact.
Thus we have succeeded in imbedding the map
f#: [X,A; Y,B] ~ [X',A'; Y,B]
(Cl,Cr)
SEC.
1
367
EXACT SEQUENCES OF SETS OF HOMOTOPY CLASSES
in an exact sequence. We shall show that the pairs (Ci"Cd and (Cj',C;u) in
sequence 4 can be replaced by other pairs more explicitly expressed in terms
of (X',A'), (X,A), and f.
'"
LEMMA
Let (Y,B) be a pair and let Y' be a closed subset of Y. Assume
that there is a homotopy H: (Y,B) X I ~ (Y,B) such that
(a) H(y,O) = y, for y E Y.
(b) H(Y' X 1) C Y'.
(c) H(Y' X 1) = yo.
Then the collapsing map k: (Y,B)
~
(Y,B)/Y' is a homotopy equivalence.
Define a map f: (Y,B)/Y'
~
(Y,B) by the equation
PROOF
= H(y,I) Y E Y
[this is well-defined, because H(Y' X 1) = yo]. We show that f is a homotopy
f(k(y))
inverse of k. By definition of f, we see that H is a homotopy from I(y,B) to f
On the other hand, because H(Y' X 1) C Y', there is a homotopy
H': ((Y,B)/Y') X I
~
0
k.
(Y,B)/Y'
such that H'(k(y),t) = k(H(y,t)) for y E Yand t E 1. Then
k(f(k(y)))
= k(H(y,I)) = H'(k(y),I)
y EY
Therefore H' is a homotopy from the identity map of (Y,B)/Y' to k
is a homotopy inverse of k. •
0
f, and f
6
COROLLARY
Let f: (X',A') ~ (X,A) be a map and let i: (X,A) C (C{"C",).
Then CX C Ci', (Ci"Cd/CX = (C{"C",)/X, and the collapsing map
k: (Ci"Cd ~ (Ci"Cd/CX
is a homotopy equivalence.
Ci' is the quotient space of CX'v CX with the identifications [x',I] =
[f(x'),I] for all x' E X', hence CX C Ci" Since Ci' is the union of the closed
subspaces CX and C{" it follows that
PROOF
Ci,/CX
Similarly, Ci,,/CA
= Cr/(C{,
n CX)
= C",/A, and because Ci"
(Ci"Cd/CX
= Cr/X
n CX = CA,
= (C{"C",)/X
This proves the first two parts of the corollary.
Let F: C(X,A) X I ~ C(X,A) be the contraction defined by F([x,t], t')
[x, (1 - t')t] and let g: C(X',A') ~ (Ci"Cd be the composite
C(X',A') C C(X',A') v C(X,A)
~
(Ci"Cd
where the second map is the canonical map. The composite
(X',A') X I ~ (X,A) X I C C(X,A) X 1.4 C(X,A) C (Ci"Cd
=
368
HOMOTOPY THEORY
CHAP.
7
is a homotopy G: (X' ,A') X I ~ (Ci"Cd such that G(x',O) = [f(x'),l] = g[x',l].
By lemma 2, there is a homotopy F': C(X',A') X I ~ (Ci,Cd such that
F'I (X',A') X I = G and F'([x',t], 0) = g([x',t]). Then a homotopy
H·• (G,1., G,,)
X I ~ (C·,"" C·,,)
1.
1.
is defined by the equations
H([x',t], t')
H([x,t], t')
= F'([x',t], t')
= F([x,t], t')
x' EX'; t, t' E I
x E X; t, t' E I
[this is well-defined because F'([x',l], t') = G(x',t') = F([f(x'),l], t')]. Then H
satisfies a, b, and c of lemma 5 with (Y,B) = (Ci"Cd and Y' = CX. Therefore
k: (Ci"Cd ~ (Ci"Cd/CX is a homotopy equivalence. •
Recall from Sec. 1.6 that the suspension SX is defined as the space
X X II(X X 0 U Xo X I U X X 1) (therefore SX = CXIX). For a pair (X,A)
we define S(X,A)
(SX,SA). Then, for any map f: (X' ,A') ~ (X,A), we have
(Cr,C")IX = S(X',A'), and we let k: (Cr,C,,) ~ S(X',A') be the collapsing map.
=
7
For any map f: (X',A')
LEMMA
(X,A) the sequence
(X',A').4 (X,A) ~ (Cr,C,,) ~ S(X',A') §4 S(X,A)
is coexact.
PROOF
~
We shall use the coexact sequence 4,
(X',A')
-4 (X,A) ~ (Cr,C,,) -4 (Ci"Cd ~ (Cl,Cr)
By corollary 6, there is a homotopy equivalence
(Ci"Cd ~ (Cr,C")IX = S(X',A')
and the composite (Cr,C{,,) -4 (Ci"Cd 14 S(X',A') is seen to be the collapsing
map k: (Cr,C,,) ~ S(X',A'). Also by corollary 6, there is a homotopy equivalence
k"
(Cl,Cr) ~ (Cl,Cr)ICCr
= (Ci"Cd/Cr = S(X,A)
and the composite (Ci"Cd ~ (Cl,Cr) !4 S(X,A) is easily seen to be the
collapsing map k: (Ci"Cd ~ (Ci"Cd/C" = S(X,A). Let g: S(X',A') ~ S(X,A)
be the map defined by g([x',t]) = [f(x'), 1 - t]. The triangle
(Ci',Cd
\k
kl
S(X',A')
~
S(X,A)
is homotopy commutative because a homotopy
H: (Ci"Cd X I
from k to g
0
~
S(X,A)
k' is defined by
H([x',t], t') = [f(x'), 1 - tt']
H([x,t], t') = [x, (1 - t')t]
x' EX'; t, t' E I
x E X; t, t' E I
SEC.
1
369
EXACT SEQUENCES OF SETS OF HOMOTOPY CLASSES
=
[this is well-defined because H([x',l], t')
[f(x'), 1 - t']
Therefore there is a homotopy-commutative diagram
= H([f(x'),l], t')].
.
-4 (Ci"Cd ~ (Cj',Cr)
(Cr,C,,)
k\
k'l
kul
S(X',A')
S(X,A)
~
in which k' and k" are homotopy equivalences. From the coexactness of the
sequence 4, the coexactness of the sequence
-4 (X,A)
(X',A')
~ (Cr,C,,) ~ S(X',A') ~ S(X,A)
follows. Let h: S(X,A) ~ S(X,A) be the homeomorphism defined by h([x,t]) =
[x, 1 - t]. The coexactness of the above sequence implies the coexactness of
the sequence
(X',A')
Because hog
8
LEMMA
L
(X,A) ~ (Cr,C,,) ~ S(X',A') ~ S(X,A)
= Sf, this is the desired result.
-
If the sequence
(X',A')
-4 (X,A) ~ (X",A")
is coexact, so is the suspended sequence
S(X',A') ..§4 S(X,A) ~ S(X" ,A")
=
For any pair (Y,B) let Q(Y,B)
(QY,QB). By theorem 2.8 in the
Introduction, there is a commutative diagram (in which the vertical maps are
equivalences of pointed sets)
PROOF
[S(X",A"); Y,B]
(Sg)#)
[S(X,A); Y,B]
t
(Sf)#)
[S(X',A'); Y,B]
t
t
[X",A"; Q(Y,B)] ~ [X,A; Q(Y,B)]
L
[X',A'; Q(Y,B)]
=
Hence im (Sg)# ker (Sf)# in the top sequence is equivalent to im g#
in the bottom sequence. We define Sn(x,A) inductively for n
SO(X,A)
Sn(X,A)
9
THEOREM
(X',A')
L
Snf)
0 so that
= (X,A)
= S(Sn-l(X,A))
For any map f: (X',A')
(X,A) ~ ...
~
= ker f#
Sn(X,A)
~
Sni)
n~l
(X,A) the sequence
Sn(C"C,,)
Snk)
sn+1(X',A')
Sn+lf) •.•
is coexact.
PROOF
From lemmas 7 and 8, for n
Sn(X' ,A')
Snf)
Sn(X,A)
Sni)
~
0 there is a coexact sequence
Sn(Cf"C,,)
Snk)
Sn+1(X',A')
Sn+1f)
Sn+1(X,A)
370
HOMOTOPY THEORY
CHAP.
7
Since every three-term subsequence of the sequence in the theorem is contained in one of these five-term co exact sequences, the result follows. In the coexact sequence of theorem 9 all but the first three pairs are
H cogroup pairs, and all but the first three of these are abelian. Furthermore,
all maps between H cogroup pairs are homomorphisms. Thus, for any (Y,B)
the coexact sequence of homotopy classes of maps of the sequence of
theorem 9 into the fixed pair (Y,B) (with B not necessarily closed in Y) consist of groups and homomorphisms except for the last three pointed sets, and
all but three of the groups are abelian.
We now show how the last group in the sequence, namely [S(X',A'); Y,B],
acts as a group of operators on the left on the next set in the sequence,
namely [Gr,G r ,; Y,B], in such a way that the orbits are mapped injectively by
i# into [X,A; Y,B]. If a: S(X',A') ~ (Y,B) and /3: (Gr,G",) ~ (Y,B), we define
a T
(Gr,G",)
{ a[x',2t]
/3)[ x']
,t = /3[x', 2t _
( T
a
by
/3:
and
~
(Y,B)
o : : ; t ::::; Ih, x' E X', tEl
1]
Ih ::::; t ::::; 1, x' EX', tEl
x EX
= /3(x)
/3) 1(X,A) = /31 (X,A), and the following statements
(a T /3)(x)
It is then clear that (a T
are easily verified.
lOa
~
a T
a' and
/3 ~ a'
/3
T
~ /3' (or /3
/3' reI X). -
~
/3'
reI X) implies a T
I I If ao is the constant map, then ao T
12 (a1
* (2) T /3 ~ a1
13 a1 T (a2
0
k) ~ (a1
/3) reI X.
T (a2 T
* (2)
0
/3
k reI X.
~
/3 reI X.
1,
~
a' T
/3'
(or
-
-
-
Given maps /31,/32: (Gr,G",) ~ (Y,B) such that
define d(/31,/32): S(X',A') ~ (Y,B) by
d(/3
/3
/31 1(X,A)
= /321 (X,A), we
o ::;
t ::::; Ih, x' EX', tEl
Ih ::::; t ::::; 1, x' EX', tEl
/3)[
'] {/31[x' ,2t]
2 x ,t = /32[X', 2 _ 2t]
The following results are easily verified.
14
/31
~
/31 reI X and /32
IS d(/31,/33) ~ d(/31,/32)
16 d(a T /3,j3) ~ a.
17
/31
~
d(/31,/32) T
~
/32 rel X imply d(/3b/32)
* d(/32,/33)'
~
d(/31,/32)'
-
-
-
/32 reI X.
-
From statements 17, lO, and 11, it follows that if d(/31,/32) is null homotopic, then /31 ~ /32 reI X. Conversely, if /31 ~ /32 reI X, it follows from
statements 11, 14, and 16 that
d(/31,/32)
~
d(ao
T /31,/31)
~
ao
SEC.
2
371
HIGHER HOMOTOPY GROUPS
Therefore we have f3l = f3z reI X if and only if d(f3l,f3z) is null homotopic.
By statements 10, 11, and 12, there is an action of [S(X',A'); Y,B] on the
left on [C("C(,,; Y,B] defined by [a] T [f3] = [a T f3].
18 THEOREM Given [f3l], [f3z] E [C("C(,,; Y,BJ, then i#[f3l] = i#[f3z] if and
only if there is [a] E [S(X',A'); Y,B] such that [f3l] = [a] T [f3z].
PROOF
By the definition of a T f3z we see that
i#[a T f3z] = [(a T f3z) I (X,A)] = [f3zl (X,A)] = i#[f3z]
showing that i#([a] T [f3z]) = i#[f3z]. Conversely, if i#[f3d = i#[f3z], we can
choose representatives f3l and f3z such that f3l I (X,A) = f3z I (X,A) [because
the map i: (X,A) C (C(',C(") is a cofibration]. Then, by statement 17,
[f3l]
= [d(f3l,f3z)
T f3z]
= [d(f3l,f3z)]
T [f3z]
•
19 THEOREM Given [al], [az] E [S(X',A'); Y,B], then k#[ al] = k#[ az] if and
only if there is [y] E [S(X,A); Y,B] such that [az] = [all + (Sf)#[y].
PROOF
By statement 13, if f3o: (C",Cd
k#[al
* (y
0
Sf)]
= [all
~
(Y,B) is the constant map
T (k#Sf#[y])
= [all
T [f3o]
= [al] T k#[ ao] = k#[ al * ao]
+ (Sf)#[y]) = k#[al]' Conversely, if k#[al] = k#[az],
Therefore k#([al]
by statements 10 and 13,
0= k#[al- l
* al]
= [aI-I] T k#[al] = [aI-I] T k#[az] = k#[al- l
Therefore there is [y] E [S(X,A); Y,B] such that [a1- l
[az] = [all
2
+ [al- l * az]
= [all
then
* az]
* az] = (Sf)#[y], and so
+ (Sf)#[y]
•
HIGHER HOMOTOPY GROUPS
The higher homotopy groups of a space or pair are covariant functors defined
to be sets of homotopy classes of maps of fixed spaces or pairs into the
given one. In the absolute case these are the functors already defined in
Sec. 1.6. The exactness property established in ~e last section implies an
important exactness property relating relative and absolute homotopy groups.
This section contains definitions of the homotopy groups, some of their
elementary properties, and a proof of the exactness of the homotopy sequence
of a fibration.
We shall use 0 as base point for I and for the subspace j C 1. Let X be a
space with base point Xo. For n ;;::: 1 the homotopy group 7T n(X) [or 7Tn(X,XO),
when it is important to indicate the base point] is the group [sn(i );X] [this being
equivalent to the definition given in Sec. l.6, because Sn is homeomorphic to
Sn(SO) :::::: Sn(i )]. For n = 0 the homotopy set 7To(X) is defined to be the
pointed set [i;X] (that is, the set of path components of X with the path com-
372
HOMOTOPY THEORY
CHAP.
7
ponent of Xo as distinguished element). Then 'TTn is a covariant functor from
the category of pointed spaces to the category of abelian groups if n ~ 2, the
category of groups if n = 1, and the category of pointed sets if n = 0.
Let (X,A) be a pair with base point Xo E A. For n ~ 1 the nth relative
horrwtopy group (or homotopy set for n = 1), denoted by 'TTn(X,A) or
'TTn(X,A,xo), is defined to equal [sn-l(I,i); X,A]. Then 'TTn is a covariant functor
from the category of pairs of pointed spaces to the category of abelian groups
if n ~ 3, the category of groups if n = 2, and the category of pointed sets if
n=1.
There is a homeomorphism of S(1) with Iji which sends [O,t] E s(i) to
the base point of Iji and [I,t] E S(i) to that point of Iji determined by the
point t E 1. Therefore, for n ~ 1, sn(i) and sn-l(Iji) = Sn-l(I)jsn-l(i) are
homeomorphic. This induces a natural one-to-one correspondence between
[Sn-l(I,i); X, {xo}] and [Sn(i );X]. By means of this correspondence we identify the relative homotopy group 'TT n(X, {xo}) for n ~ 1 with the absolute
homotopy group 'TTn(X). Then the inclusion map j: (X,{xo}) C (X,A) induces a
homomorphism
n
~
1
Because sn(i) is path connected for n ~ 1, it follows that if X' is the path
component of X containing xo, the inclusion map X' C X induces isomorphisms
'TTn(X') :::::; 'TTn(X) for n ~ 1. Similarly, if A' is the path component of A containing xo, the inclusion map (X' ,A') C (X,A) induces isomorphisms 'TTn(X',A') :::::;
'TTn(X,A) for n ~ 1.
We present an alternative description of t~e relative homotopy groups.
For n ~ 1 there is a homeomorphism of Sn-l(I,I) with (I X In-I, i X In-l)j
(I X jn-l U X In-l) sending [ ... [t,tl], . . . ,tn-I] to [t,tl, . . . ,tn-I]
(10 is a single point and i o is empty). Therefore, for n ~ 1, 'TTn(X,A,xo) is in
one-to-one correspondence with the set of homotopy classes of maps
°
(In, in, I X i n- 1 U
Since I X i n- 1 U
map
°
X In-I) ~ (X,A,xo)
°X In-l is contractible, if zo = (0,0, . . . ,0), the inclusion
(In,in,zo)
(In, in,) i n° In-l)
C
X
1
U
X
is a homotopy equivalence. Hence, for n ~ 1, 'TTn(X,A,xo) is in one-to-one
correspondence with the set of homotopy classes of maps
(In,in,zo) ~ (X,A,xo)
Since (In,in,zo) is homeomorphic to (En,Sn-l,po) for n ~ 1, 'TTn(X,A,xo) is in
one-to-one correspondence with the set of homotopy classes of maps
(En,Sn-l,po)
~
(X,A,xo)
The following condition for a map (En,Sn-l,po) ~ (X,A,xo) to represent
the trivial element of 'TT n(X,A,xo) is a relative version of theorem 1.6.7.
I
THEOREM
Given a map a: (En,Sn-l,po)
~
(X,A,xo), then [a]
= ° in
SEC.
2
373
HIGHER HOMOTOPY GROUPS
'lTn(X,A,xo) if and only if a is homotopic relative to Sn-1 to some map of En to A.
PROOF
Assume [a]
= 0 in 'lTn(X,A,xo). Then there is a homotopy
H: (En,Sn-l,po) X I
~
(X,A,xo)
from a to the constant map En ~ Xo. A homotopy H' relative to Sn-1 from a
to some map En to A is defined by
H'(z,t)
=
H( 1 - z t/2 ' t)
o : :;: Ilzll :::;: 1 - ~
HCI:II ' 2 - 211 Z II)
1-
"2t :::;: Ilzll :::;: 1
Conversely, if a is homotopic relative to Sn-1 to some map a' such that
a'(En) C A, then [a] = [a'] in 'lTn(X,A,xo), and it suffices to show that [a'] = 0
in 'lTn(X,A,xo). A homotopy H: (En,Sn-l,po) X I ~ (X,A,xo) from a' to the
constant map En ~ Xo is defined by
H(z,t)
= a'((l
- t)z
+ tpo)
•
A pair (X,A) is said to be n-connected for n ~ 0 if for 0 :::;: k :::;: n every
map a: (Ek,Sk-1) ~ (X,A) is homotopic relative to Sk-1 to some map of
Ek to A. For k = 0, (EO,S-l) is a pair consisting of a single point and the
empty set, and the condition that every map a: (EO,S-l) ~ (X,A) be homotopic to a map EO ~ A is equivalent to the condition that every point of X
be joined by a path to some point of A. From theorem 1 we obtain the following relation between n-connectedness of (X,A) and the vanishing of relative homotopy groups of (X,A).
2
COROLLARY
A pair (X,A) is n-connected for n ~ 0 if and only if every
path component of X intersects A and for every point a E A and every
1 :::;: k :::;: n, 'lTk(X,A,a) = O. •
For n ~ 1 there is a map (which is a homomorphism for n ~ 2)
0: 'lTn(X,A,xo)
~
'lTn-1(A,xo)
defined by restriction. That is, given a: sn-1(I,i) ~ (X,A), then
ora]
= [a I Sn-1(i)]
It is trivial that if f: (X',A',x&) ~ (X,A,xo) is
square
a map,
there is a commutative
'lTn(X',A',x&) ~ 'lT n_1(A',x&)
'lTn(X,A,xo) ~ 'lTn(A,xo)
In other words, 0 is a natural transformation between covariant functors
'lTn(X,A) and 'lT n_1(A) on the category of pairs (X,A) of pOinted spaces. Thus
the homotopy-group functors and the natural transformation 0 are in analogy
374
HOMOTOPY THEORY
CHAP.
7
with the constituents of a homology theory. We shall show that they also
satisfy many of the axioms of homology theory. It is obvious that the homotopy axiom and the dimension axiom are satisfied for the homotopy-group
functors.
We shall now investigate the exactness property. Given a pair (X,A) of
pointed spaces, let i: A C X and i: (X,{xo}) C (X,A). The homotopy sequence
of (X,A) [or of (X,A,xo)] is the sequence of pointed sets (all but the last three
being groups)
... ~ '7Tn+l(X,A) ~ '7T n(A) ~ '7Tn(X) ~ '7Tn(X,A) ~ ... ~ '7To(X)
3
THEOREM
The homotopy sequence of a pair is exact.
PROOF
Letf: (i,{0}) C (i,i) andletf': i C iand!,,: {O} c
7.1.9, there is a coexact sequence
(i,{0})
-4
i.
By theorem
(i,i) ~ (Cr,Cd ~ S(i,{O}) §4 S(i,i) ~ .. ,
=
Let g: (Cr,C",) ~ (I,i) be the homeomorphism defined by g([O,t])
0 and
g([l,t])
t. Then the composite g i is the inclusion map i': (i,i) C (I,i), and
the composite k g-l equals the composite
=
0
0
(I,i) ~ (Iji,{O}) ~ (S(i),{O})
where k' is the collapsing map and h is the homeomorphism used in identifying '7Tn(X,{XO}) with '7Tn(X). Therefore there is a coexact sequence
(i,{O})
-4
(i,i) ~ (I,i) ~ S(i,{O}) §4
This yields an exact sequence
...
~
'7Tn+l(X,A)
(Sni')#)
'7T n(A)
(Snf)#)
'7Tn(X)
(Sn-l(h o k'»#)
'7Tn(X,A)
~
...
~
'7To(X)
The proof is completed by the trivial verification that
(Sni')# =
4
COROLLARY
a, (snf)#
For n
~
= i#, and
(Sn-l(h
0
k'))# = i# •
0, (En+l,Sn) is n-connected.
PROOF
For n ~ 0, En+l is path connected and Sn is nonempty; therefore
every path component of En+l meets Sn. If x E Sn, then '7Tk(En+l,x) = 0 for
o ::; k, because En+! is contractible. By theorem 3.4.11, '7Tk(Sn,X) = 0 if
o ::; k n. It follows from theorem 3 that '7Tk(En+l,Sn,x) = 0 for 1 ::; k ::; n.
The result follows from corollary 2. •
<
We shall see that the excision property fails to hold for the homotopy
group functors. There is, however, a different property possessed by the
homotopy group functors but not by homology functors. This property is the
existence of an isomorphism between the absolute homotopy groups of the
base space of a fibration and the corresponding relative homotopy groups of
the total space modulo the fiber. This is true for a more general class of maps
than fibrations, and we now present the relevant definition.
A map p: E ~ B is called a weak fibration (or Serre fiber space in the
SEC.
2
375
HIGHER HOMOTOPY GROUPS
literature) if P has the homotopy lifting property with respect to the collection
of cubes {In }n;>o. E is called the total space and B the base space of the weak
fibration. For b E B, p-l(b) is called the fiber over b.
If s is a simplex, lsi is homeomorphic to some cube, and so a map
p: E ~ B is a weak fibration if and only if it has the homotopy lifting property with respect to the space of any simplex. We shall show that, in fact, a
weak fibration has the homotopy lifting property with respect to any
polyhedron.
It is clear that a fibration is a weak fibration. If p: E ~ B is a weak
fibration and f: B' ~ B is a map, let E' be the fibered product of B' and E.
Then there is a weak fibration p': E' ~ B', called the weak fibration induced
from p by f
:.
LEMMA
Let p: E ~ B be a weak fibration and let g; In X 0 U jn X I ~ E
and H: In X I ~ B be maps, with n ;:::: 0, such that H is an extension of
p g. Then there is a map G: In X I ~ E such that p G = Hand G is an
extension of g.
0
0
PROOF
The lemma asserts that the dotted arrow in the diagram
In X 0 U jn X I ~ E
~P
n~
In X I
..14B
represents a map making the diagram commutative. This follows from the
homotopy lifting property of p since the pair (In X I, In X 0 U jn X I) is
homeomorphic to the pair (In X I, In X 0). •
6
THEOREM
Let (X,A) be a polyhedral pair and let p: E ~ B be a weak
fibration. Given maps g: X X 0 U A X I ~ E and H: X X I ~ B such that
H is an extension of p g, there is a map G: X X I ~ E such that p G = H
and G is an extension of g.
0
0
PROOF
The method of obtaining G involves a standard stepwise-extension
procedure over the successive skeleta of a triangulation of X. Let (K,L) be a
triangulation of (X,A) and identify (X,A) with (IKI,ILI). For q ;:::: -1 set
Xq
IKI X 0 U (IKq U LI X 1), so that X_l X X 0 U A X I and Xq- l C Xq
for q ;:::: O. By induction on q, we shall define a sequence of maps Gq: Xq ~ E
such that
=
=
(a) G_ l = g
(b) Gq I Xq - l = Gq- l for q ;:::: 0
(c) po Gq = HI Xq for q ;:::: -1
Once a sequence {G q } with these properties is obtained, a map G: X X I ~ E
with the desired properties is defined by the conditions G I Xq
Gq, for
q ;:::: -1. Thus the problem is reduced to the construction of such a
sequence {G q }.
=
376
HOMOTOPY THEORY
CHAP.
7
<
Condition (a) defines G- l . Assume G q defined for q
n, where n 2:: o.
To define Gn to satisfy conditions (b) and (c), for every n-simplex s E K - L
let gs: lsi X 0 U lsi X I --,) E and Hs: lsi X I --,) B be the maps defined by
gs
Gn - l I (lsi X 0 U lsi X 1) and Hs
H I (lsi X 1). Because (lsl,181) is
homeomorphic to (In,in), it follows from lemma 5 that there is a map
Gs: lsi X I --,) E such that Gs I (lsi X 0 U lsi X 1) = gs and po Gs = Hs.
Then a map Gn: Xn --,) E satisfying conditions (b) and (c) is defined by the
conditions Gn I Xn - l = G n - l and Gn I (lsi X 1) = G s for s an n-simplex of
K - L. •
=
=
Taking A to be empty in theorem 6, we see that a weak fibration has the
homotopy lifting property with respect to any polyhedron.
7 COROLLARY Let (X',A') be a polyhedral pair such that A' is a strong
deformation retract of x' and let p: E --,) B be a weak fibration. Given maps
g': A' --,) E and H': X' --,) B such that H' I A' = P g', there is a map
G': X' --,) E such that p G' = H' and G' I A' = g'.
0
0
Let D: X' X I --,) X' be a strong deformation retraction of X' to A'.
Then D(X' X 1 U A' X 1) C A', and we define g: X' X 1 U A' X I --,) E to
be the composite
PROOF
X' X 1 U A' X I ~ A' ~ E'
Let H: X' X I --,) B be the composite
X'XI~X'~B
Then H is an extension of p g, and it follows from theorem 6 that there is a
map G: X' X I --,) E such that p 0 G = Hand G is an extension of g. Let
G': X' --,) Ebe defined by G'(x') = G(x',O). Then G' has the desired properties. •
0
The following theorem is the main result relating the homotopy groups
of the base and total space of a weak fibration.
8 THEOREM Let p: E --,) B be a weak fibration and suppose b o E B' C B.
Let E' = p-l(B') and let eo E p-l(bo). Then p induces a bijection
p#: 7Tn(E,E',eo) :::::: 7Tn(B,B',b o)
n 2:: 1
PROOF
To show that p# is surjective, let a: (In,in,zo) --,) (B,B',b o) represent
an element of 7T n(B,B',b o). Because Zo is a strong deformation retract of In, we
can apply corollary 7 to the pair (In,{zo}) and to maps g': {zo} --,) E and
H': In --,) B, where g'(zo) = eo and H' = a I In. We then obtain a map
G': In --,) E such that po G' = H' and G'(zo) = eo. Then
G'(in) C p-l(H'(in)) C p-l(B')
= E'
Therefore G' defines a map a': (In,in,zo) --,) (E,E',eo) such that p a' = a.
Then a' represents an element [a'] E 7T n(E,E',eo) and p#[a'] = [a].
To show that P# is injective, let lXO,al: (In,in,zo) --,) (E,E',eo) be such that
p 0 ao ~ p 0 al. Let X' = In X I and A' = (In X 0) U (zo X I) U (In Xl).
0
SEC.
2
377
HIGHER HOMOTOPY GROUPS
Then (X',A') is a polyhedral pair, and because X' and A' are both contractible,
A' is a strong deformation retract of X'. Let g': A' ~ E be defined by
g'(z,O) = (l'o(z), g'(z,l) = (l'l(Z), and g'(zo,t) = eo and let H': X' ~ B be a map
which is a homotopy from p (1'0 to P (1'1 in (B,B',b o). By corollary 7, there
is a map G': X' ~ E such that p G' = H' and G' I A' = g'. Since
0
0
0
G'(in X 1) C p-1(H'(in X 1)) C p-1(B') = E'
G' is a homotopy from
(1'0
to
(1'1
in (E,E',eo); hence
[(1'0]
=
[(1'1]
in 7Tn(E,E',eo).
9
COROLLARY
Let p: E ~ B be a weak fibration, b o E B, and eo E F
p-1(b o). Then p induces a biiection
n>
PROOF
This follows from theorem 8 on taking B'
canonical identification 7Tn (B,{b o},bo) = 7T n (B,b o). •
If p: E
~
•
=
1
= {b o}
and using the
B is a weak fibration with F = p-1(b o) and eo E F, we define
n> 1
to be the composite
7T n(B,b o)
p#-\
7T n (E,F,eo)..z... 7T n-1(F,eo)
The homotopy sequence of the weak fibration is the sequence
... ~ 7T n (F,eo) ~ 7Tn(E,eo) ~ 7Tn (B,b o) ~ 7T n-1(F,eo) ~
~ 7To(B,bo)
where i: (F,eo) C (E,eo).
10
THEOREM
The homotopy sequence of a weak fibration is exact.
PROOF
Exactness at 7To(E,eo) is an easy consequence of the homotopy lifting
property. Exactness at any set to the left of 7To(E,eo) is a consequence of the
exactness of the homotopy sequence of the pair (E,F). •
I I COROLLARY Let p: E ~ B be a weak fibration with unique path lifting.
Then p induces an isomorphism
Because F = p-1(p(eo)) has no nonconstant paths (by theorem 2.2.5),
7Tq(F,eo) = 0 for q ~ 1. The result then follows from theorem 10. •
PROOF
12
COROLLARY
For q
~
2, 7Tq(Sl) = O.
This follows from application of corollary 11 to the covering projection p: R ~ 51 and the fact that because R is contractible, 7T q (R) = 0 for all
q ~ O. •
PROOF
13
COROLLARY
an isomorphism
Let p: 52n + 1 ~ Pn(C) be the Hopf fibration. Then p induces
378
HOMOTOPY THEORY
CHAP.
7
PROOF
Because F = 51 for the Hopf fibration, this follows from corollary 12
and theorem 10. •
14
COROLLARY
7T3(52) =1=
o.
PROOF
Because the identity map (5 3,po) C (5 3,po) induces a nontrivial
homomorphism of H3(5 3,po), it is not homotopic to the constant map. Therefore 7T3(53) =1= 0, and the result follows from corollary 13, with n = 1 (since
P1(C) :::::: 52). •
This last result shows that, unlike the homology groups, the homotopy
groups of a polyhedron need not vanish in degrees larger than the dimension
of the polyhedron.
If H is a closed hemisphere of 52 and a is the pole in H, then the pair
(52 - a, H - a) has the same homotopy type as (E2,5 1). Therefore
7T3(52 - a, H - a) :::::: 7T3(E2,51)
?
7T2(5 1) = 0
On the other hand, (5 2,H) has the same homotopy type as (5 2 ,{a}). Therefore
7T3(52,H) :::::: 7T3(52,{ a}) = 7T3(52) =1= 0
Hence we see that the excision map;: (52 - a, H - a) C (5 2,H) does not
induce an isomorphism of 7T3(52 - a, H - a) with 7T3(52,H). Therefore the
excision property does not hold for homotopy groups.
Recall the path fibration p: P(X,xo) ---7 X with fiber p-1(XO) = QX (see
corollary 2.8.8). 5ince P(X,xo) is contractible (by lemma 2.4.3), 7T n(P(X,xo)) = 0
for n ~ 0, and by theorem 10, there is an isomorphism
n> 1
This result can also be deduced directly from the canonical one-to-one correspondence [5 n U);X] :::::: [5 n - 1U);QX] given by the exponential law. We shall
use the path space to prove the exactness of the homotopy sequence of a triple.
Given a triple (X,A,B) with base point Xo E B, let i: (A,B) C (X,B) and
j: (X,B) C (X,A) and let j': (A,{xo}) C (A,B). Define
n~2
to equal the composite
7Tn(X,A,xo) ~ 7T n-1(A,xo) ~ 7T n_1(A,B,xo)
The homotopy sequence of the triple (X,A,B) is defined to be the sequence
• . . ---7
15
7Tn+1(X,A) ~ 7T n(A,B) ~ 7Tn(X,B) ~ 7Tn(X,A)
THEOREM
---7 . • . ---7
7Tl(X,A)
The homotopy sequence of a triple is exact.
Let p: P(X,xo) ---7 X be the path fibration and let X' = P(X,xo),
A' = p-1(A), and B' = p-l(B). Then (X',A',B') is a triple, and it follows from
theorem 8 that P# maps the homotopy sequence of (X',A',B') bijectively to the
homotopy sequence of (X,A,B). Therefore it suffices to prove that the homotopy sequence of the triple (X',A',B') is exact.
PROOF
SEC.
3
379
CHANGE OF BASE POINTS
Let i: (A',B') C (X',B'), i: (X',B') C (X',A'), i': B' C A', and 1': A' C (A',B').
There is a commutative diagram
...
~
7Tn+1(X',A') ~ 7T n(A',B')
21
.. .
~
7Tn(A')
S
7T n(X',B') ~ 7Tn(X',A') ~
1=
S
10
"1
7Tn(A',B') ~ 7Tn-1(B')
i"
~
7Tn_1(A')
~
...
in which each vertical map is a bijection (because X' is contractible). Therefore
the exactness of the homotopy sequence of the triple (X',A',B') follows from
the exactness of the homotopy sequence of the pair (A',B'). •
This result can also be derived from the exactness of the homotopy sequence of a pair and functorial properties of the homotopy groups (as was the
case with the corresponding exactness property for homology, theorem 4.8.5).
3
CHANGE OF BASE POINTS
The absolute and relative homotopy groups are defined for pointed spaces and
pairs. This section is devoted to a study of the extent to which these groups
depend on the choice of base point. By generalizing the methods of Sec. 1.8,
we shall see that these groups based at different base points in the same path
component are isomorphic, but the isomorphism between them is not usually
unique. Much of these considerations apply to more general homotopy sets,
and we begin with this.
Let (X,A) be a pair with base point Xo E A and let (Y,B) be a pair. Two
maps aD, a1: (X,A) ~ (Y,B) are said to be freely homotopic if they are homotopic as maps of (X,A) to (Y,B) (that is, no restriction is placed on the base
point during the homotopy). If w is a path in B from ao(xo) to a1(xo), an
w-homotopy from 0'0 to a1 is a homotopy
H: (X,A) X I
~
(Y,B)
such that H(x,O) = ao(x), H(x,l) = a1(x), and H(xo,t) = w(t). If such a homotopy exists, we say that aD is w-homotopic to a1. It is clear that aD and a1 are
freely homotopic if and only if there is some path w in B such that aD and a1
are w-homotopic. In particular, two maps aD, a1: (X,A,xo) ~ (Y,B,yo) are
freely homotopic if and only if there is some closed path w in B at yo such
that aD is w-homotopic to a1.
Although the relation of free homotopy is an equivalence relation in the
set of maps from (X,A) to (Y,B), for a fixed w the relation of w-homotopy is
not generally an equivalence relation. For example, if w is not a closed path,
it is impossible for any map aD to be w-homotopic to itself.
(a) Given a map f: (X',A',xo) ~ (X,A,xo), maps aD, a1: (X,A) ~
(Y,B), and a path w in B such that aD is w-homotopic to a1, then aD f is whomotopic to a1 f.
I
LEMMA
0
0
380
HOMOTOPY THEORY
CHAP.
7
(b) Given a map g: (Y,B) ~ (Y',B'), maps 0:0, 0:1: (X,A) ~ (Y,B), and a
path win B such that 0:0 is w-homotopic to 0:1, then g 0:0 is (g w)-homotopic to g 0:1.
(c) Given maps 0:0, 0:0: (SX,SA,xo) ~ (Y,B,w(O)) and 0:1, o:i: (SX,SA,xo) ~
(Y,B,w(l)) such that 0:0 is w-homotopic to 0:1 and 0:0 is w-homotopic to 0:1,
then 0:0 * 0:0 is w-homotopic to 0:1 * o:i.
0
0
0
If H: (X,A) X I
the composite
PROOF
(Y,B) is an w-homotopy from
~
0:0
to
0:1,
then for (a)
(X',A') X I ~ (X,A) X I ~ (Y,B)
is an w-homotopy from
0:0
0
f to
0:1
0
f, and for (b) the composite
(X,A) X I ~ (Y,B) 14 (Y',B')
is a (g w)-homotopy from g 0:0 to g 0:1.
In (c), if H, H': (SX,SA) X I ~ (Y,B) are w-homotopies from
to 0:1 and 0:1, respectively, the map
0
0
H
defined by
(H
*H
')([
* H':
0
(SX,SA) X I
~
0:0
* 0:0 to 0:1 * 0:1.
and
0:0
(Y,B)
]')
{H([x,2t], t')
x,t, t = H'([x, 2t - 1], t')
is an w-homotopy from
0:0
O<t<1h
1h<t<l
-
The base point Xo for a pair (X,A) is said to be a nondegenerate base point
if the inclusion map (xo,xo) C (X,A) is a co fibration [that is, if, given a map
0:0: (X,A) ~ (Y,B) and a homotopy w: Xo X I ~ B, there is a homotopy
H: (X,A) X I ~ (Y,B) such that H(x,O) = o:o(x) and H(xo,t) = w(t)]. It follows
from lemma 3.8.1 and corollary 3.2.4 that any point of a polyhedral pair is a
nondegenerate base point.
2
LEMMA
Let (X,A) be a pair with nondegenerate base point and let (Y,B)
be an arbitrary pair.
(a) Given a path w in B and a map 0:1: (X,A,xo) ~ (Y,B,w(1)), there is a
map 0:0: (X,A,Xo) ~ (Y,B,w(O)) such that 0:0 is w-homotopic to 0:1.
(b) If 0:0, 0:0: (X,A,xo) ~ (Y,B,w(O)) are both w-homotopic to 0:1, then
[0:0] = [0:0] in [X,A,xo; Y,B,w(O)].
(c) If 0:0 is w-homotopic to 0:1 and 0:0 c:::: 0:0 as maps from (X,A,xo) to
(Y,B,w(O)), 0:1 c:::: o:i as maps from (X,A,xo) to (Y,B,w(l)), and w c:::: w' as paths
in B, then 0:0 is w' -homotopic to 0:1.
(a) Given 0:1 and w, it follows from the non degeneracy of Xo that
there is a map H': (X,A) X I ~ (Y,B) such that H'(x,O) = O:l(X) and
H'(xo,t) = w(l - t). Define 0:0: (X,A,xo) ~ (Y,B,w(O)) by o:o(x) = H'(x,l). Then
H: (X,A) X I ~ (Y,B) defined by H(x,t) = H'(x, 1 - t) is an w-homotopy
from 0:0 to 0:1.
(b) Because Xo is a non degenerate base point, there is a retraction
PROOF
SEC.
3
381
CHANGE OF BASE POINTS
r: (X,A) X 1 ~ (xo X 1 U X X 1, Xo X 1 U A X 1) (by theorem 2.8.1), and
we let rt: (X,A) ~ (xo X 1 U X X 1, Xo X 1 U A X 1) be defined by rt(x) = r(x,t).
Let G: (xo X 1 U X X 1, Xo X 1 U A X 1) X 1 ~ (X,A) X 1 be the homotopy
relative to (xo,O) defined by G(x,t,t') = (x,tt') and define
Gt< (xo X 1 U X X 1, Xo X 1 U A X 1)
~
(X,A) X 1
by Ge-(x,t) = G(x,t,t'). Then Go 0 ro ::::::: G 1 0 ro reI Xo, and because Go(xo X 1) =
(xo,O), Go ro ::::::: Go r1 reI Xo. Let H: (X,A) X 1 ~ (Y,B) be an w-homotopy
from lXo to lX1. Then H G 1 ro ~ H Go r1 reI Xo. Clearly, H Go r1 = lXo,
and so lXo ::::::: H G 1 ro reI Xo. Similarly, if H': (X,A) X 1 ~ (Y,B) is an
w-homotopy from lXo to lXl. then lXo ::::::: H' 0 G 1 0 ro reI Xo. Because
0
0
0
0
0
0
0
0
0
0
HI (xo X 1 U X X 1) = H' I (xo X 1 U X X 1)
H
G 1 ro = H' G 1 ro, and so lXo ~ lXo reI Xo.
(c) First we observe that the inclusion map
0
0
0
(X X
0
i U Xo X 1, A X i U Xo X 1) C (X,A) X 1
is a cofibration. In fact, let h: (1 X 1, 1 X 0 U i X 1) ~ (1 X 1, 0 X 1) be a
homeomorphism. Then there is a homeomorphism
1 X h: (X X 1 X 1, A X 1 X 1) ::::::; (X X 1 X 1, A X 1 X 1)
which maps
X X 1 X 0 U X X i X 1 U Xo X 1 X 1 to
X X 0 X 1 U Xo X 1 X 1
and
A X 1 X 0 U A X i X 1 U Xo X 1 X 1 to
A X 0 X 1 U Xo X 1 X 1.
Thus we need only show that (X X 0 U Xo X 1, A X 0 U Xo X 1) X 1 is a
retract of (X X 1, A X 1) X 1, which follows from the fact that (X X 0 U Xo X 1,
A X 0 U Xo X 1) is a retract of (X X 1, A X 1).
Now let F, F': (X X j U Xo X 1, A X j U Xo X 1) ~ (Y,B) be defined by
F(x,O)
F'(x,O)
= lXO(X)
= lXO(X)
F(x,l)
F'(x,l)
= lX1(X)
= lXl(X)
F(xo,t)
F'(xo,t)
= w(t)
= w'(t)
Because lXo ~ lXo, lX1 ::::::: lXI, and w ::::::: w', it follows that F ::::::: F'. Because lXo is
w-homotopic to lX1, F can be extended to a map H: (X,A) X 1 ~ (Y,B). By
the cofibration property established above, F can be extended to a map
H': (X,A) X 1 ~ (Y,B). Then H' is an w'-homotopy from lXo to lXl. It follows from lemmas 2a and 2b that, given wand lX1: (X,A,xo) ~
(Y,B,w(l)), the set of all maps lXo: (X,A,xo) ~ (Y,B,w(O)) which are w-homotopic
to lX1 belong to a single homotopy class of maps (X,A,xo) ~ (Y,B,w(O)).
It follows from lemma 2c that this set of maps equals a homotopy class of
maps (X,A,xo) ~ (Y,B,w(O)) which depends only on the homotopy class
[lX1] E [X,A,xo; Y,B,w(l)] and the path class [w]. Therefore, if (X,A) has a nondegenerate base point, there is a map
382
HOMOTOPY THEORY
CHAP.
7
h[w): [X,A,xo; Y,B,w(I)] ___ [X,A,xo; Y,B,w(O)]
characterized by the property h[w)[ a1] = [ao] if and only if ao is w-homotopic
to a1. It follows from lemmas la and Ib that this map is functorial in (X,A)
and in (Y,B) and from lemma lc that if (X,A) is a suspension, the map is a
homomorphism.
3 THEOREM Let (X,A) be a pair with nondegenerate base point Xo. For any
pair (Y,B) there is a covariant functor from the fundamental groupoid of B to
the category of pointed sets which assigns to a point yo E B the set
[X,A,xo; Y,B,yo] and to a path class [w] in B the map h[w)' If (X,A) is a suspension, this functor takes values in the category of groups and homomorphisms.
We need only verify the two functorial properties. If a: (X,A,xo) ___
(Y,B,yo) is arbitrary and to is the constant path at yo, the constant homotopy
is an to-homotopy from a to a proving that h[d = 1.
Given paths wand w' in B such that w(l) = w'(O), an w-homotopy H
from ao to a1, and an w'-homotopy H' from a1 to a2 [where ao, a1, a2 are
maps of (X,A) to (Y,B)], an (w * w')-homotopy H * H' from ao to a2 is defined by
PROOF
(H
*H
')(
) {H(x,2t)
x,t = H'(x, 2t - 1)
This shows that h[w*w') = h[w)
0
h[w')'
o -:::
t -::: 1;2
1;2<t<1
•
4
COROLLARY
If BeY is path connected and (X,A) has a nondegenerate
base point Xo, then for any yo, Y1 E B the pointed sets [X,A,xo; Y,B,yo] and
[X,A,xo; Y,B,Y1] are in one-to-one correspondence. Furthermore, '7T1(B,yo) acts
as a group of operators on the left on [X,A,xo; Y,B,yo], and the one-to-one
correspondence above is determined up to this action of '7Tl(B,yo).
PROOF
If [w] is any path class in B, h[w) is a one-to-one correspondence.
If [w] E '7Tl(B,yo), then h[w) is a permutation of [X,A,xo; Y,B,yo], and in this
way '7Tl(B,yo) acts as a group of operators. If yo and Yl are points in B, the
set of one-to-one correspondence h[ro) determined by path classes [w] in Y from
Yo to Yl is the same as the set of maps h[roo) h[ro'), where [wo] is a fixed path
class from yo to Y1 and [w'] E '7T1(B,yo). •
0
In all of the above, by taking B = Y, we get the corresponding results
for the absolute case. Thus, if X is a space with non degenerate base point Xo
and yo E Y, then '7Tl(Y,YO) acts as a group of operators on [X,xo; Y,yo]. If Y is
path connected and yo, Yl E Y, then [X,xo; Y,yo] and [X,xo; Y,Y1] are in
one-to-one correspondence by a bijection determined up to the action of
'7Tl(Y,YO).
In case Y is an H space and BeY is a sub-H-space, there is the following
result, which can be regarded as a generalization of theorem 1.8.4.
5 THEOREM Let (X,A) have a nondegenerate base point Xo and let (Y,B)
be a pair of H spaces. If yo E B is the base point, '7T1(B,yo) acts trivially on
[X,A,xo; Y,B,yoJ.
SEC.
3
383
CHANGE OF BASE POINTS
PROOF
Let p,: (Y X Y, B X B) ~ (Y,B) be the multiplication. Given
a: (X,A,xo) ~ (Y,B,yo) and a closed path w: (I,i) ~ (B,yo), define an w'homotopy H: (X,A) X I ~ (Y,B) from a' to a' (where w' ~ wand a' ~ a) by
H(x,t)
Therefore h[w)[a]
= [a] for all [a]
= p,(a(x),w(t))
E [X,A,xo; Y,B,yo] and all [w] E 7T1(B,yo). •
There is an interesting relation between the action of 7T1(B,yo) on
[X,A,xo; Y,B,yo] and the action of 7T1(B,yo) as covering transformations on a
universal covering space of B. We assume that Band Yare path connected
and locally path connected, that 7T1(B,yo) ;::::; 7T1(Y,YO), and that Y is a simply
connected covering space of Y with covering projection p: Y ~ Y. Then
B = p-1(B) is a simply connected covering space of B [because 7T1(B,yo) ;::::;
7T1(Y,YO)]. Let yo E p-1(yO). There is a canonical map
(): [X,A,xo; Y,B,yo] ~ [X,A; Y,B]
from base-point-preserving homotopy classes to free homotopy classes.
Because B is simply connected, this map is a bijection [recall that two maps
ao, a1: (X,A) ~ (Y,B) are freely homotopic if and only if there is a path win
B from ao(xo) to a1(xO) such that ao is w-homotopic to a1].
6 LEMMA With the notation above, let g: (Y,B,yo) ~ (Y,B,Y1) be a covering transformation and let wbe a path in B from yo to Y1. There is a commutative diagram
[X,A,xo; Y,B,yo] ~ [X,A,xo; Y,B,yo]
P#
[X,A,xo; Y,B,yo]
~
:!?
[X,A; Y,B]
- - 0
- [X,A,xo; Y,B,yo]
-:;;!
[X,A; Y,B]
PROOF
Because g is a covering transformation, p = p g and p# = p# ~.
The commutativity of the left-hand square follows from this and from
lemma lb. Since () h[w)
the commutativity of the right-hand side follows
from the trivial verification that () ~ = g#
0
0
0
= (),
0
0
().
•
Recall the isomorphism 1/;: G(B I B) ;::::; 7T1(B,yo) of corollary 2.6.4, which
assigns to g the element [p w] E 7T1(B,yo). Therefore lemma 6 expresses a
relation between the action of G(B I B) ;::::; G(Y I Y) on the free homotopy
classes [X,A; Y,B] and the action of 7T1(B,yo) on [X,A,xo; Y,B,yo].
0
7
COROLLARY
Let X be a simply connected locally path-connected space
with nondegenerate base point and let Y be a simply connected covering
space of a locally path-connected space Y. There is a bijection from the free
homotopy classes [X; Y] to the pointed homotopy classes [X,xo; Y, Yo] compatible with the action of G(YI Y) on the former, the action of 7T1(Y,YO) on
the latter, and the isomorphism 1/;: G(YI Y) ;::::; 7T1(Y,YO).
PROOF
This follows from lemma 6, with B = Y and A = X, and from the
observation that because X is simply connected, it follows from the lifting
384
HOMOTOPY THEORY
CHAP.
7
theorem 2.4.5, the homotopy lifting property of p: Y ~ Y, theorem 2.2.3, and
the unique-lifting property, theorem 2.2.2, that p#: [X,xo; Y,Yo] ~ [X,xo; Y,Yo]
is a bijection. We now specialize to the homotopy groups. Because
'7Tn(X,xo)
= [sn(i ),0; X,xo] = [sn(i ),sn(i),O; X,X,xo]
we obtain the following result.
8 THEOREM For any space X and any n ~ 1, there is a covariant functor
from the fundamental groupoid of X to the category of groups and homomorphisms which assigns to x E X the group '7T n(X,x) and to a path class [w] in X
the map h[wj: '7Tn(X,w(I)) ~ '7Tn(X,w(O)). In this way, '7T1(X,XO) acts as a group
of operators on the left on '7Tn(X,xo), by conjugation if n = 1, and if X is path
connected and xo, Xl E X, then '7T n(X,xo) and '7Tn(X,X1) are isomorphic by an
isomorphism determined up to the action of '7T1(X,XO).
PROOF
Everything follows from theorem 3 and corollary 4 except for the
statement that '7T1(X,XO) acts on '7T1(X,XO) by conjugation. For this let
H: s(i) X I ~ X be on w-homotopy from ao to a!, where w, ao, and a1 are
closed paths in X at Xo. Define H': I X I ~ X by
H'(t,t')
= H([I,t], t')
Then H' lOx I = H' 11 X I = wand H' I I X 0 = ao and H' I I X 1 = a1.
It follows from lemma 1.8.6 that (w * (1) * (w- 1 * ao- 1) is null homotopic.
Therefore [aD] = [w][a1][w]-1, and so h[wj[a1] = [w][a1][w]-1. Theorem 8 shows that the action of '7T1(X,XO) on itself by conjugation, as
in theorem 1.8.3, is extended to an action of '7T1(X,XO) on '7Tn(X,xo) for every
n~1.
A path-connected space X is said to be n-simple (for n ~ 1) if for some
Xo E X (and hence all base points x E X) '7T1(X,XO) acts trivially on '7Tn(X,xo).
Thus a simply connected space is n-simple for every n ~ 1, and a pathconnected space X is I-simple if and only if '7T1(X,XO) is abelian. For n-simple
spaces there is a unique canonical isomorphism '7Tn(X,xo) ::::: '7Tn(X,X1), any map
a: Sn ~ X determines a unique element of '7Tn(X,xo) (whether or not a maps the
base point po E Sn to xo), and '7Tn(X,xo), is in one-to-one correspondence with
the free homotopy classes of maps Sn ~ X. The latter is a useful property,
and for n-simple spaces X we shall usually omit the base point and merely
write '7Tn(X). From theorem 5 we obtain the following generalization of
theorem 1.8.4.
9
A path-connected H space is n-simple for every n
Similar consideration apply to the relative homotopy groups.
THEOREM
~
1.
-
10 THEOREM For any pair (X,A) and any n ~ 1 there is a covariant functor
from the fundamental groupoid of A to the category of pointed sets if n = 1
and the category of groups if n
1 which assigns '7T n(X,A,x) to x E A and to
a path class [w] in A the map
>
SEC.
3
385
CHANGE OF BASE POINTS
In this way, 7TI(A,xo) acts as a group of operators on the left on 7Tn(X,A,xo),
and if A is path connected and xo, Xl E A, then 7T n(X,A,xo) and 7Tn(X,A,XI) are
isorrwrphic by an isomorphism determined up to the action of 7TI(A,xo). •
If w is a path in A, it follows from lemma 1a that there is a commutative
square for n
1,
>
7T n(X,A,w(1)) ~ 7Tn_I(A,w(1))
h["ll
lh["l
7T n(X,A,w(O)) ~ 7T n_I(A,w(O))
Thus there is also a covariant functor from the fundamental groupoid of A to
the category of exact sequences which assigns to X E A the homotopy sequence
of (X,A,x).
A pair (X,A) with A path connected is said to be n-simple (for n 2: 1) if
7TI(A,xo) acts trivially on 7Tn(X,A,xo) for some (and hence all) base points Xo E A.
If A is simply connected, (X,A) is n-simple for every n 2: 1.
I I THEOREM Let (X,A) be a pair of H spaces with A path connected. Then
(X,A) is n-simple for all n 2: 1.
PROOF
This is immediate from theorem 5.
•
If (X,A) is n-simple and xo, Xl E A, then 7T n(X,A,xo) and 7T n(X,A,XI)
are canonically isomorphic. Therefore any map a: (En,Sn-l) ---7 (X,A) determines a unique element of 7Tn(X,A,xo) (whether or not a maps the base point
po E Sn-l to XO), and 7Tn(X,A,xo) is in one-to-one correspondence with the free
homotopy classes [En,sn-l; X,A]. If (X,A) is n-simple, we shall frequently omit
the base point and write 7Tn(X,A).
The action of 7TI(A,xo) on 7T2(X,A,xo) is closely related to conjugation, as
shown by the next result.
12
THEOREM
If a, b E 7T2(X,A,xo), then
aba- 1
= haa(b)
Let X' = P(X,xo) and let p: X' ---7 X be the path fibration. Let
A' = p-I(A) and let Xo E A' be the constant path at Xo. By theorem 7.2.8,
there is an isomorphism
PROOF
p#: 7T2(X',A',xo) :::::: 7T2(X,A,xo)
= p#-l(a) and b' = p#-l(b) and observe that, by lemma Ib,
haa(b) = p~haa,(b'))
Hence it suffices to prove that a'b'a'-l = haa,(b'). Because X' is contractible,
Let a'
it follows from the exactness of the homotopy sequence of (X' ,A',xQ) that
0: 7T2(X',A',xQ) :::::: 7TI(A',xo)
386
HOMOTOPY THEORY
CHAP.
7
So to complete the proof we need only prove that
o(a'b'a'-l)
= o(hca,(b'))
The left-hand side equals (oa')(ob')(oa')-l, and because a commutes with hi'a',
the right-hand side equals hua,(ob'). The result now follows from the fact that
the action of 7Tl(A',xb) on itself given by h is the same as conjugation. This again implies that '7Tz(X,xo) ;::::: '7Tz(X, {xo },xo) is abelian. Together with
the exactness of the homotopy sequence, it yields the next result.
13
COROLLARY
The inclusion map j: (X,xo)
j#: '7Tz(X,xo)
--c>
C
(X,A) induces a homomorphism
'7Tz(X,A,xo)
-
whose image is in the center of '7Tz(X,A,xo).
The following result is a generalization of theorem 1.8.7 to the higher
relative homotopy groups.
14 THEOREM Let f: (X,A,xo) --c> (Y,B,yo) and g: (X,A,xo) --c> (Y,B,Yl) be
freely homotopic. Then there is a path w in B from yo to Yl such that
f# = h[w]
~:
0
'77
n(X,A,xo)
--c> '77 n(Y,B,yo)
n
>2
PROOF
Let F: (X,A) X I --c> (Y,B) be a homotopy from f I (X,A) to g I (X,A)
and let w(t) = F(xo,t). Then w is a path in B from yo to Yl, and if
a: (In,in,po) --c> (X,A,xo) represents an element of '77 n(X,A,xo), then the composite
(In,in) X I ~ (X,A) X I ~ (Y,B)
is an w-homotopy from f
f#[aJ
a to goa. Therefore
0
= [J
0
a]
= h[w]([g
0
aJ)
= (h[w]
0
~)[aJ
-
This yields the following analogue of theorem 1.8.8.
15 COROLLARY Let f: (X,A)
x E A, f induces isomorphisms
--c>
(Y,B) be a homotopy equivalence. For any
f#: 7Tn(X,A,x) ;::::: '77 n(Y,B,f(x))
Let g: (Y,B) --c> (X,A) be a homotopy inverse of f. By theorem 14,
there are paths w in A from gf(x) to x and w' in B from fgf(x) to f(x) sueh that
the following diagram is commutative
PROOF
'7T n(X,A,x)
f#~
'77
n(Y,B,f(x))
h r",)
'77
y
hlwi)
n(X,A,gf(x))
~f#
'77
n(Y,B,fgf(x))
Since the maps h[w] and h[w'] are isomorphisms, all the maps in the diagram
are isomorphisms. -
SEC.
4
4
387
THE HUREWICZ HOMOMORPHISM
TilE III'REWU'Z
1I0MO~IORPIIISM
There are no algorithms for computing the absolute or relative homotopy
groups of a topological space (even when the space is given with a triangulation). One of the few main tools available for the general study of homotopy
groups is their comparison with the corresponding integral Singular homology
groups. Such a comparison is effected by means of a canonical homomorphism
from homotopy groups to homology groups. The definition and functorial
properties of this homomorphism are our concern in this section. A theorem
asserting that in the lowest nontrivial dimension for the homotopy group this
homomorphism is an isomorphism will be established in the next section.
We shall be working with the integral singular homology theory throughout this section. Let n :::0: 1 and recall that Hq(In,1n) = 0 for q =1= nand
Hn(In,1n) is infinite cyclic. To consider relations among the homology groups
of certain pairs in In, for n :::0: 1 we define
lIn = {(t 1, ...
lIn = (lIn n in)
12 n = {(h, ...
12n = (I2n n In)
,tn) E In I tn ~ lh}
{(tl, . . . ,tn) E In I tn = Vz}
,tn) E In I tn :::0: lh}
U {(tl, . . . ,tn) E In I tn = Vz}
U
Then lIn U 12n = In and (lIn U 12n) n (11 n U 12n) = lIn U 12n. By the exactness
of the Mayer-Vietoris sequence of the excisive couple {lIn U 12n, lIn U 12n},
we have
Hq(I l n U 12n, lIn U 12n) EB Hq(1 1n U 12n, lIn U 12n) :::::: Hq(In, lIn U 12n)
By excision, we also have isomorphisms
Hq(I1n,11n) :::::: Hq(I l n U 12n, lIn U 12n)
Hq(I2n,12n) :::::: Hq(i1 n U 12n, tIn U 12n)
Combining these, we see that if we let i 1 : (I1 n,11 n) C (In, lIn U 12n) and we
let i 2 : (I2n,12n) C (In, lIn U 12n), then we have the following result.
I
LEMMA
The inclusion maps
i1
and
i2
define a direct-sum representation
h* ED i 2*: Hq(I1n,11n) ED Hq(I2n,12n) :::::: Hq(In, lIn
U 12n)
•
Let VI: (In,1n) ---'? (I1 n,11 n) be defined by V1(t1, ... ,tn) = (t 1, ... ,tn-1,tn/2)
and define V2: (In,1n) ---'? (I2n,12n) by v2(h, ... ,tn) = (t1, ... ,tn-I, (tn + 1)/2).
Let i: (In,1n) C (In, lIn U 12n).
2
COROLLARY
For any z E Hn(In,1n)
i* z
=
h*
V1* Z
+ i 2* V2* Z
Let i1: (In, lIn U 12n) C (In, lIn U 12n) and i2: (In, lIn U 12n) C
(In, tIn U 12n). Then h*i 1* = 0 and i1*i 2* is an isomorphism of Hq(I2n,12n)
PROOF
388
HOMOTOPY THEORY
CHAP.
7
onto Hq(In, lIn U j2 n) (induced by the inclusion map, which is an excision).
Similarly, j2*i2* = 0 and j2*h* is an isomorphism of Hq(I1n,iln) onto
Hq(In, jln U 12n). It follows from lemma 1 that
ker 11*
n ker 12* =
0
Therefore, to prove the corollary it suffices to prove that
is in the kernel of ir* and in the kernel of Iz* .
We first prove that ir* (i* z - il* /11* Z - i z* /1z*z) = O. Because 11* il* = 0,
we must show that iI* i* z = il* i z* /1z* z. Clearly iIi is the inclusion map
(In,in) C (In, h n U jzn) and iIiz/1z is the map f: (In,in) --,) (In, lIn U jzn)
defined by f(tr, . . . ,tn) = (tl, . . . ,tn-I, (tn + 1)/2). A homotopy H from
iIi to f is defined by
H((tl, . . . ,tn), t)
Therefore ir* i* = f*
= (tl,
= jl* iz* /1z*.
... ,tn-I, (tn
+ t)/(l + t))
A similar argument shows that
jz* (i* z - i1* /11* Z
iz* /1Z* Z)
-
=0
•
For n ~ 1 the subset I X jn-l U 0 X In-l C jn is contractible. Therefore Hq(In, I X jn-l U 0 X In-I) = 0 for all q. By exactness of the homology
sequence of the triple (In, jn, I X jn-l U 0 X In-I), it follows that the map
0: Hq(In,in) --,) Hq_l(jn, I X jn-l U 0 X In-I)
is an isomorphism for all q. For n ~ 2 let
j: (In-l,in-l) --,) (jn, I X jn-l U 0 X In-I)
be defined by i(tl, ... ,tn-I) = (1, tl, . . . ,tn-I). Then j is the composite of
a homeomorphism from (In-l,in-l) to (1 X In-I, 1 X jn-l) and the excision
map
(1 X In-I, 1 X jn-l) C (in, I X jn-l U 0 X In-I)
Therefore the homomorphism
i*: Hq(In-l,in-l) --,) Hq(in, I X jn-l U 0 X In-I)
is an isomorphism for all q.
We define canonical generators Zn E Hn(In,in) for n ~ 1 by induction on
n as follows:
(a) Zl E HI(I,i) is the unique element with OZI = (I) - (0) in Ho(i).
(b) For n ~ 2, Zn E Hn(In,in) is the unique element such that
oZn = t* Zn_l in Hn_l(in, I X jn-l U 0 X In-I).
Given a map a: (In,in) --,) (X,A), then a* Zn E Hn(X,A). If a
= /3* Zn. Therefore there is for n ~ I a well-defined map
a* Zn
cp:
7T n(X,A,xo)
--,) Hn(X,A)
c::::::
/3,
then
SEC.
4
389
THE HUREWICZ HOMOMORPHISM
such that <p[a) = a* Zn, where a: (In,in) ---'> (X,A) maps Zo to Xo and represents
an element of 7Tn(X,A,xo). By identifying 7T n(X,XO) with 7T n(X,{Xo},xo), we also
have a map <p: 7T n(X,XO) ---'> Hn(X,xo). Some of the basic properties of <p are
summarized in the next result.
3 THEOREM If n ::::: 2 or if n = 1 and A = {xo}, the map <p is a homomorphism. It has the following functorial properties:
(a) For n ::::: 2 commutativity holds in the square
7Tn(X,A,xo) ~ 7Tn-l(A,xo)
'Pt
t'P
Hn(X,A) ~ Hn-l(A,xo)
(b) Given f: (X,A,xo)
---'>
(Y,B,yo), commutativity holds in the square
7T n(X,A,xo)
~ 7T n(Y,B,yo)
'Pt
t'P
Hn(X,A) ~
PROOF
Let aI, a2: (In,in)
---'>
Hn(Y,B)
(X,A) be such that
al(tl, ... ,tn-I, 1)
= a2(tl, ... ,tn-I, 0)
[any two maps of (In,in) to (X,A) are homotopic to such maps if n ::::: 2 or if
n = 1 and A = {xo}). Then al * a2 = {3o i, where i: (In,in) C (In, i l n U i2n)
and {3: (In, i l n U i2n) ---'> (X,A) is defined by
Then <p[al * a2J = {3* i* Zn = {3* (i~l*Zn + i2*V2*Zn) , the last equality by
corollary 2. Since {3hvl = al and {3i2v2 = a2, we see that
<p[al
* (2) = al*Zn + a2*Zn = <p[al) + <p[(2)
which shows that <p is a homomorphism whenever 7T n(X,A,xo) is a group.
To prove (a), let a: (In,in) ---'> (X,A) represent an element of 7Tn(X,A) for
n ::::: 2 and suppose that a(I X i n- l U 0 X In-I) = Xo. Then a[a) = [a'), where
a': (In-l,in-l) ---'> (A,xo) is defined by a' = (a I (in, I X i n- l U 0 X In-l))
Then
0
cpa[a]
;.
= a~ z,,-l = (al(1n, I X !n-l U 0 X In-l))*j*z,,_l
= (a I (in, I X i n- l U 0 X In-l))* aZ n
aa* Zn
a<p[ a)
=
=
Finally, (b) follows from the fact that (fa)* = f* a*.
•
The map <p is called the Hurewicz homomorphism. The next result follows
from theorem 3.
390
HOMOTOPY THEORY
CHAP.
7
4
COROLLARY
The Hurewicz homomorphism maps the homotopy sequence
of (X,A,xo) into the homology sequence of (X,A,xo). -
Our next objective is to show that the Hurewicz homomorphism commutes with the actions of the appropriate fundamental group on the homotopy
set. We consider the relative case first.
:;
LEMMA
Let [a] E 'lTn(X,A,xo) for n
<p(h[w][a])
~
2 and let [w] E 'lTl(A,xo). Then
= q;[a]
Let [a] be represented by a: (In,in) ~ (X,A) and let h[w][a] be represented by a': (Injn) ~ (X,A). Then a and a' are freely homotopic [that is,
a and a' are homotopic as maps of (Injn) to (X,A)]. Therefore
PROOF
<p[a]
= a* Zn = a~ Zn = <p[a'] = <p(h[w][a])
-
Next we prove the corresponding result for the absolute case.
6
LEMMA
Let [a] E 'lTn(X,xo) and [w] E 'lTl(X,XO). Then
<p(h[w][a])
= <p[a]
PROOF
Let Y be the space obtained from In by collapsing in to a single
point, this point to be the base point of Y, denoted by yo. The collapsing map
g: (Injn) ~ (Y,yo) induces a one-to-one correspondence between [Y,yo; X,xo]
and [Injn; X,xo]. Therefore 'lTn(X,xo) can be identified with [Y,yo; X,xo].
Furthermore, g*: Hn(In,i n) :::::: Hn(Y,yo), and we let g* Zn = Z~ E Hn(Y,yo).
In these terms, if an element of 'lTn(X,xo) is represented by a: (Y,yo) ~ (X,xo),
then <p[a] = a* Z~. Let h[w][a] be represented by a': (Y,yo) ~ (X,xo). Then a
and a' are homotopic as maps of Y to X. Therefore, if Z~ E Hn(Y) is the
unique element such that i~ Z~ = Z~ [where i': Y C (Y,yo)], then
(a I Y)* Z~
Let
1': X C
= (a' I Y)* Z~
(X,xo). Then
<p[ a]
Similarly, <p[ a'] =
i~ (a'
= a* Z~ = a* i~ Z~ = i~ (a I Y)* Z~
I Y)* Z~,
<p[a]
and
= <p[a'] = <p(h[w][a])
-
We define 'lT~(X,A,xo) for n ~ 2 to be the quotient group of 'lTn(X,A,xo)
by the normal subgroup G generated by
{(h[w][a])[a]-l I [a] E 'lTn(X,A,xo), [w] E'lTl(A,xo)}
By lemma 5, <p maps G to 0 and there is a homomorphism
<p':
'lT~(X,A,xo) ~
Hn(X,A)
whose composite with the canonical map 1/: 'lTn(X,A,xo) ~ 'lT~(X,A,xo) is <po
Note that, by theorem 7.3)2, 'lT~(X,A,xo) is abelian for all n ~ 2.
Similarly, we define 'lT~(X,xo) for n ~ 1 to be the quotient group of
SEC.
4
391
THE HUREWICZ HOMOMORPHISM
'lTn(X,xo) by the normal subgroup H generated by
{(h[wJ[a])[a]-ll [a] E 'lTn(X,xo), [w] E 'lT1(X,XO)}
By lemma 6, cp maps H to 0, and there is a homomorphism
cp':
'IT~(X,xo)
--?
Hn(X,xo)
whose composite with the canonical map 1): 'lTn(X,xo) --? 'IT~(X,xo) is cpo Note
that 'lTl(X,xo) is the quotient group of 'lT1(X,XO) by its commutator subgroup.
In particular, 'IT~(X,xo) is abelian for all n :::::: l.
Because the groups 'IT~(X,A,xo) and 'IT~(X,xo) are abelian, we shall find
them easier to compare with the homology groups (which are abelian) than
the homotopy groups themselves. For the comparison it will be convenient to
replace the triple (In,in,zo) , which is the antecedent triple used to define
'lTn(X,A,xo), by the homeomorphic triple (Lln,~n,vo), where Lln is the standard
n-simplex used in Sec. 4.1 to define the singular complex (vertices of Ll n will
be denoted by Vo, V1. . . . ,vn). To achieve this replacement we need only
choose a homeomorphism of (Lln,~n,vo) onto (In,in,zo). Any homeomorphism
h: (Lln,~n) --? (In,in) will induce an isomorphism
h*: Hn(Lln,~n) ::::: Hn(In,in)
The identity map
~n:
Ll n C Ll n is a singular simplex which is a cycle modulo
~n and whose homology class {~n} is a generator of the infinite cyclic group
Hn(Lln,~n). Since Zn is a generator of Hn(In,in) and h* is an isomorphism,
=
=
either h* gn}
Zn or h* {~n}
-Zn. We want to choose h so that the
former holds. If n = 1, the choice of Zl is such that the simplicial homeomorphism h: Ll 1 --? I with h(vo) =
and h(V1) = 1 will have the desired
property (that is, h* {~d = Zl). If n
1, we choose an arbitrary homeomorphism h: (Lln,~n) --? (In,in) such that h(vo) = ZOo If h*{~n} = -Zn, we
replace h by hA., where A. is a simplicial homeomorphism of Ll n to itself such
that A.(vo) = Vo and A.* {~n} = - {~n} (for example, A. is the simplicial map
which interchanges V1 and V2 and leaves all other vertices of Ll n fixed). Therefore, in any event, we can find a homeomorphism h: (Lln,~n,vo) --? (In,in,zo)
such that h* {~n} = Zn. Using such a homeomorphism to represent elements
of 'lTn(X,A,xo) by maps a: (Lln,~n) --? (X,A) such that a(vo) = Xo, we see that
cpr a] = a* {~n} = {a}, the latter being the homology class in (X,A) of the
singular simplex a.
For any pair (X,A) with base point Xo E A and any n :::::: 0, let Ll(X,A,xo)n
be the subcomplex of Ll(X) generated by singular simplexes 0: Llq --? X having
the property that 0 maps each vertex of M to Xo and maps the n-dimensional
skeleton (Llq)n of Llq into A. Then Ll(X,A,xo)n+1 C Ll(X,A,xo)n, and these two
chain complexes agree in degrees :::; n. Thus we have a decreasing sequence
of sub complexes Ll(X,A,xo)n (where n :::::: 0) of Ll(X) whose intersection is contained in Ll(A). If X is path connected and (X,A) is n-connected for some
n :::::: 0, we shall see that the inclusion map Ll(X,A,xo)n C Ll(X) is a chain
equivalence. The following lemma will be used for this purpose.
°
>
392
HOMOTOPY THEORY
CHAP.
7
7
LEMMA Let C be a subcomplex of the free chain complex ~(X) such that
C is generated by the singular simplexes of X in it. Assume that to every singular
simplex cr: ~ q -> X there is assigned a map P( cr): ~ q X I -> X such that
(a) P(o)(z,O) = o(z) for z E /1q.
(b) Define a: /1q ~ X by a(z) = P(o)(z,l). Then a is a singular simplex in
C, and if 0 is in C, a = o.
(c) If eqi : /1q-1 ~ /1q omits the ith vertex, then P(o) (e q i X 1) = P(O(i»).
0
Then the inclusion map C
C
/1(X) is a chain equivalence.
PROOF
Let;: C C /1(X) be the inclusion chain map and let 7': /1(X) ~ C be
the chain map defined by 7'(0) = a [(c) implies that 7' is a chain map]. By (b),
7' a i = Ie, hence to complete the proof we need only verify that i
7' c::-:' 1,,(X).
For any space Y let h o, hI: Y ~ Y X I be the maps ho(y) = (y,O) and
hl(y) = (y,l). In the proof of theorem 4.4.3 it was shown (by the method of
acyclic models) that there exists a natural chain homotopy D: /1(Y) ~ /1(Y X 1)
from /1(ho) to /1(hl)' Define a chain homotopy
0
D': /1(X)
~
/1(X)
by D'(o) = /1(P(o))(D(~q)), where 0: f1q ~ X and ~q: /1q C /1q. By (c), D' is a
chain homotopy, and by (a) and the definition of a, D' is a chain homotopy
from 1,,(X) to i 7'. •
0
8 THEOREM Let Xo E A C X and assume that X is path connected and
(X,A) is n-connected for some n :::;, O. Then the inclusion map /1(X,A,xo)n C /1(X)
is a chain equivalence.
PROOF
For 0: f1q ~ X we define P(o) by induction on q to satisfy the properties oflemma 7, and to have the additional property that if 0 is in /1(X,A,xo)n,
then P(o) is the composite
/1q X 14 /1q
~
X
where p is projection to the first factor.
If q = 0, then 0: /10 ~ X is a point of X, and because X is path connected,
there is a map P(o): /10 X I ~ X such that P(o)(/1° X 0) = 0(/10) and
P(o)(/1° X 1) = Xo [and if 0(/10) = Xo, we take P(o) to be the constant map to
xo]. This defines P(o) for all 0 of degree 0 to have the desired properties.
Assume 0
q ::;: n and that P(o) has been defined for all 0 of degree
q
to have the properties stated above. Given a singular simplex 0: /1q ~ X, if 0
is in /1(X,A,xo)n, define P(o) = 0 p. If 0 is not in /1(X,A,xo)n, (a) and (c) of
lemma 7 define P(o) on /1q X 0 u I1q X I, and we letf: /1q X 0 u I1q X I ~ X
be this map. There is a homeomorphism h: Eq X I::::; f1q X I such that
<
<
0
h(Eq X 0)
= f1q
X 0 u I1q X I,
and
h(Sq-1 X I U Eq X 1)
h(Sq-1 X 0)
= /1q
X 1
= I1q
X 1
SEC.
5
393
THE HUREWICZ ISOMORPHISM THEOREM
Let f': (Eq,Sq-l) - ? (X,A) be defined by f'(z) = f(h(z,O)). Because q ~ nand
(X,A) is n-connected, there is a homotopy H: (Eq,Sq-l) X I - ? (X,A) from f'
to some map of Eq into A (in fact, by the definition of n-connectedness, there
is even such a homotopy relative to Sq-l). Then the composite
!:,.q X I ~ Eq X I ~ X
can be taken as P(a).
In this way P(a) is defined for all degrees q ~ n. Note that a singular
simplex of degree> n is in !:"(X,A,xo)n if and only if every proper face is in
!:"(X,A,xo)n. Therefore, if P(a) has been defined for all degrees
q, where
q > n, and if a: !:,.q - ? X, then we define P(a) = a p if a is in !:"(X,A,xo)n and
to be any map !:,.q X I - ? X satisfying (a) and (c) of lemma 7 (such maps exist
by the homotopy extension property). Then P(a) will necessarily satisfy (b) of
lemma 7, and we have shown that P(a) can be defined for all a to satisfy
lemma 7. •
<
0
For n
>
°
we define
There are canonical homomorphisms
. . . -?
Hq(n)(X,A,xo)
-?
Hq(n-l)(X,A,xo)
-? . . . -?
Hq(O)(X,A,xo)
-?
Hq(X,A)
9
COROLLARY
Assume that A is path connected and for some n :2 0,
(X,A) is n-connected. Then the canonical map is an isomorphism for all q
Hq(n)(X,A,xo) ::::::: Hq(X,A)
PROOF
For any n :2 0, !:"(X,A,xo)n n !:"(A) is generated by the set of singular
simplexes of A all of whose vertices are at Xo. This is independent of n, and
because A is path connected, (A,{ xo}) is O-connected, and it follows from
theorem 8 that the inclusion map !:"(X,A,xo)n n !:"(A) C !:"(A) is a chain
equivalence for all n :2 0.
Since (X,A) is n-connected, where n :2 0, and A is path connected, X is
also path connected, and by theorem 8, the inclusion map !:"(X,A,xo)n C !:"(X)
is a chain equivalence. The result follows from these facts, using exactness
and the five lemma. •
5
TilE IIfTREWJ('Z ISOMORPIIIS!\<1 TIIEOREM
The main result of this section asserts that if X and A are path connected and
for some n :2 1, (X,A) is n-connected, then the Hurewicz homomorphism cp
induces an isomorphism cp' of 7T~+l(X,A,xo) with Hn+l(X,A). This result is
equivalent to a homotopy addition theorem which asserts that the sum of the
(n + I)-dimensional faces of an (n + 2)-simplex is the homotopy boundary of
the identity map of the simplex. We prove both these theorems Simultaneously
by induction on n.
394
HOMOTOPY THEORY
CHAP.
7
In the proof we shall make essential use of the complexes A(X,A,xo)n and
of corollary 7.4.9. Let ex: (An,Lin,(An)O) ~ (X,A,xo) represent an element of
'1Tn(X,A,xo). Then ex is a singular simplex in A(X,A,xo)n-l and represents a
homology class {ex} E Hn(n-l)(X,A,xo). Since any element of '1Tn(X,A,xo) can be
represented by such a map ex, the Hurewicz homomorphism cp': '1T~(X,A,xo) ~
Hn(X,A) factors into the composite
'1T~(X,A,xo) ~ Hn(n-l)(X,A,xo) ~ Hn(X,A)
and there is a commutative diagram
'1Tn(X,A,xo) ~
q>~
V
Hn(X,A)
~
Hn(n-l)(X,A,xo)
We now formulate the propositions corresponding to the relative and
absolute Hurewicz isomorphism theorems.
I
PROPOSITION
tP n (n 2:: 2). Let A be path connected and let (X,A) be
(n - I)-connected. Then cp' is an isomorphism
cp':
2
PROPOSITION
'1T~(X,A,xo) ~
Hn(X,A)
<l>n (n 2:: 1). Let X be (n - I)-connected. Then cp' is an
isomorphism
cp':
'1T~(X,xo) ~
Hn(X,xo)
We shall prove both these propositions simultaneously by induction on n,
together with a third proposition, which we now formulate. For n 2:: 2, each
face map eA+l is a map of triples
eli+l: (An,Lin,vo) ~ (Lin+1,(An+l)n-l,Vl)
eA+l: (An,Lin,vo) ~ (Lin+1,(An+l)n-l,vo)
0
<i ~ n + 1
For vertices v and v' of An+l we use [vv'] to denote the path class of the
linear path in An+1 from v to v'. We define an element b l E '1Tl(Li2,VO) and, for
n 2:: 2, an element bn E '1Tn(Lin+l,(An+l )n-l,vo) by
hi = [VoVI] * [VI V2] * [V2 VO]
b 2 = (h[vovd[e3oJ)[e32J[e3l]-l[e33]-l
b n = h[vovd[e~+l] +
~
(-I)i[eJ,+l]
Ods;n+l
n 2:: 3
For n = 1 let i: (Li2,VO) C (A2,VO) and for n 2:: 2 let 1: (Lin+l,(An+l)n-l,vo) C
(An+1,(An+l)n-l,vo). The following proposition corresponds to the homotopy
addition theorem.
3
PROPOSITION
Bn (n 2::
1).
l#Jn = O.
The simultaneous proof of propositions 1, 2, and 3 will consist of the following five parts:
SEC.
5
395
THE HUREWICZ ISOMORPHISM THEOREM
(a) Proof of B1
(b) Proof that B1
1>1
(c) Proof that 1>1, 1>2,.
, 1>n-1
(d) Proof that Bn
<Pn for n ::;, 2
(e) Proof that <Pn
1>n for n ::;, 2
=
=
=
= Bn for n ::;, 2
(a) PROOF OF B1 We must prove that i#b 1 = O. But
7Tl(il2,VO) = 0 because il 2 is contractible. -
i~l
E 7T1(il2,VO), and
=
(b) PROOF THAT B1
1>1 Let X be path connected. We must prove that
cp': 7Ti(X,xo) :::::; H1(X,XO). Because X is path connected, the inclusion map
il(X,{xo},xo)O C il(X) is a chain equivalence, and we need only show that
cp": 7Ti(X,xo):::::; H1(0)(X,{xo},xo)
If a: (il1,Li1) ~ (X,xo) represents an element raJ' E 7Ti(X,xo), then
= {a}, where {a} is the homology class in H 1(0)(X,{xo},xo) of the
singular cycle a. Given a singular I-simplex a: (ill,Li1) ~ (X,xo) in il(X,{xo},xo)O,
it determines an element [a] E 7T1(X,XO), and therefore an element
[aJ' E 7Ti(X,xo). If a is the constant singular I-simplex at Xo, then clearly,
[a], = O. Because 7Ti(X,xo) is abelian and il1(X,{XO},xo)0 is the free abelian
group generated by the singular simplexes in it, there is a homomorphism
cp"[a]'
1/;: il1(X,{XO},xo)0/il1(XO)
~
7Ti(X,xo)
such that 1/;(a) = [aJ'. We shall show, by using B1, that the composite
il2(X,{xo},xo)0/il 2(xo) ~ il 1(X,{xo},xo)0/il 1(xo)'±' 7Ti(X,xo)
is trivial. Given a: (il2,(il2)0)
as usual. Then
1/;o[a]
~
(X,xo), let a(O), a(1), and a(2) be the faces of a,
= [a(2)]' + [a(O)]'
=
- [a(1)]'
TJ(U' I A2)#([voVt]
= [(a(2) * a(O)) * (a(1)t1J'
* [V1V2] * [V2VO]) =
rjU' #J#b1 =
0
Therefore 1/; defines a homomorphism
1/;': H1(0)(X,{xo},xo)
~
and this is easily seen to be an inverse of cp".
(c)
PROOF THAT
1>1, ... , 1>n-1
diagram
= Bn
FOR
7Ti(X,xo)
-
n ::;, 2
Consider the commutative
7T n(il n+l,(il n+1 )n-1, vol
7T n+1( iln+l,Lin+l,vo)
\.'"
j"J"
7Tn(Li n+1,( il n+1)n-1 ,vol
The top row, being part of the homotopy sequence of the triple
(iln+1,Lin+1,(iln+1)n-1), is exact. The bottom row, being part of the homotopy
396
HOMOTOPY THEORY
CHAP.
7
sequence of the pair (~n+l,(~n+1)n-l), is also exact. From the exactness of the
homotopy sequence of the pair (~n+l,~n+l) and the fact that ~n+l is contractible, it follows that a is an isomorphism. Therefore
ker i#
= im a' = im (i#
a) = im
0
i#
= ker a"
Thus Bn is equivalent to the equation a"(b n) = O. We prove the latter, giving
one proof for n = 2 and another for n
2.
If n = 2, we have
>
a" (b 2) =
(h[vovd a"[ e3 0]) a"[ e3 2]a"[ e3 1]-la"[ e3 3]-1
To calculate a"[e3 i ], let ~: (~2,~2,VO) C (~2,~2,VO) be the identity map. Then
[~] E '1T2(~2,~2,VO), and because '1Tl(~2,VO) is infinite cyclic (since ~2 is homeomorphic to SI), it follows from <PI that cp: '1Tl(~2,VO) ::::: H1(~2,VO)' There is a
commutative square
'1T2(~2,~2,VO) ~ '1Tl(~2,VO)
'1'1
;:::;~'P
-4
H2(~2,~2)
and
acp[~]
= a{o = W2) + ~(O)
H1(~2,VO)
-
= {w} = cp[w]
W 2) * ~(O») * (~(1»)-1.
~(1)}
where w: (~1,~1) --') (~2,VO) is the path w =
(The 2-chain
~(2) + ~(O) _ ~(1) is homologous to w because it is easy to find singular
2-simplexes 01 and 02 in ~2 such that
01(0) =
02(0) =
Then a( 01 follows that
01(1) =
02(1) =
~(O)
~(1)
02) =
~(2)
+
* ~(O)
~(2) * ~(O)
01(2) = ~(2)
02(2) = W2)
~(2)
~(O) -
~(1)
* ~(O») * (~(1»)-1
w.) Because cp is an isomorphism, it
-
To return to the calculation of a"[e3 i ], we have
a"[e3 i ]
= a"(e3 i )#W = (e3 i I ~2)#a[~]
= [e3 1(vO)e3 1(v1)] * [e3 1(V1)ea l (V2)] * [e3 i (v2)e3 t (vO)]
Using this, direct substitution into the right-hand side of the equation for
a"(b 2) shows that a"(b 2) = O.
For n
2 note that (~n+1)n-l contains the two-dimensional skeleton of
~n+1. Therefore (~n+l )n-l is simply connected (because ~n+l is simply connected). Similarly, for q .:s; n - 2, Hq((~n+1)n-l,vo)::::: Hq(~n+1,vo) = O. By
<PI, ... , <P n- 2, it follows that (~n+1)n-l is (n - 2)-connected, and by <P n- 1,
there is an isomorphism
>
cp:
'1Tn_l((~n+l)n-l,vo)
:::::
Hn_l((~n+l)n-l,vo)
Hence, to complete the proof it suffices to show that cpa"(b n) = O. This
follows from the equalities
SEC.
5
qJo"(bn)
(d)
397
THE HUREWICZ ISOMORPHISM THEOREM
= o"qJ(bn) = 0" {~ (-I)ieh+d = 0"0' {~n+d = o"i*o{ ~n+d = 0
Bn
= <Pn
•
n ~ 2 The argument is similar to the proof of
part (b) above. The map qJ' factors into the composite
PROOF THAT
FOR
'7T~(X,A,xo) ~ Hn(n-l)(X,A,xo) :? Hn(X,A)
If a: (Lln,6 n,vo) ----,) (X,A,xo) is a map such that a maps all the vertices to Xo,
then qJ"[a], = {a} E Hn(n-l)(X,A,xo). To define an inverse of qJ", if
a: (Lln,6 n,(Lln)0) ----,) (X,A,xo) is a singular simplex in Lln(X,A,xo)n-l, then
[a) E '7T n(X,A,xo) and 1/[a) = [a], E '7T~(X,A,xo). If a(Lln) C A, then [a)' = 0,
and because '7T~(X,A,xo) is abelian, there is a homomorphism
1/;: Lln(X,A,xo)n-l/(Lln(X,A,xo)n-l n Lln(A)) ----,)
'7T~(X,A,xo)
such that 1/;(a) = [a)'.
We show that the composite
1/;
0
0: Lln+l(X,A,xo)n-l/(Lln+l(X,A,xo)n-l
n
Lln+l(A)) ----,)
'7T~(X,A,xo)
is trivial. This follows from Bn , because if
a: (Lln+1 ,(Lln+l )n-l ,(Lln+l )0) ----,) (X,A,xo)
then
1/;o(a)
= ~ (-I)i[a(i)]' = 1/(a I (6 n+1,(Lln+l)n-l)#(bn))
= 1/a#i#(bn) = 0
Therefore 1/; defines a homomorphism
1/;': Hn(n-l)(X,A,xo) ----,)
such that 1/;' {a}
= [a)',
=
'7T~(X,A,xo)
and 1/;' is easily seen to be an inverse of qJ".
•
(e) PROOF THAT <Pn
<I>n FOR n ~ 2 For n ~ 2, if X is (n - I)-connected,
then the pair (X,{xo}) is (n - I)-connected and '7T~(X,{xo},xo) is canonically
isomorphic to '7T~(X,xo) = '7T n(X,xo). Then <l>n results from <Pn applied to the
pair (X, {xo}). •
This completes the proof of propositions 1, 2, and 3. From proposition 1
we obtain the following relative Hurewicz isomorphism theorem.
4
THEOREM
Let Xo E A C X and assume that A and X are path connected.
If there is an n ~ 2 such that '7Tq(X,A,xo) = 0 for q < n, then Hq(X,A) = 0
for q < nand qJ' is an isomorphism
qJ':
'7T~(X,A,xo)
;::::; Hn(X,A)
Conversely, if A and X are simply connected and there is an n ~ 2 such that
Hq(X,A) = 0 for q < n, then '7Tq(X,A,xo) = 0 for q < nand qJ is an isomorphism
qJ: '7Tn(X,A,xo) ;::::; Hn(X,A)
•
Similarly, from proposition 2 we obtain the following absolute Hurewicz
isomorphism theorem.
398
li
HOMOTOPY THEORY
THEOREM
'1Tq(X,xo)
CHAP.
7
Let Xo E X and assume that there is n ~ 1 such that
= 0 for q < n. Then Hq(X,xo) = 0 for q < nand cp' is an isomorphism
cp':
'1T~(X,xo)
::::::; Hn(X,xo)
Conversely, if X is simply connected and there is n ~ 2 such that Hq(X,xo)
for q < n, then '1Tq(X,xo) = 0 for q < nand cp is an isomorphism
cp: '1Tn(X,xo) ::::::; Hn(X,xo)
=0
•
In the absolute case when X is simply connected and in the relative case
when X and A are simply connected, each of these theorems asserts that the
first nonvanishing homotopy group is isomorphic to the first nonvanishing
homology group.
6
COROLLARY
For n
~
1 there is a commutative diagram of isomorphisms
'1T n+1(En+1,Sn,po) ~ '1Tn(Sn,po)
~~
~~
Hn+l(En+l,Sn)
~ Hn(Sn,po)
PROOF
The diagram is commutative, by theorem 7.4.3a, and both horizontal
maps are isomorphisms because En+1 is contractible [and because the homotopy and homology sequences of (En+1,Sn,po) are exact]. The right-hand vertical map is an isomorphism, by proposition 2 and the fact that (in the case
n = 1) '1Tl(Sl,PO) is abelian. •
The following useful consequence of corollary 6 is called the Brouwer
degree theorem.
7 COROLLARY For n ~ 1 two maps f, g: Sn ~ Sn are homotopic if and
only if f* = g* : Hn(Sn) ~ Hn(Sn). Similarly, two maps f, g: (En+1,Sn) ~
(En+1,Sn) are homotopic if and only if f* = ~: Hn+l(En+l,Sn) ~ Hn+l(En+l,Sn).
We consider the absolute case first. Given maps f, g: Sn ~ Sn, there
exist homotopic maps f' and g', respectively, such that f'(po) = g'(po) = po
(because Sn is path connected). Because Sn is n-simple, f' and g' are freely
homotopic if and only if they are homotopic as maps from (Sn,po) to (Sn,po).
Therefore f ~ g if and only if [f'] = [g'] in '1Tn(Sn,po). By corollary 6,
[f']
[g'] if and only if cp[f']
cp[g'], and from the definition of cp,
cp[f'] = cp[g'] if and only if
PROOF
=
=
f~
= ~:
Hn(Sn,po)
~
Hn(Sn,po)
Since there are commutative squares
Hn(Sn) ?
f.~
Hn(Sn) ?
the result follows.
Hn(Sn,po)
~f*
Hn(Sn,po)
Hn(Sn) ?
g.~
Hn(Sn) ?
Hn(Sn,po)
19,;
Hn(Sn,po)
SEC.
5
399
THE HUREWICZ ISOMORPHISM THEOREM
For the relative case note that because En+l is contractible, it follows
from the homotopy extension property of (En+1,Sn) that two maps
f, g: (En+1,Sn) -? (En+1,Sn) are homotopic if and only if fl Sn, g I Sn: Sn -? Sn
are homotopic. Since there are commutative squares
Hn+l(En+l,sn) ~ Hn(Sn)
1.1
Hn+l(En+l,sn)
1(/18")*
COROLLARY
Hn(Sn)
g.l
the relative case follows from the absolute case.
8
b
l(glsn).
•
For Xo E X the map
l{;: [Sn,po; X,xol-? Hom (7Tn(Sn,po), 7Tn(X,XO))
sending [0'] to 0'# is an isomorphism.
PROOF
This follows from corollary 6, because the fact that 7T n(Sn,po) is infinite
cyclic implies that there is an isomorphism
[3: Hom (7Tn(Sn,po), 7Tn(X,XO)) :::::: 7Tn(X,XO)
sending a homomorphism A to A(a), where a E 7T n(sn,po) is the homotopy
class of the identity map. Then, ([30 l{;)[O'l = O'#(a) = [0'], and so l{; is an
isomorphism. •
The following useful consequence of the relative Hurewicz isomorphism
theorem is known as the Whitehead theorem.
9
f:
Let X and Y be path-connected pointed spaces and let
(Y,yo) be a map. If there is n ~ 1 such that
THEOREM
(X,xo)
-?
f#: 7Tq(X,XO)
is an isomorphism for q
-?
7Tq(Y,yO)
< n and an epimorphism for q = n, then
f* : Hq(X,xo)
-?
Hq(Y,yo)
<
is an isomorphism for q
n and an epimorphism for q = n. Conversely, if
X and Yare simply connected and f* is an isomorphism for q
n and an
epimorphism for q = n, then f # is an isomorphism for q
n and an epimorphism for q = n.
<
PROOF
<
Let Z be the mapping cylinder of f. There are inclusion maps
i: X C Z and i: Y C Z and a deformation retraction r: Z - ? Y such that
f = r i. Then r: (Z,yo) -? (Y,yo) induces isomorphisms r#: 7Tq(Z,yO) :::::: 7Tq(Y,yO)
0
and T*: Hq(Z,yo) :::::: Hq(Y,yo) for all q. Because X and Yare path connected,
so is Z, and 7Tq(Z,XO) :::::: 7Tq(Z,yO)' Therefore r: (Z,xo) -? (Y,yo) also induces
isomorphisms r#: 7Tq(Z,XO) :::::: 7Tq(Y,yO) and r*: Hq(Z,xo) :::::: Hq(Y,yo) for all q.
It follows that we can replace (Y,yo) in the theorem by (Z,xo) and the conditIuns un f# ana f* by the corresponding conditions on i# aud i*. From the
exactness of the homotopy sequence of (Z,X,xo), it follows that i# is an
400
HOMOTOPY THEORY
CHAP.
7
<
isomorphism for q
n and an epimorphism for q = n if and only if
1T.q(Z,X,xo) = 0 for q ::::: n. Similarly, from the exactness of the homology
sequence of the triple (Z,X,xo), it follows that i* is an isomorphism for q
n
and an epimorphism for q = n if and only if Hq(Z,X) = 0 for q ::::: n. The
result now follows from the relative Hurewicz isomorphism theorem 4. •
<
6
CW COMPLEXES
For homotopy theory the most tractable family of topological spaces seems to
be the family of CW complexes (or the family of spaces each having the same
homotopy type as a CW complex). CW complexes are built in stages, each
stage being obtained from the preceding by adjoining cells of a given dimension. The cellular structure of such a complex bears a direct connection with
its homotopy properties. Even for such nice spaces as polyhedra it is useful to
consider representations of them as CW complexes, because such complexes
will frequently require fewer cells than a simplicial triangulation.
In this section we shall investigate CW complexes and related concepts.
In Sec. 7.8 we shall show that any topological space can be approximated by
a CW complex which is unique up to homotopy. We begin with some results
about a space X obtained from a subspace A by adjoining n-cells (defined in
Sec. 3.8).
I
LEMMA
If X is obtained from A by ad;oining n-cells, then X X 0 U A X I
is a strong deformation retract of X X I.
PROOF
For each n-cell ejn of X - A let
J+J"..
(En Sn-l)
~
(e·Jn, e·Jn)
be a characteristic map. Let D: (En X I) X I ~ En X I be a strong deformation retraction of En X I to En X 0 U Sn-l X I (which exists, by corollary
3.2.4). There is a well-defined map Dj : (ejn X 1) X I ~ ejn X I characterized
by the equation
Dj((fiz),t), t') =
(fi
X lI)(D(z,t,t'))
z E En; t, t' E I
Then there is a map D': (X X 1) X I ~ X X I such that D' I (ej X 1) X I = D j
and D'(a,t,t') = (a,t) for a E A, and t, t' E I, and D' is a strong deformation
retraction of X X I to X X 0 U A X I. •
2
COROLLARY
If X is obtained from A by ad;oining n-cells, then the
inclusion map A C X is a cofibration. •
3
LEMMA
Let X be obtained from A by ad;oining n-cells and let (Y,B) be
a pair such that 7T n(Y,B,b) = 0 for all b E B if n 2 1 and such that every
point of Y can be ;oined to B by a path if n = O. Then any map from (X,A)
to (Y,B) is homotopic relative to A to a map from X to B.
SEC.
6
401
CW COMPLEXES
PROOF
This follows from theorem 7.2.1 by a technique similar to that in
lemma 1 above. •
A relative CW complex (X,A) consists of a topological space X, a closed
subspace A, and a sequence of closed subspaces (X,A)k for k 2 0 such that
(a)
(b)
(c)
(rI)
(X,A)O is obtained from A by adjoining O-cells.
For k 2 1, (X,A)k is obtained from (X,A)k-l by adjoining k-cells.
X = U (X,A)k.
X has a topology coherent with {(X,A)k}k.
In this case (X,A)k is called the k-skeleton of X relative to A. If X = (X,A)n
for some n, then we say dimension (X - A) :::;; n. An absolute CW complex X
is a relative CW complex (X, 0), and its k-skeleton is denoted by Xk.
Following are a number of examples.
4 If (K,L) is a simplicial pair, there is a relative CW complex (IKI,ILI),
with (IKI,ILl)k = IKk U LI.
:. If (X,A) is a relative CW complex, for any k the pair (X, (X,A)k) is a relative CW complex, with
Similarly, the pair ((X,A)k, A) is a relative CW complex, with
As in example 3.8.7, for i = 1, 2, or 4 let Fi be R, C, or Q, respectively,
6
and for q 2 0 let Pq(Fi) be the corresponding projective space of dimension q
over Fi . Then Pq(Fi) is a CW complex, with
k :::;; iq
> iq
for k < n k
7 En is a CW complex, with (En)k
and (En)k = En for k 2 n.
8
I is a CW complex, with (1)0
= po
1, (En)n-l
= i and (1)k = I for k 2
= Sn-l,
1.
9 If (X,A) and (Y,B) are relative CW complexes and either X or Y is locally
compact, then (X,A) X (Y,B) is also a CW complex,l with
((X,A) X (Y,B))k
= Ui+j=k (X,A)i
X (Y,B)j U X X B U A X Y
10 If (X,A) is a relative CW complex, so is (X,A) X I, with
(X X I, A X 1)k
= (X,A)k
X
i
U (X,A)k-l X I U A X I
1 It is not true that the product of two CW complexes is always a CW complex. For a counterexample, see C. H. Dowker, Topology of metric complexes, American Journal of Mathematics,
vol. 74, pp. 555-577, 1952.
402
HOMOTOPY THEORY
CHAP.
7
I I If (X,A) is a relative CW complex, then XI A is a CW complex, with
(XI A)k = (X,A)k I A.
A subcomplex (Y,B) of a relative CW complex (X,A) is a relative CW
complex such that Y is a closed subset of X and (Y,B)k = Y n (X,A)k for all k.
If (Y,B) is a subcomplex of (X,A), then (X, A U Y) is a relative CW complex,
with (X, A U Y)k = (X,A)k U Y for all k. In particular, if X is a CW complex
and A is a subcomplex of X, then (X,A) is a relative CW complex. A CW pair
(X,A) consists of a CW complex X and subcomplex A (hence a CW pair is a
relative CW complex).
The definition of relative CW complex suggests its inductive construction.
We start with a space A, attach O-cells to A to obtain a space Ao, attach I-cells
to A o to obtain AI, and continue in this way to define Ak for all k ::;> 0.
Letting X be the space obtained by topologizing U Ak with the topology
coherent with {Akh?o, then (X,A) is a relative CW complex, with (X,A)k = A k.
12 THEOREM If (X,A) is a relative CW complex, then the inclusion map
A C X is a cofibration.
PROOF
This follows from corollary 2, using induction and the fact that X X I
has the topology coherent with {(X,A)k X Ih. •
13 THEOREM Let (X,A) be a relative CW complex, with dimension
(X - A) S; n, and let (Y,B) be n-connected. Then any map from (X,A) to
(Y,B) is homotopic relative to A to a map from X to B.
This follows, using induction, from corollary 7.2.2, lemma 3, and
theorem 12. •
PROOF
14 COROLLARY Let (X,A) be a relative CW complex and let (Y,B) be
n-connected for all n. Then any map from (X,A) to (Y,B) is homotopic relative to A to a map from X to B.
Let f: (X,A) ~ (Y,B) be a map. It follows from theorems 12 and 13
that there is a sequence of homotopies
PROOF
H k : (X,A) X I
~
(Y,B)
constructed by induction on k such that
(a)
(b)
(c)
(d)
Ho(x,O) = f(x) for x E X.
H k(x,l) = Hk+1(X,O) for x E X.
Hk is a homotopy relative to (X,A)k-l.
Hk((X,A)k X 1) C B.
Then a homotopy H: (X,A) X I
defined by
H(x,t)
= Hk- 1( x,
~
(Y,B) with the required properties is
t - (1 - 11k) )
(11k) _ I/(k + 1)
1
1
I--<t<I--k- k+I
x E (X,A)k
•
15 LEMMA If X is obtained from A by adjoining n-cells, then for n ::;> 1,
(X,A) is (n - I)-connected.
SEC.
6
403
CW COMPLEXES
PROOF
For k ~ n - 1 let f: (Ek,Sk-l) --') (X,A) be a map. Because f(Ek) is
compact, there exist a finite number, say, e}, . . . , em, of n-cells of X - A
such that f(Ek) C el U ... U em U A. For 1 ~ i ~ m let Xi be a point of
ei - ei. Each of the sets Y = A U (el - Xl) U ... U (em - xm) and ei - ei
for 1 ~ i ~ m intersects f(Ek) in a set open in f(Ek). There is a simplicial
triangulation of Ek, say K, such that (identifying IKI with Ek) for every
simplex 8 E K either f(181) C Y or for some 1 ~ i ~ m, f(181) C ei - ei' Let A'
be the subpolyhedron of Ek which is the space of all simplexes 8 E K such
that f(181) C Y, and for 1 ~ i ~ m let Bi be the subpolyhedron which is the
space of all simplexes 8 of K such that f(181) C ei - ei. Then Sk-l C A',
Ek = A' U BI U ... U Bm, and if i =1= f, then Bi - A' is disjoint from
Bj - A'. Let 13i = Bi n A' and observe that (Bi,13 i) is a relative CW complex,
with dim (Bi - 13i ) ~ k ~ n - 1.
For 1 ~ i ~ m the pair ((ei - ei), (ei - ei) - Xi) is homeomorphic to
(En - Sn-l, (En - Sn-l) - 0) and has the same homotopy groups as (En,Sn-l).
By corollary 7.2.4, (En,Sn-l) is (n - I)-connected. It follows from theorem 13
that f 1 (B;,13 i ) is homotopic relative to 13i to a map from Bi to (ei - ei) - Xi.
Because Bi - 13i is disjoint from Bj - 13j for i =1= f, these homotopies fit
together to define a homotopy relative to A' of f to some map f' such that
f'(Ek) C Y. Clearly, A is a strong deformation retract of Y. Therefore f' is
homotopic relative to Sk-l to a map f" such that f"(Ek) C A. Then
f ~ f' ~ f", all homotopies relative to Sk-l. Therefore (X,A) is (n - 1)connected. •
16 COROLLARY If (X,A) is a relative CW complex, then for any n 2:: 0,
(X, (X,A)n) i8 n-connected.
PROOF
We prove by induction on m that ((X,A)m, (X,A)n) is n-connected for
m> n. Since (X,A)n+1 is obtained from (X,A)n by adjoining (n + I)-cells,
it follows from lemma 15 that ((X,A)n+1, (X,A)n) is n-connected. Assume
m
n + 1 and that ((X,A)m-l, (X,A)n) is n-connected. By lemma 15, the pair
((X,A)m, (X,A)m-l) is (m - I)-connected, and since n
m - 1, it is also
n-connected. Then '1To((X,A)n) --') '1To((X,A)m-l) and '1To((X,A)m-l) --') '1To((X,A)m)
are both surjective, whence '1To((X,A)n) --') '1To((X,A)m) is also surjective.
Furthermore, for any X E (X,A)n, it follows from the exactness of the homotopy
sequence of the triple ((X,A)m, (X,A)m-l, (X,A)n), with base pOint x, that
'1Tk((X,A)m, (X,A)n, x) = 0 for 1 ~ k ~ n. By corollary 7.2.2, ((X,A)m, (X,A)n)
is n-connected.
To show that (X, (X,A)n) is n-connected, if 0 ~ k ~ n and a: (Ek,Sk-l) --')
(X, (X,A)n), then because a(Ek) is compact and X has a topology coherent
with the subspaces (X,A)m, there is m
n such that a(Ek) C (X,A)m. Hence
a can be regarded as a map from (Ek,Sk-l) to ((X,A)m, (X,A)n) for some m
n.
Because ((X,A)m, (X,A)n) is n-connected, a is homotopic relative to Sk-l to
some map of Ek to (X,A)n. •
>
<
>
>
Given relative CW complexes (X,A) and (X',A'), a map f: (X,A) --') (X',A')
is said to be cellular if f((X,A)k) C (X',A')k for all k. Similarly, a homotopy
F: (X,A) X I --') (X',A') is said to be cellular if F((X,A) X I)k C (X',A')k for
404
HOMOTOPY THEORY
CHAP.
7
all k. Analogous to the simplicial-approximation theorem is the following
cellular-approximation theorem.
17 THEOREM Given a map f: (X,A) ~ (X',A') between relative CW complexes which is cellular on a subcomplex (Y,B) of (X,A), there is a cellular map
g: (X,A) ~ (X',A') homotopic to f relative to Y.
PROOF
It follows from corollary 16, theorem 13, and theorem 12 that there
is a sequence of homotopies H k: (X,A) X I ~ (X' ,A') relative to Y, for k ~ 0,
such that
(a)
(b)
(c)
(d)
Ho(x,O) = f(x) for x E X.
Hk(x,I) = Hk+l(X,O) for x E X.
Hk is a homotopy relative to (X,A)k-l.
Hk((X,A)k X 1) C (X',A')k.
Then a homotopy H: (X,A) X I
defined by
H(x,t)
= Hk- 1 ( x,
~
(X',A') with the desired properties is
t - (1 - 11k) )
(11k) _ I/(k + 1)
H(x,I)
= Hk(x,I)
1
1
< t < 1 - -k+I
-k- -
1- -
x E (X,A)k
-
18 COROLLARY Any map between relative CW complexes is homotopic to
a cellular map. If two cellular maps between relative CW complexes are
homotopic, there is a cellular homotopy between them. -
A continuous map f: X ~ Y is called an n-equivalence for n ~ 1 if f
induces a one-to-one correspondence between the path components of X and
of Y and if for every x E X, f#: 7Tq(X,X) ~ 7Tq(Y,f(x)) is an isomorphism for
q n and an epimorphism for q = n (the condition concerning the case
q
n is sometimes omitted in the definitions occurring in the literature).
A map f: X ~ Y is called a weak homotopy equivalence or oo-equivalence if
f is an n-equivalence for all n ~ 1. The following results are immediate from
the definition and from corollary 7.3.15.
°<
<
=
19 A composite of n-equivalences is an n-equivalence.
-
20 Any map homotopic to an n-equivalence is an n-equivalence.
2 I A homotopy equivalence is a weak homotopy equivalence.
-
-
Let f: X ~ Y be a map and let Z, be the mapping cylinder of f. Then
f = r i, where r: Z, ~ Y is a homotopy equivalence. Therefore f is an
n-equivalence if and only if i: X C Z, is an n-equivalence. It follows from the
exactness of the homotopy sequence of (Z"X) and from corollary 7.2.2 that i
is an n-equivalence if and only if (Z"X) is n-connected.
0
22 THEOREM Let f: X ~ Y be an n-equivalence (n finite or infinite) and
let (P,Q) be a relative CW complex, with dim (P - Q) ~ n. Given maps
g: Q ~ X and h: P ~ Y such that h I Q = fog, there exists a map g': P ~ X
such that g' I Q g and fog' ~ h relative to Q.
=
PROOF
Let Z, be the mapping cylinder of f, with inclusion maps i: X C Z,
SEC.
6
405
cw COMPLEXES
and j: Y C Zr, and retraction r: Zr ~ Ya homotopy inverse of j. Then in
x
~ Zr
a homotopy i g ~ j h I Q can be found whose composite with r is constant.
By theorem 12, there is a map h': P ~ Zr such that h' I Q = i g and such that
r h' ~ r j h relative to Q. We regard h' as a map from (P,Q) to (Zr,X).
Since (Zr,X) is n-connected and dim (P - Q) ~ n, it follows from theorem 13
that h' is homotopic relative to Q to some map g': P ~ X. Then g' I Q = g and
0
0
0
0
0
0
fog'
=r
i
0
0
g'
~
r h'
0
~
r
0
j h
0
=h
all the homotopies being relative to Q. Hence g' has the desired properties. •
23
Let f: X
COROLLARY
~ Y
be an n-equivalence (n finite or infinite) and
consider the map
f#: [P;X]
~
If P is a CW complex of dimension
~ n - 1, it is injective.
~
[P;Y]
n, this map is surjective, and if
dim P
The first part follows from theorem 22 applied to the relative
CW complex (P, 0).
For the second part, we apply theorem 22 to the relative CW complex
(P X I, P X j). Given go, gl: P ~ X such that fogo ~ f gl, there is a map
g: P X j ~ X such that g(z,O) = go(z) and g(z,I) = gl(Z) for z E P and a map
h: P X I ~ Y such that hiP X i = fog. Since dim (P X I) ~ n, by theorem 22
there is a mapping g': P X I ~ X such that g' I P X j = g. Then g' is a
homotopy from go to gl, showing that [go] = [gl]' •
PROOF
0
24 COROLLARY A map between CW complexes is a weak homotopy equivalence if and only if it is a homotopy equivalence.
It follows from statement 21 that a map which is a homotopy equivalence is always a weak homotopy equivalence. Conversely, if f: X ~ Y is a
weak homotopy equivalence between CW complexes, it follows from corollary 23 that f induces bijections
PROOF
= [lv]' then fog ~ lv, and also
fl = [lv fl = [f Ix] = f#[lx]
If g: Y ~ X is any map such that f#[g]
f#[g
Therefore [g
0
0
fl = [f
fl = [Ix]
0
or g
g
0
0
f
0
~
0
lx, and so f is a homotopy equivalence. •
Thus, for CW complexes the concepts of homotopy equivalence and weak
homotopy equivalence coincide. The following theorem is a direct consequence
of the Whitehead theorem 7.5.9.
406
HOMOTOPY THEORY
CHAP.
7
25 THEOREM A weak homotopy equivalence induces isomorphisms of the
corresputLding integral singular homology groups. Conversely, a map between
simply connected spaces which induces isomorphisms of the corresponding
integral singular homology groups is a weak homotopy equivalence. •
7
HOMOTOPY
FUNCTORS
In this section we shall study a general class of functors on the homotopy
category of path-connected pointed spaces. The main result characterizes, on
the subcategory of CW complexes, those functors of the form 'TTy for some Y
in terms of simple properties. In the next section we shall apply this result to
prove the existence of approximations to any space by a CW complex. 1
In a category 8, given objects A and X and morphisms fo: A ---) X and
fl: A ---) X, an equalizer of fo and it is a morphism ;: X ---) Z such that
(a) ; fo = ; it(b) If;': X ---) Z' is a morphism in 8 such that
morphism g: Z ---) Z' such that i' = go;.
0
0
i'
0
fo =
i' it,
0
there is a
Note that it is not asserted in condition (b) that g is unique.
We define 80 to be the homotopy category of path-connected pointed
spaces having nondegenerate base points.
I
LEMMA
The category
80
has equalizers.
PROOF
Let A and X be arbitrary objects of 80 and let fo: A ---) X
and it: A ---) X be maps preserving base points. Let Z be the space obtained
from the topological sum X v (A X 1) by identifying (a,O) E A X I with
fo(a) E X, (a,l) E A X I with fl(a) E X for all a E A, and (ao,t) E A X I with
(ao,O) (ao the base point of A) for all tEl. Then Z is an object of 80 and the
inclusion map ;: X C Z has the property that; fo ':':0 ; it [in fact, the composite A X I C X v (A X I) ---) Z is a homotopy from ; fo to ; fll.
Furthermore, if ;': X ---) Z' is a map such that i' fo ':':0 i' fl, there is a map
G: X v (A X I) ---) Z' such that G I X = i' and G I A X I is a homotopy
from i' fo to i' it- Then G is compatible with the collapsing map
k: X v (A X I) ---) Z, so there is a map g: Z ---) Z' such that G = g k. Then
i' = go;, and therefore [; l: X ---) Z is an equalizer of [fol and [itl in t'O. •
0
0
0
0
0
0
0
0
0
LEMMA
Let {Yn }n2 0 be ob;ects of 80 that are subspaces of a space Yin
such that Y n C Yn+l is a cafibration for all n Z 0, Y = Un Y n, and Y has
the topology coherent with {Yn }. Let in: Y n C Y n+1 , In: Y n C Y n, and
in: Yn C Y be the inclusion maps. Then the homotopy class [{in} l: V Yn ---) Y
is an equalizer in 80 of the homotopy classes
2
80
The techniques of this section are based on E. Brown, Cohomology theories, Annals of
Mathematics, vol. 75, pp. 467-484, 1962.
1
SEC.
7
407
HOMOTOPY FUNCTORS
[V in]: V Y n ---7 V Y n and
[V In]: V Y n ---7 V Y n
Since in+1 in = in In, it follows that Un} V in = Un} V In.
Given a map i': V Y n ---7 Z' such that i' V in ~ i' V In, let i~: Y n ---7 Z' be
defined by i~ = i'l Y n. Then i~+l in ~ i~, and using the fact that Y n C Y n+1
is a cofibration and by induction on n, there is a sequence of maps gn: Y n ---7 Z'
such that gn ~ i~ and gn+1 in = gn' Let g: Y ---7 Z' be the map such that
g I Yn = gn' If i = Un}: V Yn ---7 Y, then g i ~ i' completing the proof. •
PROOF
0
0
0
0
0
0
0
0
0
A homotopy functor is a contravariant functor H from
of pointed sets such that both of the following hold:
to to the category
(a) If [i]: X ---7 Z is an equalizer of [fol, [f11: A ---7 X and if u E H(X) is
such that H([fo])u = H([h])u, there is v E H(Z) such that H([ i J)v = u.
(b) If {X"h is an indexed family of objects in
and i,,: X" C V XI"
there is an equivalence
to
{H[i"lh: H(V X,,):::::: X H(X,,)
If f: X ---7 Y is a base-poi nt-preserving map and H is a homotopy functor,
we shall also use H(f) for H([fJ). If X C X' and u E H(X'), we use u I X for
H(i )u, where i: X eX'.
If X is a one-point space, and Xl and Xz are both equal to X, then
Xl v Xz is also equal to X, and the equivalence of condition (b)
{H(i1),H(i z)}: H(XI v X z ) :::::: H(Xl) X H(X z)
corresponds to the diagonal map of H(X) to H(X) X H(X). Because this is a
bijection, H(X) consists of a single element.
Following are some examples.
a
Let Y be a pOinted space. Then the functor '7T Y on Co defined as in
Sec. 1.3 (that is, '7TY(X) = [X; Yl for an object X in Co) is a homotopy functor.
>
0 and an abelian group G. Then the functor
4 Fix an integer n
H(X) = Hn(X,xo; G) (singular cohomology) on t'o is a homotopy functor called
the nth cohomology functor with coefficients G.
5 Let G be an arbitrary group (possibly nonabelian). There is a homotopy
functor H such that H(X) is the set of all homomorphisms '7Tl(X,XO) ---7 G with
the trivial homomorphism as base point.
An important result of this section is that on the subcategory of pointed
path-connected CW complexes every homotopy functor is naturally equivalent
to '7T Y for a suitable pointed space Y.
6 LEMMA Let v: SX ---7 SX V SX be the comultiplication map. If X is in 20
and H is a homotopy functor, the composite
H(SX) X H(SX)
(H(il),H(i2)j-l)
H(SX v SX)
H(v\
H(SX)
is a group multiplication on H(SX), which is abelian if X is a suspension. If H
408
HOMOTOPY THEORY
CHAP.
7
is a homotopy functor taking values in the category of groups, the two group
structures on H(SX) agree.
PROOF
Each of the group properties for this multiplication follows from the
corresponding H cogroup property of P. The final statement of the lemma
follows from theorem 1.6.8, because the two multiplications in H(SX) are
mutually distributive. •
In particular, for any homotopy functor H, H(Sq) is a group for q ~ 1
and abelian for q ~ 2 and is called the qth coefficient group of H. Thus the
qth coefficient group of the functor 'TTy of example 3 is 'TT q(Y). The qth coefficient group of the nth cohomology functor with coefficients G of example 4
is 0 if q =I=- n and isomorphic to G if q = n. The qth coefficient group of the
functor of example 5 is G if q = 1 and 0 if q
1.
If Y is an object of <?o and H is a homotopy functor, any element
u E H(Y) determines a natural transformation
>
Tu: 'TTy ~ H
defined by Tu([f]) = H([f])(u) for [f] E [X; Y]. For a suspension SX, Tu is a
homomorphism from 'TTY(SX) = [SX; Y] to the group H(SX), with the multiplication of lemma 6 (because both group multiplications are induced by the
co multiplication P: SX ~ SX v SX). An element u E H(Y) is said to be
n-universal for H, where n ~ 1, if the homomorphism
Tu: 'TTY(Sq)
~
H(Sq)
<
is an isomorphism for 1 S q
n and an epimorphism for q = n. An element
u E H(Y) is said to be universal for H if it is n-universal for all n ~ 1, in which
case Y is called a classifying space for H.
7 THEOREM Assume that H is a homotopy functor with universal elements
u E H(Y) and u' E H(Y') and let f: Y ~ Y' be a map such that H(f)u' = u.
Then f is a weak homotopy equivalence.
PROOF
Since Yand Y' are path connected, this is a consequence of the commutativity of the diagram (for q ~ 1)
[Sq; Y]
f#
~
[Sq; Y']
7T
TS
u'
H(Sq)
•
The same kind of argument establishes the next result.
LEMMA
Let Y be an object of <?o and let Y' be an arbitrary pathconnected space. A map f: Y ~ Y' is a weak homotopy equivalence if and
only if [f] E [Y; Y'] = 'TTY'(Y) is universal for 'TTY'. •
8
We are heading toward a proof of the existence of universal elements for
any homotopy functor. The following two lemmas will be used in this proof.
SEC.
7
HOMOTOPY FUNCTORS
eo,
409
9
LEMMA
Let H be a homotopy functor, Y an object in
and u E H(Y).
There exist an object Y' in
obtained from Y by attaching I-cells, and a
I-universal element u' E H(Y') such that u' I Y = U.
eo,
For each "1\ E H(S1) let SA l be a I-sphere and define Y' = Yv V ASA 1.
Then Y' is an object of ~o obtained from Y by attaching I-cells. If gA is the
composite Sl:? SAl C Y', it follows from condition (b) on page 407 that there
is an element u' E H(Y') such that u' I Y = u and H(gA)U' = "1\ for "1\ E H(S1).
Since TU,([gA]) = "1\, Tu ,([S1; Y'J) = H(S1), and u' is I-universal. PROOF
10 LEMMA Let H be a homotopy functor and u E H(Y) an n-universal
element for H, with n :;:: 1. There exist an object Y' in
obtained from Y by
attaching (n + I)-cells, and an (n + I)-universal element u' E H(Y') such
that u' I Y = U.
eo,
For each "1\ E H(Sn+1) let SA n+1 be an (n + I)-sphere, and for each map
= 0 attach an (n + I)-cell ean+1 to Y by 0'. Let Y'
be the space obtained from Yv V ASA n + 1 by attaching the (n + I)-cells {e an +1}.
Then Y' is an object of
obtained from Y by attaching (n + I)-cells.
If gA: Sn+l ~ Y V V ASA n+1 is the composite Sn+1 0::7 SA n+1 C Y V V ASA n+l, it
PROOF
0': Sn ~ Y such that H(a)u
eo
follows from condition (b) on page 407 that there is an element
u E H(Yv VA SA n+l ) such that u I Y = u and H(gA)U ="1\ for "1\ E H(Sn+1).
For each map 0': Sn ~ Y such that H(a)u = 0 let San be an n-sphere
and let fo: Va San ~ Yv V ASA n+1 be the constant map and let f1: Va San ~
Yv V ASA n+1 be the map such that San is mapped by 0'. Then
j: Yv V A SAn+1
C Y'
is a map such that [j] is an equalizer of [fo] and [/1]. Since H(fo)u = 0 =
H(f1)U, by condition (a) on page 407 there is an element u' E H(Y') such that
H(f)u' = u. Then u' I Y = u and to complete the proof we need only show
that u' is (n + I)-universal.
There is a commutative diagram
?Tq+1(Y',y) ~ ?Tq(Y) ~ ?Tq(Y') ~ ?Tq(Y',Y)
IT..
T~
H(Sq)
with the top row exact. Since Y' is obtained from Y by attaching (n + I)-cells,
it follows from lemma 7.6.15 that ?T q( Y', Y) = 0 for q :::; n. Therefore i# is an
isomorphism for q
n and an epimorphism for q = n. Since u is n-universal,
Tu is an isomorphism for q
n and an epimorphism for q = n. It follows
that Tu ' is also an isomorphism for q
n and an epimorphism for q = n.
Furthermore, if a E [Sn; Y] is in the kernel of Tu, then a is represented by a
map 0': Sn ~ Y and
<
a
= [0']
<
<
E 3(?Tn+l(ean+l,ean+1)) C 3(?Tn+1(Y',Y))
= ker i#
410
HOMOTOPY THEORY
CHAP.
7
Therefore, for q = n, ker Tu = ker i#, and so Tu' is an isomorphism from
'lTn(Y') to H(Sn). For any A E H(Sn+1) the map j 0 gA: Sn+l ~ Y' has the
property that
Tu'([ j gAl)
0
= H(gA)it = A
showing that Tu ' is an epimorphism for q = n
universal. -
+
1, and so u' is (n
+ 1)-
I I THEOREM Let H be a homotopy functor, let Y be an object in 2 0 , and
let u E H(Y). Then there are a classifying space Y' for H containing Y such
that (Y',Y) is a relative CW complex and a universal element u' E H(Y') such
that u' I Y = u.
PROOF
Using lemmas 9 and 10, we construct, by induction on n, a sequence
and elements Un E H(Yn) such that
of objects {Yn}n~O in
eo
(a)
(b)
(c)
(d)
Yo = Y and Uo = u.
Yn + 1 is obtained from Yn by attaching (n
Un+l I Yn = Un·
Un is n-universal for n ~ l.
+
I)-cells, where n ~ O.
It follows from (b) above that Y' = U Yn topologized with the topology
coherent with {Yn } is a path-connected pointed space containing Y such that
(Y',Y) is a relative CW complex. By lemma 2, the homotopy class
[{in}]: V Yn ~ Y' is an equalizer of the homotopy classes [V in]: V Yn ~ V Y n
and [V In]: V Yn ~ V Yn . By condition (b) on page 407 there is an element
it E H(V Y n) such that it I Y n = Un. It follows from (c) above that H(V in)it =
H( V In)it, and by condition (a) on page 407 there is an element u' E H(Y')
such that H({jn})u' = it (that is, u' I Y n = Un for n ~ 0). Then u' I Y = u, and
it remains to show that u' is universal.
By the definition of Y' and u', there is a commutative diagram for q ~ 1
lim~
{'lTq(Yn)} =? 'lTq(Y')
ITu'
(Tu.l\
H(Sq)
>
Since Un is n-universal, TUn is an isomorphism for n
q, and so the left-hand
map is an isomorphism. Therefore Tu ' is also an isomorphism, and u' is
universal. 12 COROLLARY For any homotopy functor there exist classifying spaces
which are CW complexes.
PROOF
of H(Y).
Apply theorem 11 to a one-point space Y, with u the unique element
-
13 COROLLARY Let u E H(Y) be a universal element for a homotopy
functor H. Let (X,A) be a relative CW complex, where A and X are objects
SEC.
7
411
HOMOTOPY FUNCTORS
in t'o. Given a map g: A ----) Y and an element v E H(X) such that v I A = H(g)u,
there exists a map g': X ----) Y such that g = g' I A and v = H(g')u.
PROOF
Let i: X C X v Y and i': Y c X v Y and let ;: X v Y ----) Z be a map
such that [il is an equalizer of [i fl (where f: A C X) and [i' g]. By condition (b) on page 407, there is an element v E H(X v Y) such that v I X = v
and 13 I Y = u. Since H(f)v = H(g)u, it follows that H(i f)13 = H(i' g)v,
and by condition (a) on page 407, there is an element u E H(Z) such that
H(iJu = v. We now apply theorem 11 to 11 to obtain a Y' containing Z and a
universal element u' E H( Y') such that 11 = u' I Z. Let i': Y ----) Y' be the
composite
0
0
0
r
.
0
h
Y C Xv Y!...c, Z C Y'
Then H(;')u' = u, and by theorem 7,
Since the composite
f
i'
is a weak homotopy equivalence.
i
.
h
A C X C X v Y ~ Z C Y'
is homotopic to i' g, it follows from the fact that f is a cofibration that there
is a map g: X ----) Y' such that g I A = i' g and g is homotopic to h
i.
Since i' is a weak homotopy equivalence, by theorem 7.6.22, there is a map
g': X ----) Y such that g' I A = g and i' g' ~ g. Then
0
0
0
;
0
0
H(g')u
= H(g')H(j')u' = H(i)H(i)H(h)u' = vi X = v
showing that g' has the requisite properties.
•
14 THEOREM If Y is a classifying space and u E H(Y) is a universal
Tu is a
element for a homotopy functor H, then for any CW complex X in
natural equivalence of 7TY(X) with H(X).
eo,
PROOF
Given v E H(X), apply corollary 13, with A = Xo and g the constant
map, to obtain a map g': X ----) Y such that H(g')u = v. Then Tu[g'] = v,
proving that Tu is surjective.
If go, gl: X ----) Yare maps such that Tu[go] = Tu[gl], let X' be the CW
complex X X I/xo X I, with (X')q = [(Xq X 1) U (Xq-l X 1)JI(xo X 1) for
q ::::: O. Let v E H(X') be defined by v = H(h)H(go)u, where h: X' ----) X is the
map h([x,tJ) = x. Let A = X X j/xo X j and let g: A ----) Y be the map such
that g([x,OJ) = go(x) and g([x,lJ) = gl(X). Then H(g)u = v I A, and by corollary 13, there is a map g': X' ----) Y such that g' I A = g. Then the composite
X X I ----) X X I/xo X I ~ Y
is a homotopy relative to Xo from go to gl, showing that Tu is injective.
•
15 COROLLARY If Y and Y' are classifying spaces which are CW complexes
and u E H( Y) and u' E H( Y') are universal elements for a homotopy functor H,
there is a homotopy equivalence h: Y ----) Y', unique up to homotopy, such
that H(h)u' = u.
412
HOM9TOPY THEORY
CHAP.
7
By theorem 14, there exists a unique homotopy class [g]: Y ~ Y'
such that H(g)u' = u. By theorem 7, g is a weak homotopy equivalence. By
corollary 7.6.24, g is a homotopy equivalence. •
PROOF
8
WE.~K
HOMOTOPY TYPE
In this section we shall show that any space can be approximated by CW
complexes. This leads to an equivalence relation based on weak homotopy
equivalence which is weaker than homotopy equivalence. We shall also consider the same equivalence relation in the category of maps. This will be used
in defining and analyzing the general relative-lifting problem.
A relative CW approximation to a pair (X,A) consists of a relative CW
complex (Y,A) and a weak homotopy equivalence f: Y ~ X such that f(a) = a
for all a E A. A CW approximation to a space X is a relative CW approximation to (X, 0).
I
THEOREM
Any pair has relative CW approximations, and two relative
CW approximations to the same pair have the same homotopy type.
First we consider the case where X is path connected. Let xo E X
and let {Aj} j E J be the set of path components of A, and for each i E J choose
a point aj E A j. There is a relative CW complex (A',A) with (A',A)O = A U eO,
where eO is a single point and
PROOF
A'
= (A',A)! = (A',A)O
U U e/
JEJ
where ejl is a I-cell such that e/ = eO U aj for i E J. Let g: A' ~ X be a map
such that g(a) = a for a E A, g(e O) = Xo, and g I ejl is a path in X with end
points Xo and aj for each i E J. Then A' is a path-connected space with nondegenerate base point eO and [g] E '/TX(A'). It follows from theorem 7.7.11
that there is a relative CW complex (Y,A') [which can be chosen such that
(Y,A')! = A' v V S"l] and a universal element [g'] E '/TX(Y) for '/TX such that
g' I A' ~ g. Since A' C Y is a cofibration, there is a map f: Y ~ X such that
[f] E '/TX(Y) is universal for '/TX and f I A' = g. By lemma 7.7.8, f is a weak
homotopy equivalence. Since (Y,A) is a relative CW complex [with (Y,A)O =
(A',A)O and (Y,A)q = (Y,A')q for q 2': 1] and since f(a) = a for a E A, (Y,A)
and f constitute a relative CWapproximation to (X,A).
Next we consider the case where X is not path connected and we let
{X,,} be the set of path components of X. By the case already considered, for
each a there is a relative CWapproximation f,,: (Y", X" n A) ~ (X", X" n A).
Let Y be the space obtained from the disjoint union A U U Y" by identifying
x E Xa n A c Y" with x E A for each a and let k: A U U Y a ~ Y be the
collapsing map. Then k I A: A ~ Y is an imbedding and (Y,A) is a relative
CW complex with (Y,A)q = k(A U U (Y", X" n A)q) for all q 2': O. There is a
map f: Y ~ X such that fk(a)
a for a E A and f 0 (k I Y,,)
f" for all a.
=
=
SEC.
8
413
WEAK HOMOTOPY TYPE
Since {k( Ya )} is the set of path components of Y and f induces a weak
homotopy equivalence of each of these with the corresponding path component Xa of X, f is a weak homotopy equivalence from Y to X. Identifying A
with k(A), we see that (Y,A) and f constitute a CWapproximation to (X,A).
Given two relative CW approximations to (X,A), say f1: (Y 1,A) ~ (X,A)
and fz: (Y2,A) ~ (X,A), it follows from theorem 7.6.22 that there are maps
gl: (Y1,A) ~ (Y2,A) and g2: (Y2,A) ~ (Y1,A) such that fz gl ~ it and
it g2 ~ fz, both homotopies relative to A. Then fz (gl g2) ~ fz 1 rel A,
and by theorem 7.6.22 again, gl g2 ~ 1 rel A. Similarly, g2 gl ~. 1 reI A,
and so (Y 1 ,A) and (Y 2 ,A) have the same homotopy type. •
0
0
0
0
0
0
0
Two spaces Xl and X2 will be said to have the same weak homotopy type
if there exists a space Yand weak homotopy equivalences f1: Y ~ Xl and
fz: Y ~ X 2 • By replacing such a space Y with a CWapproximation to it, we
see that Xl and X2 have the same weak homotopy type if and only if they
have CW approximations by the same CW complex.
2 LEMMA The relation of having the same weak homotopy type is an
equivalence relation.
The relation is reflexive and symmetric by its definition. To prove it
transitive, let Xl, X 2 , and X3 be spaces and let Y1 and Y2 be CW complexes
such that there exist weak homotopy equivalences
PROOF
Y1
tIl
\12
gt
Y2
Then fz: Y1 ~ X2 and g2: Y2 ~ X2 are both CWapproximations to X 2 , and
by theorem 1, there is a homotopy equivalence h: Y1 ~ Y2 such that
fz ~ g2 h. Then g3 h: Y1 ~ X3 , being the composite of weak homotopy
equivalences, is a weak homotopy equivalence. Therefore Xl and X3 have the
same weak homotopy type. •
0
0
We are interested in applying these ideas to weak fibrations. The main
result is that any two fibers of a weak fibration with path-connected base
space have the same weak homotopy type.
3 LEMMA Let p: E ~ B be a weak fibration with contractible base space B.
For any b o E B the inclusion map i: p-1(b o) C E is a weak homotopy
equivalence.
=
=
PROOF
Let F
p-1(b o). Since B is contractible, 'lTq(B,b o) 0 for q 2:: O.
From the exactness of the homotopy sequence of p, it follows that for any
e E F, i induces an isomorphism i#: 'lTq(F,e) :::::: 'lTq(E,e) for q 2:: 1 and
i#('lTo(F,e))
'lTo(E,e).
It only remains to verify that i# maps 'lTo(F,e) injectively into 'lTo(E,e).
Assume that e, e' E F are such that there is a path w in E from e to e'.
Since B is simply connected and pow is a closed path in B at bo, there is a map
=
414
HOMOTOPY THEORY
H: I X I ~ B such that H(t,O) = pw(t) and H(O,t')
Let g: I X 0 u i X I ~ E be the map defined by
and g(l,t') = e'. By lemma 7.2.5, there is a map
po G = Hand G I I X 0 u i X I = g. Let w': I ~
w'(t) = G(I,t). Then w' is a path in F from e to
showing that i#: 'lTo(F,e) ~ 'lTo(E,e) is injective. -
CHAP.
7
= H(I,t') = H(t,l) = boo
g(t,O) = w(t), g(O,t') = e,
G: I X I ~ E such that
E be the path defined by
e' [because pw'(t) = bo],
4 COROLLARY Let p: E ~ B be a weak fibration and let w be a path in B.
Then p-l(W(O)) and p-l(w(I)) have the same weak homotopy type.
PROOF
Let p':. E' ~ I be the weak fibration induced from p by w: I ~ B.
Then p-l(w(O)) and p-l(w(I)) are homeomorphic to p'-l(O) and p'-l(I),
respectively. By lemma 3, each of the inclusion maps p'-l(O) C E' and
p'-l(l) C E' is a weak homotopy equivalence. The corollary follows from this
and lemma 2. -
This result implies the following analogue of corollary 2.8.13 for weak
fibrations.
is COROLLARY If p: E ~ B is a weak fibration with path-connected base
space, any two fibers have the same weak homotopy type. -
We now consider the category whose objects are continuous maps
a: P" ~ P' between topological spaces and whose morphisms (also called
map pairs) f: a ~ {3 are commutative squares
P" ~ Q"
aJ
P'
Jf1
4
In this category a homotopy pair H: fo
mutative square
Q'
~
/1,
P" X I
H")
P' X I
H'
where fo,
/1:
a ~ {3, is a com-
Q"
Jf1
such that H": fg
~
f'{ and H': fo
~
~Q'
f1
(note that H is a map pair from
a X II to {3). If such a homotopy pair exists, fo is said to be homotopic to fl.
This is an equivalence relation in the set of map pairs from a to {3, and the
corresponding equivalence classes are called homotopy classes. We use [a;{3]
to denote the set of homotopy classes of map pairs from a to {3, and if
f: a ~ {3 is a map pair, its homotopy class is denoted by [f]. It is trivial to
verify that the composites of homotopic map pairs are homotopic, so there is
a homotopy category of maps whose objects are maps a: P" ~ P' and whose
morphisms a ~ {3 are homotopy classes [f], where f: a ~ {3 is a map pair.
A map pair f: a ~ {3 is called a homotopy equivalence from a to f3 if [f] is
an equivalence in the homotopy category of maps. Two maps a and f3 are
SEC.
8
415
WEAK HOMOTOPY TYPE
said to have the same homotopy type if they are equivalent in the homotopy
category of maps.
Given a map pair g: a' ~ a (or a map pair h: {3 ~ {3') there is an induced
map g#: [a;{3] ~ [a';{3] (or h#: [a;/3l ~ [a;{3'l) such that g#[fl = [f g) (or
h#[fl = [h fl). Since g# h# = h# g#, the function which assigns [a;{3] to
a and {3 and g# and h# to [g) and [h], respectively, is a functor of two
variables from the product of the homotopy category of maps by itself to the
category of sets that is contravariant in a and covariant in {3.
If a: P" ~ P' and {3: Q" ~ Q' are maps, given a map f: P' ~ Q", there
is a map pair p(f): a ~ {3 consisting of the commutative square
0
0
0
0
P"~Q"
at
tf3
P' ~Q'
[that is, (p(f))" = f a and (p(f))' = {3 j). Given a map pair f: a ~ {3, a
lifting of f is a map f: P' ~ Q" such that p(f) = f. Two liftings fo, h P' ~ Q"
of f: a ~ {3 are homotopic relative to f if there is a homotopy H: P' X I ~ Q"
from fo to fl such that H (a X II) and {3 H are both constant homotopies
[that is, p(H) is the constant homotopy pair from f to fl. Such a map H is
called a homotopy relative to f, and we write H: fo ~ fl reI f. Homotopy
relative to f is an equivalence relation in the set of liftings of f, and the set of
equivalence classes is denoted by [P';Q"]f. The relative-lifting problem is the
study of [P';Q"]f (for example, do liftings of f exist, and if so, how many
homotopy classes relative to f of liftings of f are there?).
0
0
0
0
If P" is empty, then a map pair f: a ~ {3 consists of a map
~ Q" of f is a lifting of f' to Q" in the sense
defined in Sec. 2.2. In this case, if {3 is a fibration, two liftings fo, fl: P' ~ Q"
of f' are homotopic relative to f if and only if they are fiber homotopic in the
sense of Sec. 2.8. Thus the absolute-lifting problem is a special case of a
relative-lifting problem.
6
EXAMPLE
f': P'
~
Q', and a lifting f: P'
7 EXAMPLE If a is an inclusion map and Q' is a one-point space, then a
map pair f: a ~ {3 corresponds bijectively to a map f": P" ~ Q" and a
lifting f: P' ~ Q" of f corresponds bijectively to an extension of f" to P'. In
this case two extensions fo, fl: P' ~ Q" are homotopic relative to f (as liftings)
if and only if they are homotopic relative to P". Thus the extension problem
is a special case of a relative-lifting problem.
8 EXAMPLE Let fo, h P' ~ Q" be liftings of a map pair f: a ~ {3. Let
R' = P' X I and let R" be the quotient space of the disjoint union of P' X j
and P" X I by the identifications (z",O) E P" X I equals (a(z"),O) E P' X j and
(z",I) E P" X I equals (a(z"),I). Define a map y: R" ~ R' by y(z",t) = (a(z"),t)
for (z",t) E P" X I and y(z',t) = (z',t) for (z',t) E P' X i. There is a map pair
g: y ~ {3 consisting of the maps g": R" ~ Q" and g': R' ~ Q' such that
416
HOMOTOPY THEORY
CHAP.
7
g"(z",t) = f"(zll) for (z",t) E P" X I, g"(z',O) = lo(z') and g"(z',I) = 11(z') for
z' E P', and g'(z',t) = f'(z') for (z',t) E P' X 1. Then 10 and 11 are homotopic
relative to f if and only if there exists a lifting of g.
We are particularly interested in the relative-lifting problem in case a is
the inclusion map of a relative CW complex and f3 is a weak fibration. Thus,
if i: A C X is an inclusion map and p: E ~ B is a weak fibration, a map pair
f: i ~ P consists of a map f': X ~ B and a lifting f": A ~ E of f' I A.
A lifting I of f is a lifting of f' to E, which is an extension of f". Two liftings
of f are homotopic relative to f if and only if there is a fiber homotopy relative to A between them. The following relative homotopy extension theorem
is the main reason for giving particular attention to this case.
9 THEOREM Let (X,A) be a relative CW complex, with inclusion map
i: A C X, and let p: E ~ B be a weak fibration. Given a map f: X ~ E and
a homotopy pair H: i X I] ~ P consisting of a homotopy H': X X I ~ B
starting at pol and a homotopy H": A X I ~ E starting at I i, there is a
homotopy H: X X I ~ E starting at f such that H' = P Hand
H" = H (i X I]).
0
0
0
=
PROOF
Let g: X X 0 U A X I ~ E be the map defined by g(x,O)
f(x) for
x E X and g(a,t) = H"(a,t) for a E A and t E 1. Then H' is an extension of
p g, and by the standard stepwise-extension procedure over the successive
skeleta of (X,A) (applied to polyhedral pairs in the proof of theorem 7.2.6 and
equally applicable to any relative CW complex), there is a map H: X X I ~ E
such that p Ii = H' and Ii IX X 0 U A X I = g. Then H has the desired
properties. •
0
0
Let us reinterpret this last result. A map pair f: i
square
~
P is a commutative
A4E
i~
~P
X4B
Therefore, if we let EX X' EA denote the fibered product of the map EX ~ BA
induced by restriction and the map EA ~ BA induced by p, the pair (f',!")
is a point of EX X' EA. In this way the set of map pairs f: i ~ P is identified
with the fibered product EX X' EA. The map p corresponds to a map
p: EX ~ EX X' EA, and [X;E]f is the set of path components of p-l(f).
10 COROLLARY Let (X,A) be a relative CW complex with X locally compact Hausdorff, with inclusion map i: A C X, and let p: E ~ B be a weak
fibration. Then p: EX ~ EX X' EA is a weak fibration.
Given a map g: In ~ EX and a homotopy H: In X I ~ EX X' EA
starting with p(g), the exponential correspondence assigns to g a map
g: X X In ~ E and to H a homotopy pair HI from (i X Id X I] to p, startPROOF
8
SEC.
417
WEAK HOMOTOPY TYPE
ing with p(g). By theorem 9, there is a homotopy HI: X X In X I ~ E
starting with g such that p(H1 ) = HI. Then the exponential correspondence
associates to HI a map G: In X I ~ EX starting with g such that p G
H. •
0
=
It follows from corollaries 10 and 4 that if fo, fl: i ~ P are homotopic
map pairs with X locally compact Hausdorff, then [X;E]ro and [X;E]rl are in
one-to-one correspondence. Thus the relative-lifting problem for fo is equivalent to the relative-lifting problem for ft.
Given weak fibrations PI: El ~ Bl and P2: E2 ~ B 2, a map pair
g: PI ~ P2 is called a weak homotopy equivalence if gil: El ~ E2 and
g': Bl ~ B2 are weak homotopy equivalences. We shall show that a weak
homotopy equivalence in the category of maps has much the same properties as
a weak homotopy equivalence in the category of spaces. The following analogue of theorem 7.6.22 is our starting point.
I I LEMMA Let (X,A) be a relative CW complex, with inclusion map
i: A C X, and let g: PI ~ P2 be a weak homotopy equivalence between weak
fibrations. Given a map pair f: i ~ PI and a lifting Ii: X ~ E2 of the map
pair g 0 f, there is a lifting X ~ El of f such that gil 0 f and Ii are homo-
f:
topic relative to g f.
0
PROOF
The proof involves two applications of theorem 7.6.22 and then two
applications of theorem 9. We shall not make specific reference to these
when they are invoked.
We have a commutative diagram
A
4
El ~ E2
X
4
Bl ~ B2
in which gil and g are weak homotopy equivalences, and we are given a map
X ~ E2 such that Ii i
g" f" and P2 Ii
g f'. The!! there is a
map X ~ E1_such that 0 i = f" and a homotopy Gil: g" 0 f ~ Ii reI A.
The maps PI f and f' agree on A and P2 Gil is a homotopy relative to A
from g'.o PI
P2 g"
to g' f'
P2 Ii. Therefore there is a homotopy
F': PI
~ f' reI A and a homotopy H': g' F' ~ P2 G" reI A 0 I U X X i.
Let F": X X I ~ El be a lifting of F' such that F"(x,O)
f(x) for x E X
and F"(a,t)
f"(a) for a E A and tEl. Define f: X ~ El by j(x)
F"(x,l).
We show that f has the desired properties. It is clearly a lifting of f.
_
The maps g" 0 F" and G" are homotopies relative to A from gil 0 f to
gil f and to Ii, respectively, and H' is a homotopy from P2 g" F" to P2 G"
reI A X I U X X 1. Since there is a homeomorphism of (X X I X I, A X I X I)
onto itself taking X X (I X i U 0 X I) onto X X I X 0, there is a lifting H"
of H' which is a homotopy from g" F" to G" reI X X 0 U A X I. Then
the map H: X X I ~ E2 defined by H(x,t) = H"(x,l,t) is a homotopy from
gil f to h relative to g f. •
Ii:
1:
0
0
0
J
0
J=
0
0
=
J
J
0
0
0
=
0
0
=
0
0
0
=
=
0
0
0
0
0
This gives us the following important result.
0
=
0
418
HOMOTOPY THEORY
CHAP.
7
12 THEOREM Let (X,A) be a relative CW complex, with inclusion map
i: A C X, and let g: PI -7 P2 be a weak homotopy equivalence between weak
fibrations. Given a map pair f: i -7 PI, the map pair g induces a biiection
g'#: [X;Ell f
:::::
[X;E 2 1g o f
The fact that g,# is surjective follows immediately from lemma 11.
The fact that g,# is injective follows from application of lemma 11 to the relative CW complex (X,A) X (I,i). •
PROOF
EXERCISES
A EXACTNESS OF HOMOTOPY SETS
1 Assume that i: (X,N) C (X,A) is a cofibration, where A and X' are closed subsets of
X and N = A n X. Prove that the collapsing map
(Cy,Cd ~ (Cy,Cd/CX
= (X,A)/X = (X/X, A/A')
is a homotopy equivalence.
2 With the same hypotheses as in exercise 1, let g': (X,A) ~ C(X,A') be any map such
that g'(x')
x' for x' E X and let g: (X/X',A/ A') ~ S(X,A') be the map such that the
following square is commutative, where k' and k" are the collapsing maps:
=
(X,A)
£.
C(X',A')
k'l
lk"
(X/X,A/A') -4 S(X,A')
Prove that there is a co exact sequence
(X',A') ~ , . , ~ sn(X',A') §:'4 sn(X,A) ~ Sn(x/x', A/A') .§:.4 , ..
3
If (X,A) is a relative CW complex, prove that there is a coexact sequence
A C X ~ X/A
B
1
~
SA C SX ~ ...
~
SnA C SnX
~
...
HOMOTOPY GROUPS
If A is a retract of X, prove that there is an isomorphism
n>2
2
If X is defonnable into A relative to
Xo
E A, prove that there is an isomorphism
'1Tn(A,xo) :::::: '1Tn(X,xo) EB '1Tn+l(X,A,xo)
3 If p: E ~ B is a weak fibration such that the fiber F
relative to eo E F, prove that there is an isomorphism
n
2':
2
= p-l(bo) is contractible in E
n2':2
4 If p: E ~ B is a weak fibration which admits a section, prove that there is an isomorphism for eo E F
p-l(bo)
=
'1Tn(E,eo) :::::: '1Tn(B,b o) EB '1Tn(F,eo)
n 2': 2
419
EXERCISES
:; Let {Xj} be an indexed family of spaces with base points Xj E Xj. Prove that there is
an isomorphism
n;::::O
6
Given X v Y
= X X yo
U Xo X Y C X X Y, prove that there is an isomorphism
7Tn (X V Y, (xo,yo)) ::::: 7Tn(X,XO) EB 7Tn(Y,yO) EB 7Tn+1(X X Y, X v Y, (xo,yo))
C
I
BASE POINTS 1
Give an example of a degenerate base point.
2 If X and Y have nondegenerate base points, prove that also X v Y, X X Y, and
X X YIX v Y have nondegenerate base points.
3 If (X,xo) and (Y,yo) have the same homotopy type, prove that Xo is a nondegenerate
base point of X if and only if yo is a nondegenerate base pOint of Y.
" Prove that any space has the same homotopy type as some space with a nondegenerate base point.
:; Let X and Y be path-connected spaces with nondegenerate base points Xo and yo,
respectively. Prove that X and Y have the same homotopy type if and only if (X,xo) and
(Y,yo) have the same homotopy type.
D THE WHITEHEAD PRODUCT
Let p ;:::: 1 and q ;:::: 1 and let h: (l p+q,ip+q) ~ (lP,ip) X (JQ,iq) be the homeomorphism
h(tt, ... ,tP+Q)
((t1, ... ,tp),(tv+t, ... ,tp+q)). Then h determines an element
[h] E 7Tp+q((lP,iP) X (Iq,iq), (0,0)) and an element
=
1/p,q
= o[h] E 7Tp+q_1(IP X jq U jp
Given maps a: (IP,iP) ~ (X,xo) and
y: (IP X jq U jp X Iq, (0,0)) ~ (X,xo) by
y(z,z')
X Iq, (0,0))
13: (Iq,jq) ~ (X,xo),
define
a
map
z' E jq, (z,z') E Ip X Iq
z E jp, (z,z') E Ip X Iq
a(z)
= { f3(z')
I Prove that Y#(1/p,q) E 7Tp+Q_1(X,XO) depends only on [a] and [13], It is called the
Whitehead product of [a] and [13] and is denoted by [[a],[f3]] E 7Tp+Q_1(X,XO)'
2
3
"
:;
6
7
Prove that if p
> 1 and q = 1, prove that [[a],[f3]] = [a]h[~]([a]-l).
If P + q > 2, prove that [[a],[f3]] = (-l)pq[[f3],[a]].
Iff: (X,xo) ~ (Y,yo), prove thatf#[[a],[f3]] = U#[a],f#[f3]]·
If w is a path in X, prove that h[OJ][[a],[I3]] = [h[OJ][a],h[OJ][I3]].
Prove that [[a],[f3]] = 0 if and only if there is a map f:
If p
f(t
8
= q = 1, then [[a],[f3]] = [a][f3][a]-l[f3]-l.
h
h···, p+q
)={a(t1" "
f3(tp+h'"
Ip X Iq
~
X such that
,tp)iff!=Oorlforsomep+1:S;i:S;p+q
,tp+q)iff!=Oorlforsomel:S;i:S;p
If X is an H space, prove that [[a],[f3]]
= 0 for all [a] and [13].
1 See D. Puppe, Homotopiemengen und ihre induzierten Abbildungen. I, Mathematische
Zeitschriften, vol. 69, pp. 299-344, 1958.
420
9
HOMOTOPY THEORY
Prove that Sn is an H space if and only if [[a],[,8]]
= 0 for all [a], [,8]
CHAP.
7
E 'lTn(sn).
E CW COMPLEXES
I If (X,A) is a relative CW complex, prove that X has a topology coherent with the
collection {A} u {e I e a cell of X - A}.
2 If (X,A) is a relative CW complex, prove that X is compactly generated if and only
if A is compactly generated.
3
If (X,A) is a relative CW complex and A is paracompact, prove that X is paracompact.
4 If (X,A) is a relative CW complex and A has the same homotopy type as a CW complex, prove that X has the same homotopy type as a CW complex.
:;
Prove that a CW complex is locally contractible.
6
Prove that a CW complex has the same homotopy type as a polyhedron.
F
ACTION OF THE FUNDAMENTAL GROUP
I
Prove that the real projective n-space pn is simple if and only if n is odd.
2 For 1 < n < m show that pzn+1 X S2m+1 and pzm+1 X S2n+1 are simple compact
polyhedra having isomorphic homotopy groups in all dimensions, but are not of the same
homotopy type.
3 Let (Z,Z) be an (n - I)-connected CW pair, with n ;:: 2, such that Z is simply connected. Let (X* ,X) be the adjunction space obtained by adjoining Z to a CW complex X
by a map f: (Z,zo) ~ (X,xo) and let g: (Z,Z,zo) ~ (X* ,X,xo) be the canonical map. Prove
that (X* ,X) is (n - I)-connected and that the map
(fl
['lTn(Z,Z,zo)hw] ~ 'lTn(X* ,X,xo)
[wlE"'l (X,Xo)
sending [aJrw] to h[w](g#[a]) for [a] E 'lTn(Z,Z,Zo) is an isomorphism. [Hint: Let X be the
universal covering space of X and let {f[w]: Z ~ X}[w] E"" (X,Xo) be the set of liftings of f.
Show that the space X* obtained by attaching a copy of Z to X for each map few] is the
universal covering space of X*. Then use the fact that 'lTq(X* ,X) ;::::; 'lTq(X* ,X) and compute 'lTn(X* ,X) by the Hurewicz theorem.]
4 Let X be the CW complex obtained from Sl v S2 by attaching a 3-cell by a map
representing 2[a] - h[w][a], where [a] is a generator of 'lT2(S2) and [w] is a generator of
'lT1(Sl). Prove that the inclusion map Sl C X induces an isomorphism of the fundamental
groups and all homology groups but not of the two-dimensional homotopy groups.
G CW APPROXIMATIONS
I If (X,A) is an arbitrary pair, prove that there is a CW pair (X',N) and a map
f: (X',A') ~ (X,A) such that f I X': X' ~ X and f I A': A' ~ A are both weak homotopy
equivalences.
2
h
If h: Xl ~ Y1 and fz: X2 ~ Y2 are weak homotopy equivalences, prove that
X fz: Xl X X2 ~ Y1 X Y2 is also a weak homotopy equivalence.
3 If h: Xl ~ Y1 and fz: X2 ~ Y2 are weak homotopy equivalences, show by an
example that f1 v fz: Xl v X2 ~ Y1 V Y2 need not be a weak homotopy equivalence.
4 Show by an example that a weak homotopy equivalence need not induce isomorphisms of the corresponding Alexander cohomology groups.
:; If X is simply connected and H. (X) is finitely generated, prove that X has the same
weak homotopy type as some finite CW complex.
421
EXERCISES
A space X is said to be dominated by a space Y if there exist maps f: X ~ Y and
f ~ Ix. Prove that a space is dominated by a CW complex if and
only if it has the same homotopy type as some CW complex.
6
g: Y ~ X such that g
0
H GROUPS OF HOMOTOPY CLASSES
Throughout this group of exercises it is assumed that Y is (n - I)-connected, where
n 2 2, with base pOint Yo, and that X is a CW complex of dimension ::;: 2n - 2.
I Prove that any map X ~ Y is homotopic to a map sending Xn-l to yo and that if
f, g: (X,xn-l) ~ (Y,Yo) are homotopic as maps from X to Y, they are homotopic relative
to Xn-Z.
2 Prove that the diagonal map d: X ~ X X X is homotopic to a map d' such that
d'(X) C (X X xn-z) U (xn-Z X X). Prove that maps d', d": X ~ (X X Xn-Z) U (Xn-Z X X)
which are homotopic in X X X are homotopic in (X X xn-l) U (xn-l X X). (Hint: Use
the cellular-approximation theorem.)
Let d': X ~ (X X xn-Z) U (xn-Z X X) be homotopic in X X X to the diagonal
map. Given f, g: X ~ Y, let!" g': (X,Xn-l) ~ (Y,yo) be homotopic to f and g, respectively. Then (f' X g') d': X ~ Y X Y maps X into Yv Y. Let y: Yv Y ~ Y be defined
by y(y,yo) = y = Y(Yo,Y)·
0
Prove that [y (f' X g') d'] depends only on [fl and [g] and that the operation
(f' X g') d'] is associative, commutative, and has a unit element,
making [X;Y] into a commutative semigroup with unit.
3
[fl + [g] = [y
0
0
0
0
4 Prove that if g: Y ~ Y', where Y' is also (n - I)-connected (or if h: X' ~ X, where
X' is a CW complex of dimension::;: 2n - 2), then ~: [X; Y] ~ [X; Y'] is a homomorphism (or h#: [X; Y] ~ [X'; Y] is a homomorphism).
5 The semigroup [X;Y] is a group. (Hint: Use induction on the dimension of X, the
fact that [Xk+I/Xk;Y] is a group for any k and any Y, because Xk+l/Xk, being a wedge
of (k + I)-spheres, is a suspension, and the exactness of the sequence of homomorphisms
[Xk+1/Xk; Y]
[Xk+l; Y]
~
~[Xk; Y]
where X' is a disjoint union of k spheres, one for each (k
[X'; Y]
+
I)-cell of X.)
In case Y = Sn and dimension X::;: 2n - 2, the group [X;Sn] is called the nth
cohomotopy group of X,l denoted by 'IT"(X).
I
I
MISCELLANEOUS
Let 0': '1Tn+1(~n+1,.in+1,vo) ~ '1Tn(.in+l,(~n+1)n-l,vo) if n
2 2 and let
0': '1T2(~z,.iz,vo) ~ '1TI(.i z,VO)
if n
= 1. Prove that O'[~n+l] = bn for n 2
I (see page 394 for definition of bn).
2 Let H be a homotopy functor and let f: X ~ Y be a base-point-preserving map
between path-connected spaces, with nondegenerate base points. Prove that the sequence
H(C,)
~
H(Y)
~
H(X)
is exact.
3 If H is a homotopy functor and (X,A) is a CW pair, prove that there is an exact
sequence
H(A)
~
H(X)
~
H(X/A)
~
H(SA)
~
...
~
H(SnA)
~
...
For more details see E. Spanier, Borsuk's cohomotopy groups, Annals of Mathematics, vol. 50,
pp. 203-245, 1949.
1
CHAPTER EIGHT
OBSTRUCTION THEORY
IN THIS CHAPTER WE DEVELOP OBSTRUCTION THEORY FOR THE GENERAL LIFTING
problem. A sequence of obstructions is defined whose vanishing is necessary
and sufficient for the existence of a lifting. The kth obstruction in the sequence
is defined if and only if all the lower obstructions are defined and vanish, in
which case the vanishing of the kth obstruction is a necessary condition for
definition of the (k + l)st obstruction.
We begin by applying the general theory of homotopy functors to study
the set of homotopy classes of maps from a CW complex to a space with
exactly one nonzero homotopy group and we show that a suitable cohomology
functor serves to classify maps up to homotopy in this case. This result is then
used to obtain a solution, in terms of cohomology, of the lifting problem for a
fibration whose fiber has exactly one nonzero homotopy group.
With this in mind, we then consider the problem of factorizing an arbitrary fibration into simpler ones each of which has a fiber with exactly one
nonzero homotopy group. We show that such factorizations do exist for a
large class of fibrations, and that when they exist, a sequence of obstructions
can be associated to the factorization. These obstructions are subsets of coho423
424
OBSTRUCTION THEORY
CHAP.
8
mology groups, and we apply the general machinery to some special cases
where, because of dimension restrictions, the only obstructions which enter
are either the first one or the first two. For the case of only one obstruction
we obtain the Hopf classification theorem.
Finally, we prove the suspension theorem, which we use to compute the
(n + l)st homotopy group of the n-sphere. Combining this with the technique
of obstruction theory, we obtain a proof of the Steenrod classification theorem.
Section 8.1 is devoted to spaces with exactly one nonzero homotopy
group. We prove tqat a suitable cohomology functor serves both to classify
maps from a CW complex to such a space and to provide a solution for the
extension problem for maps involving a relative CW complex and such a space.
We use this result to derive the Hopf extension and classification theorems for
maps of an n-dimensional CW complex to Sn. Section 8.2 deals with fibrations
whose fiber has exactly one nonzero homotopy group, and again it is shown
that a suitable cohomology functor serves to provide a solution for the lifting
problem and to classify liftings of a given map.
In Sec. 8.3 we prove that many fibrations can be factored as infinite
composites of fibrations each of which has a fiber with exactly one nonzero
homotopy group. The corresponding lifting problem is then represented as an
infinite sequence of simpler lifting problems. In Sec. 8.4 we show how to
define obstructions inductively for such a sequence of fibrations, and how to
apply the resulting machinery.
In Sec. 8.5 we shall study the suspension map and prove the exactness
of the Wang sequence of a fibration with base space a sphere. This result is
used to prove the suspension theorem, which is applied to compute 17n+l(Sn)
for all n. We then prove the Steenrod classification theorem for maps of an
(n + I)-dimensional CW complex to Sn.
I
EILENBERG-MACLANE SPACES
This section is devoted to a study of spaces with exactly one nonzero homotopy
group. Such spaces are classifying spaces for the cohomology functors, and
because of this, there is an important relation between the cohomology of
these spaces and cohomology operations. At the end of the section we shall
apply the results to derive the Hopf classification and extension theorems.
Then, later in the chapter, we shall study arbitrary spaces by representing
them as iterated fib rations whose fibers are spaces with exactly one nonzero
homotopy group. Thus, these homotopically simple spaces serve as building
blocks for more complicated spaces.
Let 17 be a group and let n be an integer ~ 1. A space of type (17,n) is a
path-connected pointed space Y such that 17q (Y,yo) = 0 for q =1= nand
17n(Y,yo) is isomorphic to 17. An Eilenberg-MacLane space 1 is a path-connected
pointed space all of whose homotopy groups vanish, except possibly for a
1 See S. Eilenberg and S. MacLane, On the groups H(7T,n), I, Annals of Mathematics, vol. 58,
pp. 55-106, 1953.
SEC.
1
425
EILENBERG-MACLANE SPACES
single dimension. Thus a space of type ('7T,n) is an Eilenberg-MacLane space.
Conversely, if Y is an Eilenberg-MacLane space and '7T q (Y,Yo) = 0 for q 0:/== n,
then Y is a space of type ('7T n(Y,yo), n). Let us consider a few examples.
I
It follows from corollary 7.2.12 that Sl is a space of type (Z,I).
2
Let px be the CW complex which is the union of the sequence
p1 C p2 C . .. topologized by the topology coherent with the collection
{Pi}i21' Then '7T q (PX) ~ lim~ {'7T q(Pi)}, and it follows from application of corollary 7.2.11 to the covering Sn ~ pn that poc is a space of type (Z2,l).
3
Let P x(C) be the CW complex which is the union of the sequence
P 1(C) C P2(C) c ... topologized by the topology coherent with the collection {Pi (C)}i21. Then '7T q (P x(C)) ~ lim~ {'7T q (PiC))}, and it follows from
corollary 7.2.13 that P x(C) is a space of type (Z,2).
Let '7T be an abelian group and Y a path-connected pointed space.
An element v E Hn( Y, yo; '7T) is said to be n-characteristic for Y if the composite
'7Tn(Y,Yo) ~ Hn(Y,yo) ~ '7T
is an isomorphism (where <p is the Hurewicz homomorphism and h is the
homomorphism defined in Sec. 5.5). If Y is (n - I)-connected, it follows from
the absolute Hurewicz isomorphism theorem and the universal-coefficient
theorem for cohomology that there is an n-characteristic element
v E Hn(Y,yo; '7T) if and only if '7T ~ '7T n(Y,yo). Such an element is unique up to
automorphisms of '7T. In particular, a space Y of type ('7T,n) with '7T abelian has
n-characteristic elements v E Hn(Y,yo; '7T).
4
LEMMA
Let u E Hn(Y,yo; G) be a universal element for the nth cohomology functor with coefficients G, where n ;:::: 1. Then Y is a space of type
(G,n) and u is n-characteristic for Y.
PROOF
By theorem 7.7.14, there are isomorphisms
q;::::
1
Therefore '7T q (Y,yo) = 0 if q 0:/== n, and Tu: '7Tn(Y,yo) ~ Hn(Sn,po; G). If
a: (Sn,po) ~ (Y,yo), then Tu([a]) = a* (u), and there is a commutative diagram
'7Tn(Sn,po) ~ Hn(Sn,po)
\ . h(a*(u))
"=1
= h(Tu[a])
G
/ ' h(u)
'7Tn(Y,yo) ~ Hn(Y,yo)
Let v: Hn(Sn,po; G)
~
v(v)
G be the isomorphism defined by
= h(v)(<p[lsn])
From the commutativity of the diagram above,
426
It follows that h(u)
OBSTRUCTION THEORY
0
CHAP.
8
cp equals the composite
'TTn(Y,yo) ~ Hn(Sn,po; G)
it
G
and so is an isomorphism. Therefore Y is a space of type (G,n) and u is
n-characteristic for Y. •
:;
COROLLARY
Given n ;::0: 1 and a group 'TT (abelian if n
a space of type ('TT,n).
> 1), there exists
PROOF
If 'TT is abelian, it follows from lemma 4 that any classifying space for
the nth cohomology functor with coefficients 'TT is a space of type ('TT,n).
If n = 1 and 'TT is arbitrary, it is easy to see that a classifying space for the
homotopy functor of example 7.7.5 which assigns to a pointed path-connected
space X the set of homomorphisms 'TTl(X,XO) ~ 'TT is a space of type ('TT,l). In
either case, since any homotopy functor has a classifying space by corollary
7.7.12, the result follows. •
6
COROLLARY
Let {'TTn}n~l be a sequence of groups which are abelian for
n > 2. There is a space X, with base point Xo, such that 'TTn(X,xo) ::::; 'TTn
for n ;::0: 1.
fROOF
By corollary 5, for each n ;::0: 1 there is a space Yn, with base point Yn,
such that 'TTq(Yn,Yn) = 0 for q =1= nand 'TTn(Yn,Yn) ::::; 'TTn. Then the product
space X Y n with base point (Yn) has the desired properties. •
The last result can be strengthened so that if 'TTl acts as a group of operators on 'TTn for every n ;::0: 2, then the sequence is realized as the sequence of
homotopy groups of a space X in such a way that the action of 'TTl on 'TT n corresponds to the action of 'TTl(X,XO) on 'TTn(X,xo) of theorem 7.3.8.
7
LEMMA
Let F: H ~ H' be a natural transformation between homotopy
functors which induces an isomorphism of their qth coefficient groups for
n and a surjection of their nth coefficient groups (where 1 :::; n :::; 00). For
q
any path-connected pointed CW complex W the map
<
F(W): H(W)
~
H'(W)
is a bijection if dim W :::; n - 1 and a surjection if dim W :::; n.
Let u E H(Y) and u' E H'(Y') be universal elements for Hand H',
respectively, and let f: Y ~ Y' be a map such that H'(f)(u') = F(Y)(u). For
any CW complex W there is a commutative square
PROOF
[W;Y] ~ [W;Y']
r"l
lr"
H(W) ~ H'(W)
in which, by theorem 7.7.14, both vertical maps are bijections. Since
F(Sq): H(Sq) ~ H'(Sq) is an isomorphism for q
n and a surjection for q = n,
it follows that f #: 'TT q( Y) ~ 'TT q( Y') is an isomorphism for q
n and a surjection for q = n. Since Y and Y' are path-connected pointed spaces, the map f
<
<
SEC.
I
427
ElLENBERG-MAC LANE SPACES
is an n-equivalence. The result follows from corollary 7.6.23 and the commutativity of the above square. We use this last result to obtain the following classification theorem,
which is a converse of lemma 4.
THEOREM
Let 'TT be an abelian group, Y a space of type ('TT,n), and
E Hn(Y,yo; 'TT) an n-characteristic element for Y. Let 1/;: 'TTy ~ Hn(. ;'TT) be the
natural transformation defined by I/;[fl = f* t for [fl E [X; Y]. Then I/; is a
natural equivalence on the category of path-connected pointed CW complexes.
8
t
By lemma 7, it suffices to verify that I/; induces an isomorphism of all
coefficient groups of the two homotopy functors 'TTy and Hn( • ;'TT). The only
nonzero coefficient groups are 'TTn(Y,yo) and Hn(Sn,po; 'TT), and we need only
verify that
PROOF
I/;(Sn): 'TTn(Y,yo)
~
Hn(Sn,po; 'TT)
is an isomorphism. If v: Hn(Sn,po; 'TT) ;:::; 'TT is defined by v(v) = h(v)(<p[lsn]) (as
in the proof of lemma 4), then v 0 I/;(Sn) = h(t) 0 <po Because tis n-characteristic
for Y, v I/;(Sn) is an isomorphism, and thus so is I/;(Sn). 0
9
THEOREM Let Y be a space of type ('TT, 1) and let H be the functor which
assigns to a pointed s/,Jce X the set of homomorphisms from 'TTl(X,XO) to
'TTl(Y,YO). Let;j;: 'TTy ~ H be the natural transformation defined by ~[fl = f#
for [fl E [X; Y]. Then ~ is a natural equivalence on the category of pathconnected pointed CW complexes.
PROOF
By lemma 7, it suffices to verify that
~(Sl):
'TTl(Y,YO) ~ H(Sl,PO)
is an isomorphism. Let ii: H(Sl,po) ;:::; 'TTl(Y,YO) be the isomorphism defined by
ii(y) = y([lsl]) for y: 'TTl(Sl,PO) ~ 'TTl(Y,YO). Then ii is an inverse of ~(Sl),
showing that ~(Sl) is an isomorphism. Note that if 'TTl(Y,YO) is abelian in theorem 9, the set of homomorphisms
from 'TTl(X,XO) to 'TTl(Y,YO) is in one-to-one correspondence with the group
Hom (?fl(X,xo), 'TTl(Y,YO)) ;:::; Hom (H1 (X,xo), 'TTl(Y,YO)) ;:::; H1(X,xo; 'TTl(Y,YO))
and so theorems 8 and 9 agree in this case.
We now consider the free homotopy classes of maps from X to Y. Since
any O-cell Xo of a CW complex X is a nondegenerate base point (because, by
theorem 7.6.12, the inclusion map Xo C X is a cofibration), it follows from
corollary 7.3.4 that there is an action of 'TTl(Y,YO) on the set [X,xo; Y,yo].
Furthermore, if Y and X are path connected and this action is trivial, then
the map from base-point-preserving homotopy classes to free homotopy classes
[X,xo; Y,yo]
~
[X;Y]
is a bijection. In case Y is a space of type ('TT,n), with n
and so there is a bijection
> 1, then 'TTl(Y,YO) = 0,
428
OBSTRUCTION THEORY
CHAP.
8
[X,xo; Y,yol :::::; [X;Yl
In case Y is a space of type ('IT,1), the action of 'lTl(Y,YO) on [X,xo; Y,yol corresponds under the bijection ;J; of theorem 9 to the action of 'lTl(Y,YO) on H(X,xo)
by conjugation. Thus, if 'IT is abelian, there is a bijection
[X,xo; Y,yol:::::; [X;Yl
10 THEOREM If 'IT is an abelian group, Y is a space of type ('IT,n), and
t E Hn(y,yo; 'IT) is n-characteristic for Y, then for any relative CW complex
(X,A) the map
!f;: [X,A; Y,yol
--7
Hn(X,A; 'IT)
is a bi;ection.
In case A is empty and X is path connected, it follows from theorem 8
and the observation above that there is a commutative square
PROOF
[X,xo; Y,yol
~
[X;Yl
ty
vt==
Hn(x,xo; 'IT)
~
Hn(x;'IT)
and so !f;: [X; Y 1:::::; Hn(x,'IT). In case A is empty and X is not path connected,
let {XI.} be the set of path components of X. The result follows from the first
case on observing that [X; Yl :::::; X [XI.; Yl and Hn(x;'IT) :::::; X Hn(XA;'IT). In
case A is not empty, let k: (X,A) --7 (X/ A,xo) be the collapsing map. Then the
result follows from the already established bijection!f;: [X/A;Yl:::::; Hn(X/A;'IT)
and the commutative diagram
[X,A; Y,yol /';; [X/A,xo; Y,Yol?
.;-t
vt
Hn(X,A; 'IT) ~ Hn(x/ A,xo; 'IT) ?
[X/A;Yl
==t~
Hn(X/ A; 'IT) •
I I THEOREM Let Y be a space of type ('IT,1). For any path-connected CW
complex X the set of free homotopy classes of maps from X to Y is in
one-to-one correspondence with the set of con;ugacy classes of homomorphisms
'lTl(X,XO) --7 'lTl(Y,YO) under the map [fJ --7 f#·
This follows from theorem 9 and the remark above covering the
action of 'lTl(Y,YO) on [X,xo; Y,yo]. •
PROOF
12 THEOREM Let Y be a space of type ('IT,n), with n:::;' 1 and 'IT abelian,
and let t E Hn(y,yo; 'IT) be n-characteristic for Y. If (X,A) is a relative
CW complex, a map f: A --7 Y can be extended over X if and only if
of* (t) = 0 in Hn+1(X,A;'IT)
PROOF
Assume f = g
because
oi * g* (t) = 0,
of*(t) = O.
0
i, where i: A C X and g: X --7 Y. Then of* (t) =
Hence, if f can be extended over X, then
oi * = O.
SEC.
I
429
EILENBERC·MACLANE SPACES
Conversely, assume of* (l) = 0. To extend f over X we need only extend
over each path component of X, and therefore there is no loss of generality
in assuming X to be path connected (and A to be nonempty). Let Y' be the
space obtained from the disjoint union X U Y by identifying a E A with
f(a) E Y for all a E A. Then Y is imbedded in Y', the pair (Y', Y) is a relative
CW complex, and there is a cellular map i: (X,A) ----7 (Y', Y) which induces an
isomorphism i*: H*(Y',Y);:::::; H*(X,A) such that there is a commutative
square
f
Hn(Y,yo) ~ Hn+1(Y',Y)
f* ~
:::~j*
Hn(A) ~ Hn+1(X,A)
Since of* (l) = 0, it follows that O(l) = 0, and there is v E Hn(Y',yo; 7T) such
that v I (Y,Yo) = l. Since X and Yare path connected and A is nonempty, Y'
is path connected.
Let Y = Y' v I (that is, yo E Y' is identified with
E I) and let
yo = 1 E Y. Then Y is a path-connected space with nondegenerate base
point yo. Let r: (Y,I) ----7 (Y',yo) be the retraction which collapses I to yo and
let is = 1* (v) I (Y,yo) E Hn(y,yo; 7T). By theorem 7.7.11, there is an imbedding
of Y in a space Y" which is a classifying space for the nth cohomology functor
with coefficients 7T and which has a universal element 11 E Hn(y",yo; 7T) such
that 11 I (Y,!/o) = is. Then Y" is a space of type (7T,n), and there is a unique
n-characteristic element u E Hn(y",yo; 7T) such that u I Y" = 11 I Y". Then
u I (Y,yo) = l, and it follows from theorem 8 and the commutativity of the
diagram
°
[Sq,po; Y,Yo]
----7
[sq,po; Y",Yo]
.;,\.:::
:::JC'.;,«
Hn(Sq,po; 7T)
that Y c Y" is a weak homotopy equivalence. Since the composite
X ~ Y' c Y" is an extension of the composite A -4 Y c Y", it follows from
theorem 7.6.22 that f can be extended to a map X ----7 Y. •
We now show that cohomology operations are closely related to the
cohomology of Eilenberg-MacLane spaces. Let 8(n,q; 7T,G) be the group of
all cohomology operations of type (n,q; 7T,G). Thus 7T and G are abelian
groups and an element () E 8(n,q; 7T,G) is a natural transformation from the
Singular cohomology functor Hn(" ;7T) to the singular cohomology functor
Hq(" ;G).
Let 7T be an abe lian group and let Y be a space of type (7T, n),
with an n-characteristic element l E Hn(Y,yo; 7T). There is an isomorphism
13
THEOREM
y: 8(n,q; 7T,G) ;:::::; Hq(Y,yo; G)
defined by y(()) = ()(l) for () E 8(n,q; 7T,G).
430
OBSTRUCTION THEORY
CHAP.
8
PROOF
Since, by theorem 7.8.1, every pair has a relative CWapproximation,
a cohomology operation corresponds bijectively to a cohomology operation on
the category of relative CW complexes. To define an inverse to y, given
u E Hq(Y,yo; G), let 8u be the cohomology operation of type (n,q; 'IT,G) defined
for a relative CW complex (X,A) by
8u (v) =f~(u)
where fv: (X,A) ----> (Y,yo) is a map such that f: (t)
up to homotopy, by theorem 10). Then
y(8u )
= v (fv exists and is unique
= 8u (t) = It(u) = u
showing that the map u ----> 8u is a right inverse of y. To show that it is also a
left inverse of y, let (X,A) be a relative CW complex and let v E Hn(X,A; 'IT).
We must show that 8Y (B)(V) = 8(v). Let fv: (X,A) ----> (Y,yo) be such that
f'; (t) = v. Then we have
8(v)
= 8(f:f(t)) = f~(8(t)) = f:f(y(8)) = 8Y(B)(V)
•
We present one application of this result.
14 COROLLARY Let 8 be a cohomology operation of type (n,q; 'IT,G). For any
relative CW complex (X,A) the map
8: Hn((X,A) X (U); 'IT) ----> Hq((X,A) X (1,1); G)
is a homomorphism.
PROOF
The collapsing map
k: (X X I, A X I U X X 1) ----> X X I/(A X I U X X
i)
induces isomorphisms in cohomology. Furthermore, X X I/(A X I U X X 1)
is homeomorphic to S(X/ A) (where X/A is understood to be the disjoint
union of X and a base point Xo in case A is empty). Thus it suffices to show
that if X' is any pointed CW complex, then the map
8: Hn(SX',xo; 'IT) ----> Hq(SX',xo; G)
is a homomorphism.
Let Y be a CW complex of type ('IT,n), with n-characteristic element t,
and let Y' be a space of type (G,q), with q-characteristic element t'.
Let f: Y ----> Y' be a map such that f* t' = 8(t). There is then a commutative
diagram
[SX',xo; Y,yol ~ [SX',xo; Y',y6l
Hn(SX',xo; 'IT) ~ Hq(SX',xo; G)
It is trivial that f# is a homomorphism when the top two sets are given group
structures by the H cogroup structure of SX'. By lemma 7.7.6, it follows that
SEC.
1
431
EILENBERG-MACLANE SPACES
both vertical maps are homomorphisms. Hence the bottom map () is a
homomorphism. •
Let I E Hl(I,i; Z) be a generator and define an isomorphism
r:
Hr(X,A; G') ;:::; Hr+1((X,A) X (I,i); G')
by r(u) = u X 1. Given a cohomology operation () of type (n,q; 7T,G), its
suspension S() is the cohomology operation of type (n - 1, q - 1; 7T,G)
defined by (S())(u) = r- 1 ()r(u) for u E Hn-l(X,A; 7T). Then corollary 14 implies
that the suspension of any cohomology operation is an additive cohomology
operation.
We now extend theorems 10 and 12 to other spaces Y by restricting the
dimension of the relative CW complex (X,A). Let Y be an n-simple (n - 1)connected pointed space for some n :2: 1 [if n = 1 then 7Tl(Y,YO) is abelianJ.
If t E Hn(y,yo; 7T) is an n-characteristic element for Y, an argument similar to
that in theorem 12 shows that Y can be imbedded in a space Y' of type (7T,n)
having an n-characteristic element u E Hn(Y',yo; 7T) such that u I Y = t. It
follows that the inclusion map Y C Y' is an (n + I)-equivalence. Then
theorems 7.6.22 and 10 yield the following generalization of theorem 10.
15 THEOREM Let t E Hn( Y, yo; 7T) be n-characteristic for an n-simple (n - 1)connected pointed space Y and let (X,A) be a relative CW complex. The map
1/;,: [X,A; Y,yoJ
~
Hn(X,A;
7T)
defined by 1/;,[fl = f* (t) is a bijection if dim (X - A) :::; n and a surjection
if dim (X - A) :::; n + 1. •
For the special case Y = Sn let s* E Hn(sn,po; Z) be a generator. Then
s* is an n-characteristic element of Sn, and we obtain the following Hopf
classificati(J.n theorem. 1
16
COROLLARY
where n
Let (X,A) be a relative CW complex, with dim (X - A) :::; n,
:2: 1. If s* E Hn(sn,po; Z) is a generator, there is a bijection
1/;8*: [X,A; sn,poJ ;:::; Hn(X,A; Z)
defined by 1/;8* ([fl) = f* (s*).
•
Similarly, we obtain the following generalization of theorem 12.
17 THEOREM Lett E Hn(y,yo; 7T) be n-characteristicforann-simple(n - 1)connected pointed space Y and let (X,A) be a relative CW complex, with
dim (X - A) :::; n + 1. A map f: A ~ Y can be extended over X if and only
if 8f* (t) = 0 in Hn+l(X,A; 7T). •
This specializes to the following Hopf extension theorem.
1 See H. Hopf, Die Klassen der Abbildungen der n-dimensionalen Polyeder auf die n-dimensionale Sphiire, Commentarii Mathematici Helvetici, vol. 5, pp. 39-54, 1933, and H. Whitney,
The maps of an n-complex into an n-sphere, Duke Mathematical Journal, vol. 3, pp. 51-55, 1937.
432
OBSTRUCTION THEORY
CHAP.
8
18 COROLLARY Let (X,A) be a relative CW complex, with dim (X - A) :::;;
n + 1, and let s* E Hn(Sn,po; Z) be a generator. A map f: A ~ Sn can be
extended over X if and only if 13f* (s*) = 0 in Hn+1(X,A; Z). •
2
PRINCIPAL FIBRATIONS
This section is concerned with fibrations whose fiber is an Eilenberg-MacLane
space. We shall develop an obstruction theory for the lifting problem of maps
of relative CW complexes to such fibrations. In the next section we shall show
that many maps can be factored up to weak homotopy type as infinite composites of such fibrations. In this way the obstruction theory for these special
fib rations leads to an obstruction theory for arbitrary maps.
For any pointed space B' there is the path fibration PB' 14 B', where PB'
is the space of paths in B' beginning at the base point boo Under the exponential correspondence there is a one-to-one correspondence between homotopies H: X X I ~ B' such that H(x,O) = boand maps H': X ~ PB', the correspondence defined by H'(x)(t) = H(x,t). This easily implies the following result
(which is dual to lemma 7.1.1).
I
LEMMA A map X ~ B' is null homotopic if and only if it can be lifted
to the path fibration PB' ~ B'. •
If (}: B ~ B' is a base-point-preserving map, there is a fibration po: Eo ~ B
induced from the path fibration PB' ~ B'. This induced fibration is called the
principal fibration induced by (} and has fiber po -l(bo) bo X QB'. A
straightforward verification shows that there is a covariant functor from the
category of base-point-preserving maps between pointed spaces to the subcategory of fibrations which assigns to (} the principal fibration induced by (}.
Let (X,A) be a pair and let i: A c X be the inclusion map. Let po: Eo ~ B
be the principal fibration induced by (}: B ~ B'. Recall that a map pair
f: i ~ po (defined in Sec. 7.8) is a commutative square
=
A ~ Eo
i~
~P'
X-4B
The set of homotopy classes [i;PoJ of map pairs from i to po is the object function
of a functor of two variables contravariant in pairs (X,A) and covariant in basepoint-preserving maps (}. We are interested in studying in more detail the
relative-lifting problem (that is, the map p: [X;EoJ ~ [i;po]) for this situation.
Because po is an induced fibration, the relative-lifting problem is equivalent
to an extension problem, as shown below.
Let po: Eo ~ B be induced by (}: B ~ B'. For any space W a map
f: W ~ Eo consists of a pair /1: W ~ Band fz: W ~ PB' such that
p' fz = (} fl. By the exponential correspondence, fz corresponds to a
homotopy F: W X I ~ B' from the constant map to (} /1. Thus, given a map
f1: W ~ B, there is a one-to-one correspondence between liftings f: W ~ Eo
0
0
0
SEC.
2
433
PRINCIPAL FIB RATIONS
of f1 and homotopies F: W X I --c> B' from the constant map to B h.
Let (X,A) be a pair with inclusion map i: A C X and let f: i --c> po be a
map pair consisting of maps f": A --c> Eo and 1': X --c> B such that po f" =
l' i. We define a map
0
0
0
B(f): (A X I U X X
i,
X X 0)
--c>
(B',b&)
by the conditions B(f)(x,O) = b&, B(f)(x,l) = B1'(x), for x E X, and
B(f) I A X I is the homotopy from the constant map A --c> b& to the map
B l' i corresponding to the lifting f" of l' i. There is then a one-to-one
correspondence between liftings of f and extensions of B(f) over X X 1.
We now specialize to the case where B' is a space of type (w,n), with
n ~ 1 and w abelian, and we let t E Hn(B',b&; w) be n-characteristic for B'.
In this case, if B: B --c> B' is a base-point-preserving map, the induced fibration
po: Eo --c> B is called a principal fibration of type (w,n). If (X,A) is a relative
CW complex, then (X,A) X (I,i) is also a relative CW complex, and given a
map g: A X I U X X i --c> B', it follows from theorem 8.1.12 that g can be
extended over X X I if and only if 8g* (t) = 0 in Hn+1((X,A) X (I,i); w).
In particular, given a map pair f: i --c> po, there is a lifting of f if and only if
8B(f)* (t) = O. The obstruction to lifting f, denoted by c(f) E Hn(X,A; w),
is defined by
0
0
0
8B(f)* (t) = (-l)nT(c(f))
where T: Hn(X,A; w) :::::: Hn+1((X,A) X (1)); w) is the map T(U) = U X 1, defined in Sec. 8.1 [1 E H1( I,i; Z) is the generator such that if 0 E HO( {O}; Z)
and I E HO( {1 }; Z) are the respective unit integral cohomology classes, then,
identifying HO(i;Z) :::::: HO({O};Z) EB HO({l};Z), we have 81 = 1 = -80J.
2
EXAMPLE
In case A is empty, a map pair f: i --c> po is just a map
1': X --c> B. In this case B(f): X X i --c> B' is such that B(f)(x,O) = b& and
B(f)(x,l) = B1'(x). Then B(f)* (t) = l' * B* (t) X 1, and so, by statement 5.6.6,
= (-l)n1'*B*(t)
Therefore, in this case c(f) = l' * B* (t).
8B(f)*(t)
X 1
= (-l)nT1'*B*(t)
It is clear from the definition that the obstruction to lifting f is functorial
in i and B and that it vanishes if and only if there is a lifting of f. We obtain
a similar cohomological criterion for the existence of a homotopy relative to f
of two liftings of f.
Let J: i --c> po be a map pair, where (X,A) is a relative CW complex, with
i: A C X, and po is a principal fibration of type (w,n). Given two liftings
!o, !1: X --c> Eo of f, let g: i' --c> po be the map pair consisting of the commutative square
A X I U X X
i'J
XXI
i
g"
~ Eo
434
OBSTRUCTION THEORY
CHAP.
8
where g' is the composite X X I ~ X 4 Band g" is the map such that
g"(x,O) = Io(x) and g"(x,l) = Il(x) for x E X and g"(a,t) = f"(a) for a E A and
tEl. Then 10 and 11 are homotopic relative to f if and only if g can be lifted.
The obstruction to lifting g is an element c(g) E Hn( (X,A) X (I,i); 'TT), and we
define the difference between 10 and h denoted by d(fo,fl) E Hn-l(X,A; 'TT), by
c(g) = (- 1 )nT( d(fo,fl))
[so 88(g)* (t) = T2(d(fo,fl))]. Then 10 and 11 are homotopic relative to f if and
only if d(fo,fl) = O. The difference d(fo,fl) is functorial and has the following
fundamental properties.
3
LEMMA
Given a map pair f: i ~ po and liftings 10, 11, fz: X ~ Eo, then
d(fo,f2) = d(fo,fl)
+ d(fd2)
Let II = [O,lh], 11 = {O,~}, 12 = [IJ2,l], and 12 = {~,l} and define a
map pair G: i ~ po consisting of the commutative square
PROOF
A X I U X X
(il
U
i2) ~
XXI
Eo
J4B
where G'(x,t) = f'(x), G"(a,t) = f"(a), G"(x,O) = Io(x), G"(x,1f2) = Il(x), and
G"(x,l) = fz(x). Then c(G) E Hn((X,A) X (1,1 1 U 12 ); 'TT), and by the naturality
of c( G) and the definition of d, we see that
c(G) I (X,A) X (I,i) = (-1)nT(d(fo,f2))
c(G) I (X,A) X (11,11) = (-l)nTl(d(fo,fl))
c(G) I (X,A) X (Zz,1 2 ) = (-1)nT2(d(fd2))
where
Tl: Hn-l(X,A) ;:::::: Hn((X,A) X (11,11))
and
are defined analogously to T. From these properties, an argument similar to
that used in proving that the Hurewicz homomorphism is a homomorphism
(d. theorem 7.4.3) shows that
T(d(fo,f2)) = T(d(fo,fl))
Since
T
is an isomorphism, this is the result.
+ T(d(fd2))
-
4
THEOREM
Given a map pair f: i ~ po, a lifting 10: X ~ Eo off, and an
element v E Hn-l(X,A; 'TT), there is a lifting fr: X ~ Eo of f such that
d(fo,fl) = v.
The map (J(f): A X I U X X 1 ~ B' used in defining c(f) admits an
extension ho: X X I ~ B' which corresponds to the lifting 10: X ~ Eo. We
seek another extension of (J(f) which will correspond to the desired lifting 11
of f. Let F: (A X I X I U X X (0 X I U I X 1), X X I X 0) ~ (B',b o) be the
map defined by F(a,t,t') = (J(f)(a,t') for a E A and t, t' E I, and
F(x,O,t) = ho(x,t), F(x,t,O) = bo, and F(x,t,l) = ho(x,l) for x E X and t E 1.
PROOF
SEc.2
435
PRINCIPAL FIBRATIONS
Because X X [ X 0 is a strong deformation retract of the space
A X [ X [ U X X (0 X [ U [ X i), there is a homotopy relative to X X [ X 0
from F to the constant map F from A X [ X [ U X X (0 X [ U [ X i) to boo
Let G: (X X 1 X [, A X 1 X [ U X X 1 X i) ~ (B',b o) be a map such
that G*(L) = (_l)n-lv X I X IE Hn((X,A) X {l} X ([,i); 'IT) [such a map
exists, by theorem 8.1.10, because (X,A) X {I} X ([,1) is a relative CW com-
plex]. There is a well-defined map
H': (A X
[2
u
X X
F,
such that H' I X X 1 X [
H'I A
A X [ X [ U X X (0 X [ U [ X 1)) ~ (B',b o)
= G. Then
X [X [ U X X (0 X [ U [X j)
=F
and because (X,A) X ([ X [,OX [ U [ X j) is a relative CW complex, the
homotopy F ::::0 F reI X X [ X 0 extends to a homotopy H' ::::0 H reI X X [ X 0,
where
H: (A X [ X [ U X X j X [ U X X [ X i, X X [ X 0) ~ (B',b o)
=
is an extension of F. Let hI: X X [ ~ B' be defined by h 1 (x,t)
H(x,l,t).
Since H is an extension of F, hI is an extension of (}(f), and hence hI corresponds to a lifting /1 of f.
We now show that /1 has the desired properties. The definition of the
map pair g: i' ~ po used to define d(jo,fl) is such that (}(g)
H. Therefore
=
= ~H* (L) = ~H' * (L)
T 2 (d(jo,fl))
H' is a map from (A X [2 U X X j2, A X [2 U X X (0 X [ U [ X 1)) to
(B',b o) whose restriction to X X 1 X [is G. From the commutativity of the
diagram [where the map p, is given by p,(w X I X I)
w X I for
wE H*(X,A)]
=
Hn(A X [2 U X X
F, A
X [2 U X X (0 X [ U [ X
i))
"-so:::::
:::::j('
Hn(A X [2 U X X j2, X X [ X 0)
~~XlX~AXlX[UXXlX~
st
fLt:::::
Hn+l((X.A) X ([2,F))
Hn((X.A) X ([,~)
«_1)n-l,.
it follows that
8H'*(L)
Since
T2
= (_l)n-l T p,G*(L) = T(V X 1) = T2(V)
is an isomorphism, dUo,/!) =
V.
•
5
THEOREM
Let (X,A) be a relative CW complex and let (X',A) be a subcomplex, with inclusion maps i: A C X, i': A C X', and i": X' C X. Given a
map pair f: i ~ po (consisting off": A ~ Eo and f': X ~ B) and two liftings
go, gl: X' ~ Eo offl i': i' ~ po, let go, gl: i" ~ po be the map pairs consisting,
respectively, of the commutative squares
436
OBSTRUCTION THEORY
X' ~Ee
i"l
X
CHAP.
8
X' ~ E8
Ip,
Ip,
i"l
LB
LB
X
Then
where 8: Hn-l(X',A; 'TT)
PROOF
---'>
Hn(x,X'; 'TT).
Let h: T ---'> po be the map pair defined by the commutative square
A X I U X' X
i 4
Eo
X' X I U X X i ~ B
where h"(a,t) = f"(a) for a E A and tEl, h"(x',O) = go(x') and h"(x',l) =
gl(X') for x' E X', and h'(x,t) = f'(x) for (x,t) E X' X I U X X i. Then
c(h) E Hn(X' X I U X X i, A X I U X' X i; 'TT). There is an isomorphism
Hn(X' X I U X X
i,
A X I U X' X
i;
'TT) ;::::;
Hn((X',A) X (I,i); 'TT) (fl Hn((x,X') X
i;
'TT)
induced by restriction. By the naturality of the obstruction, c(h) corresponds
to (-l)n7"d(go,gl) = (-l)nd(go,gl) X i in the first summand and to
c(go) X 0 + C(gl) X 1 in the second summand.
Let h: i ---'> po be the map pair defined by the commutative square
A X I U X' X
XXI
i4
Eo
h'
~B
where h'(x,t) = f'(x) for x E X and tEl. Then
c(h) E Hn(x X I, A X I U X' X
i;
'TT)
and by the naturality of the obstruction again,
c(h) I (X' X I U X X i, A X I U X' xi) = c(h)
From the exactness of the sequence
Hn(X X I, A X I U X' xi)
it follows that 8c(h)
theorem 5.6.6)
o=
---'>
Hn(X' X I U X X i, A X I U X' xi)
~ Hn+1(X X I, X' X I U X xi)
= o. Therefore, in Hn+l((X,A)
X (I)); 'TT) we have (using
8[( -l)nd(go,gl) X i + c(go) X 0 + C(gl) X 1J
X 1 - (-l)nc(go) X 1 + (_l)nC(gl) X i
= (-1)n8d(go,gl)
Therefore 7"( 8d(go,gl) - c(go)
result follows. •
+ C(gl))
= 0, and since 7" is an isomorphism, the
SEc.3
437
MOORE·POSTNIKOV FACTORIZATIONS
We compute the obstruction c(f) explicitly for the case of a fibration
pI: QB' ~ bo, where B' is a space of type ('7T,n), with n> 1. Then QB' is a
space of type ('7T, n - 1), and if II E Hn-l(QB',wo; '7T) is (n - I)-characteristic
for QB' and l E Hn(B',b o; '7T) is n-characteristic for B', then Oll and p* l [where
0: Hn-l(QB',wo) :::::: Hn(PB',QB') and p: (PB',QB') ~ (B',b o)] are both elements
of Hn(PB',QB'; '7T). The characteristic elements land II are said to be related
if &1 = p* l. Given one of l or ll, it is always possible to choose the other one
(uniquely) so that the two are related.
6
THEOREM
Let l E Hn(B',b o; '7T) and II E Hn-l(QB',wo; '7T) be related
characteristic elements. Let (X,A) be a relative CW complex, with inclusion
map i: A eX. Given a map pair f: i ~ pI, where pI: QB' ~ bo, then
c(f) = - of" * (l/), where f": A ~ QB' is part of f
PROOF
Let f: (A X I, A X i) ~ (PB',QB') be the map defined by f(a,t)(t')
f"(a)(tt' ). Then
=
8(f): (A X I U X X t, X X 0) ~ (B',b o)
=
=
i~ the map such that 8(f) I A X I
P f and 8(f)(X X t)
boo Let
(A X I U X X t, X X t) ~ (B',b o) be the map defined by 8(f) and let
f': (A X t, A X 0) ~ (QB',WO) be the map defined by There is then a com-
f:
0
f.
mutative diagram [in which i and l' are appropriate inclusion maps and
hI: A ~ (X X t, A X 0) is defined by hl(a) = (a,I)]
Hn(A X I U X X 1, X X 0)
if
8(f),,/
1*
--'--?
Hn(B',b o)
~
Hn(A X I U X X 1, X X 1) -4 Hn+l((X,A) X (I,i))
:: I
/*1
pol
Hn(PB',QB') ~
Hn(A X I, A X 1)
Hn-l(QB' ,wo) ~
Hn(X,A)
( _l)n-IT
81
~
Hn-l(A X 1, A X 0)
h*
~
81
(_l)n-IT
18
Hn-l(A)
Furthermore, 0 r- 1 l' * = r- 1 0: Hn(A X I U X X t, X X i) ~ Hn(X,A).
Since f" = f' hI, then f" * = h! f' * , and we have
0
0
0
0
0
(_I)n-lr- 10(8(f))* (l) = of" * (l/)
By definition, the left-hand side above equals - c(f).
3
•
MOORE-POSTNIKOV FACTORIZATIO:\,S
This section is devoted to a method of factorizing a large class of maps up to
weak homotopy type as infinite composites of simpler maps, the simpler maps
438
OBSTRUCTION THEORY
CHAP.
8
being of the same weak homotopy type as principal fibrations of type ('IT,n)
for some 'IT and n. The cohomological description of the lifting problem for
these fibrations, given in the last section, will lead us ultimately to an iterative
attack on general lifting problems.
Given a sequence of fibrations Eo .EJ El ~ ... , we define
Ex;
= lim~ {Eq,pq} = {(eq) E
X Eq I pq(eq)
= eq-d
and we define aq: Eoo ~ Eq to be the projection of Kxo to the qth coordinate.
Then each map aq is a fibration and aq = Pq+l aq+l for q ;:-:: O. For any
space X a map f: X ~ KfO corresponds bijectively to a sequence of maps
{fq: X ~ Eq}q:>o such thatfq = Pq+l fq+l for q ;:-:: 0 (givenf, the sequence
{fq} is defined by fq = aq f). In particular, given a pair (X,A) with inclusion
map i: A C X and a map pair f: i ~ ao consisting of the commutative square
0
0
0
A ~ Ex
il
X
lao
L
Eo
a lifting f: X ~ Eoo corresponds bijectively to a sequence of maps
{fq: X ~ Eq} q:>O such that
(a) fo = f': X ~ Eo
(b) For q ;:-:: 1 the map fq: X ~ Eq is a lifting of the map pair from ito pq
consisting of the commutative square
A ~ Eq
i1
1
pQ
X ~ E q_ 1
In this way the relative-lifting problem for a map pair f: i ~ ao corresponds
to a sequence of relative-lifting problems for map pairs from i to pq. In many
cases the relative-lifting problems for the fibrations pq may be simpler to deal
with than the original relative-lifting problem for the fibration ao.
A sequence of fib rations Eo J!! El ~ ... is said to be convergent if for
any n
00 there is N n such that pq is an n-equivalence for q
Nn •
Let f: Y' ~ Y be a map. A convergent factorization of f consists of a
sequence {pq,Eq,fq}q:>1 such that
>
<
(a) For q > 1, pq: Eq ~ Eq- 1 is a fibration, and for q = 1, PI: El ~ Y
fibration.
For q ;:-:: 1, fq: Y' ~ Eq is a map, fq = Pq+l fq+l for q ;:-:: 1, and
PI h
For any n < 00 there is N n such that fq is an n-equivalence for
q>Nn .
is a
(b)
f =
(c)
0
0
Conditions (a) and (b) imply that for
q;:-::
1,
f
equals the composite
SEc.3
MOORE-POSTNIKOV FACTORIZATIONS
439
P1 a ••• a pq a fq. The convergence condition (c) implies that, in a certain
sense, the infinite composite P1 a P2 a ••• exists.
If {pq,Eq,fq}q;,1 is a convergent factorization of a map f: Y' --,) Y, then
the sequence of fib rations Y?' E1 ?!: .. , is convergent. The following
theorem shows that any convergent sequence of fibrations is obtained in this
way from a convergent factorization of some map.
If Eo .f!-:- E1 jl3 . .. is a convergent sequence of fibrations,
then {pq,Eq,aq}q;,1 is a convergent factorization of the map ao: Eoo --,) Eo.
I
THEOREM
Conditions (a) and (b) for a convergent factorization are clearly
satisfied. To prove that the convergence condition (c) is also satisfied, given
00, choose N so that pq is an (n + I)-equivalence if q ~ N. We
I :::;; n
prove that aq is an n-equivalence for q ~ N. Because aq = Pq+1 a aq+1, and
Pq+1 is an (n + I)-equivalence for q ~ N, it suffices to prove that aN is
an n-equivalence.
Let (P,Q) be a polyhedral pair such that dim P :::;; n and let lX: Q --,) Ex
and {3H: P --,) EN be maps such that {3fv I Q = aN a lx. We now prove that
there is an extension {3: P --,) KfO of lx such that aN a {3 = {3/V. The map lx
corresponds to a sequence lXq = aq a lx: Q --,) Eq such that lXq = Pq+1 a lXq+1,
and to define a map {3: P --,) Eoo with the desired properties, we must obtain
a sequence of maps {3q: P --,) Eq such that {3q I Q = lx q, {3q = Pq+1 a {3q+1, and
{3N = {3N. Such a sequence of maps {{3q} is defined for q :::;; N by {3q =
Pq+1 a • . • a PN a {3N, and for q ~ N it is defined by induction on q as follows.
Assuming {3q defined for some q ~ N, we use theorem 7.6.22 to find a map
{3~+1: P --,) Eq+1 such that {3~+1 I Q = lXq+1 and such that {3q ~ Pq+1 a /3'q+1
reI Q. We use the fact that Pq+1 is a fibration (and theorem 7.2.6) to alter {3~+1 by
a homotopy relative to Q to obtain a map {3q+1: P --,) Eq+1 such that
{3q+1 I Q = lXq+1 and such that {3q = Pq+1 a {3H1. Thus the sequence {{3q} can
be found, and hence a map {3: P --,) Eoo with the requisite properties exists.
Taking P to be a single point and Q to be empty, we see that aN is
surjective, and so aN maps 'lTo(E"J surjectively to 'lTO(EN)' Taking (P,Q) = (I,i),
we see that aN maps 'lTo(E"J injectively to 'lTO(EN)' Then aN induces a one-toone correspondence between the set of path components of Ex and the set of
path components of EN.
Let e* = (eq) E Ex be arbitrary and let I :::;; k :::;; n. Taking (P,Q) = (Sk,ZO)
it follows that aN# maps 'lTk(Eoo,e*) epimorphically to 'lTk(EN,eN). For I :::;; k
n,
taking (P,Q) = (Ek+l,Sk), it follows that lXN# maps 'lTk(Eoo,e*) monomorphically
to 'lTk(EN,eN). Hence aN is an n-equivalence. PROOF
<
<
COROLLARY
Let {pq,Eq,fq} q;, 1 be a convergent factorization of a map
Y' --,) Y and let f': Y' --,) Ex be the map such that aq a f' = fq for q ~ I
and ao a f' = f. Then f' is a weak homotopy equivalence.
2
f:
PROOF
+ I)-equivalences
aq
0
f'
<
For any I :::;; n
00 there is q such that aq and fq are both
(by theorem 1). Then f' is also an n-equivalence (because
= fq). Since this is so for all n, f' is a weak homotopy equivalence. -
(n
440
OBSTRUCTION THEORY
CHAP.
8
In particular, given a convergent factorization {pq,Eq,fq}q'21 of a weak
fibration p: E ---'? B, there is a weak homotopy equivalence g: p ---'? ao consisting of the commutative square
B-4B
If (X,A) is a relative CW complex, with inclusion map i: A C X, it follows
from theorem 7.8.12 that the relative-lifting problem for a map pair h: i ---'? P
is equivalent to the relative lifting problem for the map pair g h: i ---'? ao.
We shall now add hypotheses which will ensure that the sequence of fibrations into which the fibration ao is factored (namely, the fibrations {pq}) leads
to relative-lifting problems which can be settled by the methods of the last
section.
A Moore-Postnikov sequence of fib rations Eo ~ El J!.: ... is a convergent
sequence of fibrations such that pq: Eq ---'? Eq_1 is a principal fibration of type
(Gq,nq) for q ;::: 1. A Moore-Postnikov factorization of a map f: Y' ---'? Y is a
convergent factorization {pq,Eq,fq}q'21 of f such that Eo <f!1 El .j!1 ... is a
Moore-Postnikov sequence of fibrations. A Postnikov factorization of a space
Y' is a Moore-Postnikov factorization of the map f: Y' ---'? Y, where Y is the
set of path components of Y' topologized by the quotient topology and f is
the collapsing map. Thus, if Y' is path connected, a Postnikov factorization of
Y' is a Moore-Postnikov factorization of the constant map Y' ---'? yo.
A Moore-Postnikov factorization of a map is a factorization of the map
(up to weak homotopy type) as an infinite composite of elementary maps.
The relative-lifting problem associated to this sequence is thereby factored
into an infinite sequence of elementary relative-lifting problems. We shall
show that Moore-Postnikov factorizations exist for a large class of maps
between path-connected spaces.
Let f: Y' ---'? Y be a map between path-connected pointed spaces. For
n ;::: 1 an n-factorization off is a factorization of f as a composite Y'14 E' -4 Y
such that
0
(a) E' is a path-connected pointed space, p' is a fibration, and h' is a
p' h')
lifting of f (that is, f
(h) h#: '7Tq(Y') ---'? '7Tq(E') is an isomorphism for 1 ::;: q n and an epimorphism for q = n (that is, h' is an n-equivalence)
(c) p#: '7Tq(E') ---'? '7Tq(Y) is an isomorphism for q n and a monomorphism
for q = n
=
0
<
>
A map f: Y' ---'? Y between path-connected pointed spaces is said to be
simple if f#('7Tl(Y')) is a normal subgroup of '7Tl(Y) and the quotient group is
abelian, and if (Z" Y') is n-simple for n ;::: 1 (as defined in Sec. 7.3). We are
heading toward a proof of the result that a simple map admits Moore-Postnikov
factorizations. We need one more auxiliary concept.
SEc.3
MOORE-POSTNIKOV FACTORIZATIONS
441
Given a pointed pair (X,A) of path-connected spaces, a cohomology
class v E Hn(X,A; 1T) is said to be n-characteristic for (X,A) if either of the following conditions hold:
(a) n = 1 and i#( 1Tl(A)) is a normal subgroup of 1Tl(X) whose quotient
group is mapped isomorphically onto 1T by the composite
1Tl(X)/i#(1Tl(A)) ~ H1(X)/i* (Hl(A)) ~ Hl(X,A) ~ 1T
(b)
n> 1 and the composite
1Tn(X,A) ~ Hn(X,A) ~ 1T
is an isomorphism
In case A = {xo}, the concept of n-characteristic element for the pair
(X,{xo}) agrees with the concept of n-characteristic element for the space X
as defined in Sec. 8.1.
3
LEMMA
Let i: A c X be a simple inclusion map between path-connected
pointed spaces such that the pair (X,A) is (n - I)-connected, where n ~ l.
Then there exist cohomology classes v E Hn(X,A; 1T) which are n-characteristic
for (X,A), where 1T = 1Tl(X)/i#(1Tl(A)) for n = 1 and 1T = 1Tn(X,A) for n
l.
>
If n = 1, it follows from the absolute Hurewicz isomorphism theorem
applied to A and to X that there are isomorphisms
PROOF
1Tl(X)/i#(1Tl(A))
~
H1(X)/i* (Hl(A)) i. Hl(X,A)
By the universal-coefficient formula for cohomology, there is also an
isomorphism
h: Hl(X,A; 1T) :::::; Hom (Hl(X,A),1T)
Hence, if 1T = 1Tl(X)/i#(1Tl(A)), there exist I-characteristic elements
v E Hl(X,A; 1T).
If n
1, it follows from the relative Hurewicz isomorphism theorem and
the universal-coefficient formula for cohomology that there are isomorphisms
cp: 1Tn(X,A) :::::; Hn(X,A) and h: Hn(X,A; 1T) :::::; Hom (Hn(X,A),1T). Therefore, if
1T = 1Tn(X,A), there are n-characteristic elements v E Hn(X,A; 1T). •
>
4
LEMMA
Let (X,A) be a pointed pair of path-connected spaces (n - 1)connected for some n ~ 1 and such that the inclusion map i: A C X is simple.
Then there is an n-Jactorization A .!4 E' .4 X of i such that p' is a principal
fibration of type (1T,n), where 1T = 1Tl(X)/i#(1Tl(A)) if n = 1 and 1T = 1Tn(X,A)
if n > 1.
By lemma 3, there is a class v E Hn(X,A; 1T) which is n-characteristic
for (X,A). Let CA be the cone (nonreduced) over A and observe that {X,CA}
is an excisive couple in X U CA. Therefore there is an element
v' E Hn(X U CA; 1T) corresponding to v under the isomorphisms
PROOF
Hn(X U CA; 1T)
~
Hn(x U CA, CA; 1T) -:;? Hn(X,A; 1T)
442
OBSTRUCTION THEORY
CHAP.
8
It is possible to imbed X U CA in a space X' of type (7T,n) having an
n-characteristic element L' such that L' I X U CA = v'. Let p': E' ~ X be the
principal fibration induced by the inclusion X C X' and let PA: EA ~ A be
the restriction of this fibration to A. There is a section s: A ~ EJ. such that
s(a) = (a,w a ) for a E A, where Wa is the path from Xo to the vertex of CA
followed by the path from the vertex of CA to a (that is, wa(t) = [xo,1 - 2t]
for 0::; t::; lh and wa(t) = [a, 2t - 1] for lh ::; t::; 1). We define h': A ~ E'
iA
I
to be the composite A -4 EJ. C E' and shall prove that A 14 E' ~ X is an
n-factorization of i.
The fiber of P' (and hence also of pJ.) is [2X', and we define g: EJ. ~ [2X'
by g(a,w)
w * (s(a))-l. Then g I [2X': [2X' ~ [2X' is homotopic to the identity
map. If i": [2X' C EJ. is the inclusion map, it follows from the exactness of the
homotopy sequence of the fibration pJ.: E1 ~ A that there is a direct-sum
decomposition
=
q
1
~
(This is a direct-product decomposition for q = 1, but we shall still write it
additively.) We define a homomorphism A: 7Tq(X,A) ~ 7Tq _l([2X'), where
q ~ 1, to be the composite
7Tq(X,A) P~\ 7Tq (E',EA) ~ 7Tq_l(EA) ~ 7Tq_l([2X')
We show that the following diagram commutes up to sign:
7Tq (A) ~ 7Tq(X) ~ 7Tq(X,A) --4 7Tq_l(A)
~
7T q(E')
..!!.4
~
b
~
--4 7Tq_l([2X') .!4 7Tq_l(E')
7T q(X)
In fact, the left-hand and middle squares are easily seen to be commutative.
We shall show that h# 0 = -i# A.
For q = 1 this is so because 7To(A) = 0 implies that h# 0 is the trivial
map and the fact that i# is surjective and i# 0 A 0 i# = i# 0 a = 0 implies
that i# A is also the trivial map. For q
1 we have
0
0
0
>
0
a
= i~ + s#pJ.~
Since the composite 7Tq(E',El) -4 7Tq_l(E;')
for f3 E 7Tq (E',EA)
a E 7Tq -l(EJ.)
~ 7Tq_l(E')
0= iA#of3 = i~i'~of3
is trivial, it follows that
+ iA~#pJ.#of3
= i~of3 + h#oP#f3
= ~of3. Therefore
i#-\P#f3 + h#op#f3 = 0
By definition of A, we see that Ap#f3
Since P#: 7Tq (E',E.J.) ;:::; 7Tq(X,A), this proves h# 0 = -i# A.
A straightforward verification shows that A is also the composite
0
0
SEc.3
443
MOORE-POSTNIKOV FACTORIZATIONS
The construction of X' and
diagram
7Tn(X,A)
E Hn(X','rr) shows that there is a commutative
7T n(X U CA, CAl
--'>
~
7T n(X U CAl
--'>
Hn(X U CA, CAl
",.
h(u~
--'>
~
Hn(X U CAl
h(VY
7T n(X')
~lqo
qcl
qcl
qol~
Hn(X,A)
t'
--'>
Hn(X')
~<')
7T
Therefore A: 7T n(X,A) ;::::; 7T n _l(r2X').
In case n = 1,
7Tl(X) -+ 7To(flX') is surjective [because 7To(A) = 0], and
so E' is path connected. If n
1, E' is path connected because 7To(r2X') = O.
Therefore E' is a path-connected pointed space. Since 7T q(r2X') = 0 for q :?: n,
it follows from the exactness of the homotopy sequence of the fibration
p': E' --'> X that p#: 7T q(E') --'> 7T q(X) is an isomorphism for q
n and a
monomorphism for q = n.
Because A: 7Tq(X,A) --'> 7Tq_l(r2X') is a bijection for q ::;: n (the only nontrivial case in these dimensions being q = n), it follows from the five lemma
and the commutativity up to sign of the diagram on page 442 that
b#: 7Tq(A) --'> 7Tq(E') is an isomorphism for 1 ::;: q n and an epimorphism for
q = n. Therefore b' and p' have the properties required of an n-factorization
of i. -
a:
>
>
<
5
COROLLARY
Let g: X' --'> X be a simple map between path-connected
pointed spaces such that for some n :?: 1 the map g#: '7T q(X') --'> 7Tq(X) is an
isomorphism for 1 ::;: q < n - 1 and an epimorphism for q = n - 1. Then
there is an n-factorization X' .!4 E' 4 X of g such that p' is a principal
fibration of type (7T,n) for some abelian group 7T.
PROOF
Let Z be the reduced mapping cylinder of g (that is, the mapping
cylinder of g I xo: Xo --'> Xo has been collapsed to a point). Then (Z,X') is a
pOinted pair of path-connected spaces (n - l)-connected and with simple
inclusion map i: X' C Z. By lemma 4, there is an n-factorization X' 14 E" 4 Z
of i such that p" is a principal fibration of type (7T,n). Let p': E' --'> X be the
restriction of p" to X. Then E' C E" is a homotopy equivalence, so there is a
map h": X' --'> E' such that b" is homotopic to the composite X' 14 E' C E".
Then p' h" is easily seen to be homotopic to g. By the ho~otopy lifting
property of p', there is a map b': X' --'> E' homotopic to b" such that
0
p' b' = g. Then X'.!4 E'
properties. 0
.4
X is easily verified to have the requisite
444
OBSTRUCTION THEORY
CHAP.
8
We are now ready to prove the existence of Moore-Postnikov factorizations of a simple map between path-connected pointed spaces.
6
THEOREM
Let f: Y' ~ Y be a simple map between path-connected
pointed spaces. There is a Moore-Postnikov factorization {pq,Eq,fq}Q?l of f
such that for n ~ 1 the sequence
Y' ~ En PI
Pn) Y
0
••.
0
is an n-factorization of f.
By induction on q, we prove the existence of a sequence {pq,Eq,fq}q?l
such that
PROOF
(a) For n = 1 the sequence Y'
.4 E1 ~
Y is a I-factorization of f.
(b) For n
> 1 the sequence Y' b. En ~ En- 1 is an n-factorization offn-1'
(c) For n
~
1, pn is a principal fibration of type (7Tn,n) for some 7T n.
Once such a sequence {pq,Eq,fq} has been found, it is easy to verify that
it is a Moore-Postnikov factorization of f with the desired property. Therefore
we hmit ourselves to proving the existence of such a sequence.
By corollary 5, with n = 1, there is a I-factorization Y' .4 E1 ~ Y of f
with P1 a principal fibration of type (7T1,I) for some 7T1. This defines P1, E 1,
andh Assume {pq,Eq,fq} defined for 1 :s:; q
n, where n
1, to satisfy (a),
<
>
(b), and (c) above. By corollary 5, there is an n-factorization Y' b. En ~ En- 1
of fn-1 such that pn is a principal fibration of type (7Tn,n) for some 7T n. Then
pn, En, and fn have the desired properties. •
Let Y' be a simple path-connected pointed space. Then Y'
has a Postnikov factorization {pq,Eq,fq}q?l in which 7Tq(En) = 0 for q ~ n
(lnd fn: Y' ~ En is an n-equivalence.
7
COROLLARY
PROOF
If Y' is a simple space, the constant map Y'
The result follows from theorem 6. •
~
yo is a simple map.
In the above the spaces En approximate Y' in low dimensions. We now
present an alternate method of approximating a space in high dimensions by
kilhng low-dimensional homotopy groups.
8
COROLLARY
Let Y be a simple path-connected pointed space. There is a
Moore-Postnikov sequence of fibrations Y ~ E1 .J!-2. .. such that En is
n-connected and P1
pn: En ~ Y induces isomorphisms 7Tq(En) :::::: 7Tq(Y)
for q > n.
0
•••
0
If Y is a simple space, the inclusion map yo C Y is a simple map.
The result then follows from theorem 6. •
PROOF
In the last result the fibration P1: E1 ~ Y has the homotopy properties of a
universal covering space of Y. The fibration P1
pn: En ~ Y is a kind
of "n-covering space."
0
•••
0
SEC.
4
4
445
OBSTRUCTION THEORY
OBSTRL'CTION THEORY
In this section we show how to use Moore-Postnikov factorizations to study
the relative-lifting problem. A sequence of obstructions to the existence of a
lifting (or to the existence of a homotopy between two liftings) is defined
iteratively, and we apply the general machinery to the special case where
either the first one or the first two obstructions are the only ones that enter.
Let p: E -0 B be a fibration between path-connected pointed spaces and
assume that p is a simple map. By theorem 8.3.6, there exist Moore-Postnikov
factorizations {pq,Eq,fq}q"l of p. By corollary 8.3.2, there is a map p': E -0 E"
which is a weak homotopy equivalence. Since p = ao p', where ao: Ex -0 B,
if (X,A) is a relative CW complex, with i: A C X, it follows from theorem 7.8.12
that the relative-lifting problem for a map pair from i to p is equivalent to the
relative-lifting problem for a corresponding map pair from i to ao. Thus we
are led to consider the relative-lifting problem for a map pair from i to ao.
0
Let Eo 'p-l E1 .j!3 ... be a sequence of fibrations with limit Ex and maps
a q: K" -0 Eq and let (X,A) be a relative CW complex, with inclusion map
i: A C X. A map pair f: i -0 ao is a commutative square
A~K"
X
L
Eo
where f" corresponds to a collection {f~: A
for q ;::: O. For q ;::: 1 let fq: i -0 P1
the commutative square
0
A
f"
~
i1
X
f'
~
-0
•••
0
Eq}q"o such that Pq+1 f~~l = f~'
pq be the map pair consisting of
0
Eq
1pl
0
.OPq
Eo
>
If fq: X -0 Eq is a lifting of fq, then pq fq is a lifting of fq-1 for q
1 and a
lifting f: X -0 Ex of f corresponds to a sequence {fq: X -0 Eq} q" 1 such that
0
(a) fq is a lifting of fq for q ;::: l.
(b) Pq+1 fq+1 = fq for q ;::: l.
0
Given a lifting fq: X -0 Eq of fq for q ;::: 1, let g(fq): i
pair consisting of the commutative square
A f;;+l) Eq+1
i1
X
lpq+l
L
Eq
-0
Pq+1 be the map
446
OBSTRUCTION THEORY
CHAP.
8
A map jq+1: X ~ Eq+1 is a lifting of g(jq) if and only if it is a lifting of fq+1
such that Pq+1 jq+1 :::: jq. Thus a sequence of maps {jq: X ~ Eq}q:>l satisfies
conditions (a) and (b) above if and only if it has the following properties:
0
(c) j1 is a lifting of h
(d) For q ~ 1, jq+1 is a lifting of g(jq).
We now add the hypothesis that Eo <fl.1 E1 ~ ... is a Moore-Postnikov
sequence of fibrations. For each q ~ 1, pq is then a principal fibration of type
(7T q,nq). It follows from Sec. 8.2 that h can be lifted if and only if
c(h) E Hnl(X,A; 7T1) is zero. The class C(1) is called the first obstruction to
lifting f.
Assume that for some q
1 there exist liftings jq-1: X ~ Eq- 1 of the
map pair fq-1: i ~ P1
Pq-1. We then obtain map pairs g(jq-1): i ~ pq
and corresponding elements C(g(jq-1)) E Hnq(X,A; 7Tq). The collection
{C(g(jq_1))} corresponding to the set of allliftings jq-1: X ~ Eq_1 of fq-1 is
called the qth obstruction to lifting f. It is a subset of Hnq(X,A; 7T q) and is defined if and only if fq-1 can be lifted. It is clear that there is a lifting of fq if
and only if the qth obstruction to lifting f is defined and contains the zero
element of Hnq(X,A; 7Tq).
Corresponding to a Moore-Postnikov sequence of fibrations we have been
led to a sequence of successive obstructions. The first obstruction is a single
cohomology class, while the higher obstructions are subsets of cohomology
groups. In some cases these obstructions can be effectively computed in terms
of the given map pair f: i ~ ao, and this computation provides a solution of
the lifting problem in these cases. In general, however, the determination of
the successive obstructions involves an iterative procedure of increasing complexity and has not been effectively carried out in each case.
>
0
•••
0
We illustrate this technique by applying it to the Postnikov factorization
of a simple path-connected pointed space Y, given in corollary 8.3.7. There is
a Postnikov factorization {pq,Eq,fq}q:>l of Yin which 7Tq(Em) :::: 0 for q ~ m and
fm: Y ~ Em is an m-equivalence. We call this the standard Postnikov factorization of Y. By corollary 8.3.2, there is a weak homotopy equivalence
f': Y ~ Eoo, and so we consider the lifting problem for a map i ~ ao, where
i: A C X and ao: Eoo ~ yo. Since yo is a point, this is equivalent to the extension problem for a map ftl: A ~ EooThus we seek a sequence of maps jq: X ~ Eq such that j1: X ~ E1 is an
extension of a1 f" and jq+1: X ~ Eq+1 for q ~ 1 is a lifting of the map pair
g(jq): i ~ Pq+1 consisting of
0
l!q+l
X _--,-l--,-q~) Eq
Since Pq+1 is a principal fibration of type (7T q(Y,yo), q + 1), the obstruction to
lifting g(jq) is an element of Hq+1(X,A; 7Tq(Y,yo)). Hence there is defined a
SEC.
4
447
OBSTRUCTION THEORY
sequence of obstructions to extending f": A -') Y, the (q + I)st obstruction
being a subset of Hq+1(X,A; 'TTq(Y,yo)). If Y is (n - I)-connected for some
n ::::: 1, the lowest-dimensional nontrivial obstruction is in Hn+1(X,A; 'TTn(Y,yo)). If
l E Hn(Y,yo; 'TT) is n-characteristic for such a space Y, it follows easily from
theorem 8.2.6 that this lowest obstruction is -+-8f" * L This gives us the following generalization of theorem 8.1.17. 1
I THEOREM Let l E Hn(Y,yo; 'TT) be n-characteristic for a simple (n - 1)connected pointed space Y, where n::::: 1, and let (X,A) be a relative
CW complex such that Hq+l(X,A; 'TTq(Y,yo)) = 0 for q
n. A map f: A -') Y
can be extended over X if and only if 8f* (l) = 0 in Hn+1(X,A; 'TT).
>
PROOF
We use the standard Postnikov factorization of Y. This leads to a sequence of obstructions to extendingfwhich are subsets of Hq+1(X,A; 'TTq(Y,Yo)).
Since these are all zero except Hn+1(X,A; 'TTn(Y,yo)) ;:::::; Hn+l(X,A; 'TT), the only
obstruction to extending f is an element of Hn+1(X,A; 'TT). By the remarks
above, this obstruction vanishes if and only if 8f* (l) = O. •
Let fo, it: X -') Y be maps and define g: X X j -') Y by g(x,O) = fo(x)
and g(x,I) =h(x). For any u E Hq(Y), 8g*(u) = (-I)qT(f!u -f~u) in
Hq+1(X X I, X X
Therefore 8g*(u) = 0 if and only iff~(u) = f!(u), and
we obtain the following partial generalization of theorem 8.1.15 by applying
theorem 1 to the pair (X X I, X X i}
h.
2
THEOREM
Let l E Hn(Y,yo; 'TT) be n-characteristic for a simple (n - 1)connected space Y, where n ::::: 1, and let X be a CW complex such that
Hq(X; 'TTq(Y,Yo)) = 0 for q
n. Then fo, it: X -') Yare homotopic if and only
iff~(l) = f!(l). •
>
This last result gives a condition that the map 1/;,: [X; Y 1 -') Hn(X, 'TT)
be injective. The condition that 1/;, be surjective is that if {pq,Eq,fq}q?1 is the
standard Postnikov factorization of Y, then any map X ~ En+l can be lifted.
The obstructions to lifting such a map lie in Hq+l(X; 'TTq(Y,yo)) for q
n.
Therefore, by combining these, we have the following result.
>
3 THEOREM Let l E Hn(Y,yo; 'TT) be n-characteristic for a simple (n - 1)connected space Y, where n ::::: 1, and let X be a CW complex such that
Hq(X;'TTq(Y)) = 0 and Hq+1(X;'TTq(Y)) = 0 for all q n. Then there is a bijection
>
1/;,: [X; Yl
;: : :; Hn(X;'TT)
•
These last results have been derived by assuming hypotheses which ensure
that the lowest-dimensional obstruction is the only nontrivial one. In this case
we are essentially studying maps to a space of type ('TT,n). The case where the
two lowest-dimensional obstructions are the only nontrivial obstructions is
essentially the study of maps to a fibration E -') B of type (G,q), where B is a
1 See S. Eilenberg, Cohomology and continuous mappings, Annals of Mathematics, vol. 41,
pp. 231-251, 1940.
448
OBSTRUCTION THEORY
CHAP.
8
space of type (w,n). Before we consider this, let us establish some cohomology
properties of X X I.
Define inclusion maps
jl
i1
A X I U X X 1 C A X I U X X j C (A X I U X X i, A X I U X X 1)
There is a weak retraction r: A X I U X X j ~ A X I U X X 1 defined by
r(x,t) = (x,l) for (x,t) E A X I U X X j (that is, roil is homotopic to the
identity map of AX I U X X 1). Using the exactness of the cohomology
sequence of (A X I U X X i, A X I U X X 1), it follows that for an arbitrary
element u E Hq(A X I U X xi) there is an associated unique element
u' E Hq(A X I U X X i, A X I U X X 1) such that
u
= iT u' + r* i Tu
Let h: (X,A) ~ (A X I U X X j, A X I U X X 1) be defined by
h(x)
= (x,O) for x E X. Then h induces an isomorphism
h*: Hq(A X I U X X j, A X I U X X 1) ;::::; Hq(X,A)
and we define an epimorphism
il: Hq(A X I U X X j) ~ Hq(X,A)
by il(u)=h*u', where u'EHq(AXIUXxi,AxIUXX1) is the
unique element associated to u. Then il is a natural transformation on
the category of pairs (X,A).
4
LEMMA
Commutativity holds in the triangle
Hq(A X I U X X i) ~ Hq+l((X,A) X (I,i))
Hq(X,A)
=
PROOF
Let 1': X X I ~ A X I U X X 1 be defined by 1'(x,t)
(x,l). Then
1'1 (A X I U X X i)
r, and so r*itu
(1'*itu) 1(A X I U X X i) for
u E Hq(A X I U X X i). For any v E Hq(X X 1), 8(v 1 (A X I U X X i))
O.
Therefore, 8r i Tu
0, and to complete the proof it suffices to show that for
u' E Hq(A X I U X X i, A X I U X Xl), 8iT (u')
(-l)q+Lrh* (u'). This
=
*
=
=
=
=
follows from the commutativity of a diagram analogous to the one used in the
proof of theorem 8.2.4. •
:.
COROLLARY
Let (X,A) be a relative CW complex, with inclusion map
i: A C X, and let p': QB' ~ b o be the constant map, where B' is a space of
type (w,n + 1). Given a map pair f: i ~ p' and two liftings fo, /1: X ~ QB'
of f, let g": A X I U X X I ~ QB' be defined by g"(x,O) = fo(x), g"(x,l) =
/1 (x), and g"(a,t) fo(a). If t' E Hn(QB',wo; w) and t E Hn+l(B',b o; w) are
related characteristic elements, then d(fo,/1) = - ilg" (t').
=
*
SEC.
4
PROOF
449
OBSTRUCTION THEORY
Let g: i'
~
p' be the map pair consisting of the commutative square
A X I U X X j
it
L
~B'
lp'
~ bo
X Xl
From the definition of d(fo,h) we have d(fo,h) = ( -l)n+Lr -l(c(g)). By theorem 8.2.6 c(g) = - 8g"* (l'), and therefore d(fo,fl) = (-1 )n'J"-18g"* (l').
The result follows from this and lemma 4. •
6 LEMMA Let ho, h 1 : (X,A) ~ (A X I U X X t A X 1) be defined by
ho(x) = (x,O) and h1(x) = (x,l). For any u E Hq(A X I U X X t A X I)
Ll(u I (A X I U X X i)) = h~(u) - h!(u)
PROOF
There are inclusion maps
~
(A X I U X X 1, A X 1) C (A X I U X X
t
h
A X 1) C
(A X I U X X
t
A X I U X X 1)
and a weak retraction r': (A X I U X X t A X I) ~ (A X I U X X 1, A X I)
defined by r'(x,t) = (x,l). For v E Hq(A X I U X X A X 1) there is an
associated unique element v' E Hq(A X I U X X t A X I U X X 1) such that
t
+ r' * ii * v
v = ii * v'
If k: A X I U X X j C (A X I U X X j, A X I), we then have
k* v
= k* ii * v' + k* r' * ii * v = it v' + r* i ! k* v
Therefore Llk* v = h* v'. Since h =
i1
0
ho and hl = i 1 r'
0
0
ho, we have
Llk* v = h~ ii * v' = h~ (v - r' * i 1* v) = h~ v - h! v
•
7
COROLLARY
Given a map pair g: i' ~ p, where (X,A) is a relative CW
complex, i': A X I c A X I U X X t and p: E ~ B is a principal fibration
of type (G,q) induced by a map 0: B ~ B', let fo, h: i ~ P be the map pairs
from i: A C X to p defined by restriction of g to (X,A) X and (X,A) X 1,
respectively. Then
°
Llg' * 0 * (l) = c(fo) - c(h)
where g': A X I U X X j ~ B is part of the map pair g.
PROOF
The obstruction c(g) E Hq(A X I U X X i, A X I; G) has the property that c(g) I (A X I U X xi) is the obstruction to lifting g'. Therefore
c(g) I (A X I U X X i) = g'*O*(l)
By the naturality of the obstruction,
result now follows from lemma 6. •
h~
c(g) = c(fo) and h! c(g) = c(h). The
450
OBSTRUCTION THEORY
CHAP.
8
Let () be a cohomology operation of type (n,q; '7T,G). Given a cohomology
class u E Hn(x;'7T), we define a map 11((),u): Hn(X,A; '7T) ~ Hq(X,A; G) by
11((),u)(v)
= M(jfh*-1(v) + k*u)
v E Hn(X,A; '7T)
where k: A X I U X X i ~ X is defined by k(x,t)
tive cohomology operation, we have
11((),u)(v)
= x. In case () is an addi-
= l1(jf h* -1()(V) + k* ()(u)) = ()(v)
Therefore 11((),u) = () if () is additive.
Given a cohomology operation () of type (n,q; '7T,G) and a cohomology
class u E Hn(x;'7T), we define a map SI1((),u): Hn-1(X,A; '7T) ~ Hq-1(X,A; G) by
the equation SI1((),u) = 'T- 1 11((),u') 'T, where u' E Hn(X X I; '7T) is the
image of u under the homomorphism induced by the projection X X I ~ X.
If () is an additive operation, then SI1((),u) = S(). In any case, we have the
following analogue of corollary 8.1.14.
0
0
8
LEMMA
If () is a cohomology operation of type (n,q; '7T,G) and
u E Hn(X;'7T), the map
SI1((),u): Hn-1(X,A; '7T)
~
Hq-1(X,A; G)
is a homomorphism.
PROOF
Let 11 = [o,~], 11 = {o,~}, 12 = [~,l], and 12 = {~,1}, and let
V1, V2 E Hn-1(X,A; '7T). Let vl = 'T1(V1) E Hn((X,A) X (11.11)) and let
vz = 'T2(V2) E Hn((X,A) X (1 2,12)), and let v E Hn((X,A) X (I, 11 U 12)) be the
unique class such that v I (X,A) X (1 1,11) = vl and v I (X,A) X (12,12) = vz.
Then v I (X,A) X (1,1) = 'T(V1) + 'T(V2). Since () and 11 are both natural,
= 'TSI1((),U)(V1 + V2)
11((),u')(v) I (X,A) X (1,1)
and
= 'T1 SI1((),U)(V1)
11((),u')(v) I (X,A) X (11.11)
11((),u')(v) I (X,A) X (12,12)
= 'T2 SI1((),U)(V2)
Therefore, as in the proof of lemma 8.2.3,
'TSI1((),U)(V1
+ V2) =
'TSI1((),U)(V1)
Since 'T is an isomorphism, this gives the result.
+ 'TSI1((),U)(V2)
•
Let B be a space of type ('7T,n) and let p: E ~ B be a principal fibration
of type (G,q) induced by a map 0: B ~ B'. Let ()' = O*(t') E Hq(B,b o; G)
correspond to a cohomology operation () of type (n,q; '7T,G) (that is, ()(t) = ()').
Given a CW complex X, a map f: X ~ B can be lifted to E if and only if
()(f*(t)) = O. For any element U E Hn(x;'7T) such that ()(u) = 0 it follows that
there are liftings f: X ~ E such that (p f) * (t) = u. We shall determine how
many homotopy classes of such liftings there are.
0
9
LEMMA
Let fo, II: X ~ E be maps such that po fo = po II (that is,
fo and II are liftings of the same map X ~ B). Then fo ~ II if and only if
there is d E Hn-1(X;'7T) such that dUo,f1)
SI1((),u)(d), where u (p 10)* (t).
=
=
0
SEC.
4
451
OBSTRUCTION THEORY
PROOF
Let Fo: i' ---> P be the map pair consisting of
.
XXI
P'
~
E
where Fo(x,O) = fo(x), F6'(x,l) = h(x), and Fo(x,t) = pfo(x). Then d(fo,fl) =
(-1)qT-1(c(Fo)). It is clear that fo c--:: h if and only if there is a homotopy
Pi: X X I ---> B from p fo to p h such that for the corresponding map pair
F1: i' ---> P we have c(F1) = O. Let G': (X X 1) X I U (X X 1) X i ---> B be
defined by G'(x,O,t) = G'(x,l,t) = pfo(x), G'(x,t,O) = Fo(x,t) and G'(x,t,l) =
Fi(x,t). By corollary 7,
0
0
6.G' * (0') = c(Fo) - C(Fl)
Thus fo c--:: fl if and only if there is a map Fi: X X I ---> B such that for the
corresponding map G' we have
d(fo,h) = (_l)qT-l(6.G'* (0'))
It is easily verified that G'*(t) = Hh*-l6.G'*(t) + k*u', where
u' E Hn(x X I; 'TJ) is the image of u = (p fo)* (t) under the projection
X X I ---> X. By definition,
0
6.G' * (0') = 6.G' * O(t)
= MG' * (t) = 6.(O,u')(6.G' * (t))
Since Fa, Pi: X X 1---> B are two liftings of the map pair
XXi--->B
1
1
X X I ---> b o
it follows from corollary 5 that d(Fo,Fl) = - 6.G' * (t), and by theorem 8.2.4,
given dE Hn-l(X;'TJ), there is a homotopy Pi: X X 1---> B from po fo to p h
such that ~G'*(t) = (-l)Qr(d). Combining all of these, we see that fo = ji.
if and only if there is d E Hn-l(X;'TJ) such that
0
d(fo,fI) = T-l6.(O,U')T(d) = S6.(O,u)(d)
•
We summarize these results in the follOwing classification theorem.
Let p: E ---> B be a principal fibration of type (G,q) over a
space B of type ('TJ,n) induced by a map B: B ---> B' such that 8*(t') = O(t).
Given a CW complex X, there is a map 1/;: [X;E] ---> Hn(X;'TJ) defined by
1/;[fl = (p f)* (t). Then im 1/; = {u E Hn(x;'TJ) 10(u) = O}, and for every
u E im 1/; the set 1/;-l(U) is in one-to-one correspondence with
10 THEOREM
0
Hq-l(X;G)jS6.(O,u)Hn-l(X;'TJ)
PROOF We have already seen that im 1/; is as described in the theorem.
Given u E im 1/;, let fo: X ---> E be such that 1/;[fo] = u. Given any map
452
OBSTRUCTION THEORY
CHAP.
8
X ----7 E such that 1/;[/1] = u, there is a map f1: X ----7 E homotopic to /1
such that p f1 = p fo (by the homotopy lifting property of p). To such a
map f1 we associate the element d(fo,fi) E Hq-l(X;G). In this way the set of
maps X ----7 E which are liftings of p fo is mapped into Hq-l(X;G), and by
theorem 8.2.4, this map is surjective.
Two maps /1,fz: X ----7 E such that p /1 = po fo = po fz, are homotopic
by lemma 9 if and only if d(fdz) E S6.(B,u)Hn-l(X;7T). By lemma 8.2.3,
d(fo,fz) = d(fo,/1) + d(fdz) , and so Jl = fz if and only if d(fo,fl) and
d(fo,fz) belong to the same coset of S6.(B,u)Hn-l(X;7T) in Hq-l(X;G). Hence
the function which assigns the coset d(fo,/1) + S6.(B,u)Hn-l(X;7T) to a map
fl: X ----7 E with po /1 = p fo induces a bijection from l/;-l(U) to
/1:
0
0
0
0
0
Hq-l(X;G)/S6.(B,u)Hn-l(X;7T)
•
We now apply this to the complex projective space Pm(C) for m 2': 1.
There is a map Pm(C) ----7 Px(C) and P",(C) is a space of type (Z,2), by
example 8.1.3. Furthermore, if t is a characteristic element for P",(C) and B'
is a space of type (Z, 2m + 2), there is a map 0: P",(C) ----7 B' such that
0* (t') = (l)m+1. For the principal fibration p: E ----7 Px(C) induced by 0 there is
a map Pm(C) ----7 E which is a (2m + 2)-equivalence. In this case the operation
B is the (m + l)st-power operation, and therefore
S6.(B,u)(v)
= 7-16.Uth*-1('r(v)) + k*u']m+l
= 7- 16.[(m
+
l)k*(u')m
v
ith*-1(7(v))] =
(m
+
l)u m v v
because 7(V) v 7(V) = O. This gives the following application of theorem 10.
I I THEOREM Let t E HZ(Pm(C);Z) be 2-characteristic for Pm(C) and let X
be a CW complex. Define 1/;: [X;Pm(C)] ----7 HZ(X;Z) by I/;[f] = f* (t). If
dim X S; 2m + 2, then im I/; = {u E HZ(X;Z) I u m+1 = O}. If dim X S; 2m + 1,
then I/; is suryective, and for a given u E H2(X;Z), l/;-l(U) is in one-to-one
correspondence with H2m+1(X;Z)/[(m + l)u m v Hl(X;Z)]. •
:;
THE SUSPENSION :MAP
One of the most useful tools for the study of the homotopy groups of spaces
is the suspension homomorphism from 7Tq(X) to 7Tq+l(SX). Iteration of this
homomorphism yields a sequence of groups and homomorphisms
7T
q(X)
----7 7T
q+l(SX)
----7 7T
q+z(S2 X)
----7 •••
This sequence has the stability property that from some point on, all the
homomorphisms are isomorphisms. For a fixed X and q, therefore, there are
only a finite number of different groups in the above sequence.
In this section we shall study the suspension map in some detail and
establish the stability property. This will enable us to compute 7Tn+l(Sn) for
all n. Knowledge of these groups, combined with obstruction theory, will lead
SEC.
5
453
THE SUSPENSION MAP
to the Steenrod classification theorem, which closes the section.!
We consider the category of pointed spaces and maps. There is a
functorial suspension map S: [X; Y] ~ [SX;SY] such that S[f] = [Sf]. The
exponential correspondence defines a natural isomorphism
[SX;SY] ;:::;
[XJ~SY]
and we define S: [X;Y] ~ [X;QSY] to be the functorial map which is the
composite of S with this isomorphism. The following result shows that S is
induced by a map Y ~ QSY.
I
LEMMA Let p: Y ~ QSY be the map defined by p(y)(t)
and tEl. Then for any space X
S = p#: [X; Y]
= [y,t] for
yEY
~ [X;QSY]
The exponential correspondence takes the identity map SY C SY to
the map p: Y ~ QSY. Because of functorial properties of the exponential
correspondence, it takes the composite
PROOF
SX ~ SY C SY
to the composite
X
-4
Y ~ QSY
•
Thus, to study the suspension map S, we study the map p. To do
this we use the fibration PSY ~ SY, which has fiber QSY. With this
in mind, let us investigate homology properties of fibrations over SY.
We assume that yo E Y is a nondegenerate base point. We define
C_Y
{[y,t] E SY 10 ::; t::; Ih} and C+Y
{[y,t] E SY IIh ::; t::; I}. Then
SY = C_ Y U C+ Y, and there is a homeomorphism Y;:::; C_ Y n C+ Y (sending
y to [y,Ih]) by means of which we identify Y with C_ Y n C+ Y. Let S'Y be
the unreduced suspension defined to be the quotient space of Y X I in which
Y X is collapsed to one pOint and Y X 1 is collapsed to another point and
let C~ Y,C+ Y be analogous subspaces of S'Y (so C~ Y n C~ Y = Y). The map
collapsing S'yo in S'y is a collapsing map k: S'y ~ SY such that k(C~ Y) = C_ Y
and k(C~Y) = C+Y.
=
=
°
2
LEMMA
If yo is a nondegenerate base point, the collapsing map
k: S'y ~ SY defines a homotopy equivalence from any pair consisting of the
spaces S'Y, C~ Y, C~ Y, and Y to the corresponding pair consisting of SY, c_ Y,
C+Y, andY.
PROOF
Because yo is a non degenerate base point of Y, it follows, as in the
proof of lemma 7.3.2c, that Y X j U yo X IcY X I is a cofibration. Let
[y,t], E S'Y denote the point of S'Y determined by (y,t) E Y X I under the
quotient map k/: Y X I ~ S'Y. Let H': (Y X j U yo X 1) X I ~ S'Y be the
homotopy defined by H'(y,O,t) = [Yo,t/2]" H'(y,l,t) = [Yo, (2 - t)/2]', and
H'(Yo,t',t) = [Yo, (1 - t)t' + t/2]'. Then H' can be extended to a homotopy
1 The first detailed study of the suspension map appears in H. Freudenthal, tiber die Klassen
der Spharenabbildungen I, Compositio Mathematica, vol. 5, pp. 299-314, 1937.
454
OBSTRUCTION THEORY
CHAP.
8
H": Y X I X I ~ S'Y such that H"(y,t,O) = k'(y,t). Since H"(y,O,t) = H"(y',O,t)
and H"(y,l,t) = H"(y',l,t) for all y, y' E Y, it follows that there is a
homotopy H: S'Y X I ~ S'Y such that H([y,t)', t') = H"(y,t,t'). Then H is a
homotopy from the identity map of S'Y to a map which collapses S'yo to a
single point such that H(S'yo X 1) C S'yo. Since H(B X 1) C B if B = C~Y,
C~Y, or Y, the result follows from lemma 7.1.5. •
3
COROLLARY
If Y is a path-connected space with nondegenerate base
point, then SY is simply connected.
By lemma 2, S'Y and SY have the same homotopy type, so it suffices
to prove that S'Y is simply connected. It is clearly path connected, being the
quotient of the path-connected space Y X 1.
Let U_
{[y,t), E S'Y It
I} and U+
{[y,t)' E S'Y 10
t}. Then
U _ and U+ are each open and contractible subsets of S' Y. If w is any closed
path in S'Yat [Yo,Ih)" there is a partition of I, say,
to
t1
tn
1,
such that for each 1 ::::; i ::::; neither W([ti_1,ti)) C U_ or W([ti_1,ti)) C U+.
Furthermore, it can be assumed that w( ti) E U _ n U+ for all 0 ::::; i ::::; n (if
some W(ti) is not in U_ n U+, ti can be omitted from the partition to obtain
another partition of I satisfying the original hypothesis, and iteration of this
procedure will lead to a partition having the additional property demanded).
Since U_ n U+ is homeomorphic to Y X R, it is path connected. For each i
let Wi be a path in U_ n U+ from w( ti-1) to w( ti) and let w' be the closed path at
[Yo,lh)' defined by w'(t) = Wi((t - ti-1)/(ti - ti-1)) for ti-1::::; t ::::; ti·
Because U_ and U+ are each simply connected, wi [ti-hti) is homotopic to
w'l [ti-1,ti) relative to {ti-1,td. Therefore w ::::: w' reI i. Since w' is a closed
path in U+, it is null homotopic. Therefore w is null homotopic and S'Y is
simply connected. •
PROOF
<
=
<
°= < < ... < =
=
4
COROLLARY
Let Y have a nondegenerate base point and let p: E
be a fibration. Then {p-1( C_ Y ),p-1( C+ Y)} is an excisive couple in E.
~
SY
PROOF
Let p': E' ~ S'Y be the fibration induced from p by k: S'Y ~ SY and
let k: E' ~ E be the associated map. It follows from lemma 2 that k induces
vertical isomorphisms in the commutative diagram
H*(p'-l(C~Y),p'-l(Y)) ~
~l
H*(p-1(C+Y),p-1(Y))
H*(E',p'-l(C'-Y))
l~
~
H*(E,p-1(C_Y))
Since C~ Y is a strong deformation retract of U+ (with U+ as defined in
corollary 3) and Y is a strong deformation retract of U+ n c'- Y, it follows that
p'-l(C~Y) and p'-l(Y) are strong deformation retracts of p'-l(U+) and
p'-l(U+ n C~Y), respectively. This implies that {p'-l(C~Y),p'-l(C~Y)} is an
excisive couple. From the commutative diagram above, the result follows. •
Because C+ Y and C_ Yare contractible relative to Yo, it follows, as in
Sec. 2.8, that for any fibration p: E ~ SY with fiber F p-1(yO) there are
=
SEC.
5
455
THE SUSPENSION MAP
fiber homotopy equivalences f-: C_ Y X F ~ p-1(C_ Y) and g+: p-1(C+ Y) ~
C+ Y X F such that f-I yo X F is homotopic to the map (yo,z) ~ z and
g+ I F is homotopic to the map z ~ (Yo,z). The corresponding clutching
function p,: Y X F ~ F is defined by the equation
g+f-(Y'z)
= (y,
y E Y, z E F
p,(y,z))
Then p, I yo X F is homotopic to the map (yo,z)
~
z.
:;
THEOREM
Let p: E ~ SY be a fibration with F = p-1(yO), where yo is
a nondegenerate base point of Y. If p,: Y X F ~ F is a clutching function
for p, there are exact sequences (any coefficient module)
... ~ Hq(E) ~ Hq(C_ Y X F, Y X F) ~ Hq-1(F) ~ Hq_1(E) ~ ...
...
PROOF
~
"* Hq(F)
Hq(E) ~
8 *
~
Hq+1(C_ Y X F, Y X F)
~
Hq+1(E)
~
...
Consider the exact homology sequence of (E,F)
... ~ Hq(F) ~ Hq(E) ~ Hq(E,F) -4 Hq-1(F) ~ ...
U sing homotopy properties and corollary 4, there are isomorphisms induced
by inclusion maps
Hq(E,F)::? Hq(E,p-1(C+Y))
~
Hq(p-1(C_Y),p-l(Y))
There is also a homotopy equivalence
and a commutative diagram
Hq(E,F) ?
Hq(E,p-1(C+Y)) ~ Hq(p-l(C_Y),p-1(Y)) /~* Hq((C_Y,Y) X F)
,1
c1
a1
01
There is also a homotopy equivalence g+: p-1(C+Y)
isomorphisms
~
c+Y X F and
Hq_1(p-1(C+ Y)) g::.*) Hq_1(C+ Y X F) ::? Hq-1(F)
where the right-hand homomorphism is induced by projection to the second
factor. Because g+ I F is homotopic to the map z ~ (Yo,z), the above composite
equals i. -1. By definition, p, is the composite
YX F
f_IYXF)
p-1(Y) C p-1(C+Y) ~ C+Y X F~ F
Therefore there is a commutative diagram
Hq(E,F)
01
~
Hq((C_Y,Y) X F)
1
0
456
OBSTRUCTION THEORY
CHAP.
8
The desired exact sequence for homology follows on replacingHq(E,F) by
a by f.L* a in the homology sequence of (E,F). A similar
argument establishes the exactness of the cohomology sequence. •
Hq((C Y, Y) X F) and
Specializing to the case where Y = Sn-l, by lemma 1.6.6, S(Sn-l) is
homeomorphic to Sn, and we obtain the following exact Wang sequence of a
fibration over Sn.
6
COROLLARY
Let p: E --,) Sn be a fibration with fiber F. There are exact
sequences
... --,) Hq(F)
Hq(E) --,) Hq_n(F) --,) Hq-1(F) --,) .. .
~
. . . --,) Hq(E) 4 HG(F) !4 Hq-n+1(F) --,) Hq+1(E) --,) .. .
If the second sequence has coefficients in a commutative ring with a unit,
then
B(u vv)
PROOF
= B(u) vv + (_l)(n-l)
degu
uv B(v)
Letting Y = Sn-l in theorem 5, we have (C_ Y, Y) homeomorphic to
(En,Sn-l). Therefore
Hq((C_Y,Y) X F):::::: Hq((En,Sn-l) X F) :::::: Hq-n(F)
and the exact sequences result from the exact sequences of theorem 5 on
replacing Hq(CY X F, Y X F) and Hq(C_Y X F, Y X F) by Hq-n(F) and
Hq-n(F), respectively. The additional fact concerning B results from the observation that for the map f.L*: Hq(F) --,) Hq(Sn-l X F) the definitions are such that
f.L* (u)
= 1 X u + s*
X B(u)
where s* E Hn-l(Sn-l) is a suitable generator. Then, since s* v s*
1 X (u v v)
+ s*
= 0,
X B(u v v)
= f.L*(u v
= [1 X u
=1X
v)
+ s*
(u v v)
X B(u)] v [1 X v
X [B(u) v v
+ s*
This implies the multiplicative property of B.
+ s* X B(v)]
+ (_l)(n-l) deg Uu
V
B(v)]
•
We now specialize to the path fibration p: PSY --,) SY with fiber QSY.
In this case there is the following simple expression for a clutching function.
7
LEMMA
Let L: C_Y --,) p-l(C_Y) and s+: C+Y --,) p-l(C+Y) be sections
of the fibration p: PSY --,) SY such that s_(yo) and s+(yo) are both null
homotopic loops. Then the map f.L: Y X QSY --,) QSY defined by
f.L(y,w)
= (w * s_(y)) * S+(y)-l
is a clutching function for p.
Such sections exist because C_ Y and C+ Yare contractible relative
to yo. We define fiber-preserving maps
PROOF
SEC.
5
457
THE SUSPENSION MAP
f+: C+ Y X QSY ----? p-l(C+ Y)
g+: p-l(C+ Y)
----?
C+ Y X QSY
by f-(z,w) = w * s_(z) and g_(w) = (p(w), w * (s_p(W))-l) andf+(z,w) = w * s+(z)
and g+(w) = (p(w), w * (s+p(w))-l), respectively. It is easy to verify that
g_ 0 f- is fiber homotopic to the identity map of C_ Y X QSY and f- 0 g_ is
fiber homotopic to the identity map of p-l(C_ Y). Therefore f- is a fiber
homotopy equivalence. Similarly, g+ is a fiber homotopy equivalence. Furthermore, f-(Yo,w) = w * s_(yo) is homotopic to the map (yo,w) ----? w because
s_(yo) is null'homotopic. Similarly, for w E QSY, g+(w) = (Yo, w * s+(YO)-l) is
homotopic to the map w ----? (yo,w). Therefore the composite
Y X QSY ~ p-l(Y) ~ Y X QSY ----? QSY
is a clutching function for p. This composite is the map
(y,w)
----?
(w
* s_(y)) * S+(y)-l
•
Let s_ and s+ be sections as in lemma 7 and let IL': Y ----? QSY be defined
by 1L'(y) = s_(y) * S+(y)-l. IL' is called a characteristic map for the fibration
p: PSY ----? SY.
8
COROLLARY
Let IL': Y ----? QSY be a characteristic map for the fibration
p: PSY ----? SY. The map Y X QSY ----? QSY sending (y,w) to w * 1L'(y) is homotopic to a clutching function for p.
PROOF
This follows from lemma 7, because the map
(y,w)
----?
(w
* s_(y)) * S+(y)-l
is clearly homotopic to the map (y,w)
----?
w * (s_(y)
* S+(y)-l) = W * 1L'(y).
•
The following theorem is the main part of the proof of the suspension
theorem.
9
THEOREM
Let Y be n-connected for some n ~ 0 and let yo be a nondegenerate base point of Y. If IL': Y ----? QSY is a characteristic map for the
fibration p: PSY ----? SY, then IL' induces an isomorphism
q
~
2n
+I
PROOF
By corollary 3, SY is simply connected. By corollary 4, {C_ Y,C+ Y}
is an excisive couple, and from the exactness of the reduced Mayer-Vietoris
sequence, iiq(SY) :::::: iiq_1(Y). Combining these with the absolute Hurewicz
isomorphism theorem, SY is (n + I)-connected. Therefore QSY is n-connected.
Because PSY is contractible, it follows from the version of theorem 5, using
reduced modules, that there is an isomorphism
~o:
is (n
Hq((C_Y,Y) X QSY):::::: iiq_1(QSY)
If Wo is the constant loop, then because QSY is n-connected and (C_ Y, Y)
+ I)-connected, it follows from the Kiinneth theorem that the inclusion
458
OBSTRUCTION THEORY
CHAP.
8
map (C_ Y, Y) X Wo C (C_ Y, Y) X QSY induces an isomorphism
q ::; 2n
+2
Let p,: Y X Q5Y ~ QSY be a clutching function which is homotopic to
the map (y,w) ~ w * p,'(y) (such a p, exists, by corollary 8). Since p,(y,wo) is
homotopic to the map y ~ p,'(y), there is a commutative diagram
Hq(CY,Y)
-:::7
Hq((CY,Y) X wo)
01
~
Hq- 1 (y) -.::!
Hq_1 (Y X wo)
I'~
~
Hq((C_Y,Y) X QSY)
-
01
Hq_1 (Y X QSY)
~*
Hq _ 1 (QSY)
The result follows from the commutativity of this diagram.
10
•
Let Y have a nondegenerate base point. If Y is n-connected
0, the map p: Y ~ QSY induces an isomorphism
COROLLARY
for n
~
q ::; 2n
+I
PROOF
Let s_: C_Y ~ p-l(C_Y) and s+: C+Y ~ p-l(C+Y) be the sections
defined by L[y,t](t') = [y,tt'] and s+[y,t](t') = [y, I - t' + tt']. The corresponding characteristic map is equal to the map p: Y ~ QSY. The result
follows from theorem 9. •
We are now ready for the following suspension theorem. 1
I I THEOREM Let Y be n-connected for n ~ I with a nondegenerate
base point and let X be a pointed CW complex. Then the suspension map
S:
is surjective if dim X ::; 2n
+
[X;Y]~
[SX;SY]
I and biiective if dim X ::; 2n.
PROOF
Because Y and QSY are simply connected, it follows from corollary 10
and the Whitehead theorem that p is a (2n + I)-equivalence. The result
follows from corollary 7.6.23 and lemma 1. •
Let Y be a space with a nondegenerate base point. Then SY also has a
nondegenerate base point and is path connected, S2Y is simply connected,
and Smy is (m - I)-connected. If X is a CW complex, so is SmX, and
dim (Smx) = m + dim X. Hence, if X is finite dimensional and m ~ 2 + dim X,
it follows from theorem 11 that S: [Smx; Smy] ::::: [Sm+1X; Sm+1Y]. Therefore,
for any finite-dimensional CW complex X the sequence
[X;Y] ~ [SX;SYJ ~ ... ~ [SmX;SmY] ~ ...
1 For a general relative form of this theorem see E. Spanier and J. H. C. Whitehead, The theory
of carriers and S-theory, in "Algebraic Geometry and Topology" (a symposium in honor of
S. Lefschetz), Princeton University Press, Princeton, N.J., 1957, pp. 330-360.
SEC.
5
459
THE SUSPENSION MAP
=
=
consists of isomorphisms from some point on. Taking X
Sn+k and Y Sn
and recalling that the suspension of a sphere is a sphere, we see that there is
a sequence
7Tn+k(Sn) ~ 7Tn+k+l(Sn+1) ~ ...
consisting of isomorphisms from some point on. The direct limit of this
sequence is called the k-stem. It follows from theorem 11 that the k-stem is
isomorphic to 7T2k+2(Sk+2). In particular, the O-stem is infinite cyclic. The following result determines the I-stem.
12
THEOREM
7T4(S3);:::::; Z2.
PROOF
Let Uo E HO(nS3) be the unit integral class and define generators
Ui E H2i(nS3), by induction on i from the exactness of the Wang sequence in
corollary 6 for the fibration PS3 ~ S3, by the equation
O(Ui+l)
= Ui
i
2
0
Because 0 is a derivation, O(UI v Ul) = 2Ul, whence Ul v Ul = 2U2. We
know 7T2(nS3);:::::; 7T3(S3) is infinite cyclic. It follows that nS 3 can be
imbedded in a space X of type (Z,2) such that the inclusion map nS 3 c X
induces an isomorphism 7T2(nS3) ;:::::; 7T2(X), Since P",(C) is also a space of type
(Z,2), it follows that H* (X) ;:::::; H* (P "'(C)) ;:::::; lim_ {H* (Pj(C))} is a polynomial
algebra with a single generator v E H2(X), and v can be chosen so that
v I nS 3 = Ul·
An easy computation using the exact cohomology sequence of (X,nS3)
establishes that Hq(X,nS3) = 0 for q
5 and H5(X,nS3);:::::; Z2. By the
universal-coefficient formula, Hq(X,nS3) = 0 for q
4 and H4(X,nS3) ;:::::; Z2.
By the relative Hurewicz isomorphism theorem, 7T4(X,nS3);:::::; Z2. Because
a
7T3(X)
0
7T4(X)' we have 7T4(X,nS3) ;:::::; 7T3(nS3) ;:::::; 7T4(S3). •
<
<
= =
The (n - 2)-fold suspension of a generator of 7T3(S2) is a generator of
7Tn+l(Sn) (because S: 7T3(S2) ~ 7T4(S3) is an epimorphism, by theorem 11).
Attaching a cell to Sn by this map must, therefore, kill 7Tn+l(Sn). The resulting
CW complex has the same homotopy type as the (n - 2)-fold suspension of
the complex projective plane P2 (C). Therefore we have proved the following
result.
n>
2
•
We want to classify maps of an (n + I)-complex into Sn. For n = 2 this
is given by the case m = 1 of theorem 8.4.11. By using the standard
Postnikov factorization of Sn, we are reduced to classifying maps of an
(n + I)-complex into E, where p: E ~ B is a principal fibration of type
(Z2, n + 2), with base space B a space of type (Z,n). This fibration determines
a cohomology operation On of type (n, n + 2; Z,Z2).
14 LEMMA For n > 2 the cohomology operation On is Sq2 0 Il*, where
Il*: Hn(X;Z) ~ Hn(X;Z2) is induced by the coefficient homomorphism
Il: Z ~ Z2.
460
OBSTRUCTION THEORY
CHAP.
8
PROOF
Sn C Sn-2(P2(C)) is not a retract, by theorem 12 and corollary 13.
Therefore On: Hn(Sn-2(P2(C));Z) ~ Hn+2(Sn-2(P2(C));Z2) is nontrivial (if On
were trivial, there would be a map f: Sn-2(P2(C)) ~ Sn such that
f*: Hn(Sn;Z) ;:::; Hn(Sn-2(P2(C));Z)
is inverse to the restriction map Hn(Sn-2(p2(C));Z);:::; Hn(Sn;Z), and such a
map f would be homotopic to a weak retraction). Since Sq2 0 fL* is also nontrivial, it follows that On
Sq2 ~ in the space Sn-2(P2(C)).
The rest of the argument follows by showing that Sn-2(P2(C)) is universal
for On and Sq2 fL*. Let X be any CW complex of dimension ::; n + 2 and let
u E Hn(X;Z). Because ?T n+1(Sn-2(P2(C))) = 0, there is a map f: X ~ Sn-2(P 2(C))
such that f* v = u, where v is a generator of Hn(Sn-2(P 2(C))). By the naturality of On and Sq2 ~, it follows that
=
0
0
0
On(u) = Onf*v = f*Onv =
f*Sq2~v
= Sq2 fL*(U)
Since this is true for every CW complex of dimension ::; n + 2 and On and
Sq2 0 ~ are operations of type (n, n + 2; Z,Z2), it is true for every CW
complex. •
Combining lemma 14 with theorem 8.4.lO yields the following Steenrod
classification theorem. 1
15 THEOREM Let s* E Hn( Sn; Z) be a generator, where n > 2, and let 'X
be a CW complex. Then the map 1/;: [X;Sn] ~ Hn(X;Z) has image equal to
{u E Hn(x;Z) I Sq2 fL* (u) = O} if dim X ::; n + 2, and if dim X ::; n + 1,
1/;-1(U) is in one-to-one correspondence with Hn+1(X;Z2)/Sq2 fL* Hn-1(X;Z). •
EXERCISES
A SPACES OF TYPE (?T,n)
n
I For P an integer let Ln(p) be the generalized lens space Ln(P) = L(p, ~).
Show that Ln(P) C Ln+l(p) and that Lx(p) = Un Ln(P) topologized with the topology
coherent with {Ln(P)} is a space of type (Zp,l).
2
If X is a CW complex of type (?T,n) for n
> 1 and Y is a CW complex, prove that
7Tn(XV Y)::::: 7Tn(Y) ffi
where 7Th
= 7T for each A E 7Tl(Y).
EEl
7Th
>
3 Given a sequence of groups {7Tq}q~l' with 7Tq abelian for q
1, and given an action
of 7Tl as a group of operators on 7Tq for q 1, prove that there is a space Y which realizes
this sequence (that is, 7Tq(Y) ::::: 7Tq and 7Tl(Y) acting on 7Tq(Y) corresponds to the action
of 7Tl on 7T q).
>
1
See N. E. Steenrod, Products of cocycles and extensions of mappings, Annals of Mathematics,
vol. 48, pp. 290-320, 1947.
461
EXERCISES
B EXACT SEQUENCES CONTAINING ~
Let g: (Y,B) - ? (Y',B') be a base-point-preserving map and let g'
g" = g I B: B - ? B'.
I
Prove that Eg" is a subspace of Eg' and Po"
3
-4
Y - ? Y' and
= Pg' lEg",
2 Define p: (Eo"E o") - ? (Y,B) so that p lEg' = Pg' and
i(w) = (Yo,w). Prove that there is an exact sequence
(QY,QB) ~ (QY',QB')
= g I Y:
i:
(Q