LOCALIZATION OF NILPOTENT GROUPS AND SPACES
LOCALIZATION OF NILPOTENT GROUPS AND SPACES
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NORTH-HOLLAND
MATHEMATICS STUDIES
15
Notas de Matematica (55)
Editor: Leopoldo Nachbin
Universidade Federal do Rio de Janeiro
and University of Rochester
Localization of Nilpotent Groups
and Spaces
PETER H I L T O N
Battelle Seattle Research Center, Seattle, and Case Western Reserve University,
Cleveland
GUIDO M l S L l N
Eidgenossische Technische Hochschule. Zurich
JOE ROITBERG
Institute for Advanced Study, Princeton, and Hunter College, New York
1975
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Table of Contents
v11
Introduction
Chapter I.
Chapter 11.
Localization of Nilpotent Groups
Introduction
1
1. Localization theory of nilpotent groups
3
2. Properties of localization in N
19
3.
23
Further properties of localization
4. Actions of a nilpotent group on an abelian group
34
5.
43
Generalized Serre classes of groups
Localization of Homotopy Types
Introduction
47
1. Localization of 1-connected CW-complexes
52
2.
Nilpotent spaces
62
3.
Localization of nilpotent complexes
72
4.
Quasifinite nilpotent spaces
79
5. The main (pullback) theorem
6.
82
90
Localizing H-spaces
7. Mixing of homotopy types
94
Chapter 111. Applications of Localization Theory
Introduction
101
1. Genus and H-spaces
104
2. Finite H-spaces, special results
122
3.
133
Non-cancellation phenomena
Bibliography
14 7
Index
154
V
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Introduction
Since Sullivan first pointed out the availability and applicability
of localization methods in homotopy theory, there has been considerable work
done on further developments and refinements of the method and on the study
of new areas of application. In particular, it has become quite clear that
an appropriate category in which to apply the method, and indeed--as first
pointed out by Dror--in which to study the homotopy theory of topological
spaces in the spirit of J. H. C. Whitehead and J.-P. Serre, is the (pointed)
homotopy category NH of nilpotent CW-complexes. Here a pointed space X
is said to be nilpotent if its fundamental group is a nilpotent group and
operates nilpotently on the higher homotopy groups.
For a given family P ofrationalprimes, the concept of a P-local
space is based simply on the requirement that its homotopy groups be P-local.
Thus a localization theory for the category NH requires, or involves, a
localization theory of nilpotent groups and of nilpotent actions of nilpotent
groups on abelian groups. This latter theory could be obtained as a by-product
of the topological theory (this is, in fact, the approach of Bousfield-Kan)
but we have preferred to make a purely algebraic study of the group-theoretical
aspects of the localization method.
Thus this monograph is devoted toanexposition
of the theory of localization of nilpotent groups and homotopy types.
Chapter I, then, consists of a study of the localization theory of
nilpotent groups and nilpotent actions. It turns out that localization
methods work particularly well in the category N of nilpotent groups, in
the sense that we can detect the localizing homomorphism
e:
G
+
Gp by
meansof effective properties of the homomorphism e, and that localization
does not destroy the fabric of a nilpotent group. For example, the nilpotency
.
embeds in HG
Chapter I
P P
also contains some applications of localization methods in nilpotent group theory.
class of Gp never exceeds that of G, and G
v11
Introduction
Vlll
Chapter I1 takes up the question of localization in homotopy theory.
We first work in the (pointed) homotopy category
H1 of 1-connected CW-complexes,
and then extend the theory to the larger category NH of nilpotent CW-complexes.
This extension is not only justified by the argument that we bring many more
spaces within the scope of the theory (for example, connected Lie groups are
certainly nilpotent spaces); it also turns out that even to prove fundamental
theorems about localization in H1,
NH
it is best to argue in the larger category
.
One may represent the development of localization theory as
presented in this monograph--as distinct from an exposition of its applications
to problems in nilpotent group theory and homotopy theory--as follows; here
Ab
is the category of abelian groups.
Thus we start from the (virtually elementary) localization theory
in the category Ab
of abelian groups. The arrow from Ab
to N
represents the generalization of localization theory from the category Ab
to the category N of nilpotent groups. The arrow from Ab
to H1 represents
the application of the localization theory of abelian groups to that of
1-connected CW-complexes. The remaining two arrows of the diagram indicate
that the localization theory in NH is a blend of application of the
localization theory in N and generalization of the localization theory in
H1. The diagram (L) which, as we say, representsschematically our approach
to the exposition of the localization theory of nilpotent homotopy types, is,
of course, highly non-commutative!
Introduction
1x
In Chapter 111, we describe some important applications of
localization methods in homotopy theory. Naturally, our choice of application
is very much colored by our particular interests. We have concentrated, first,
on the theory of connected H-spaces, and, second, on non-cancellation
phenomena in homotopy theory. Localization methods have proved to be very
powerful in the construction of new H-spaces and in the detection of obstructions
to H-structure. We give a fairly comprehensive introduction to the methods
used and obtain several results. Again, it has turned out that there is a
close connection between concepts based on localization methods and the
situation,already noted by the authors and others, of compact polyhedra
exhibiting either the phenomenon
XVANYVA,
X+Y,
XxA=YxA,
X$Y;
or the phenomenon
we describe this connection in some detail.
Given a localization theory in some category C
(and a reasonable
finiteness condition imposed on the objects under consideration, for reasons
of practicality), one can introduce the concept of the genus G(X)
object X
of C.
Thus we would say that X, Y
of an
in C belong to the same
genus, or that Y € G(X),
if X
is equivalent to Y
for all primes p.
P
P
It turns out that in the category Ab (confining attention to finitely-
generated abelian groups), objects of the same genus are necessarily isomorphic;
however, no such corresponding result holds in the categories N, H1,
N we again confine attention to finitely-generated groups; in
H1
NH.
(In
and NH,
we confine attention to spaces with finitely-generated homotopy groups in each
dimension.)
Thus localization theory naturally throws up questions of the
nature of generic invariants; we embark on a study of these questions in this
X
Introduction
menograph. We do not describe explicitly any algebraic invariants (beyond
the fundamental group) capable of distinguishing homotopy types in NH of
the same genus. We remark that all known examples of the non-cancellation
phenomenon referred to above concern spaces X, Y of the same genus; this
explains the connection with localization theory to which we have drawn
attention.
Each chapter is Surnished with its own introduction describing the
purpose and background of the chepter, and detailing its contents. We will
therefore not need to offer a more comprehensive description of the section
contents in this overall introduction.
It is a pleasure to acknowledge the encouragement of Professor
Leopoldo Nachbin, who first proposed the writing of this monograph; the
excellent cooperation which we have received from the editorial staff of the
North-Holland Publishing Company; the assistancereceived from many friends
working in or close to the area covered by the monograph; and, last but
certainly not least, the truly wonderful assistance of Ms. Sandra Smith,
who succeeded in converting a heterogenous manuscript reflecting the many
divergences of style and handwriting of its three authors into a typescript
which could be transmitted with a clear conscience to the publisher.
Battelle Seattle Research Center and
Case Western Reserve University, Cleveland
Peter Hilton
Eidgenlissische Technische Hochsahule, Ztirich
Guido Mislin
Institute for Advanced Study, Princeton and
Hunter College, New York
Joe Roitberg
June, 1974
Chapter I
L o c a l i z a t i o n of N i l p o t e n t Groups
Introduction
Our o b j e c t i n t h i s c h a p t e r is t o d e s i r i b e t h e t h e o r y of l o c a l i z a t i o n
of n i l p o t e n t groups a t a s e t of primes
P.
This t h e o r y was f i r s t developed
i n an important s p e c i a l c a s e by Malcev [52] and was l a t e r reworked and extended
by Lazard [ 5 0 ] and o t h e r s
(cf.
Baumslag
I 6 1,
H i l t o n [34 1 , Q u i l l e n [66 1,
Warfield [ 8 6 1 ) .
With t h e advent of S u l l i v a n ' s t h e o r y of l o c a l i z a t i o n of homotopy
t y p e s [ 8 3 ] , i t was observed by t h e a u t h o r s [ 4 2 , 431
and independently by
Bousfield-Kan [12, 1 3 , 1 4 1 , t h a t t h i s a l g e b r a i c t h e o r y of l o c a l i z a t i o n of n i l p o t e n t
groups e n t e r e d q u i t e n a t u r a l l y and s i g n i f i c a n t l y i n t o c e r t a i n q u e s t i o n s of
homotopy t h e o r y .
Our approach i n t h i s c h a p t e r i s , i n f a c t , i n s p i r e d by t h e
homotopy-theoretical c o n s i d e r a t i o n s of [ 4 3 ] , and f o l l o w s r a t h e r c l o s e l y t h e
s y s t e m a t i c t r e a t m e n t of H i l t o n [ 3 4 , 3 5 1 .
It should b e mentioned t h a t
Bousfield-Kan have a l s o given a t r e a t m e n t of t h e t h e o r y of l o c a l i z a t i o n of
n i l p o t e n t groups from a homotopy-theoretical p o i n t of view, b u t t h e i r approach
rests on t h e t h e o r y of l o c a l i z a t i o n of n i l p o t e n t homotopy t y p e s , whereas i n
our approach
a l l t h e i n e s s e n t i a l topology h a s been s t r i p p e d away, and w e
make u s e only of s t a n d a r d homological a l g e b r a t o g e t h e r w i t h elementary group
t h e o r y ; s e e Hilton-Stammbach
[47].
The c h a p t e r i s organized a s f o l l o w s .
In Seetionlweintroduce thebasic
n o t i o n s and terminology and prove t h e e x i s t e n c e of a P - l o c a l i z a t i o n f u n c t o r
on t h e c a t e g o r y of n i l p o t e n t groups, where
( r a t i o n a l ) primes.
is an a r b i t r a r y c o l l e c t i o n of
P
Our proof proceeds by i n d u c t i o n on t h e n i l p o t e n c y c l a s s
of t h e group and i s based on t h e c l a s s i c a l i n t e r p r e t a t i o n of t h e second
cohomology group of a group.
I n c o r p o r a t e d i n t o t h e e x i s t e n c e theorem is t h e
v e r y c r u c i a l f a c t t h a t a homomorphism
iff
K
is P-local and
$
$: G
-f
is a P-isomorphism;
K
of n i l p o t e n t groups P - l o c a l i z e s
s e e D e f i n i t i o n s 1.1 and 1.3 below.
Localization of nilpotent groups
2
Section 2 contains some immediate consequences of t h e methods and
r e s u l t s of Section 1, t h e most notable a s s e r t i o n s being t h e exactness of
P - l o c a l i z a t i o n and t h e theorem t h a t a homomorphism
0:
G
+
K
of n i l p o t e n t
groups P-localizes i f f t h e corresponding homology homomorphism
g,($)
: H*(G)
+ fi,(K)
P-localizes.
I n Section 3 , we prove a number of r e s u l t s on l o c a l i z a t i o n of n i l p o t e n t
groups which t u r n out t o be t h e a l g e b r a i c precursors of corresponding r e s u l t s
on t h e l o c a l i z a t i o n of n i l p o t e n t homotopy types.
f i n i t e l y generated n i l p o t e n t group
its localizations
G
G
may be i d e n t i f i e d with t h e pullback of
over i t s r a t i o n a l i z a t i o n
P
For example, we show t h a t a
G o , The homotopy-theoretical
counterparts of t h e r e s u l t s of Section 3 w i l l be discussed i n t h e l a t t e r p a r t
of Chapter 11.
I n Section 4 , we present r e s u l t s concerning n i l p o t e n t a c t i o n s of
groups on a b e l i a n groups, which play an important r o l e i n t h e c o n s t r u c t i o n , i n
the f i r s t p a r t of Chapter 11, of t h e l o c a l i z a t i o n f u n c t o r on t h e category of
n i l p o t e n t homotopy t y p e s .
F i n a l l y , i n Section 5, we introduce a generalized version of t h e
notion of "Serre c l a s s " , which provides t h e c o r r e c t a l g e b r a i c s e t t i n g f o r
general Serre-Hurewicz-Whitehead theorems f o r n i l p o t e n t spaces.
A s mentioned e a r l i e r , we s h a l l follow, f o r t h e most p a r t , t h e
exposition i n [ 3 4 , 3 5 1 .
I n f a c t , much of Chapter I is a revised and
somewhat condensed version of [ 3 4 , 351,
f o r t h e f i r s t time.
though some m a t e r i a l appears here
Since our primary concern i n t h i s monograph is r e a l l y
with t h e l o c a l i z a t i o n of n i l p o t e n t homotopy types, we have l a r g e l y r e s t r i c t e d
ourselves i n Chapter I t o a discussion of those a s p e c t s of t h e theory of
l o c a l i z a t i o n of n i l p o t e n t groups which a r e r e l e v a n t t o homotopy theory.
reader may consult [ 3 4 , 351
l o c a l i z a t i o n theory.
The
f o r purely g r o u p - t h e o r e t i c a l a p p l i c a t i o n s of
Localization theory of nilpotent groups
3
1. L o c a l i z a t i o n t h e o r y of n i l p o t e n t groups
G , we d e f i n e t h e lower c e n t r a Z s e r i e s of
For a group
G,
by s e t t i n g
r 1(G)
Recall t h a t
c
G
= G,
r i+l(G)
is nilpotent i f
r j ( G ) = {l}
i s t h e l a r g e s t i n t e g e r f o r which
class
c
and w r i t e
nil(G) = c.
= [G,r i (G)], i 2 1.
f(G)
for
j
sufficiently large.
# {l}, we s a y t h a t
If
hasnizpotency
G
(Dually, we d e f i n e t h e upper c e n t r a l series of
G,
by r e q u i r i n g t h a t
zi+l
i
(G)/Z (G) = c e n t e r of
so t h a t
1
Z (G)
i s t h e c e n t e r of
Zc(G) = G , ZC-'(G)
# G.)
G.
G/Z
i
Then
(G)
,
G
has nilpotency c l a s s
i 2 0,
The f u l l subcategory of t h e c a t e g o r y
N
groups c o n s i s t i n g of a l l n i l p o t e n t groups i s denoted by
subcategory of
Nc.
G
In particular,
G
c
of a l l
and t h e f u l l
c o n s i s t i n g of a l l n i l p o t e n t groups w i t h n i l (G) 5 c
N
1
=
iff
by
Ab, t h e c a t e g o r y of a l l a b e l i a n groups.
W
e s h a l l be concerned w i t h c o l l e c t i o n s of r a t i o n a l primes and s h a l l
denote such c o l l e c t i o n s by
c o l l e c t i o n of a l l primes.
denote by
P'
P , Q , e t c . ; we r e s e r v e t h e n o t a t i o n
I n general, i f
P
for the
i s a c o l l e c t i o n of primes, w e
t h e complementary c o l l e c t i o n of primes.
a product of primes i n
Il
If the integer
PI, we (somewhat a b u s i v e l y ) w r i t e
n
is
n C P'.
*It would seem t o b e more r e a s o n a b l e t o r e n o r m a l i z e and w r i t e ro(G) = G , e t c .
b u t we f o l l o w t h e convention most f r e q u e n t l y employed i n t h e l i t e r a t u r e .
Localization of nilpotent groups
4
Definition 1.1.
A group G
is said to be P-local if x
-
xn, x € G, is
If H is a full subcategory of G, then a
bijective for all n
€
homomorphism e: G
Gp in H is said to be P-universal (with respect to
-+
P'.
H), or to be a P-localizing
map if Gp i s P-local and if
e*: Hom(G ,K) 2 Hom(G,K) provided K € H , with K P-local.
P
instead of
If P = &, we speak of 0-local, 0-universal,
...,
..., and write
&local, &universal,
Go instead of G4.
We also sometimes
speak of rationalization instead of 0-localization. If P = fp), we speak
..., and write
G instead of G
{PI.
P
Assume now that each group in H admits a P-localizing map. Then,
of p-local, p-universal,
for any $: G
+
K in H, we have a unique map
": Gp
-+
5
rendering the
diagram
comutative. Thus we have a functor L: H
functor and we may view e as
a
-+
H which we call a P-localizing
natural transformation e: 1
-+
L having
the universal (initial) property with respect to maps to P-local groups in H.
We regard the pair
H.
(L,e)
as
providing a P-localization theory on the category
It is clear that if a P-localization theory exists on H, it is essentially
unique, Our main goal in this section is, in fact, to construct such a theory
on the categories Nc, N.
We note, for later use, the following Proposition.
Proposition 1.2. Let
G'
>-b
G % G" be a central extension of groups.
Then G is P-local if G',
Proof.
Let x E
y € G. Thus x = ynu(y'),
GI'
G,
are P-local.
n E PI. Then
y' € G'.
x = ynu(x')"
EX
* yttn= ~y~ for some y"€ G " ,
But y' = xtn for some x' € G'
5
(YP(X'))",
so
Localization theory of nilpotent groups
since
is central i n
uG'
Suppose now t h a t
ex =
in
G,
x q n = 1,
SO
xn = yn, x , y € G , n € P ' .
XI
Then
xn = ynu(x'">
Then
cxn = eyn, so
s i n c e uG'
is c e n t r a l
= 1, x = y .
A homomorphism
D e f i n i t i o n 1.3.
ker
G.
x = yu(x'), x' € G'.
EY,
5
G
@:
-f
is s a i d t o be P - i n j e c t i v e
K
if
c o n s i s t s of P I - t o r s i o n elements; and is s a i d t o b e P - s u r j e c t i v e if,
C$
given any
y € K, t h e r e e x i s t s
P-isomorphism
or
n € P'
with
yn € i m
@.A homomorphism
is a
P - b i j e c t i v e i f i t is b o t h P - i n j e c t i v e and P - s u r j e c t i v e .
It i s p l a i n t h a t a composite of P - i n j e c t i v e
is again P-injective
(P-surjective).
(P-surjective)
homomorphisms
I n addition, the following r e s u l t s w i l l
be u s e f u l i n t h e s e q u e l .
Lemma 1.4.
L e t a: G
+
G2, B : G2
+
G3
be group homomorphisms.
(il
If
Ba
is P - s u r j e c t i v e ,
fii)
If
Bci
is P - i n j e c t i v e and
(iii) If
Ba
is P - i n j e c t i v e ,
i s P-surjective;
B
then
i s P-surjective,
a
then
B
is
P-injective;
(ivl
G
2
€ N,
then
Since
x1 € G1.
Then
a
is P-surjective,
Baxl = 1, s o ,
xy = 1. Then
x;
of
ax^
for some
= (ax,) y2
is P - i n j e c t i v e and if, i n a d d i t i o n ,
B
(i) and ( i i i ) a r e t r i v i a l .
Ba
with
x1 € G1, n € P ' .
and
n € P'
being P-injective,
mn
x2 = 1, mn € P ' , so
y2 € G2
To prove (ii), l e t
there exists
F i n a l l y , t o prove ( i v ) , l e t
~x: =
i s P-injective;
a
i s P-surjective.
a
Proof.
with
@a i s P - s u r j e c t i v e ,
If
Bx2 = 1.
then
y:
B
with
x2 € G2,
xn = ax
2
1'
there e x i s t s
m € P'
is P - i n j e c t i v e .
x2 € G2,
Since
Then, s i n c e
= 1, m € P ' .
B
Ba
is P-surjective
is P-injective,
But i t i s a consequence
P. H a l l ' s t h e o r y of b a s i c commutators (see [34]) t h a t we t h e n have
6
Localization of nilpotent groups
mc
.yC
provided
= (ax,)
a
nil(G ) C c.
2
Since
nmc € P ' , i t follows t h a t
is P-surjective.
Lemma 1 . 5 .
Y: G1
Let
be a homomorphism between P-local groups.
G2
(il
If
y
is P - i n j e c t i v e , then
(iil
If
y
i s P-surjective,
Proof.
(i) Since
there e x i s t s
y
is i n j e c t i v e ;
then y is s u r j e c t i v e .
ker y is PI-torsion and
has no PI-torsion,
G1
ker y = {l), proving ( i ) . To prove ( i i ) ,l e t
is c l e a r t h a t
with
+
n € P'
yy = xl.
with
Thus
Proposition 1 . 6 .
(yyl)
x;
n
= yxl,
n
= x2
so t h a t
and t h e r e e x i s t s
yyl = x 2 , i . e .
y
y1
then
Then if
Q',
Q"
a r e P - i n j e c t i v e (resp.
0 i s a l s o P - i n j e c t i v e (resp. P - s u r j e c t i v e ) .
$"EX=;Qx = 1 so t h e r e e x i s t s
x ' € G ' , and
$ ' X I = 1.
mn E P ' , and
Q
so
yn
=
so
Q
Q',
with
($x0)(;y'),
y T m= Q ' x ' .
n € P'
But then
with
EX)^
= 1. Thus
xIm = 1 f o r some
Then
xn = ux',
m € P ' , so
xmn = 1,
is P-injective.
Assume now
n E P'
G1
is s u r j e c t i v e .
Proof. Assume Q', Q" P - i n j e c t i v e and l e t x C ker Q.
x" C G",
Then
Let
be a map of c e n t r a l extensions.
p-surjective),
x1 C G1
x2 € G2.
it
Q" P - s u r j e c t i v e and l e t
y C K.
;yn = @"x". Let x" = E Xo , xo E G .
y' € K ' .
Then, s i n c e
is P-surjective.
FK'
But now
there exist
is c e n t r a l in
K,
There e x i s t
Then
x' € G',
Eyn = EQxo,
m €
P' with
Localization theory of nilpotent groups
The preceding discussion, with the exception of Lemma 1.4(iv),
of a rather general nature.
was
We now concentrate our attention on nilpotent
groups and state the main result of this section.
Fundamental Theorem on the P-Localization of Nilpotent Groups. There mists
a P-1ocaZization theory
to a P-localization theory
nil LG
S
on the category N.
(L,e)
Moreover,
Nc, for each c
(Lc,ec) on
?
L restricts
1. In particular,
G € N.
nil G if
Further,
$:
G
-+
K in N
P-ZocaZizes iff
K is P-local and
$I
is
a P-isomorphism.
The proof of the Fundamental Theorem is by induction on c = nil(G).
More precisely, given Lc-l: Nc-l
-+
Nc-l
desired properties, we construct Lc: Nc
and
-+
e
*
c-1'
Nc and
1
theory on Ab.
c
=
Lc-l having the
e : 1
the desired properties and such that moreover LclNc-l
To start the induction at
-+
=
-+
L
also having
Lc-l, ec/Nc-l = ec-l.
1, we must construct a P-localization
In addition to doing this, we shall consider the interrelationships
between P-localization on Ab
and the standard functors arising in homological
algebra, which information will be required both in the inductive step and in
Chapter 11.
Recall that
the subring of
with R € P'.
Q
Zp is the ring of integers localized at P, that is,
consisting of rationals expressible as fractions k/L
Note that
%=
72, Zo = Q.
For A € Ab, we define
L ~ =A ~p
and we define el: A
-+
%
=
A a
zP
to be the canonical homomorphism
Note that a P-local abelian group is just a Zp-module.
8
Localization of nilpotent groups
It is e v i d e n t t h a t
el: A
+
Ap
is P-universal w i t h r e s p e c t t o
Ab, so t h a t we have c o n s t r u c t e d a P - l o c a l i z a t i o n t h e o r y on
prove s h o r t l y ( P r o p o s i t i o n 1 . 9 ) t h a t
el
”: %
-+
Bp
We w i l l
is P - b i j e c t i v e , from which we
immediately deduce, u s i n g Lemmas 1.4 and 1 . 5 , t h a t
P - b i j e c t i v e i f and only i f
Ab.
6:
A
-+
is an isomorphism.
B
i n Ab
is
Localization theory of nilpotent groups
9
Before v e r i f y i n g Proposition 1 . 9 and t h u s
Ab, we e s t a b l i s h t h e following
v a l i d a t i n g t h e Fundamental Theorem f o r
Propositions.
Proposition 1.7.
The f u n c t o r
+
Ab
i s exact.
It is only necessary t o n o t e t h a t
Proof.
(flat)
L1? Ab
Zp is t o r s i o n f r e e
.
We now c o l l e c t together i n t o a s i n g l e p r o p o s i t i o n a number of u s e f u l
f a c t s about
.
(Ll, el)
P r o p o s i t i o n 1.8.
If
(i)
Tor(el,l):
A , B f Ab, then:
e
1
a91: A @ B + A p @ B ; e l m e l :
Tor(A,B)
-+
Tor(%,B),
Tor(el,el):
A@B+%@Bp;
Tor (A,B)
+
Tor(%,BP)
all
P-localize.
(ii)
A p-isomorphism
$: A
Conversely, a homomorphism $ : A
and Tor($, Z / p )
-+
B
--t
B,
induces isomorphisms $ O Z f p , Tor($, Z l p ) .
such t h a t
$ @ Z/p
i s an isomorphism
is a s u r j e c t i o n , is a p-isomorphism provided t h a t
A, B
f i n i t e l y generated.
(iii)
fi*(el): g,(A)
-+
H*(%)
P-localizes,
where
H*
i s reduced
homology with i n t e g e r c o e f f i c i e n t s .
(iv)
e f : Ext(kp,B)
(v)
If
%'
B
i s P-local,
Ext(A,B)
If
A
then
e 1x : Hom($,B)
%'
Hom(A,B),
.
is PI-torsion and
B
is P-local, then
Hom(A,B) = 0 , Ext(A,B) = 0.
Proof.
(i) The f i r s t two a s s e r t i o n s a r e obvious and t h e f o u r t h
follows from t h e t h i r d , which we prove a s follows.
Let
R >-
F ->
A
are
Localization of nilpotent groups
10
be a free abelian presentation of A.
-
Localizing this short exact sequence
s-
yields, by Proposition 1.7, a short exact sequence
Fp
%
and
Fp is flat, Thus we have a commutative diagram
Tor(A,B)
>------t
R 8B
I
I
-
F C4 B >-
A
M,
B
I
I
and we invoke Proposition 1.7 together with the fact that el 69 1 P-localizes
to infer that Tor(el,l) P-localizes.
(ii) Since
Z/p
from (i), and the factthat
is p-local, the first statement follows immediately
isan isomorphism, Toprove the converse,consider
$p
the sequence
K >-
A ---f>
obtained from I$, where K = ker $, L
L
>-
=
im
B
$,
-
C =
C,
coker
$.
We thus infer a
diagram
where the horizontal and vertical sequences are exact.
I(
c sz/p
We want to prove that
and C are p'-torsion groups. However, since A, B, and hence K, C,
are finitely generated, it suffices to prove that K @Z/p = 0, C S Z / p = 0.
Now C S Z / p = 0 since
+
@
Z / p is surjective. Thus C is p'-torsion and
hence Tor(C, Z/p) = 0. Reference to the diagram then shows that
Tor(A, Z/p) >-
Tor(L, Z/p)
and A @Z/p
>->
L O Z / p , from which the
conclusion K @Z/p = 0 immediately follows.
Note that the converse certainly requires some restriction on A, B.
For the homomorphism
Tor( $, Z/p)
$:
Q
+ 0
certainly has the property that $ @Z/p
are isomorphisms, without being a p-isomorphism.
and
Localization theory of nilpotent groups
(iii)
The a s s e r t i o n i s r e a d i l y checked i f
A
i s a c y c l i c group.
Use of t h e Kunneth formula t o g e t h e r w i t h ( i ) and P r o p o s i t i o n 1 . 7 shows t h e
a s s e r t i o n t o be t r u e f o r f i n i t e d i r e c t sums of c y c l i c groups, hence f o r
a r b i t r a r y f i n i t e l y g e n e r a t e d a b e l i a n groups.
and
H,
Finally, since both l o c a l i z a t i o n
commute w i t h d i r e c t l i m i t s , t h e a s s e r t i o n is t r u e f o r a r b i t r a r y
a b e l i a n groups.
The f i r s t isomorphism simply r e s t a t e s D e f i n i t i o n 1.1.
(iv)
t h e second, l e t
ZP-module.
I -+->
B >-
be an i n j e c t i v e p r e s e n t a t i o n of
J
S i n c e Zp is f l a t , i t f o l l o w s t h a t
an i n j e c t i v e p r e s e n t a t i o n of
B
B >--f
a s an a b e l i a n group.
I ->
J
B
For
as a
is a l s o
Thus we have a c o m u t a t i v e
diagram
-
Hom(A,J)
Hom(A,I)
Ext(A,B)
>-
S i n c e t h e f i r s t two v e r t i c a l arrows a r e isomorphisms, so i s t h e t h i r d .
(v)
Clearly,
-
i n ( i v ) , then
Hom(A,J)
Hom(A,J)
Ext(A,B)
Hom(A,B) = 0.
-
0
as
J
shows t h a t
Now, i f
J
h a s t h e same meaning a s
is P - l o c a l and t h e s u r j e c t i o n
Ext(A,B) = 0.
We now r e t u r n t o t h e proof of t h e Fundamental Theorem and complete
t h e i n i t i a l s t a g e of t h e i n d u c t i o n by means of t h e f o l l o w i n g P r o p o s i t i o n .
P r o p o s i t i o n 1.9.
P-local and
$
Proof.
el
If
is a
0: A
+
B
i s in
Ab, then
$
P-localizes
iff B is
P-isomorphism.
We f i r s t show t h a t
embeds i n t h e e x a c t sequence
el: A
+
Ap
i s a P-isomorphism.
In fact,
12
Localization of nilpotent groups
Tor(A, Zp/ Z)
and since, plainly, Z / Z
P
A @Zp/Z
@ Z P /Z
is a PI-torsion group, it follows that both
and Tor(A, Zp/ Z)
Conversely, i f
% *A
A
B
are PI-torsion groups.
is P-local and
$:
A
+
B is a P-isomorphism,
we have a commutative diagram
and the proof of Proposition 1.9 is completed by means of Lemma 1.4 (i), (ii)
and Lemma 1.5.
Assume now that we have defined
appropriately. Our objective is to extend Lc-l
extend
e
c-1
sequence
+.
Nc and to
correspondingly, to have the universal property in
NC
Proposition 1.10. Lc-l: Nc-l
GI
to Lc: Nc
Nc-l i s an exact functor. If, further,
i s a central extension i n Nc-l, then so i s the localized
EP
G -Z, GI'
PP
+
G' >-
G >-
Proof.
We will write e for ec-l.
P
. We prove:
P
.G;
Consider, then the diagram
(1.11)
in Nc-l.
Assuming the top row short exact, we must show that the bottom row
is likewise short exact.
We rely on the (inductive) fact that the vertical
arrows are P-isomorphisms. First,
E~
is surjective. For
E
P
e = eE
is
Localization theory of nilpotent groups
P-surjective, so that, by Lemma 1 . 4 ( i ) ,
Lemma 1.5(ii),
13
is P-surjective. Hence, by
E~
is surjective.
E~
Second, up
up
by Lemma 1.4(ii),
u Pe
is injective. For
=
eu
is P-injective, so that,
is P-injective. The conclusion now follows from Lemma
1.5(i).
Third, ker
ker
E~
C im up.
Let
E~ =
E
P
im up.
Since clearly
= 0, we
must prove
Then, for some n C P', yn
y = 1, y € Gp.
n
y = 1. Thus
=
ex,
m
x C G , so eEx = E ex = E
EX = 1 for some m € P', so that
P
P
m
x = ux', x' C G ' , whence ymn = upex', and mn € PI. One now argues as in
the proof of Lemma 1.5(ii) that, since
then y € im up.
ymn C im u p ,
This completes the proof of the first statement of the
proposition. Notice that the proof that up
normality of
are P-local and
Gp
G;,
uG'.
is ihjective made no use of the
Thus we may say that l o c a l i z a t i o n respects subgroups and
normal subgroups.
Assuming now that the top extension in (1.11) is central, we show that
the same is true for the bottom extension. Let x' C G I , y C Gp.
x C G , n C P'.
Thus
(UX')-~X(~X') =
has unique nth roots, (upex')-'y(p
Gp
center of
x,
Then yn = ex,
-1 n
(upex') y (upex') = yn.
so
ex') = y, so
P
Since
upex' belongs to the
GP'
Then ylm = ex: x' € G I , m t P'. Thus, upyIrn
-1
m
m
belongs to the center of G ; , so that, for any y € Gp, y (upyl) y = (upy')
-1
Since Gp has unique mth roots, y (upy')y = upy', so upy' belongs to
Now let y'
the center of Gp.
Theorem 1.12.
i
K
If
C G;.
Thus ppGi
G 6
Ni, i
is central in
5 c
-
-
1 by Proposition l.a(iii).
€
Ni-l,
2 5 i 5 c
-
Gp.
1, then G,(e)
Proof. We argue by induction on
: H,(G)
+
H,(Gp)
P-localizes.
i, the theorem being true for
Suppose the theorem true for all
1 and let G € Ni.
.
If 2
-
center of G , then
Localization of nilpotent groups
14
nil(2) = 1, nil(G/Z) 5 i
-
1 and by Proposition 1.10, we have a map of central
extensions
(1.13)
Then (1.13) induces a map of the Lyndon-Hochschild-Serre spectral sequences
{EZtI
+
I E z J , where
the coefficients being trivial i n both cases. It now follows from the inductive
hypothesis, together with Proposition 1 . 8 ( i ) , taken i n conjunction with the
natural universal coefficient sequence i n homology, that (1.13) induces
z2
e2: E2 -+
which P-localizes provided that s + t > 0. Applying
St
st
Proposition 1.7 allows us to infer that em: Ett + 6Lt also P-localizes
provided that
s
+
t > 0. Finally, since for any n, Hn(G)(Hn(Gp))
has a
finite filtration whose associated graded group is @Eit(Ezt)
with
it follows once again from Proposition 1.7 that Hn(e):
-+
provided
Hn(G)
s
+ t = n,
H (G ) P-localizes
n P
n > 0.
Corollarv 1.14.
Let
G C Nc-l and Zet
t r i v i a l G-action. Then e : G
-+
A be a P-ZocaZ abeZian group with
Gp induces
e*: H*(G
P
;A)
E
H*(G;A).
(The conclusion of Corollary 1.14 holds, more generally, if G
acts nilpotently on A; see Section 4.)
Proof.
The homomorphism
e
induces the diagram
and it follows from Theorem 1.12 and Proposition 1.8(iv)
that e'
and
e"
Localization theory of nilpotent groups
15
are isomorphisms. Thus e*, too, is an isomorphism.
Let now G E Nc.
We then have a central extension
with nil(r) C 1, nil(G/r) 5 c
corresponds t o
r
-+
1.
5
an element
Then, applying e:
-
rp, we
By the cohomology theory of groups, (1.15)
E H2(G/r;r)
obtain e,S
€
with G/r acting trivially on
HL(G/r;r,)
there exists a unique element 5, E H2((G/r)p;rp)
(1.16)
is induced by
rp >-
(1.17)
correspond to
e: G
+
and, by Corollary 1.14,
such that
e*Sp = e,S,
where now e*
5,.
e: G/r
-+
Gp
(G/T)p.
-J
Let the central extension
(G/r),
It follows from (1.16) that we can find a homomorphism
Gp yielding a commutative diagram
(1.18)
In fact, the general theory tells us that given two central extensions
of (arbitrary) groups, together with homomorphisms
then there exists
(1.19)
T:
G1
+
G2
r.
p: A1+
A,,
yielding a commutative diagram
U:
Q,
-+
Q,,
Localization of nilpotent groups
16
p r e c i s e l y when
(1.20)
Moreover, i f
then
T
T
and
(1.21)
T'
and
T'
a r e two maps y i e l d i n g commutativity i n (1.191,
a r e r e l a t e d by t h e formula
T'(x)
=
T(X).II
2K E 1 ( x ) , x € G1,
f o r some
Q,
K:
+
A2.
Returning t o our s i t u a t i o n , we see,from (1.17) and t h e i n d u c t i v e
hypothesis
that
r
rp
= {l},
Gp € N
that
G/r
and, f u r t h e r , u s i n g P r o p o s i t i o n 1.2,
= G , and we n a t u r a l l y t a k e
t h e same P - l o c a l i z a t i o n i f we d e f i n e
P r o p o s i t i o n 1 . 6 , e : G + Gp
is P-local.
-
C'
e
is P-loca1,and t h e f a c t t h a t
e*
and t o prove t h e n a t u r a l i t y of
e.
G
w i l l f o l l o w d i r e c t l y from t h e
e
Lc
Let
+
Hom(G,K)
is s u r j e c t i v e i f
is i n j e c t i v e follows immediately
is P - s u r j e c t i v e and
Thus i t remains t o d e f i n e
Then we have
By
is a n a t u r a l t r a n s f o r m a t i o n of f u n c t o r s .
For we then r e a d i l y i n f e r t h a t ex: Hom(G ,K)
P
e
Gp = (G/rIp, p r e s e r v i n g
L G = Gp, a s we propose t o do.
Then t h e u n i v e r s a l p r o p e r t y of
from t h e f a c t t h a t
then
is a P-isomorphism and hence an isomorphism i f
f a c t , s t i l l t o be proved, t h a t
K € Nc
G € Nc-l,
We also remark t h a t i f , i n f a c t ,
is P-local.
Gp
= {l},
(G/I')p C Nc-l,
K
is P - l o c a l .
on morphisms of
0:
G
+c
in
Nc
a s a functor,
Nc, and l e t
r
= rc(c).
Localization theory of nilpotent groups
17
5:
5, :
(1.22)
and o u r object is to define
i s clear that any
functoriality of
4,
"front face" of ( 1 . 2 2 ) .
e*: H2 ((G/r),;
to make ( 1 . 2 2 ) commutative.
Gp
T: G
,
-
+
Gp
It
is uniquely determined so that
i s automatic once a suitable $,
W
e shall first find
But
+
$,e = e$
yielding
Lc
=
Cp
"p:
is defined.
yielding commutativity in the
To this end, we compute
e*$&S,.
r,)
S
H2 (G/r;yp)
,$I"*?
P
P
=
by Corollary 1.14 so that, i n fact,
4
-
Thus, by ( 1 . 1 9 ) , ( 1 . 2 0 ) , we may find
1:
Gp
+
Gp
so that
Lmalization of nilpotent groups
18
as claimed. However
T
need n o t satisfy the equation re = e$,
so
we now
modify T, preserving (1.23), so that the last equation also obtains. Consider
the diagram, obtained from ( 1 . 2 2 ) ,
(1.24)
where JI' = e+' = $ie, JI" = e$" = $Fe. Clearly (1.24) commutes if we set
JI = e$ or J, = re so that, using (1.211, there exists 0 : G / r
Let Bp:
(G/rIp * Fp be given by Ope
of e in Nc-l.
= 8 ; Bp
+
rp
such that
exists by the P-universality
Define
From (1.23), it follows that
and also
$pex = (rex).(i Pe PEPex) = (Tex).(L,ecx)
=
e$x, x c G.
It remains to verify the final assertion of the Fundamental Theorem for
We already know that e:
G + Gp
i s a P-isomorphism and
Gp
Nc.
is P-local. The
converse i s proved j u s t as for Proposition 1.9, making use of Lemma 1.4(i),
and Lemma 1.5.
(ii)
Properties of localization in N
2.
19
Properties of localization in N
In this section, we deduce a number of immediate consequences of
the methods and results of §I.
G C N and
If
Theorem 2.1.
Q
i s a coZlection of primes, then the s e t
consisting of the Q-torsion elements i n G i s a (normal) subgroup of
Proof.
Since e
Let P = Q'
and consider the P-localization e: G
T
Q
G.
+
Gp.
is a P-isomorphism and Gp is P-local, it is clear that T = ker e.
Q
Suppose
Theorem 2 . 2 .
G C N has no Q-torsion.
Then i f
xn = yn, x, y € G,
n C Q , i t f o l l m s t h a t ~x = Y.
Proof. Again, let P = Q'
e:
G -+ Gp. Then e
ex = ey,
so
and consider the P-localization
is injective and
(ex)" = (ey)".
Since Gp is P-local,
x = y.
Corollarv 2 . 3 .
G € N i s P-local i f f it has no PI-torsion and
i s surjective f o r a l l
n
c
x
-
xn, x € G,
P'.
We now turn to results which make explicit mention of P-localization.
Theorem 2.4.
The P-localization functor
L: N
-+
N
i s an exact functor.
Proof. This follows from Proposition 1.10, in conjunction with the
Fundamental Theorem.
A s immediate corollaries, we have the following assertions, of
which the first is the definitive version of Proposition 1.2 and the second
--
is related to Proposition 1.6.
Corollarv 2 . 5 .
Let
Then if any two of
Corollary 2 . 6 .
G'
G', G, G"
G
G" be a short exact sequence i n N.
are P-local so i s the third.
Let
be a map of short exact sequences i n N.
so does the third.
Then i f any two of $',
$,
4'' P-localize,
Localization of nilpotent groups
20
Theorem 2.7.
ri($):
ri(G)
-+
?(K)
Proof.
and l e t
$: G
P-localizes
ri(G)
G C N
Let
-+
Then
K P - l o c a l i z e G.
f o r a21
i 2 1.
It f o l l o w s from C o r o l l a r y 2.6 t h a t i t is s u f f i c i e n t t o
prove t h a t t h e homomorphism
W
e argue by i n d u c t i o n on
i
G/T (G)
$i:
K/ri (K)
induced by
i , t h e a s s e r t i o n being t r i v i a l f o r
following from Theorem 1.12 f o r i = 2.
i ? 2 , and prove t h a t
-+
$i+l
Thus we assume t h a t
P-localizes.
P-localizes.
$
i = 1 and
$
P-localizes,
A second a p p l i c a t i o n of C o r o l l a r y
2.6 shows t h a t i t is s u f f i c i e n t t o prove t h a t t h e homomorphism
5:
ri(G)/rifl(G)
-+
ri(K)/I"+'(K),
induced by
6, P - l o c a l i z e s .
We apply t h e
5-term e x a c t sequence i n t h e homology of groups t o t h e diagram
t o obtain
where t h e s u b s c r i p t
by Theorem 1.12 and
Theorem 1.12.
ab
,,$,
refers to abelianization.
Oiab
Then
I$,, bab P - l o c a l i z e
P - l o c a l i z e by t h e i n d u c t i v e h y p o t h e s i s and
It f o l l o w s from P r o p o s i t i o n 1.7 t h a t
P-localizes
and t h e
proof of t h e theorem is complete.
There is a d u a l theorem t o Theorem 2 . 7 concerning t h e upper c e n t r a l
series of
G
which, however, r e q u i r e s
more d i f f i c u l t t o prove.
G
t o be f i n i t e l y generated and is
We c o n t e n t o u r s e l v e s h e r e w i t h a s t a t e m e n t of t h e
r e s u l t , r e f e r r i n g t o [34 ] f o r d e t a i l s .
Properties of localization in N
Theorem 2.8.
i
z
( e ) = el z (G)
i
e: G
-f
is P-localization,
Gp
i
i
z (G) i n t o z ( G ~ ) . Moreover,
carries
if G
Z (G)
P-localizes
and
G € N
If
i
21
then t h e r e s t r i c t i o n
z i ( e l : z i (GI + z i ( G ~ )
is f i n i t e l y generated.
Our n e x t r e s u l t i n t h i s s e c t i o n i s t h e d e f i n i t i v e v e r s i o n of
Theorem 1 . 1 2 .
Theorem 2.9.
Let
$: G
K
-+
N.
be i n
$
P - l o c a l i z e s iff H,($)
H,($)
P-localizes i f
Then
P-localizes.
Proof.
Theorem 1 . 1 2 asserts t h a t
We n e x t prove t h a t i f
e: K
+
%
H*(K)
P-localize.
i s P-local,
so
is P - l o c a l ,
Then
H,(e)
H*(e) : H*(K)
commutes.
that
Now l e t
8,($) P-localize.
Thus
f a c t o r s as
$
But s i n c e
isomorphism.
+
i s an isomorphism.
Stammbach Theorem ( s i n c e K , I$€N)
P-local.
then
G
fi*($), i , ( e )
H,(Kp)
P-localizes.
For l e t
P-localizes;
but
&(K)
I t f o l l o w s from t h e S t a l l i n g s -
e
Then
Gp
is P - l o c a l .
K
t$
i s an isomorphism.
H*(K)
K
i s P-local,
SO
K
and
both P-localize,
HA($)
is an
Thus t h e S t a l l i n g s - S t a m b a c h Theorem a g a i n i m p l i e s t h a t J,
an isomorphism, so t h a t
@
is
is
P-localizes.
Our f i n a l r e s u l t i n t h i s s e c t i o n p l a y s a c r u c i a l r o l e i n Chapter I1
when we come t o s t u d y (weak) p u l l b a c k s i n homotopy t h e o r y .
Theorem 2.10.
Localization c o m t e s with pullbacks.
Proof.
Suppose g i v e n
Localization of nilpotent groups
22
in
N, and form p u l l b a c k s
G
a > H
Cm
4
K-M
Of course
y: G
-f
E,
UJ
G € N, being a subgroup of
c h a r a c t e r i z e d by
~y
and we show t h a t
h a s pth r o o t s ,
a C Hp, b €
(a,b) €
5
Next,
then
and
y
€ P'
Let
($,a)'
and
-
Since
(x,y) €
G
E,
5 Hp
Mp
x'
It f o l l o w s t h a t
i s P-injective.
y
=
For i f
Kp.
Then
i s P-local,
x = a',
y = bp,
$ a = $ b,
P
SO
P
y(x,y) = 1, x C H , y E K , (x,y) € G ,
1, m,n C P ' .
is P-surjective.
= e a , y'
$a
i t s u f f i c e s t o show t h a t
(x,y> = (a,b>'.
=
= eb, a € H , b € K.
($b)u, u € M, and
C
So
For l e t
xm = e h , yn = ek, m,n € PI, h E H , k € K .
and
Kp,
x
x € Hp, y E
Since
= (UJ,b)'.
ex = 1, ey = 1, xm = 1, yn
Finally,
Then
i s P-local.
G
p € PI.
Kp,
= e a , isy = eB,
i s P-universal.
y
-
First,
that
There i s then a homomorphism
H x K.
us
(x,y) €
E,
Thus (with
Now
=
( x , ~ =) 1,
~ ~
mn € P ' .
x C Hp, y €
k = mn) we have
$px = Jlpy, so
1, s € P'.
Kp.
e$a = e$b.
We deduce t h a t i f
C
(see t h e proof of Lemma 1 . 4 ( i v ) ) .
Thus
n i l M 5 c t h e n $aS = $bs
c
c
c
c
( a s ,bS ) , w i t h Ilsc E P I . T h i s shows t h a t
(aS ,bs ) € G and ( x , ~ ) '=~e ~
y
i s P - s u r j e c t i v e and t h u s , i n view of t h e Fundamental Theorem, completes
t h e proof of Theorem 2.10.
Further properties of localization
3.
23
F u r t h e r p r o p e r t i e s of l o c a l i z a t i o n
I n t h i s s e c t i o n , we prove a number of r e s u l t s i n v o l v i n g t h e l o c a l i z a -
t i o n functor i n t h e category
N.
A s mentioned i n t h e I n t r o d u c t i o n , w e a r e
s p e c i f i c a l l y concerned w i t h r e s u l t s which have f r u i t f u l homotopy-theoretic
analogs.
W
e f i r s t examine more c l o s e l y t h e n o t i o n of P-isomorphism introduced
i n 81.
Theorem 3.1.
Let
P-localization.
$: G
be i n
K
N
and l e t
$p: Gp
5
+
lil
$
i s P - i n j e c t i v e iff $p i s i n j e c t i v e ;
fiil
$
i s P - s u r j e c t i v e iff $p i s s u r j e c t i v e .
(i)
If
$
is P-injective,
t h e n so i s
i s P - i n j e c t i v e and t h e composite of P - i n j e c t i o n s i s
P-injection.
be i t s
Then:
Proof.
e
-+
Thus
$pe = e$
is P-injective,
we may apply Lemma 1 . 4 ( i i ) t o deduce t h a t
$p
and s i n c e
e+: G
+
5
since
of c o u r s e , a
e
is P-surjective,
is injective.
The converse
is proved s i m i l a r l y , u s i n g Lemma 1 . 4 ( i i i ) .
(ii)
If
$
is P-surjective,
then so is
e$
since
e
and t h e composite of P - s u r j e c t i o n s i s , of c o u r s e , a P - s u r j e c t i o n .
$pe
= e$
is P - s u r j e c t i v e and we may apply Lemma 1 . 4 ( i )
t o deduce t h a t
$p
is s u r j e c t i v e .
is P-surjective
Thus
and Lemma 1 . 5 ( i i )
The converse i s proved s i m i l a r l y , u s i n g
Lemma 1 . 4 ( i v ) and Lemma 1 . 5 ( i ) .
Remark.
The f a c t t h a t
proved by i n d u c t i o n on
$p
surjective implies
$
P - s u r j e c t i v e may a l s o b e
n i l ( G ) , making use of Theorem 2 . 7 .
We may t h u s avoid
u s i n g Lemma 1 . 4 ( i v ) which, w e r e c a l l , was based on P. H a l l ' s commutator
calculus.
Localization of nilpotent groups
24
As a corollary of Theorem 3.1, we have the following definitive
version of Proposition 1.6.
Theorem 3 . 2 .
Let
be a map of short exact sequences i n N.
my
Then i f any t u o of
$',
$I'
are P-isomorphisms, then s o i s the t h i r d .
,.
Theorem 3 . 3 .
Let G, K E N .
li)
Gp and
Then the following assertions ore equivalent:
Kp are isomorphic;
(ii) There e x i s t M
moremer,
M
C
N and P-isomorphisms
a: G
may be chosen t o be f i n i t e l y generated i f
+
My B: K
and
G
-+
M;
are
K
f i n i t e l y generated;
l i i i ) There e x i s t M' C N and P-isomorphisms
moreover,
M'
may be chosen t o be f i n i t e l y generated i f
f i n i t e l y generated.
(In the special case P =
that G and K
4,
the equivalence (i)
-
3
(ii), let
B
M, 6: K
-+
K;
K are
(i) follow directly
M to be the maps defined by
are P-isomorphisms, since
-a , -B
-
and set M - K p ,
Kp to be the composite G %- Gp % % and $: K
e. We then define M to be the subgroup of M
-r
-
Gp 2 Kp
w:
-
a: G
-+
and K are finitely generated and
The implications (ii) = (i), (iii)
from Theorem 3.1. To prove (i)
-P
and
G , 6 : M'
(iii) amounts to the assertion
torsion-free.)
E: G
G
-+
have isomorphic rationalizations iff they are commensurable
(in the senseof [ 6 , 6 5 ] ) , at least when G
Proof.
M'
y:
-+
Kp to be simply
generated by aG U E K
., B.
It is clear that
areP-isomorphisms, and that M
is
and
a
and
Further properties of localization
f i n i t e l y generated i f
G
K
and
25
a r e f i n i t e l y generated.
F i n a l l y , t o prove ( i i ) * ( i i i ) , w e c o n s t r u c t t h e p u l l b a c k diagram
M, a
G
x
B
and
having t h e i r p r e v i o u s meanings.
M' € N
K, c e r t a i n l y
f i n i t e l y generated.
and i s f i n i t e l y generated i f
t h e argument f o r
We prove t h a t
a
Gp E
and
K
are
n € P', y € G
5
but
$: G
+
K, JI: K
$p: Gp
is a P-isomorphism;
+
Hom(K,G) = 0.
$: G
Kp
G.
i s a P-isomorphism,
6
By t h e p u l l b a c k p r o p e r t y ,
Now l e t
6.
with
6(y,xn) = xn, so t h a t
and
example of a P-isomorphism
(so t h a t
so i s
Note t h a t i t i s n o t a s s e r t e d t h a t
of P-isomorphisms
then
is P-injective,
there exist
(y,xn) C M '
Remark.
being p e r f e c t l y symmetric.
y
ker a ; a s
is P-surjective,
then
G
(We u s e h e r e t h e f a c t , coming from t h e aforementioned
a r e themselves f i n i t e l y g e n e r a t e d . )
S
i s a subgroup of
P. Hall, t h a t subgroups of f i n i t e l y g e n e r a t e d n i l p o t e n t groups
t h e o r y of
ker 6
M'
Since
6
Gp G
x € K.
Since
Bxn = ( 6 ~ =) a~y .
a
But
i s P-surjective.
Kp
implies the existence
For example, i f
G = Z
and
K = Zp,
I n f a c t , Milnor h a s even c o n s t r u c t e d an
+
of f i n i t e z y generated n i l p o t e n t groups
K
by Theorem 3 . 1 ) w i t h t h e p r o p e r t y t h a t no map
JI: K
+
see R o i t b e r g [70].
A q u i t e analogous phenomenon a r i s e s i n t h e homotopy c a t e g o r y , a s
h a s been shown by Mimura-Toda
[57
I.
(Compare [70].)
We t u r n now t o a new s e r i e s of r e s u l t s descrjhine r e l a t i o n s between
t h e o b j e c t s and morphisms i n
A s e t of primes
P
N
and t h e i r v a r i o u s l o c a l i z a t i o n s .
i s c a l l e d cofinite i f
P'
is f i n i t e .
G
26
Localization of nilpotent groups
If
Lemma 3 . 4 .
G C N
s e t of primes
i s f i n i t e l y generated, then there e x i s t s a c o f i n i t e
such t h a t
P
r a t i o n a l i z a t i o n of
Proof.
G
Gp
-c
i s i n j e c t i v e , where
Go
land hence also of
The t o r s i o n subgroup
Go i s the
Gp).
( c f . Theorem 2 . 1 ) of
T
f i n i t e l y generated and hence, a s is r e a d i l y seen, f i n i t e .
has p-torsion
s e t , then
P
Theorem 3 . 5 .
cofinite,
Let
$
Proof.
Let
e
...,x
{xl,
}
yi C K , m
Now choose a c o f i n i t e subset
factorize
generate
i
and
$
is injective.
4: G
+
P
KO
be a
such t h a t
of
K
415,
Q
e: K
G, l e t
Q
+
so that
KO
such t h a t
such t h a t
mi C P', 1 5 i C n, and
as
is P-local, we have
If
G € N
the r a t i o n a l i z a t i o n maps
elYi
= zi
i
2KO, e
2 1'
, so
i s f i n i t e l y generated, then
G
= e e
5.
l i f t s uniquely i n t o
Theorem 3 . 6 .
z 0
P
m
Kp
Since
Go
5.
and then f i n d
K
-+
By Lemma 3 . 4 , we f i r s t choose a c o f i n i t e s e t
is injective.
rationalize
Gp
P
+Go,
p C II.
G
T
i s t h e complementary
Then there e x i s t s a c o f i n i t e s e t of primes
has a unique l i f t i n t o
KQ + K O
P
be f i n i t e l y generated and l e t
G, K € N
given homomorphism.
i s t o r s i o n - f r e e and
Gp
is
I t follows t h a t
f o r only f i n i t e l y many primes, s o t h a t , i f
is
G
i s the puZZback of
Further properties of localization
Proof.
We argue by induction on
if G C Ab.
G = Z/pk
nil(G),
21
the theorem being easily proved
For, in this case, the assertion is obvious if G = 52
or
and then we infer it for any finitely generated abelian group by
remarking that, if the assertion is true for the abelian groups A , B, it is
plainly true for A
Ci;
B.
To establish the inductive step, we consider the short exact sequence
with nil(G')
<
nil(G), nil(G")
theorem is true for GI, G".
=
1, so we may assume inductively that the
Write
e : G+ G
for the localization,
P
P
r : G + G
for the rationalization. We want to prove that, given x € G
P P
0
P
P'
with r x = x
for all p, there exists a unique x € G with e x = x
P P
0
P
P
Now E x € G" and
P P
P
.
r"E x =
P P P
Hence there exists a unique
€
x"
P
EX.
Then
E
x =
P P
E
e
P P
x = (e ,IX):
P
P
!J,
up, uo
E
x
all p.
P P'
-
x, so that
P
where x' € G'
P
P'
being regarded as inclusions. Moreover,
x
where
all p.
x
00'
G" with
e"x" =
Let x" =
E
r: G
= (rG)(r'x'),
P P
all
pI
Go is the rationalization, so that the elements x' have a
P
common rationalization. Hence there exists a unique x ' < G' with e'x' = x'
P
P'
+
Then x = e (xx').
P
P
Localization of nilpotent groups
28
Uniqueness is clear, even in the event that G is not finiteZy
generated, since, e being p-injective, the map
P
component is e is always injective.
PS
&: G
-+
IIG whose p
P’
th
Remarks. A particularly simple consequence of the injectivity of
6: G
-f
i7G
P
that nil(G)
is that G
=
-
11) iff Gp=fl), all p.
Another consequence is
max nil(G ). We also note that we could generalize this theorem
P
by considering any infinite partition of II into disjoint families of primes.
There is a stronger statement (compare Theorem 3.9) i f the partition is finite.
Theorem 3 . 7 .
where
o
If G C N is finitely generated, L)e have a pullback diagram
and T is the rationaZization
P
G is abelian, then the diagram is als o a pushout.
is the rationaZizatioB map of IlG
of 6. If,further,
Proof.
We proceed as in the proof of Theorem 3 . 6 .
first the case that G
Thus we consider
is finitely generated abelian. Since localization
commutes with finite direct sums, we may assume G cyclic. For G
finite
cyclic, the conclusion is obvious. For G = Z, we have a map of short exact
sequences
leo
9
(n Ep)o--
>-
C1, C2 being the respective cokernels. Since e
so
too is the induced map y: C1
+.
C2.
c‘
c2
and u
are rationalizations
But it is readily seen that
C1 = II L / Z
P
Further properties of localization
is torsion-free, divisible, that is, 0-local, Hence
29
y:
C1 e C 2 , which is equiva-
lent toour assertionthat, inthis case, the diagramis apullback anda pushout.
We now easily complete the proof of Theorem 3 . 7 by induction on
following the pattern of proof of Theorem 3 . 6 .
nil(G)
It is certainly not true that localization commutes with infinite
Cartes an products, even where the product is nilpotent. We do have the
following special result, which will be of use to us later.
Theorem 3 . 8 .
If G C
N i s f i n i t e l y generated f o r , more generally, i f the
p-torsion subgroup
T (G) = 11) f o r p s u f f i c i e n t l y large), then the map
P
@ : (nGp)o
rIG
G
e!Go, induced by the map 8 : rIG -+ IIG
P,O’ P90
P
P90
which rationaZizes each component, i s i n j e c t i v e . If, f u r t h e r , G is abelian,
-+
then $
admits a l e f t inverse.
Proof. Of course, $
But since 0 = IIr r : G
P’ P
P
+
is injective iff
ker 8 is a torsion group.
G
the rationalization, we have
PSO
ker 8
=
II ker r
P = nTP(G)
and this is a torsion group if (and only if!)
Tp(G) = {1}
for p
sufficiently large.
The final assertion follows because
(TIGp)o and IIG
P9 0
both rational vector spaces and we may invoke the Basis Theorem.
are
It is possible to formulate a version of Theorem 3 . 6 in which an
arbitrary decomposition of I7 into mutually disjoint subsets is given.
If
the number of subsets in the decomposition is infinite, as in Theorem 3 . 6 ,
then we must impose the condition that G be finitely generated, as in Theorem
3.6.
On the other hand, if the number of subsets in the decomposition is
finite, it is unnecessary to impose a finiteness condition on G.
Since, in
the sequel, we shall be particularly concerned with the case in which rI
Localization of nilpotent groups
30
is decomposed i n t o two d i s j o i n t s u b s e t s , we s t a t e t h e r e s u l t i n t h i s form,
while r e c o r d i n g t h e f a c t t h a t t h e g e n e r a l i z a t i o n t o a f i n i t e decomposition
of
Il is v a l i d .
If
Theorem 3.9.
G E N,
then we have a puZZback diagram
eP
GG
I
IP
r p , r p l denoting the rationaZization maps.
Proof.
Consider f i r s t t h e c a s e t h a t
G
is a b e l i a n .
Since t h e
a s s e r t i o n is c l e a r f o r c y c l i c groups, i t is t r u e a l s o f o r f i n i t e l y generated
a b e l i a n groups, a s i n t h e proof of Theorem 3.6.
I n general,
G
may be expressed
as t h e d i r e c t l i m i t of i t s f i n i t e l y generated subgroups
G E
But
I& GaQ = (liln Ga)Q
%'
+G a y
GQ
G" f i n i t e l y generated
f o r any c o l l e c t i o n of primes
Q , and
I&
p r e s e r v e s p u l l b a c k diagrams, so t h e a s s e r t i o n is v e r i f i e d f o r a r b i t r a r y a b e l i a n
groups.
Again, a s i n Theorem 3.6, we argue by i n d u c t i o n on
nil(G)
t o prove
t h e theorem f o r a r b i t r a r y n i l p o t e n t groups.
Remark.
It is e a s i l y proved t h a t t h e diagram of Theorem 3.9 is a l s o a
pushout i n
Ab
if
G
is abelian.
f o r an a r b i t r a r y n i l p o t e n t group
as
This remark g e n e r a l i z e s t o t h e s t a t e m e n t ,
G , t h a t every element of
r p ( x ) r p l ( x ' ) , x E Gp, x ' € G p l .
Go
is expressible
A s i m i l a r remark a p p l i e s t o Theorem 3.7.
While, i n Theorem 3.9, no f i n i t e n e s s c o n d i t i o n is imposed on
G,
i t i s n e v e r t h e l e s s u s e f u l t o know when such a c o n d i t i o n can b e deduced from
analogous c o n d i t i o n s on
groups.
Gp, G p r .
We prove t h e f o l l o w i n g r e s u l t f o r a b e l i a n
Further properties of localization
Theorem 3.10.
4,
If
A C Ab
then A
are f i n i t e l y generated Z
Proof.
We assume t h a t
i s a f i n i t e l y generated abeZian group i f f
P
-,
Zpl-
%,
modules, respectively.
a r e f i n i t e l y generated
Z p l - modules, r e s p e c t i v e l y , and prove t h a t
A
f i n i t e l y generated R-modules.
IS1,
A a9 R
...,6 II 1
Let
$ @%I
Moreover,
be a s e t of R-generators
5
j
B
for
B >-A
which i s t o r s i o n - f r e e , w e
A
S
A
b e t h e r i n g Zp d Z p l .
A @ R
a s R-modules,
R
and w r i t e
ij
C = A/B.
A@R--
C A.
a i j ’ we get a
Tensoring w i t h R ,
C@R,
t h e i n d i c a t e d isomorphism f o l l o w i n g from (3.11).
clearly implies t h a t
Let
g e t a n e x a c t sequence
B@R>-w
C = 0, so t h a t
Thus
C @ R = 0 , which
A = B , which is f i n i t e l y g e n e r a t e d .
Theorem 3.10 admits an obvious g e n e r a l i z a t i o n , i n which we have a
decomposition of
i
7 i n t o f i n i t e l y many m u t u a l l y d i s j o i n t s u b s e t s .
g e n e r a l i z a t i o n f a i l s f o r an a r b i t r a r y ( i n f i n i t e ) decomposition of
f o r example,
but
-,
g e n e r a t e d by t h e
C, with
->
Qo
rij E R, a
= Z(aij@l)rij,
i
t o b e t h e subgroup of
s h o r t e x a c t sequence
Remark.
R
is f u r n i s h e d w i t h t h e n a t u r a l R-module s t r u c t u r e .
(3.11)
I f we d e f i n e
P
i n h e r i t n a t u r a l R-module s t r u c t u r e s and, as such, a r e
and
where
Z
is a f i n i t e l y generated abelian
group, t h e converse i m p l i c a t i o n b e i n g t r i v i a l .
Then
31
@ Z/p
P
Theorem 3.12.
: G + K
@P P
P
The
ll
since,
(eP Z / P ) ~ i s a f i n i t e l y g e n e r a t e d Z -module f o r a l l primes q
q
is n o t a f i n i t e l y generated a b e l i a n group.
Let
Q: G
+
K be i n
N.
Then Q i s an isomorphism i f f
is an isomorphism f o r a l l p .
Localizationof nilpotent groups
32
Proof.
We assume 0 is an isomorphism for all p. Thus, by the
P
Fundamental Theorem, 4 is a p-isomorphism for all p . Since ker 0 is a
torsion group, and all primes are forbidden, ker @
=
{l}.
Now let y t K .
ynp
Then, for each p, we have x(~) t G, n prime to p, and
= 4x(,).
P
Since gcd(n ) = 1, we may find integers a almost all 0, such that
P
a
P'
Ca n = 1. Set x = llx
It i s then plain that y = ox.
P P
(PI '
Theorem 3.13. Let
Proof.
map
&: K
$,$I:
G
+
K
be i n N.
Then
+
= $I
iff 0
P
-
$I
P
for aZZ
p.
This is an immediate consequence of the injectivity of the
noted in the proof of Theorem 3 . 6 .
P'
The assertion of Theorem 3 . 1 3 , whose homotopy-theoretical counterpart
-+
nK
is of considerable significance, is that the morphisms in N are completely
determined by their localizations. It is fundamental to note, however, that
Thus, if we define the genus
this is not true of the objects in N.
G(G)
of
to be the set of isomorphism classes
a finitely generated nilpotent group G
of finitely generated nilpotent groups K satisfying K S G for every prime
P
P
p, it is not necessarily the case that K % G, when K belongs to the genus
of G.
The following specific examples, to some extent inspired by similar
examples in the homotopy category, were pointed out to us by Milnor:
For
let N
be the nilpotent group of nilpotency
r1 s
class 2 which is generated by four elements xl, x2, yl, y2 subject to the
mutually prime integers r,
8,
defining relations that all triple commutators are trivial and
[x1,x21r = [Y,,Y,I.
Nr/s
Nr'/s
Then N
iff either r
~ and
/ ~ Nrt,s
3
2' (mod
s)
[Xl.X2IS
-
1,
are in the same genus but
or rr' :21 (mod s ) .
Thus, for
example, NlIl2 $ N7/12 although these groups have isomorphic p-localizations
for every prime p.
p-isomorphisms
(In fact, for every prime p, it is easy to construct
N1/12 +. N7/12' N7/12
N1/12)'
Further properties of localization
33
Subsequently, f u r t h e r examples have been d i s c o v e r e d by M i s l i n [ 611.
I t should b e noted t h a t , i n d e f i n i n g t h e genus, we have r e s t r i c t e d
o u r s e l v e s t o f i n i t e l y generated groups.
s i z e d genus sets.
For example, i f
A
T h i s i s done i n o r d e r t o avoid over-
is t h e a d d i t i v e subgroup of
c o n s i s t i n g of elements e x p r e s s i b l e as f r a c t i o n s
A(n)
f r e e " by
with square-free
A 2 Z
f o r e v e r y prime p . More g e n e r a l l y ,
P
P
i s d e f i n e d i n t h e same way a s A e x c e p t t h a t w e r e p l a c e "square-
L, t h e n
denominator
if
k/k
Q
A'$
"nth-power-free",
Z but
we o b t a i n i n f i n i t e l y many m u t u a l l y nonisomorphic
a b e l i a n groups w i t h p - l o c a l i z a t i o n s
isomorphic t o
Z
f o r e v e r y prime p .
P
With o u r d e f i n i t i o n of genus, t h e genus of a f i n i t e l y g e n e r a t e d
a b e l i a n group
A
c o n s i s t s of ( t h e isomorphism c l a s s o f )
A
a l o n e . We s t a t e
t h i s a s a theorem, even though i t i s e l e m e n t a r y , s i n c e we w i l l wish t o r e f e r
t o it l a t e r .
Theorem 3.14.
abelian and
Let
B C G(A).
Proof.
be f i n i t e l y generated nilpotent groups with A
A, B
Then B
A.
The n i l p o t e n c y c l a s s of a n i l p o t e n t group is an i n v a r i a n t
of t h e genus ( s e e t h e Remark f o l l o w i n g Theorem 3 . 6 ) .
The s t r u c t u r e theorem
f o r f i n i t e l y generated a b e l i a n groups shows t h a t any f i n i t e l y g e n e r a t e d a b e l i a n
group i n t h e genus of
A
must c e r t a i n l y b e isomorphic t o
A.
More g e n e r a l l y , i t i s known t h a t t h e genus of a f i n i t e l y g e n e r a t e d
n i l p o t e n t group is a f i n i t e s e t .
[ 651.
T h i s f a c t f o l l o w s from r e s u l t s of P i c k e l
( P i c k e l ' s u s e of t h e term "genus" d i f f e r s from o u r s . )
The
homotopy-theoretical c o u n t e r p a r t of t h e f i n i t e n e s s of t h e genus i s as y e t
unsolved i n g e n e r a l , a l t h o u g h p a r t i a l r e s u l t s a r e known.
Localization of nilpotent groups
34
4. Actions of a nilpotent proup on an abelian group
Throughout this section, we denote by A
w: Q
an arbitrary group, and by
-f
Aut(A)
an abelian group, by Q
an action of Q
on A .
We adopt
x E Q , a € A.
the customary abbreviation x - a for w(x)(a),
Define the lower central w-series of A ,
...
by setting
1
= A,
rw(A)
rF(A)
=
Observe that if I Q
group generated by
{x-a-alx € Q , a E T,(A)),
i
i
3
1.
is the augmentation ideal of the integral group ring Z Q ,
then
ri+'(A)
i
in particular, each rw(A)
We say that Q
j sufficiently large.
we say that
operates nilpotently on A
If c
A
w on A.
A >-
if rA(A) = 111
is the largest integer for which
w has nilpotency class
Proposition 4.1. Let
(IQ)i.A;
is a submodule of A.
Proposition is easily proved.
Q-action
=
G
-
Then G E N i f f
c and write nil(,)
Q
= c.
f(A)
for
# Ill,
The following
be an extension giving r i s e t o the
Q € N
and
Q
operates nizpotently on
through w. Indeed,
max{ nil ( Q ) .nil (0)1.5 nil (GI 5 nil ( 9 ) + nil (w)
In the situation of Proposition 4.1, we may define
$(A)
of A by setting
.
a
subgroup
Actions of a nilpotent group on an abelian group
35
I t i s t h e n clear t h a t
A r e s u l t c l o s e l y r e l a t e d t o P r o p o s i t i o n 4 . 1 , w i t h almost i d e n t i c a l
proof, i s t h e following.
Let
Proposition 4.3.
A'
respect t o the Q-actions
w',
then
are n i l p o t e n t .
w'l
w',
w"
B.
a b e l i a n group
Notice t h a t
and
+
w',
w,
A"
w"
be an exact sequence of Q-modules w i t h
respectively.
Then
i s niZpotent i f
w
If the sequence is short exact and i f
w
is n i l p o t e n t ,
R
on t h e
are n i l p o t e n t , and
L e t now
homomorphism
* A
A(R,B)
b e t h e s e t of a c t i o n s of t h e group
The l o c a l i z a t i o n map
Aut(A)
+
e: A
(pw)'
%
e v i d e n t l y induces a
A u t ( % ) , which i n t u r n g i v e s rise t o a map
v r e s p e c t s submodules; t h u s , i f
w' = wIA',
-t
=
IJW~G
for
A'
is a submodule of
A
w F A(Q,A), then
By analogy w i t h Theorem 2 . 7 , we now prove
Theorem 4 . 5 .
Let
Proof.
w
<
A(Q,A).
Then
I n view of ( 4 . 4 ) i t s u f f i c e s t o prove t h i s f o r
s i n c e a n e a s y i n d u c t i o n then completes t h e argument.
Q-module map i f we f u r n i s h
+
w i t h t h e Q-action
pw
Now
i = 2
e: A + +
and p l a i n l y
i s a
e
induces
Localization of nilpotent groups
36
eo: r:(A)
+
2
rllw(Ap)
by restriction. By the Fundamental Theorem, we need to prove that
is P-local and that eo is a P-isomorphism.
To prove
of
2
I';w(Ap)
P-local, it is sufficient, by the commutativity
to show that any generator x*b - b, x E Q, b t
r,,(fh),
%,
may be
n
rtw(Ap).Now
divided by n, n t P', in
b
5
nb', b' C
Ap,
since
Ap
is
P-local, so
x.b
-b
Since e
=
x'(nb')
- nb'
=
n(x*b')
-
nb' = n(x*b'
-
b').
i s obviously P-injective, it remains to prove
eo
P-surjective. But the argument here i s very similar to that in the previous
paragraph. Namely, given any element of the form x'b
n t P', a C A with nb = e(a)
then there exist
n(x+b
- b)
=
e (x-a
It follows from the commutativity of ri(A),
Let AV(Q,A)
5 A(Q,A)
-
and
- b, x
t Q, b €
Ap,
so
a).
2
rpw(fh) that
eo is P-surjective.
consist of the nilpotent Q-actions on A.
Then the following is a direct consequence of Theorem 4.5.
Corollarv 4 . 6 .
?.IA~(Q,A) 5 AV(~,%).
nil(pw) 5 niliw)
.
If now A >-+
Q-action
w,
G
-
Indeed, if w € Av(Q,A),
then
Q i s any extension corresponding to the
we have a commutative diagram
(4.7)
where the Q-action induced by the lower extension is
pw.
By ( 4 . 2 ) , we have
Moreover, if G E N, then by Proposition 4.1 and Corollary 4.6, applied to
the lower extension in ( 4 . 7 ) , we conclude that G ' t N, f
i s a P-isomorphism.
Actions of a nilpotent group on an abelian group
Q E N
Now assume
and l e t
e: Q
--t
37
P-localize
Q,
Q.
Then
e
induces
and, o b v i o u s l y ,
Theorem 4.8.
Let
be P-ZocaZ,
A
e*: A,,(Q,,A)
Proof.
Let
w
Then
Q C N.
s A,,(Q,A).
E A,,(Q,A) and l e t
e x t e n s i o n corresponding t o
A
Q
>Gt--j>
By P r o p o s i t i o n 4 . 1 ,
w.
G C N
be t h e s p l i t
s o we may l o c a l i z e
to obtain
(4.9)
Let
Since
h w C A(Q,,A)
be t h e a c t i o n o b t a i n e d from t h e lower e x t e n s i o n i n ( 4 . 9 ) .
Gp F N, hw C Av(Qp,A)
and s i n c e t h e r i g h t hand s q u a r e i n (4.9) i s a
pullback,
e*Aw = w
so t h a t
satisfies
e*h = 1.
But i f we s t a r t w i t h
s p l i t extension
A >-
E->
Q,
C A (Qp,A)
f o r this action
and form t h e
i, t h e n
C N, by
P r o p o s i t i o n 4.1, and is P-local by C o r o l l a r y 2 . 5 , s o t h a t e s s e n t i a l l y t h e
same diagram ( 4 . 9 ) shows t h a t
h e * = 1. Thus
h
is i n v e r s e t o
e*.
Localization of nilpotent groups
38
Let
C o r o l l a r y 4.10.
w C AV(Q,A)
be any extension corresponding t o
% >+
is
G
P
Q,
->
with Q C N and Zet
W.
A
-G
LocaZizing yieZds an extension
and hence an action o f
on Ap.
Qp
Then t h i s action
X ~ W . independent of the original choice of extension.
Proof.
W
e f i r s t assume
A
P-local.
Then ( 4 . 9 ) , where t h e
e x t e n s i o n s a r e no longer assumed s p l i t , again shows t h a t t h e a c t i o n
Q,
on
A , given by t h e lower e x t e n s i o n , s a t i s f i e s
Now c o n s i d e r t h e g e n e r a l c a s e .
f
Q
is a P-isomorphism.
where
ef
Xuw.
Theorem 4.11.
Let
Proof.
A
W.
Thus
T =
of
Xu.
We r e v e r t t o (4.7) and r e c a l l t h a t
W
e t h u s may amalgamate (4.7) and (4.9) t o o b t a i n
is P-localizing.
extension i s
e*r =
T
Thus t h e a c t i o n of
be P-local.
i
Then r,(A)
Q,
on
%
given by t h e lower
i
= rXw(A).
Reverting t o (4.9), w e s e e t h a t
ri, ( ~ )=
i
rG(A),
i
rxw
(A)
=
ri
(A).
GP
W
e now claim
For
i = 2 , t h i s may b e proved by an argument s i m i l a r t o t h a t of Theorem 2.7
(apply t h e 5-term homology sequence t o t h e diagram
Actions of a nilpotent group on an abelian group
i , we u s e an e a s y i n d u c t i o n .
and,for general
On t h e o t h e r hand, Theorem 4.5 i m p l i e s t h a t
Thus
i
rG(A)
=
r
39
i
i s P-local.
Ti(A)
(A).
GP
We are now i n a p o s i t i o n t o g e n e r a l i z e Theorem 2 . 9 and C o r o l l a r y 1 . 1 4 .
Theorem 4.12.
w,
Q C N.
Let
A
be a n a b e l i a n group equipped w i t h a n i l p o t e n t
Then t h e n a t u r a l homomorphism
induced by P - l o c a l i z i n g b o t h
Proof.
n = 0
Q-action
A
and
Q, P-localizes.
2
Ho(Q;A> = A / r w ( A ) , Ho(Qp;Ap)
=
2
kp/rAuw(Ap).
Thus
*
t h e case
f o l l o w s from Theorem 4.5 and Theorem 4.11.
W
e suppose
w e a l s o have
n 2 1 and a r g u e by i n d u c t i o n on
n i l ( p w ) = 1, n i l ( h p w ) = 1
Now write
e x t e n s i o n of Theorem 2 . 9 .
and so
n i l w.
For
nil w
e* P - l o c a l i z e s by an e a s y
A2 = r L ( A ) , s o t h a t w e have a s h o r t
e x a c t sequence of Q-modules
A
(4.13)
Suppose
nil w
5 c, where
Q-actions of n i l p o t e n c y 5 c
we s e t
w2 = wIA2.
2
A
>-
A/A2.
c 2 2 , and t h a t t h e theoeem i s demonstrated f o r
-
1.
Then w e have
n i l ( w 2) 5 c
Moreover, t h e induced a c t i o n of
Q
on
W
e may t h u s a p p l y (4.13) and Theorem 4.5 t o o b t a i n a diagram
-
1 where
A/A2
is t r i v i a l .
= 1,
Localization of nilpotent groups
40
where w e know t h a t
Let
Theorem 4 . 1 4 .
Q-action
0,
ek2, e , 4 , e,5
e,l,
P-localize.
e*3 P-localizes.
Thus
be a P-local abeZian group equipped with a nilpotent
A
Then
Q E N.
e*: H " ( Q ~ ; A ) H"(Q;A), n 2 0.
Proof.
Ho(Q;A) = A" = {a E A1x.a = a , a l l
Referring t o ( 4 . 9 ) , we see t h a t
Ho(Qp;A) = A'".
A"
where
)
Z(
A
=
n
z ( G ) , "'A
=
A
n
z(G~),
denotes, a s u s u a l , t h e c e n t e r .
(or Theorem 2.8)
an i n c l u s i o n
e : G+Gp
that
A" 5 A'".
sends
But p l a i n l y "'A
We suppose
Z(G)
We know from Proposition 1.10
to
C Ae*'w
Z(Gp).
=
Thus ( 4 . 9 ) induces
Am. Thus A" = ."'A
n ? 1 and again argue by induction on
= 1, t h i s is p r e c i s e l y Corollary 1 . 1 4 .
nil(")
x C Q}; s i m i l a r l y ,
from Theorem 4.5 t h a t
A2
nil(w).
For
Referring t o ( 4 . 1 3 ) , we see
is P-local and hence a l s o
A/A2.
Thus, invoking
( 4 . 1 3 ) , we o b t a i n a diagram
.. . + H n-1 ( Q ~ ; A / A+ ~H)" ( Q ~ ; A+~H) " ( Q ~ ; A-)+ H " ( Q ~ ; A / A+ H~n+l
) ( Q ~ ; A ~ ).. .
+
1.*
g
n-1
. .. -+H
&
' e*
g/e*
/e*
Rt-1
(Q;A/A~) + H " ( Q ; A ~ )-+H"(Q;A) + H ( Q ; A / A ~ )-+H ( Q ; A ~ ) -+
J,
...
and t h e Five Lema completes t h e proof.
Remark,
that
For a r b i t r a r y
(Aw), 5
<".
Q , a r b i t r a r y (abelian)
If further
Q
A
and
w E A(Q,A), i t i s t r u e
is f i n i t e l y generated, then
(Am), =
<".
For t h e proofs, we r e f e r t o Hilton [351.
Consider f i n a l l y a n i l p o t e n t a c t i o n
+J
C Av(Q,A), where
Q
i s not
Actions of a nilpotent group o n an abelian group
41
necessarily nilpotent. We have, of course, an induced action of Q on the
homology (and cohomology) of A axd we wish to assert the nilpotency of this
action. First we state a general Proposition.
Proposition 4.15. Given
F: Ab
+
Ab.
w €
AV(Q,A)
as above and a haZf-exact functor
Then the induced action Fw of
Proof.
Q on FA i s nilpotent.
Fw is simply the composition Q
Thus the conclusion is clear for nil(w)
=
Aut (A) +Aut (FA).
1.
We proceed as usual by induction on nil(w).
Applying F to
(4.13) yields an exact sequence
by the half-exactness of F. The proof is then completed with the aid of
Proposition 4.3.
Corollary 4.16.
Note that, in fact, nil (Fo)
Let
w €
AV(Q,A)
Then the induced actions of
C
nil w.
and l e t B be an arbitrary abeZian group.
Q on A 8 B, Tor(A,B)
and
H,(K;A),
K any
group & t h t r i v i a l action on A, are nilpotent.
Theorem 4.17. If
w €
Av (Q,A) , then the induced action of Q on H,(A;C)
,c
triu-ial A-moduZe, is nilpotent. (A similar statement holds for H*(A;C) .)
Proof. In case nil(w) = 1, the result is clear. The inductive
step is carried out by applying the Lyndon-Hochschild-Serre spectral sequence
to (4.13).
We have
A/A2 acting trivially on H,(A2;C).
It follows from the inductive hypothesis
2
together with Corollary 4.16 that the induced action of w on Ers is nilpotent.
Localization of nilpotent groups
42
By t h e f i n i t e convergence of t h e s p e c t r a l sequence, we conclude, by r e p e a t e d
a p p l i c a t i o n of P r o p o s i t i o n 4.3, t h a t t h e induced a c t i o n of
also on
w
on E;s,
hence
is nilpotent.
H,(A;C),
F i n a l l y , we draw a t t e n t i o n t o t h e f o l l o w i n g u s e f u l addendum t o
Theorem 4.14.
Let
Theorem 4.18.
Q
be a finitely-generated n i l p o t e n t group operating
nilpotentZy on the abelian group
H " ( Q ; A ) ~ =H"(Q;+)
Proof.
Then
A.
= H"(Q~;$).
The second isomorphism was proved i n Theorem 4.14.
The
f i r s t i s proved by t h e u s u a l i n d u c t i o n on t h e n i l p o t e n c y of t h e a c t i o n ,
once i t is proved f o r t h e c a s e of t r i v i a l Q-action.
from t h e n e x t s e c t i o n , t h e f a c t t h a t t h e homology of
I n t h a t c a s e , we t a k e ,
Q
is finitely-generated
i n each dimension ( P r o p o s i t i o n 5 . 4 ) , when t h e r e s u l t f o l l o w s from t h e
following elementary p r o p o s i t i o n .
P r o p o s i t i o n 4.19.
Let
A, B
be abelian groups with B finitely-generated.
Then
Hom(B,AIp = Hom(B,$),
Ext(B,A)p = Ext(B,Pp),
P r o p o s i t i o n 4.19 h a s obvious i m p l i c a t i o n s f o r t h e cohomology groups
of t o p o l o g i c a l spaces.
Indeed, i t e n a b l e s us (once we have e s t a b l i s h e d an
a p p r o p r i a t e l o c a l i z a t i o n theory) t o i n f e r t h e analog of Theorem 4.18 when
is r e p l a c e d by a t o p o l o g i c a l space
generated (and
A
X
whose homology groups a r e f i n i t e l y -
i s merely an a b e l i a n c o e f f i c i e n t group).
Such an analog
i s i m p l i c i t l y invoked, f o r example, i n t h e p r o o f s of Theorems 111.1.7 and
111.1.14.
Q
Generalized Serre classes of groups
5.
43
G e n e r a l i z e d S e r r e c l a s s e s of groups
The n o t i o n of " c l a s s of a b e l i a n groups" goes back t o t h e fundamental
paper of S e r r e [73].
For t h e t o p o l o g i c a l a p p l i c a t i o n s we have i n mind, i t is
n e c e s s a r y t o c o n s i d e r n o n a b e l i a n groups, so w e f i n d i t c o n v e n i e n t t o extend
S e r r e ' s theory accordingly.
D e f i n i t i o n 5.1.
We b e g i n w i t h t h e d e f i n i t i o n .
A generaZized Serre class i s a c o l l e c t i o n
C
of groups
satisfying:
(i)
That i s , i f
c l a s s i n t h e o r d i n a r y sense.
e x a c t sequence of a b e l i a n groups, t h e n
are a b e l i a n groups, t h e n
then
H.(A)
GI,
G" € C
iff
A ' >-A
A ' , A" C C
A @ B , Tor(A,B) C C ; i f
€ C , i > 0 ; and
(ii)
-
The s u b c o l l e c t i o n of a b e l i a n groups i n
G ' >-
A C C
G
forms a S e r r e
A"
i s a short
A € C; i f
iff
0 C C.
Given a c e n t r a l e x t e n s i o n
C
-
A, B C
C
i s an a b e l i a n group,
of g r o u p s , t h e n
G"
G € C.
The f o l l o w i n g are examples of g e n e r a l i z e d S e r r e classes:
(a)
The class
G
of a l l groups.
(b)
The c l a s s
N , resp.
FN, of a l l n i l p o t e n t , r e s p . f i n i t e l y
g e n e r a t e d n i l p o t e n t groups.
(c)
The class
G(p) , r e s p .
FG(p)
, of
a l l p-groups,
resp. f i n i t e
p-groups.
(Observe t h a t
FG(p)
W
e remark t h a t i f
v i r t u e of D e f i n i t i o n 5 . 1 ( i i ) ,
C
i s , i n f a c t , a s u b c l a s s of
FN.)
i s any g e n e r a l i z e d S e r r e c l a s s , t h e n , by
t h e n i l p o t e n t groups i n
C
form a s u b c l a s s .
Another consequence of D e f i n i t i o n 5 . l ( i i ) i s t h a t an o r d i n a r y S e r r e c l a s s
w i l l almost n e v e r be a g e n e r a l i z e d S e r r e c l a s s .
The f o l l o w i n g p r o p o s i t i o n i s e s s e n t i a l l y proved by S t a m b a c h [78 1 ,
i n t h e c o n t e x t of t e n s o r i a l cZasses of groups.
Localization of nilpotent groups
44
G
iff Gab
C
€
C be generalized Serre class and l e t
Let
Proposition 5 . 2 .
€
N. Then
C C.
Proof. Assume G
T 3
G
€ C.
Setting
r i = r i (G)
and letting c = nil(G),
have central extensions
from which we conclude, by Definition 5,1(ii),
-
Conversely, if we assume Gab
ri/ri+'
€
C, 1
P
i i c.
€
that Gab = G / r
2
€
C.
C, then observe first that
Indeed, the function
i factors
G X . . . X G - ~
I
which sends an i-tuple of elements of G into the corresponding i-fold
commutator clearly induces a surjective homomorphism
G~~ 8
.. . 8 G~~ ->r
i/r i+l,
It follows from De'finition 5 . 1 ( i ) that Gab 0
belongs to C .
.. . 63 Gab
and hence
ri/I?+'
But then, by appealing to \the extensions ( 5 . 3 ) (in reverse
order), we conclude that G € C.
We have also the following variant of Proposition 5 . 2 .
Proposition 5.4. Let
iff
HI(G)
€ C, i >
Proof.
C be a generalized Serre class and l e t
G
€ N.
0.
We assume G € C
and argue by Induction on c = nll(G).
If c = 1, then the conclusion is Incorporated into Definition S . l ( i ) .
c
? 2,
Then G
If
we apply the Lyndon-Hochschild-Serre spectral sequence to the central
€
C
extension Tc
Generalized Serre classes of groups
-
G ->
4s
G/rC and obtain the desired result.
The converse is a special case of Proposition 5.2 in view of
H (G)
1
%
G
ab'
If
Corollarv 5.5.
then H*(G)
%
G, K are f i n i t e l y generated nilpotent groups i n the same genus,
H*(K).
Proof.
We apply Theorems 1.12 and 3.14, since, by Proposition 5.4,
we know that the homology groups of G, K
Theorem 5.6. Let
that
the relation
are finitely generated.
C be a generalized Serre class, l e t G
G € N and operates nilpotently on the d e l i a n g r m p
€ C
A
and suppose further
through
W.
We
have :
(i) i f A
C , then Hi(G;A) € C, i
€
?
0;
(ii) i f A f C , then Ho(G;A) f C .
Proof.
(i) Referring to (4.13) for the definition of A2, we have
Ho(G;A) = A/A2, hence Ho(G;A) C C.
(not nil(G)!),
5.4.
For i
z
0, we argue by induction on nil(w)
the assertion for nil(w) = 1 being a consequence of Proposition
The short exact sequence (4.13) gives rise to the exact sequence
Hi(G;A2)
- Hi(G;A)
Hi(G;A/A2),
from which the conclusion evidently follows.
(ii)
We again argue by induction on nil(w),
clear since then, Ho(G;A)
=
A.
the case nil(w) = 1 being
There are two possibilities:
Case 1: A/A2 # C: As the induced action of G on A/A2 is trivial, we
have Ho(G;A/A2) = A/A2, thus Ho(G;A/A2) # C. But we have a surjection
Ho(G;A)
+
Ho(G;A/A 2 1,
Case 2:
so
we conclude Ho(G;A) f C.
A/A2 € C:
In this case, A2 f C, otherwise A € C. Thus,
46
Localization of nilpotent groups
by the inductive hypothesis and (i), we infer Ho(G;A2)
C, H1(G;A/A2)
f
c.
From the exact sequence
H1(G;A/A2)
----f
Ho(G;A2)
+
Ho(G;A),
6 C.
we thus conclude Ho(G;A)
We conclude by using Proposition 5.2 to prove a basic theorem on
generaffzed Serre classes.
Theorem 5 . 7 .
Let
be a generalized Serre class and l e t
C
be an arbitrary (not necessarily central) extension i n N.
G
G'
G ->
G"
Then G ' , GI' € C i f f
€ C.
Proof.
Gab € C .
Assume G', G"
€
C. By Proposition 5.2, it suffices
to show
But the given short exact sequence induces the short exact sequence
of abelian groups
Moreover, G'/G' n r 2
belongs to C since it is a quotient group of Gib which
belongs to C by Proposition 5.2.
follows that Gab
As
Gib
€ C,
again by Proposition 5.2, it
€ C.
Now assume G
€
C. Then, as previously noted, Gab maps surjectively
and two applications of Proposition 5.2 allow us to conclude that G" € C .
tob''G
To show G' € C, we argue by induction on c = nil(G), the result being obvious for
c = 1. Since nil(G/rC) = c
Girc/rc c C.
But
-
1, we have, by the inductive hypothesis, that
~'r~/r'G G'/GlnrC and
abelian, it is clear that G'
n c'I
with the two extreme groups in C .
€
rc E
c by (5.3).
Since
rc
C. We thus have a central extension
It follows from Definition 5.1(ii) that
G' € C , thereby completing the induction.
is
Chapter I1
Localization of Homotopy Types
Introduction
In this chapter we apply localization methods to homotopy theory.
We use the definitions of local groups and localization given in Chapter I,
in order to introduce the corresponding notions into homotopy theory; and
we prove the basic theorems that relate to localization in homotopy theory.
These theorems find many applications in homotopy theory, but we will reserve
the applications to Chapter 111.
Our definition of a P-local (pointed) space is simply that its
homotopy groups should be P-local groups. This definition could be made
quite generally for an arbitrary pointed space. However we are concerned
to obtain a localization theory and also to obtain useful criteria for
establishing when a given map of spaces does in fact P-localize.
It is
therefore necessary for us to work in a restricted category of (pointed)
topological spaces. It is also necessary for us to work in a homotopy category
(that is, in a category in which the morphisms are homotopy classes of
continuous maps), since our procedures for establishing the existence of a
localization theory will all operate up to homotopy. A more general treatment,
valid in the semisimplicial category, has been given by Bousfield-Kan u 4 ] .
We will always suppose that our spaces have the homotopy type of CW-complexes.
In Section 1, we present a localization theory in the homotopy
category H1
of 1-connected CW-complexes. We establish two fundamental
theorems in H1, namely that every object of the category does admit a
P-localization, and that we can detect the P-localizing map
f: X
+.
Y either
through the induced homotopy homomorphisms, which should also P-localize,
or through the induced homology homomorphisms, which should also P-localize.
In the course of establishing that there is a localization theory in H1,
Localization of homotopy types
48
we a c t u a l l y c o n s t r u c t t h e l o c a l i z a t i o n of a given CW-complex
i m i t a t i n g t h e c e l l u l a r c o n s t r u c t i o n of
t h a t of a ZocaZ c e l l .
X
by
X, r e p l a c i n g t h e i d e a of a c e l l by
The f a c t t h a t t h e l o c a l i z a t i o n can b e d e t e c t e d e i t h e r
through homotopy o r through homology h a s t h e immediate consequence t h a t we
may l o c a l i z e f i b r e and c o f i b r e sequences i n
H1,
I n S e c t i o n 2 , we d e s c r i b e a broader homotopy c a t e g o r y i n which we
w i l l a l s o b e a b l e t o e s t a b l i s h a s a t i s f a c t o r y l o c a l i z a t i o n theory.
It t u r n s
o u t t h a t we would wish t o e n l a r g e t h e c a t e g o r y t o which we apply l o c a l i z a t i o n
methods from our o r i g i n a l c a t e g o r y
H1.
confined t o
o b j e c t s of
H1,
H1,
even i f our main i n t e r e s t were
For, i n o r d e r t o prove theorems about l o c a l i z a t i o n of
i t is v e r y u s e f u l t o employ function-space methods, and
t h e function-space c o n s t r u c t i o n t a k e s u s o u t s i d e t h e c a t e g o r y .
i t is t r u e t h a t i f
and i f
into
W
X
X
However,
is a niZpotent s p a c e , i n a s e n s e defined i n S e c t i o n 2 ,
is f i n i t e , t h e n t h e f u n c t i o n space Xw
is a g a i n n i l p o t e n t .
f a c t s about t h e category
S e c t i o n 2 concerns i t s e l f w i t h some b a s i c
NH of n i l p o t e n t s p a c e s , and may be regarded i n
p a r t as propaganda f o r t h e u s e of t h i s c a t e g o r y i n homotopy theory.
i t has a l r e a d y been shown by Dror [23] t h a t
homotopy t h e o r y .
W
of pointed maps of
NH
Indeed,
is a s u i t a b l e c a t e g o r y f o r
Roughly speaking, one may s a y t h a t most of t h e t e c h n i q u e s
of homotopy t h e o r y which have been developed s i n c e t h e p u b l i c a t i o n of S e r r e ' s
t h e s i s can a l l be c a r r i e d o u t i n t h e c a t e g o r y
NH
techniques were of course o r i g i n a l l y formulated i n
although many of t h o s e
H1,
The b a s i c theorem
proved i n S e c t i o n 2 is t h a t a space is n i l p o t e n t i f and only i f i t s Postnikov
tower admits a p r i n c i p a l refinement.
t h e category
Mi
It is t h i s theorem which e x p l a i n s why
is s u i t a b l e f o r homotopy theory; f o r t h e given refinement
of t h e Postnikov tower may be used i n p l a c e of t h e Postnikov tower i n t h o s e
arguments i n which t h e c r u c i a l f a c t which is r e q u i r e d is t h a t t h e f i b r a t i o n s
which appear i n t h e tower should b e induced o r p r i n c i p a l .
Introduction
49
However, it should be pointed out that the category NH has eertain
defects over the category H1.
One of the defects is that it is not closed
under the mapping cone operation. This defect has a serious consequence in
Section 3 .
We also describe in Section 2 how to relativize the notion of a
nilpotent space to obtain that of a nilpotent map.
In Section 3 we generalize the theorems of Section 1 from the
category H1
to the category NH. Formally, we get the corresponding
formulations of the two fundamental theorems of Section 1. However there is
an important difference in the way in which we construct the localization of
an object. For, whereas in the category H1 we are able to proceed cellularly,
since the mapping cone construction respects the category H1, we cannot in
the nilpotent case proceed cellularly, since the mapping cone construction would
take us outside the category. It is therefore necessary for us to proceed
homotopically rather than cellularly in constructing the localization. In
this way, of course, we lose much of the conceptual simplicity of the
construction in Section 1.
Section 4 is a brief technical section in which we introduce the
idea of a quasifinite complex in Mi.
relative to the category H1.
Here again we see a certain disadvantage
For if X
is a 1-connected CW-complex whose
homology groups are all finitely generated, and vanish above a certain
dimension, then X itself has the homotopy type of a finite complex. If we
discard the condition of simple-connectivity, we can no longer assert this
conclusion. Indeed, we have the obstruction theory of Wall which enables
US
to discuss the question whether a CW-complex X whose homology looks like
that of a finite complex in fact has the homotopy type of a finite complex.
Thus we are led to introduce the concept of a quasifinite CW-complex,
meaning a nilpotent CW-complex X such that the homology of X is finitely
Localization of homotopy types
50
generated i n each dimension and v a n i s h e s above a given dimension.
We prove
t h a t such a q u a s i f i n i t e complex always h a s t h e homology type of a f i n i t e
complex.
I n S e c t i o n 5 we prove t h e fundamental p u l l b a c k
v a r i o u s v a r i a n t s as consequences of t h a t theorem.
theorem and
Here we l e a n v e r y h e a v i l y
on t h e r e s u l t s of Chapter I .
The fundamental p u l l b a c k theorem a s s e r t s t h a t t h e p o i n t e d set
i s t h e p u l l b a c k of t h e pointed sets
[W,Xl
set
1, provided t h a t
[W,X
W
{[W,Xpll
over t h e p o i n t e d
i s a f i n i t e connected CW-complex and
a n i l p o t e n t complex of f i n i t e type.
X
is
T h i s a s s e r t i o n f a l l s i n t o two p a r t s .
The f i r s t p a r t s t a t e s t h a t given two maps
f,g: W
-+
X
such t h a t
= e g: W + X f o r a l l primes p, t h e n f = g . T h i s p a r t of t h e a s s e r t i o n
P
P
P’
does not r e q u i r e t h a t X b e of f i n i t e t y p e . The second a s s e r t i o n s t a t e s
e f
that i f
f(p): W
-+
c l a s s of t h e map
X
P
a r e maps, f o r a l l primes
r f(p): W
P
-+
Xo
p , such t h a t t h e homotopy
i s independent of
i s t h e r a t i o n a l i z a t i o n map, t h e n t h e r e exists a map
e f
P
2
f(p)
f o r a l l primes
t h e condition t h a t
W
p.
p , where
rp: Xp
f: W
such t h a t
-+
X
-+
Xo
We show by an example t h a t we cannot omit
should be f i n i t e .
However, provided t h a t
W
is
n i l p o t e n t , we may i n f a c t weaken t h e hypotheses of t h e p u l l b a c k theorem by
simply r e q u i r i n g t h a t
W
be q u a s i f i n i t e .
I n S e c t i o n 6 we make a p r e l i m i n a r y s t u d y of t h e l o c a l i z a t i o n of
H-spaces.
Our main r e s u l t i n t h i s s e c t i o n i s a g e n e r a l i z a t i o n of t h e p a r t
of t h e fundamental theorem of Chapter I which t e l l s u s how t o d e t e c t t h e
P - l o c a l i z a t i o n of a n i l p o t e n t group i n terms of t h e P - b i j e c t i v i t y of t h e
l o c a l i z i n g homomorphism.
I n S e c t i o n 7 w e formulate t h e fundamental mixing technique of
Zabrodsky w i t h i n t h e c o n t e x t of t h e l o c a l i z a t i o n of n i l p o t e n t s p a c e s .
The
Introduction
51
p a r t i c u l a r r e s u l t which w e emphasize is t h a t , given n i l p o t e n t spaces
X, Y
with equivalent r a t i o n a l i z a t i o n s , and given a p a r t i t i o n of t h e primes
Il
= P
u
Q , then t h e r e e x i s t s a n i l p o t e n t space
2
such t h a t
2
P
= Xp
and
ZQ = YQ.
We make very considerable a p p l i c a t i o n of t h e r e s u l t s of t h e l a s t two
s e c t i o n s i n Chapter 111.
Indeed, we a r e r a t h e r l i t t l e concerned t o g i v e
e x p l i c i t examples and a p p l i c a t i o n s i n t h i s Chapter i n view of t h e f a c t t h a t
Chapter 111 is e n t i r e l y concerned with applying t h e theory of Chapter 11.
52
Localization of homotopy types
1.
Localization of 1-connected CW-complexes.
H1 of 1-connected
We work i n t h e pointed homatopy category
CW-complexes.
X
X C H1,
If
and i f
P
is a family of primes, we say t h a t
is P-zocal i f the homotopy groups of
W
e say t h a t
f: X + Y
P-localizes
H1
in
X
a r e a l l P-local a b e l i a n groups.
X
if
Y
i s P-local and*
f*: [Y,Z] z [X,Zl
f o r a l l P-local
2 C H1.
Of course t h i s u n i v e r s a l property of
c h a r a c t e r i z e s i t up t o canonical equivalence:
both P-localize
H1
with
in
H1.
X
hfl = f 2 .
if
fi: X
-+
then t h e r e e x i s t s a unique equivalence
Yi,
f
i = 1, 2 ,
h : Y1
PI
Y2
in
W e w i l l prove t h e fallowing two fundamental theorems
The f i r s t a t t e s t s t h e e x i s t e n c e of a l o c a l i z a t i o n theory i n
H1
and the second a s s e r t s t h a t we may d e t e c t t h e l o c a l i z a t i o n by looking a t
induced homotopy homomorphismor induced homology homomorphisms.
Theorem 1A.
( F i r s t fundamental theorem i n H1.)
Every
X
We T r i t e
Theorem 1B.
Let
admits a P-localization.
i n HI
e: X
+
Xp
f o r a f i x e d choice of P-localization
of
X.
(Second fundamentaZ theorem i n H1.l
f: X * Y
( i ) f P-localizes
(ii) nnf: n X
(iii) Hn f : HnX
Then the following statements are equivalent:
i n H1.
X;
nnY
P-localizes f o r a l l
n 3 1;
-+ H Y
P-localizes f o r a l l
n 2 1.
-t
We w i l l prove Theorems l A , 18 simultaneously.
*We w r i t e , a s usual, [Y,Z]
of maps from Y t o 2.
f o r H1(Y,Z),
W
e r e c a l l from
t h e s e t of pointed homotopy classes
Localization of I -connectedCW-complexes
Proposition 1.1.9 that a homomorphism
B
if and only if
is P-local and
$:
A
+
B
53
of abelian groups P-localizes
is a P-isomorphism; this latter condition
@
+
means that the kernel and cokernel of
belong to the Serre class C of
abelian torsion groups with torsion prime to P.
Thus to prove that (ii)
(iii)
in Theorem 1 B above it suffices to prove the following two propositions.
Proposition 1.1. Let Y C H1.
only if H Y
is P-locat f o r a l l n
Proposition 1.2.
all
Then
f: X
Let
-+
n E 1 if and only if
TI
Y is P-local f o r a l l
Then
nn(f)
is a €'-isomorphism f o r aZZ
for all n E 1 if and only if Hn(Y;Z/p) = 0
disjoint from P.
n 2 1.
P
is a P-local abelian group, so are the homology groups
of the Eilenberg-MacLane space K(A,m).
K(A,m-l)
-+
-+
K(A,m),
with E
if H (A,m-1; Z/p) = 0
n
Now let
...
of Y.
is a P-local abelian
1. It now follows, by induction
Hn(A,m)
E
is P-loca?
for all n 2 1 and all primes
Now, by Proposition 1.1.8, if A
group, so are the homology groups H A, n
on m, that if A
1 if and
is a P-isomorphism for
Proof of Proposition 1.1. We first observe that HnY
p
?
2 1.
Y in H1,
Hn(f)
n
For we have a fibration
contractible, from which we deduce that,
for all n E 1, then Hn(A,m; Z/p) = 0
-+
Ym
-+
Ymm1-+
...
Thus there is a fibration K(nmY,m)
Thus, if we assume that
nnY
for all n 1 1.
Y2 be the Postnikov decomposition
* Ym
+
Ym-l, and Y2 = K(n 2Y , 2 ) .
is P-local for all n P 1, we may assume inductively
that the homology groups of Ym-l
are P-local and we infer (again using homology
with coefficients in Z / p , with p disjoint from P) that the homology groups
of Ym
are P-local.
Since Y
-+
Y,
is m-connected, it follows that HnY
P-local f o r all n 1 1.
To obtain the opposite implication, we construct the 'dual'
Cartan-Whitehead decomposition
is
Localization of homotopy types
54
There i s then a f i b r a t i o n
if we assume t h a t
K(amY,m-l)
i s (m-1)-connected,
Y(m)
*
Y(m+l)
n
i s P-local f o r a l l
HnY
t h a t t h e homology groups of
Y(m)
+
T
m
and
Y(2) = Y .
Since
Y Z H Y(m)
m
Y(m) = n Y
m
m
P)
t h a t the homology groups of
s t e p i s complete and
T
Y 2 H Y(n)
n
and i s P-local.
to
C
a r e P-local,
p
disjoint
so t h a t the i n d u c t i v e
is P-local.
Proof of Proposition 1 . 2 .
mod C , where
Y(m+l)
and
a
Thus we i n f e r (again using homology w i t h c o e f f i c i e n t s i n Z / p , w i t h
from
Thus,
2 1, w e may assume i n d u c t i v e l y
a r e P-local.
i t follows t h a t
Y(m)
Since a P-isomorphism
is an isomorphism
i s t h e c l a s s of a b e l i a n t o r s i o n groups with t o r s i o n prime
P, Proposition 1 . 2 is merely a s p e c i a l case of t h e c l a s s i c a l Serre theorem.
We have thus proved t h a t ( i i )
t h a t (ii) * (i).
counterimage of
f.
(iii) in Theorem 1 B .
We now prove
The o b s t r u c t i o n s t o t h e e x i s t e n c e and uniqueness of a
g: X
-+
under
Z
Now, given ( i i ) (or ( i i i ) ) ,H,f
of t h e map
Q
f*: [Y,Z]
C C , where
-+
[X,Z]
H,f
l i e in
H*(f;n,Z).
r e f e r s t o t h e homology groups
Thus (i) follows from the u n i v e r s a l c o e f f i c i e n t theorem f o r
cohomology and Proposition 1.1.8(v).
We now prove Theorem 1 A .
of
f: X
-c
Y
in
H1
More s p e c i f i c a l l y , we prove t h e e x i s t e n c e
s a t i s f y i n g (iii). Since we know t h a t (iii) 3 (i), t h i s
w i l l prove Theorem 1 A .
Our argument is f a c i l i t a t e d by t h e following key
observation.
Proposition 1.3.
Let
we have constructed
U be a f u l l subcategory of
f: X
-+
Y
automatically y i e l d s a functor
satisfying l i i i ) .
L: U
transfornation from the embedding U
-+
H1,
5 H1
Then the assignment X * Y
f o r which
t o L.
f o r whose objects X
HI
f
provides a natural
Localization of 1-connectedCW-complexes
Proof of P r o p o s i t i o n 1 . 3 .
g: X
Let
-+
in
X'
U.
55
W
e t h u s have a
diagram
Y
in
H1
with
f, f'
Y'
satisfying ( i i i ) .
Since
f
s a t i s f i e s ( i ) and
P-local by P r o p o s i t i o n 1.1, we o b t a i n a unique ( i n
H1)
Y'
is
h 6 [Y,Y'] making
t h e diagram
I t i s now p l a i n t h a t t h e assignment
commutative.
desired functor
X
rf
Y, g* h
yields the
L.
We e x p l o i t P r o p o s i t i o n 1 . 3 t o prove, by i n d u c t i o n on
may l o c a l i z e a l l n-dimensional
CW-complexes i n
H1.
n, t h a t w e
.If n = 2, t h e n such a
complex i s merely (up t o homotopy e q u i v a l e n c e ) a wedge of 2-spheres
x = v s2 ,
a
a
where
runs
through some index s e t , and w e d e f i n e
Y = VM( ZP,2),
a
where
map
M(A,2)
f: X
+
fo: Xo + Y o
dim X = n
Y
i s t h e Moore s p a c e having
satisfying ( i i i ) .
satisfying ( i i i ) i f
+ 1, X
€
H1.
H M = A.
2
There i s t h e n an e v i d e n t
Suppose now t h a t w e have c o n s t r u c t e d
dim X
5 n , where
n 1 2 , and l e t
Then w e have a c o f i b r a t i o n
vsn R, xn i
.
x
By t h e i n d u c t i v e h y p o t h e s i s and P r o p o s i t i o n 1 . 3 , we may embed (1.5)
diagram
i n the
Localization of homotopy types
56
where fo, fl satisfy (iii) and the square in (1.6) homotopy-commutes.
embeds Yo
Thus if j
(1.6) by
f: X
+
Y
in the mapping cone Y of h, then we may complete
to a homotopy-commutative diagram and it is then easy
to prove (using the exactness of the localization of abelian groups) that
f: X
+.
Y
also satisfies (iii).
(iii) if X
Thus we may construct f: X
+
Y satisfying
is (n+l)-dimensional, and the inductive step is complete.
It remains to construct f: X
-+
Y satisfying (iii) if X
is
infinite-dimensional. We have the inclusions
x2
cx3 2 . .
and may therefore construct
where f", fn+l
satisfy (iii).
We may even arrange that (1.7)
is strictly
If we define Y = UY(n), with the weak topology,
n
and the maps fn combine to yield a map f: X -+ Y which again
commutative for each n.
then Y t H1
obviously satisfies (iii). Thus we have proved Theorem lA in the strong form
that, to each X
in H1,
there exists f : X
-+
Y
in H1 satisfying (iii).
Finally, we complete the proof of Theorem 1B by showing that
(i) = (iii).
Given f: X
-+
Y which P-localizes X, let fo: X
constructed t o satisfy (iii).
Then fo: X
+
Y
which one immediately deduces the existence of
-+
Y
be
also satisfies (i), from
a
homotopy equivalence
Localization of 1-connectedCW-complexes
u: Yo
+.
Y with uf
= f.
It immediately follows that f also satisfies
(iii).
Thus the proofs of Theorems l A , 1B are complete.
We note that our proof of the first fundamental theorem does much
more than establish the existence of a localization theory in H1;
it provides
us with a combinatorial recipe for constructing the localization of a given
CW-complex. The Moore spaces S;
=
may be called P-ZocaZ n-spheres,
M(Zp,n)
and a cone on a P-local n-sphere may be called a P-ZocaZ (n+l)-ceZz.
Then,
given a cellular decomposition of X, we may--as shown in the proof of Theorem
1A--construct a P-ZocaZ-ceZZuZar decomposition of Xp by 'imitation'; that
is, whenever, in building up X, we attach an n-cell to Y
f : Sn-'
of a map
to Yp
+.
$, we attach a P-local n-cell
Y, then, in building up
by means of the localized map
sn-l
fp:
(say) by means
P
-+
Yp.
We illustrate this
procedure by means of an example.
Example 1.8. Let Sm
-+
E
-+
Sn
be an Sm-bundle over
S",
n
z
m, and let
m
a C T ~ ~ - ~) ( be
S the characteristic (homotopy) class of the bundle.
it is well-known that E
Then
admits the cellular decomposition
E
=
sm ua
en
u
emh,
where we will not trouble to specify the attaching map f o r the top-dimensional
cell. If m 1 2 we may now localize E
Ep = S;
Ua
to obtain
nntl-n
ep U ep
P
.
Let us consider,in particular, the Stiefel manifold Vn+l , 2
vectors to
S",
and let us assume that n
over Sn with fibre Sn-',
a = 2 6 T~-~(S~-').
so
We obtain
of unit tangent
is even. Then 'n+1,2
that we may take E = V
n+1,2'
m = n
fibres
-
1, and
Localization of homotopy types
58
Suppose now t h a t
P
is t h e f a m i l y of a l l odd primes.
Sn-'
s o t h a t i t is e a s y t o s e e t h a t
n
ep
2p
Then
is i n v e r t i b l e ,
h a s t h e homotopy type o f a p o i n t .
It t h u s f o l l o w s from (1.9) t h a t
if
E = V n+1,2, n
even, and
f u r t h e r follows t h a t i f
[E,Y]
%
P
is t h e f a m i l y of odd primes.
It t h e r e f o r e
is a P - l o c a l space t h e n
Y
[I? ,Y] '2 [Sp
211-1,Y] '2 [ S 2n-1 , y ] =
P
Here t h e most s t r i k i n g f a c t is t h a t t h e s e t
[E,Y]
2n-1
y.
h a s acquired a v e r y n a t u r a l
a b e l i a n group s t r u c t u r e and h a s simply been i d e n t i f i e d w i t h a c e r t a i n homotopy
group of
Y.
We have t h e f o l l o w i n g immediate c o r o l l a r i e s of t h e second fundamental
theorem.
C o r o l l a r y 1.10.
Let
F
i s a f i b r e sequence i n
Proof.
-t
E + B
be a f i b r e sequence i n H1.
Then Fp
+
H1'
Of c o u r s e we a r e o n l y making t h e a s s e r t i o n up t o homotopy,
so t h a t our c l a i m amounts t o s a y i n g t h a t
e x a c t sequence of homotopy groups.
Fp
-+
Ep
-t
Bp
induces t h e u s u a l
However t h i s f o l l o w s immediately from
P r o p o s i t i o n 1 . 1 . 7 and t h e e q u i v a l e n c e of (i) and (ii) i n Theorem 1B.
S i m i l a r l y , r e p l a c i n g (ii) of Theorem 1B by ( i i i ) of Theorem lB,
we o b t a i n
C o r o l l a r y 1.11. Let
5 + Yp
-+
Cp
Ep
X
+
Y
-+ C
be a cofibre sequence i n H1.
i s a cofibre sequence.
Then
+
B
P
Localization of I -connected CW-complexes
59
We now introduce an important definition.
Definition 1.12.
A map
f: X
is called a P-equivalence if
in H1
Y
--f
fP
is a (homotopy) equivalence.
Theorem 1.13. Let
primes.
f: X
Y in
+
H ,and l e t
1
P
be a non-empty f a m i l y of
Then the following statements are equivalent:
(il
(iil
f i s a ?-equivalence;
f i s a p-equivalence f o r a l l
Q C
(iii) TInf i s a P-isomorphism f o r a l l n
( i v ) H f i s a P-isomorphism f o r a l l n
Proof.
By Whitehead's Theorem f
P;
2 1;
2 1.
is a P-equivalence iff n f
n P
is an isomorphism for all n 1 1. By Theorem 1B it follows that
Thus the equivalence (i)
(i)
Q
(iv).
(iii) follows from Theorem 1.3.1.
Theorem 1.3.12 ensures that
(nnf)p
TI
f = (nnf)p.
n P
Similarly
is an isomorphism iff
(nnf)
is an isomorphism for all p C P. Thus the equivalence of (i)
P
and (ii) readily follows.
A further refinement is possible in the case in which
are of f i n i t e t y p e , that is,
n
?
1. Notice that, in
IT
X and nnY
n
X and Y
are finitely generated for all
this is equivalent to asking that H X and
H1,
HnY be finitely generated.
Theorem 1.14. Let
f: X
-+
Y i n H1 with X , Y
P be a non-empty f a m i l y of primes.
f,:
Hn(X; Z/p) zz Hn(Y; Z/p) f o r
Proof. Let
Z/p
f: X
+
Y
Then
of f i n i t e type, and l e t
f i s a P-equivalence i f f
p € P.
be a P-equivalence. Then, since, for p C P,
is P-local, it follows from Theorem 1B and PropOsftiOn I.1.8(ii)
we have a comutative diagram
that
Localization of homotopy types
60
Thus
is a n isomorphism; n o t i c e t h a t t h e i m p l i c a t i o n we have proved does
f,
not require t h a t
X, Y
It is i n t h e o p p o s i t e i m p l i c a t i o n
b e of f i n i t e type.
t h a t t h i s condition plays a decisive r o l e .
induces
Hn(X; Z f p )
f,:
Hn(Y;
We w i l l prove t h a t
type,
H f
n
Consider t h e diagram, f o r each
ZIP),
Thus w e suppose t h a t
n 2 1, p 6 P , where
i s a p-isomorphism,
f: X + Y
are of f i n i t e
X, Y
Hnf: HnX
H Y , n 11.
-+
n 5 1,
$
(1.15)
f
*n
t h e v e r t i c a l homomorphisms being induced by
f.
i t f o l l o w s t h a t we must prove t h a t each
is b i j e c t i v e and each :f
It f o l l o w s immediately from (1.15) t h a t e a c h
surjective.
and each
f i
surjective.
It a l s o f o l l o w s from (1.15)
Suppose, i n d u c t i v e l y , t h a t we have shown
Then
f:
By P r o p o s i t i o n 1 . 1 . 8 ( i i ) ,
Hrf: HrX
--t
fi,
is a p-isomorphism f o r
HrY
It t h u s f o l l o w s from (1.15) t h a t
bijective.
that
..., f C 1
r 5 n
f:
-
f;
is injective
fi
is bijective.
bijective,
1, so t h a t
n Z 2.
f:-l
is
is b i j e c t i v e , and t h e i n d u c t i v e
s t e p i s complete.
Since
H f
Theorem 1.13, t h a t
i s a p-isomorphism
f
for a l l
p € P , we conclude, from
is a P-equivalence.
Remark.
Theorems 1.13 and 1 . 1 4 show t h a t we have t h r e e p r a c t i c a l ways t o t e s t
if
+
f: X
Y
p-isomorphism
i s a P-equivalence.
for a l l
We may t r y t o show t h a t
n Z 1, p € P; we may t r y t o show t h a t
II
f
Hnf
is a
is a
Localizationof 1-connected CW-complexes
p-isomorphism for all n 1 I, p t P; or, if X, Y
are of finite type, we
have the potentially most practical procedure, namely to try to show that
f,:
Hn(X; Z/p)
-+
Hn(Y; Z/p)
is an isomorphism for all n 1 1, p € P.
In the case when P is empty, we have the following evident
modification of Theorem 1 . 1 3 .
Theorem 1.16. Let
f: X
-+
Y i n H1.
Then the f o l l o w h q statements are
equivalent:
(il
(iil
f
i s a 0-equivaZence;
nnf
f i i i l Hnf
( i v ) f,:
i s a O-$somorphism f o r a12 n 2 1;
is a 0-isomorphism f o r a l l
Hn(X;Q)
2
H (Y;Q)
n 2 1;
for a12 n 2 1.
61
Localization of homotopy types
62
2.
Nilpotent spaces.
It turns out that the category
is not adequate for the full
H1
exploitation of localization techniques.
This is due principally to the fact
that it does not respect function spaces. We know, following Milnor, that
is a (pointed) CW-complex and W
if X
a finite (pointed) CW-complex, then
the function space Xw of pointed maps W
-+
X
has the homotopy type of a
CW-complex. However its components will, of course, fail to be 1-connected
even if X
Xw
is 1-connected. However, it turns out that the components of
are nilpotent if X
is nilpotent. Moreover, the category of nilpotent
CW-complexes is suitable for homotopy theory (as first pointed out by E. Dror),
and for localization techniques [14,83].
Definition 2.1.
A connected CW-complex X
and operates nilpotently on
IT
is nilpotent if nlX
X for every n
is nilpotent
>_ 2.
n
Let NH be the homotopy category of nilpotent CW-complexes. Plainly
NH
2
H1.
Moreover, thesimple CW-complexes are plainly in MH; in particular,
NH contains all connected Hopf spaces. It isalsonotdifficult tosee (Roitberg [69])
that, if G
is a nilpotent Lie group (not necessarily connected) then its
classifying space BG
is nilpotent in the sense of Definition 2.1.
We prove
the following basic theorem, which provides us with a further rich supply of
nilpotent spaces.
Theorem 2.2.
Let F
Then F C NH if
Proof.
-i
E
f
B be a fibration of connected CW-compzexes.
E €NH.
We exploit the classical result that the homotopy sequence
of the fibration is a sequence of
is
IT
IT
E-modules.
1
E-nilpotent of class icy then
1
IT
n
F
is
IT
We will prove that, if nnE
F-nilpotent of class 5c
1
+
1.
Nilpotent spaces
63
(A mild modification of the argument is needed to prove that if
nilpotent of class Zc, then
71
F is nilpotent of class 5c
1
+
IT
E
is
1
1; we will
deal explicitly with the case n 1 2.)
We will need the fact that
and that the operation of I T ~ Eon
TI
E
1
F
IT
operates on
IT
B
through f,,
is such that
It will also be convenient to write IF, IE for the augmentation ideals of
rlF,
IT
E.
1
Then the statement that anE is
IT
E-nilpotent of class 5c
1
translates into
Consider the exact sequence of
...
and let
S-CC=
,€ €
I;, a
€
a ~ B, c IT,+IB.
(i*n-.l)['a,
=
-IT
F.
Let
n+l B
IT
E-modules
1
- - a
IT
nF
Then i*(S.a)
11 €
i*
IT
=
nE
(i*c)*i,(a)
rlF. Then a((i*n-l).B)
( n - l ) c . c ~ , by ( 2 . 3 ) .
But
...
= 0
=
by ( 2 . 4 ) .
(i,n-l).aB
(i*Q-l)*B = (f,i,n-l).B
=
Thus
=
0, SO
0. This shows that IC+'*~ F = (O), and thus the theorem is
F
n
proved. Note that, in fact, our argument shows that, even if F is not
(n-1)S-a
=
connected, each component of
F is nilpotent. F e w i l l feel free to invoke
this more general statement.
Now let W be a finite connected CW-complex and let X be a
connected CW-complex. Let Xw b e the function space of pointed maps
W
W + X and let Xfr be the function space of free maps. Choose a map
W
as base point and let
g € XW (g€Xfr)
of
g.
(XW ,g)((XfrW,g))
be the component
Localization of homotopy types
64
(Compare G. Whitehead [871 , Federer [ 2 6 1 . )
Theorem 2 . 5 .
( i ) (xw,g) i s nilpotent.
W
liil
x
i s nilpotent i f
(xfr,g)
Proof. We may suppose t h a t
( i ) , ( i i ) are certainly true i f
i n d u c t i o n on t h e dimension of
i s nitpotent.
is a p o i n t .
Wo
i s 0-dimensional,
W
Thus t h e a s s e r t i o n s
and w e w i l l argue by
We w i l l be c o n t e n t t o prove ( i ) . We have
W.
a cof i b r a t i o n
v
is a wedge of
V
where
-+
wn
*
wn+l,
g i v i n g r i s e t o a f i b r a t i o n (where we d i s p l a y
n-spheres,
one component of t h e f i b r e )
where
w"+'-+
g:
Wn
(X ,go)
X
and
g
= glw".
Our i n d u c t i v e h y p o t h e s i s i s t h a t
is n i l p o t e n t , so t h a t Theorem 2 . 2 e s t a b l i s h e s t h e i n d u c t i v e s t e p .
Let
Corollary 2 . 6 .
W
be a f i n i t e CW-comptex and
W
(X ,g)
X € NH. Then
and
W
(Xfr,g) are nilpotent.
Proof. L e t
Wo,
W1,
..., Wd
b e t h e components of
W, w i t h
o € Wo.
Then
xw =
x"0
x
;*
x
...
x
Xd'f r .
Since p l a i n l y a f i n i t e product of n i l p o t e n t s p a c e s is n i l p o t e n t , i t f o l l o w s
that
W
(X ,g)
is n i l p o t e n t .
Similarly
W
(Xfr,g)
is n i l p o t e n t .
C o r o l l a r y 2 . 6 t h u s e s t a b l i s h e s ( i n view of M i l n o r ' s theorem)
t h a t we s t a y i n s i d e t h e c a t e g o r y NH when we t a k e f u n c t i o n s p a c e s
X € NH and
f i n i t e , i n t h e s e n s e t h a t each component of
W
Xw
Xw
is i n
with
NH.
W
e now proceed t o g i v e an important c h a r a c t e r i z a t i o n of n i l p o t e n t
spaces.
Let
X
be a connected CW-complex and l e t
-
...
(2.7)
Nilpotent spaces
- ...
P
xn 4 xn-l
be i t s Postnikov decomposition, so t h a t
K(nnX,n).
65
-x1-0
i s a f i b r a t i o n with f i b r e
pn
W
e s a y t h a t t h e Postnikov decomposition
refinement a t stage
x
(2.8)
n
qc.
Yc
=
n
where t h e f i b r e of
Let
decomposition of
if and only i f
Proof.
TI
pn
may be f a c t o r e d as a product of f i b r a t i o n s
- -...
91
Y1
gi: Yi-l + K(Gi,n+l),
Y
n o
.
K(Gi,n)
x
admits a principal refinement a t stage
X
operates n i l p o t e n t l y on
1
X
71
LK(Gi,n+l),
Since
IT
Y = (O),
n o
TI
and
n ? 2
Suppose conversely t h a t
W
e consider
p :X
n
n
-t
Xn-l.
IT,X
Suppose
Then we may r e g a r d
i = 1,
X(=nlYi,05i5c)
1
rnYC = rnXn =
(stage 1)
n 2 2.
..., c ,
o p e r a t e s t r i v i a l l y on
Thus, by r e p e a t e d a p p l i c a t i o n s of t h e proof of Theorem 2.2,
o p e r a t e s n i l p o t e n t l y on
qi
( T I ~ Xi s n i l p o t e n t ) .
We w i l l be c o n t e n t t o g i v e t h e argument f o r
yi
IT
'n-1'
Then the Postnikov
f i r s t t h a t we have t h e p r i n c i p a l refinement (2.8).
as a fibration.
o
1 5 i 5 c.
be a connected CW-compZex.
X
Y
is an Eilenberg-MacLane s p a c e
qi
i s induced by a map
Theorem 2 . 9 .
if
principal
admits a
nlX
X.
IT
is
IT
X-nilpotent of c l a s s 3.
1
Then, by t h e r e l a t i v e Hurewicz Theorem we have
a n a t u r a l isomorphism
where
Thus
n+l
(p )
n
Gn+l(pn)
a s a n element of
i s o b t a i n e d from
I T ~ + ~ ( P , )by k i l l i n g t h e a c t i o n of
may be i d e n t i f i e d w i t h
Hn+l
2
(pn;nnX/r rnX).
IT
2
n
X/r I T ~ X ,and h-'
Thus
h-l
nlXn.
may b e regarded
g i v e s rise t o a diagram
Localization of homotopy types
66
with
u b - 0.
If
u
induces
ql: Y
1 -+ Xn-1'
then
pn
factors a s
(2.10)
with
q1
The homofopy sequence of (2.10) reduces t o
induced a s required.
rl replacing
Thus we may r e p e a t t h e above procedure, with
p
and,
n'
continuing i n t h i s way, we reach
xn
(2.11)
each
qi
&
r
yC
Yc-l
- -
being induced by a map
...
Yi-l
However, a l l t h e homotopy groups of
-f
r
o
Y2
Gi =
K(Gl,n+l), where
r
vanish, s o t h a t
n-1'
r innX/ri+lnnX.
i s a homotopy
equivalence, and (2.11) is e s s e n t i a l l y t h e p r i n c i p a l refinement a t s t a g e
n
whose e x i s t e n c e we set out t o prove.
We would say t h a t t h e Postnikov system of
X
refinement i f i t admits a p r i n c i p a l refinement a t s t a g e
admits a principal
n
f o r every
n 2 1.
We then have the evident
Corollary 2.12.
Let
X
be a connected CW-complex. Then X
i s nilpotent i f
and only i f i t s Postnikov system a h i t s a principal refinement.
W
e p o i n t out t h a t t h e s i m p l e spaces a r e i d e n t i f i e d , by t h e
correspondence i m p l i c i t i n t h i s c o r o l l a r y , with those spaces whose Postnikov
system is i t s e l f principal.
Remark.
Once we have obtained a p r i n c i p a l refinement of t h e Postnikov system
of a space, t h e r e i s , of course, no d i f f i c u l t y i n obtaining f u r t h e r refinements,
Nilpotent spaces
which w i l l remain p r i n c i p a l .
61
Thus i f , f o r example,
i s of f i n i t e t y p e
X
and n i l p o t e n t we may r e f i n e i t s Postnikov system so t h a t each map
of t h e r e f i n e d system i s induced by some map
ZIP,
or
f o r some prime
Yi-l
+
f: E
i f a l l of
IT
+
nlF).
To t h i s end we s a y
a r e connected ( s o t h a t
f,
maps
n E
1
F
onto
n i l p o t e n t l y on t h e homotopy groups of
f.
nlB),
and
( i n c l u d i n g , of c o u r s e ,
W
e could a l s o e x p r e s s t h i s l a s t c o n d i t i o n by a s k i n g t h a t
by t a k i n g
A = Z
F , of ( p o i n t e d ) CW-complexes i s nilpotent
o p e r a t e s n i l p o t e n t l y on t h e homotopy groups of
E
1
Yi-l
p.
B, w i t h f i b r e
F, E, B
+
where
K(A,n+l)
We w i l l need a r e l a t i v e form of Theorem 2.9.
t h a t a map
Yi
n E
1
operate
Note t h a t w e r e c o v e r D e f i n i t i o n 2 . 1
t o b e a p o i n t , provided w e adopt t h e r i g h t n o t i o n of n i l p o t e n c y
B
f o r t h e o p e r a t i o n of
T
E
1
on
TI
F.
Although we w i l l n o t need t h e g e n e r a l
1
case i n t h i s t e x t , w e now d e s c r i b e t h i s n o t i o n f o r t h e a c t i o n of a group
on a group
Q
With r e s p e c t t o such a n a c t i o n w e d e f i n e a lower central series
N.
as f o l l o w s (see H i l t o n [36]):
rlN = N ,
Q
rrh
=
gp{(x.a)ba
Q
W
e t h e n s a y t h a t t h e a c t i o n of
YC+lN = 11).
Q
N
N
i s commutative.
by c o n j u g a t i o n , t h e n
r%
= r
9
nilpotent map, it f o l l o w s t h a t i f
i s nilpotent of class
b€N, i21.
5c
if
f
Note a l s o t h a t i f
i
N.
Q = N
and o p e r a t e s
T h u s , w i t h o u r d e f i n i t i o n above of a
i s n i l p o t e n t , t h e n t h e f i b r e of
f
W
e a l s o have t h e f o l l o w i n g f a i r l y e v i d e n t p r o p o s i t i o n .
nilpotent.
P r o p o s i t i o n 2.13.
connected f i b r e .
Proof.
Q =
N
r iN ,
Note t h a t t h i s d e f i n i t i o n a g r e e s w i t h t h a t g i v e n i n 1.4 i n
t h e case i n which
on
on
-1 -1
b 1 , x C Q, a €
rlE, and
Let
f: E
+
B
Then, if E , B
be a map of connected CW-complexes with
are nilpotent,
f
i s nilpotent.
We have a n e x a c t sequence of groups w i t h Q-action,
F = f i b r e of
f,
where
is
Localizationof homotopy types
68
...
+
IT
n+l
B - +IT F +
n
E+
n
IT
...
+
s B
2
-+
TI
F +
1
E+
1
TI
IT
B.
1
are nilpotent, slE operates nilpotently on nnB, TI E
n
for all n 2 1. Thus, if n 5_ 2 , the conclusion that IT E operates nilpotently
1
on n F follows from Proposition 1.4.3, since our definition of nilpotent
Now since E, B
action coincides, in the case of a commutative group, with that of 1.4. Thus
the case n = 1 remains. We have an exact sequence
IT
B-+n F+TIE
2
1
1
of Q-groups and the argument of Proposition 1.4.3 may be adapted to yield the
result in this case in view of the fact that the image of n2B lies in the
center of
IT
F.
1
It is, of course, necessary to take account of both facts
noted after the definition of a nilpotent Q-action.
The relative form of Corollary 2.12 reads
Theorem 2.14.
Let
f: E
+
B
be a map of connected CW-complexes inducing a
surjection of fundamental gruups.
Then f i s niZpotent i f and only i f i t s
Moore-Postnikou system admits a principal refinement.
be the Moore-Postnikov system of f .
Now if pn may be factored as in ( 2 . 8 ) ,
we obtain a sequence of extensions of n E-modules
Gi>where Gi
nYi
IT
-
1
nnYi-l, i
*
1,
is a trivial module. Now the fibre of
an Eilenberg-MacLane space K(Hi,n)
an extension of
nlE-modules
..., c,
is
i = ql’”qi: Yi Y
and the relation siqi+l = si+l yields
s
-+
69
Nilpotent spaces
where Ho = {O}, Hc
=
~ l ~ =
+ n~ F.
p ~ It follows from (2.15) and Proposition
1.4.3, by an easy induction, that
TI
F is a nilpotent
n
~l
1E-module.
(The
case n = 1 is again slightly special, but we will omit the details in this
case.)
The converse implication is proved exactly as in the absolute case;
see the Proof of Theorem 2 . 9 .
Before proceeding to discuss how to intr0duce.a localization theory
Here we confine
into NH we show how Serre's C-theory may be applied to NH.
attention to the absolute case, since the relative case requires stronger
axioms on a Serre class, as is already familiar in the classical case of the
Thus we will be considering generalized Serre classes in the
category H1.
sense of Definition 1.5.1.
We prove the one basic theorem which we need in
the sequel.
Theorem 2.16.
Let
X E NHand l e t
be a generalized Serre class.
C
Then
the following assertions are equivalent:
(il
T I ~ Xf
C for a l l
(ii) HnX E C f o r all n
(iii) nlx c
cover of
1
n
Z
c and H ~ Xc c
1
for a l l
n 2 1, where
X
i s the u n i v e r s a ~
X.
Proof.
We need two lemmas, which are interesting in their own right.
The first is a generalization (to general m?l) of Theorem 1.4.17, though we
here only state the result for homology with integer coefficients.
Lemma 2.17.
If
~l
acts n i l p o t m t t y on the abelian group
A, then
n
acts
nilpotently on Hn(A,m), n 1 0 .
Proof.
n-series of A
Let 0 = rC+'A
5 rCA 5
... 5 I-1A = A
(see Section 1.41, and write Ai = r iA
be the lower central
for convenience,
Note that each Ai is a nilpotent a-module, of class less than that of A
Localization of homotopy types
70
if
i 2 2.
Moreover,
a
a c t s t r i v i a l l y on
Ai/Ai+l.
We have a s p e c t r a l
sequence of a-modules,
converging ( f i n i t e l y ) t o t h e graded group a s s o c i a t e d w i t h
filtered.
I f w e assume i n d u c t i v e l y t h a t
i t o p e r a t e s n i l p o t e n t l y on
n i l p o t e n t l y on
Hn(%) where
K(nmX,m) +
that
TI
x
E2
whence i t r e a d i l y f o l l o w s t h a t
P4'
X € NH and Zet
a = nlX.
i s the universaZ cover of
Proof.
o p e r a t e s n i l p o t e n t l y on
operates
X.
m Z 2 , where
X
1
= 0.
%. We
o p e r a t e s n i l p o t e n t l y on t h e homology of
have a f i b r a t i o n
Thus we may suppose i n d u c t i v e l y
o p e r a t e s n i l p o t e n t l y on t h e homology of
IT
Hq(Ai+l,m),
Then a operates n i l p o t e n t l y on
Consider t h e Postnikov system of
+ Xm-l,
IT
suitably
completing t h e i n d u c t i v e s t e p .
Hn(Ai,m),
Let
Lemma 2.18.
a
Hn(Ai,m),
%m-l
and, by Lemma 2.17,
K(nmX,m).
We ?iave a s p e c t r a l
sequence of n-modules
converging ( f i n i t e l y ) t o t h e graded group a s s o c i a t e d w i t h
filtered.
We s e e immediately t h a t
i t r e a d i l y follows t h a t
the inductive step.
IT
Sihce
a
o p e r a t e s n i l p o t e n t l y on
o p e r a t e s n i l p o t e n t l y on
k
+
Hnk,
imi s m-connected,
Hnim.
suitably
EL
P4'
whence
This completes
the c o n c l u s i o n of t h e
lemma f o l l o w s .
We now r e t u r n t o t h e proof of Theorem 2.16.
(i)0 (iii) is c l a s s i c a l , s i n c e t h e a b e l i a n groups i n
c l a s s i n t h e o r i g i n a l sense.
(ii)
0
(iii).
Of c o u r s e , t h e e q u i v a l e n c e
c
constitute a Serre
Thus we may complete t h e proof by showing t h a t
For t h i s we invoke t h e s p e c t r a l sequence of t h e covering
I n t h i s s p e c t r a l sequence we have
k*
X.
Nilpotent spaces
71
and t h e s p e c t r a l sequence converges t o t h e graded g oup a s o c i a ed w i t h a
( f i n i t e ) f i l t r a t i o n of
HnX.
By Lemma 2 . 1 8 and Theorem 1 . 5 . 6
Assume, t h e n , t h a t ( i i i ) h o l d s .
EL C C u n l e s s p + q = 0. I t t h e r e f o r e q u i c k l y follows t h a t
P4
n 2 1. Assume now, c o n v e r s e l y , t h a t ( i i ) h o l d s . By P r o p o s i t i o n
we i n f e r t h a t
c,
H X f
1.5.2 we know t h a t
TI = II
X C C.
1
( i f s u c h e x i s t s ) such t h a t
E2
infer that
f
Pq
c
H
q
q c s
if
q = s 2 2
Let
2 fC
.
b e t h e s m a l l e s t v a l u e of
By Lemma 2.18 and Theorem 1.5.6 we
(unless
p
+
q = 0)
2
and t h a t
Eos
f C.
Consider t h e diagram, e x t r a c t e d from t h e s p e c t r a l sequence,
I
Es+l
s+l,O
Then, by t h e axioms of a S e r r e class, each of
c , while
belongs t o
3
EoS,
..., Eoss+l, EEs
E2
0s
r' c.
2
E2,s-l,
3
..., Es+l
s+l ,0
E3,s-2,
We t h u s deduce, s u c c e s s i v e l y , t h a t
do n o t belong t o
C.
But
E:s
i s a subgroup of
HsX,
which b e l o n g s t o C , s o t h a t w e have a r r i v e d a t a c o n t r a d i c t i o n .
Theorem 2.16 w i l l , i n p a r t i c u l a r , be a p p l i e d i n t h e s e q u e l t o t h e
c a s e i n which
Remark.
(2.19)
c
i s t h e class of f i n i t e l y g e n e r a t e d n i l p o t e n t groups.
It is e a s y t o see t h a t t h e converse of Lemma 2.18 h o l d s .
Let
X be a connected CW-complez.
Then X f NH i f
nlX
That i s ,
is
nilpotent and operates nilpotently on the homology groups of the universal
cover of
X.
However, no use w i l l b e made of (2.19) i n t h e s e q u e l .
q
Localization of homotopy types
12
3. Localization of nilpotent complexes.
In this section we extend Theorems 1A and 1B from the category
H1
to the category NH. To do so we need, of course, to have the notion of the
localization of nilpotent groups, which was developed in Chapter I. We are
thus able to make the following definition.
Let X ENH. Then X
Definition 3.1.
all n 2 1. A map
X is P-local for
n
in Ni P-localizes if Y is P-local and
f: X + Y
is P-local if
TI
f*: [Y,Z] s [ X , Z ]
for all P-local
in NH.
Then the main theorems of this section extend the enunciations of
Theorems lA, 1B from H1
to NH.
Theorem 3A (First fundamental theorem in NH.)EVery
X in NH admits a
P-localization.
Theorem 3B (Second fundamental theorem in NH.) Let f: X
-f
Y in NH. Then
the following statements are equivalent:
li)
f P-localizes X;
(iil vnf: snx+nnY
(iii) Hnf: HnX
+.
P-localizes f o r all n P 1;
HnY P-localizes for all n P 1.
The pattern of proof of these theorems will closely resemble that
of Theorems lA, 1B. However, an important difference is that the construction
of localization in NH
does not proceed cellularly, as in the 1-connected case,
but via a principal refinement of the Postnikov system.
We first prove that
universal covers of X, Y
so
(ii)
=3
(iii) in Theorem 3 B .
that we have a diagram
Let X, i! be the
Localization of nilpotent complexes
2
73
-X
K(nlX,l)
Y
Jfl
K(slY,l)
Ii-If-
(3.2)
Y
Since
induces localization in homotopy, it induces localization in homology
by Theorem 1B. Moreover, we obtain from ( 3 . 2 ) a map of spectral sequences
2
which i s , at the E -level,
(3.3)
By Lemma 2.18 n X operates nilpotently on H
and alY operates nilpotently
1
q
on H 9 . We thus infer from Theorem 1 . 4 . 1 2 , together with Theorem 1 . 2 . 9
q
if q = 0, that ( 3 . 3 ) is localization unless p = q = 0. Passing through the
spectral sequences and the appropriate filtrations of HnR, Hna, we infer that
Hnf localizes if n 2 1.
Now let ( i ' )be the statement: f*: [Y,Z]
Z in
Zi
[X,Z]
f o r aZZ P-zOCUZ
NH.
-
Note that this statement differs from (i) only in not requiring that
Y be P-local. We prove that (iii)
(ii)=a
(i').
This will, of course, imply that
(0.
If Z i s P-local nilpotent, then we may find a principal refinement
of its Postnikov system. Moreover this principal refinement may be chosen
that the fibre at each stage is a space K(A,n),
where A
so
is P-local abelian.
For, as we saw i n the proof of Theorem 2 . 9 , we may take A = riITnZ/ri+'anZ
for some i , and we know (Theorem 1 . 2 . 7 ) that
P-local. Given g: X
+
r iB
i s P-local if
B is
Z, the obstructions to the existence and uniqueness
of a counterimage to g under f* will thus lie in the groups H*(f;A)
and,
as in the corresponding argument in the 1-connected case (note that we have
trivial coefficients here, too), these groups will vanish if f induces
P-localization in homology.
Localization of honiotopy types
14
Next we proceed to prove Theorem 3A, via a key observation playing
the role of Proposition 1.3.
Proposition 3 . 4 .
Let
U be a f u l l subcategory of
have constructed
f: X
-+
Y
s a t i s f y i n g (ii). Then t h e assignment
automatically y i e l d s a functor
L: U
transformation from the embedding
diagram
-+
X
r+
X
we
Y
NH,f o r which f provides a natural
U LNH t o
Proof of Proposition 3 . 4 .
L.
Let g : X
+
X' in U.
We thus have a
If If
in NH,where
x
X'
Y
Y'
f, f' satisfy (ii).
,fi
(3.5)
satisfies (i) and Y'
Then f
P-local, so that there exists a unique h
commutes.
NH, f o r whose o b j e c t s
Y
is
in NH such that the diagram
If'
Y'
It i s now plain that the assignment X I + Y, g * h yields the
desired functor L.
We now exploit Proposition 3 . 4 to prove Theorem 3A.
consider spaces X
in
We first
NH yielding a f i n i t e refined principal Postnikov
system and, for those, we argue by induction on the height of the system.
Thus we may assume that we have a principal (induced) fibration
where G
is abelian even if n = 1, and we may suppose that we have constructed
f ' : X ' + Y'
satisfying (ii).
(The induction starts with X ' =
0.)
Since
Localization of nilpotent complexes
75
( 3 . 6 ) i s induced, we may, i n f a c t , assume a f i b r a t i o n
-
X
Now we may c e r t a i n l y l o c a l i z e
i s t h e l o c a l i z a t i o n of
X'
K(G,n+l); we o b t a i n
-
X'
be t h e f i b r e of
K(Gp,n+l), where
Gp
K(G,n+l)
If'-
Y'
Y
K(G,n+l)
and s o , by P r o p o s i t i o n 3 . 4 , w e have a diagram
G
x
Let
-&
h
K(Gp,n+l)
There i s then a map
h.
f: X
-+
Y
rendering t h e
diagram
X --+
X'
K(G,n+l)
Y
Y'
K(Gp,n+l)
4f -4f' A
commutative i n NH and a s t r a i g h t f o r w a r d a p p l i c a t i o n of t h e exact homotopy
sequence shows t h a t
f
satisfies ( i i ) .
It remains t o consider t h e case i n which t h e r e f i n e d p r i n c i p a l Postnikov
system of
has i n f i n i t e height ( t h i s i s , of course,the ' g e n e r a l ' c a s e ! ) .
X
Thus we have p r i n c i p a l f i b r a t i o n s
...
(3.7)
-
g
Xi
4-XiWl
and t h e r e i s a weak homotopy equivalence
- ...
X
*
0
Lim Xi.
Now w e may apply t h e reasoning already given t o embed ( 3 . 7 ) i n t h e
diagram, commutative i n
NH,
... -xi
gi.
I
- ...
0
(3.8)
where each
fi
satisfies (ii).
Moreover, w e may suppose t h a t each
hi
is
Localization of hornotopy types
16
a f i b r e map.
of
@Yi.
Let
Y
be t h e geometric r e a l i z a t i o n of the s i n g u l a r complex
Then t h e r e is a map
is homotopy-commutative.
f: X
-+
such t h a t t h e diagram
Y
Moreover, t h e construction of (3.8) shows t h a t t h e
Y -sequence is again a r e f i n e d p r i n c i p a l Postnikov system, from which i t
i
r e a d i l y follows t h a t
is i n NH.
@ fi
satisfies (ii).
So t h e r e f o r e does
f , and
f
Thus we have completed t h e proof of Theorem 3A i n t h e s t r o n g e r form
t h a t t h e r e e x i s t s , f o r each
The proof t h a t (i)
t h e category
H1.
X
=)
i n NH, a map
f: X
-+
Y
in NH s a t i s f y i n g (ii).
( i i ) proceeds exactly a s i n t h e e a s i e r case of
Thus we have e s t a b l i s h e d t h e following s e t of i m p l i c a t i o n s ,
r e l a t i n g t o Theorem 3B:
(3.9)
(ii) = (iii), ( i i i ) * (if), (ii)
=)
(i), (i)
=a
(ii).
All t h a t remains is t o prove t h e following p r o p o s i t i o n , f o r then we
w i l l be a b l e t o i n f e r t h a t , i n f a c t , (iii) =. (i)
Proposition 3.10.
is P-local f o r every n 1 1, then n Y
If Y C NH and HnY
is P-local f o r every n
?
.
n
1.
To prove t h i s , we invoke Dror's theorem, which we, i n f a c t , reprove
s i n c e i t follows immediately from (3.9).
P
- n,
where
n
Thus we consider t h e s p e c i a l case
is t h e c o l l e c t i o n of a l l primes.
Then a homomorphism of
( n i l p o t e n t , abelian) groups Il-localizes i f and only i f i t i s an isomorphism.
Moreover, every space i n NH is II-local, so t h a t , i n t h i s s p e c i a l c a s e , t h e
d i s t i n c t i o n between (if) and (i) disappears.
t h e equivalence of (ii) and ( i i i ) f o r
P =
n,
Thus (3.9) implies, i n p a r t i c u l a r ,
which i s Dror's theorem.
Localization of nilpotent complexes
We construct f: Y
Now we prove Proposition 3.10.
(ii).
I1
It thus also satisfies (iii); but HnY i s P-local,
so
+
Z satisfying
that f induces
an isomorphism in homology. By Dror's theorem, f
induces an isomorphism in
homotopy. However, the homotopy of Z i s P-local,
so
that Proposition 3.10
is proved, and, with it, the proof of Theorems 3A and 3B is complete.
Remark. Of course, we do not need the elaborate machinery assembled in this
section to prove Dror's theorem. In particular, Theorem 3A is banal for
P
-
IT, since, then, the identity X
-r
X n-localizes!
The fact that we have both the homotopy criterion (ii) and the
f
homology criterion (iii) of Theorem 3B for detecting the localizing map
enables us to derive some immediate conclusions. For example we may use
(ii) to prove
Theorem 3.11. If X i s nilpotent and
W connected f i n i t e a d i f
localizes, then fw : (Xw ,g)
(Yw,fg) localizes, where
W
and (X ,g) i s the component of xW containing g.
+-
f: X
-+
Y
w x
g i s any map
-t
Proof. We argue just as inthe proofs of Theorem 2.5 and Corollary
2.6, using Theorem 3.12 below.
A similar result holds for
W
Xfr
(Roitberg [ 6 9 ] ) ; thus we may
remove the condition that W be connected in the theorem. We also note that
-
the theorem implies that H(Fp) = E(F)p
-
where F € NH is finite and H
is
the identity component of the space of (free or pointed) self-homotopy-equivalences.
Theorem 3.12. Let
F
+-
E
+
B be a f i b r e sequence i n NH.
Then Fp
+
Ep
-+
is a f i b r e sequence i n tti.
Theorem 3.13.
Then
% + Yp -+
Let
X
+
Y
-+
C be a cofibre sequence i n NH. With
c
Cp is a cofibre sequence i n NH.
These two theorems are proved exactly in the manner of their
counterparts in H~
(Corollaries 1.10, 1.11).
Our reason for
H1-
Bp
Localization of homotopy types
I8
imposing i n Theorem 3.13 t h e condition
C
proof t h a t , i n general, t h e c o f i b r e of
5
If
7
i s t h a t w e have given no
E H1
-t
is necessarily nilpotent.
Yp
were t h i s c o f i b r e , we would, of course, have a homology equivalence
H1
k k e w i s e Theorens1.13, 1.14, and 1.16 extend from
t o NH; we
w i l l f e e l f r e e t o quote them i n t h e sequel in t h i s extended context.
Given
k
X C NH l e t
component of t h e loop space of
be t h e supension of
a l l belong t o
X.
b e the u n i v e r s a l cover of
X
X, l e t
ZX
be t h e
containing t h e constant loop, and l e t
k , PX
It i s , of course, t r i v i a l t h a t
NH ( f o r t h i s we do not even need t h a t
X
and
CX
EX
i t s e l f be n i l p o t e n t ! ) .
We then have
Theorem 3.14.
(i)
N
($)
ru
(k)p; (ii) E ( X p ) =
Proof. To prove (i)
that
B
we l i f t
e: X
3
(zX)p; ( i i i j
to
Xp
E:
s a t i s f i e s c r i t e r i o n (ii) of Theorem 3B (or 1B).
follow immediately from Theorems 3.12, 3.13 r e s p e c t i v e l y .
k
Z(%)
3
rr/
3
(ZX),.
and observe
(X,)
P a r t s (ii) and (iii)
Notice t h a t
Theorem 3.14(i) has t h e following g e n e r a l i z a t i o n ; r e c a l l t h a t l o c a l i z a t i o n
preserves subgroups (Theorem 1.2.4).
Theorem 3.15.
Let
covering space o f
of
3
X E NH and Let
be a subgroup of
X corresponding t o Q and l e t
corresponding t o Q,.
P-ZocaZizes.
Q
Then e : X
+
5
2
nlX.
Let
Y
be the
be the covering space
l i f t s to
e: Y
-+
Z
which
Quasifinite nilpotent spaces
19
4. Quasifinite nilpotent spaces.
In this short section we present a result which will enable us to
prove an important modification of the main theorem (The Pullback Theorem) of
is of f i n i t e type if anX is finitely
Section 5. Let X € hkl. We say that X
generated for all n 2 1 and that X
is q u a s i f i n i t e if X is of finite type
and moreover H X = {O}
for n
and H X = {O}
N, we will say that X has homological dimension
for n
and may write dim X
(iil
x
3-l
N.
i s of f i n i t e type;
H X i s f i n i t e Z y generated f o r
n
(iii) X
is quasifinite
X € NH. Then the following statements are equivalent:
Theorem 4.1. Let
(i)
5
sufficiently large. If X
N
Y, where
Y
n 1 1;
i s a CW-complex with f i n i t e skeleta.
Proof. The equivalence of (i) and (ii) follows from Theorem 2.16.
That (iii) implies (ii) is trivial. We prove that (i) implies (iii).
Since
nlX is finitely-generatednilpotent, the integral group ring Z[alX]
is
noetherian. Moreover, if
x
Is
the universal cover of X, Hi?
is certainly
finitely-generated over Z [ n X I , being, in fact, finitely-generated as abelian
1
group. Thus (iii) follows from Wall's Theorem (p. 61 of [ S S ] ) .
From Theorem 4.1 we deduce the result in which we will be interested
in the next section.
Theorem 4.2. Let
f:
x
-+
x
X 6 NH. Then X i s q u a s i f i n i t e i f f there e x i s t s a map
o f a f i n i t e CW-complex i n t o X inducing isomorphisms i n homology.
Proof.
It is obvious (in the light of the equivalence of (i) and
(ii) in Theorem 4.1) that the existence of such a map
quasifinite. Suppose conversely that X
f
implies that X
is
is quasifinite. By Theorem 4.1 we
Localization of homotopy types
80
may assume that each skeleton of X
is finite. If dim X 5 N, we will show
that we can attach a finite number of (N+l)-cells to XN to obtain a finite
complex X such that the inclusion XN 5 X extends to f: .+ X inducing
-
x
homology isomorphisms. We have a diagram
where the vertical arrows are Hurewicz homomorphisms. Now %+l(X,X N) , as a
N
subgroup of H# , is free abelian and finitely-generated. Let B be a basis
N
N
for s+l(X,X ) and let
be a subset of K~+~(X,X) mapped by h bijectively
to B. Attach (N+l)-cells to XN by maps in the classes ab, b € B, to form
X. It is then obvious that the inclusion XN 5 X‘ extends to a map X -+ X.
-
Let f:
x
+
X be any such extension.
N
It is plain that f induces an isomorphism s + p , x )
It follows almost immediately that %+lX
isomorphism
HN”; % €$X.
Corollary 4 . 3 .
4.2.
Let
-
=
?2 5
N
+ p , x 1.
{O}, and that f induces an
This completes the proof of the theorem.
X € NH be q u a s i f i n i t e and l e t
f:
x
.+
X be a s i n Theorem
Then
f*: [X,Y]
for all
Y
E
[X,Y]
NH.
-.Proof. Construct
a
principal refinement
... -Yi & Yi-l
-
* *.
of the Postnikov tower of Y. Then, if the fibre of
gi
is K(Gi,ni),
nil 1, the pbstructiomto the existence and uniqueness of a counterimage,
Quasifinite nilpotent spaces
under
i
=
f*, of an arbitrary element of
1, 2,
..., r
= ni
+
1
or
ni.
these cohomology groups all vanish.
[x,Y]
Since
f
will all be in Hr(f;Gi),
induces homology isomorphisms,
81
Localization of homotopy types
82
5. The Main (Pullback) Theorem.
We will denote by X
the p-localization of the nilpotent CW-complex
P
X; by e
the canonical map X + X where p E II, or p = 0; by r : X
X
P
P)
P P
0
the rationalization, p E n , and by can ('canonical map') the function
-+
[W,Xp]
[W,Xo] induced by
-+
r
P
.
We also denote by
g
P
the p-localization of
a map g.
Theorem 5.1. (The Pullback Theorem).
and
Let
W be a connected f i n i t e CW-complex
X a n i l p o t e n t CW-complex of f i n i t e type.
pullback of the diagram of s e t s
{[W,Xpl
Then the 3et
[w,x0i
IP
E
[W,X] i s the
ni.
It will follow that, under the conditions stated, X is determined
r
by the family {X a Xolp E n). Indeed, X is the unique object in the
P
homotopy category of connected CW-coriplexes which represents the functor
{[W,X 3 + [W,X ]Ip E II} from the category of connected finite
P
CW-complexes to the category of pointed sets.
W
I+
pullback
Our main theorem also implies, in the light of Corollary 4 . 3 , that,
for X
as in Theorem 5.1 and W now quasifinite nilpotent, a map
g:
W+ X
is completely determined (up to homotopy) by the family of its p-localizations
{gplp E rI),
and, conversely, a family of maps
a unique homotopy class g: W
the maps g(p)
-+
rationalize to
X with
a
{g(p):
X Ip E n) determines
P
for all p, provided that
We
-t
g N g(p)
P
common homotopy class not depending on p.
Therefore the situation is analogous to that in the theory of localization
of finitely-generated nilpotent groups (Theorem 1.3.6).
Indeed, this algebraic
fact provides one with an easy proof of Theorem 5.1 in case W
or X
is a suspension
a loop space, in view of Theorem 3B.
The method we use to prove Theorem 5.1 is the localization of function
spaces (Theorem 3.11), which enables us to prove the result by induction on
the number of cells of the CW-complex W.
The main (pullback) theoreni
Definition 5.2.
g,:
g: X
A map
-f
Y
83
i n NH i s an F-monomorphism i f
i s i n j e c t i v e f o r a l l connected f i n i t e CW-complexes
[W,X]+ [W,Y]
2: X
W.
IIX
t h e map with components { e p l p C rI].
We
P
prove one h a l f of Theorem 5.1, b u t , f o r t h i s h a l f , remove a f i n i t e n e s s
Denote by
r e s t r i c t i o n on
Theorem 5.3.
2: X
IIX
-+
P
+
(Compare Theorem 1.3.6.)
X.
Then t h e canonical map
be a n i l p o t e n t CW-complex.
X
Let
is an F-monomorphism.
Proof.
W
e have t o show t h a t
f o r an a r b i t r a r y f i n i t e CW-complex
[W,X]
If
W.
the cofibration
Sn-l
+
Z l[W,X ]
P
i s injective
i s a f i n i t e wedge of s p h e r e s ,
W
i:W
Given
3
W = V U en
W , and assume
V + W.
P
Hence we can proceed by induction
t h e theorem follows from Theorem 1.3.6.
on the number of c e l l s of
[WJX
-+
-+
X, l e t
n 2 2.
with
We consider
g = g l V ; we g e t a f i b r a t i o n ,
up t o homotopy, ( i n which we e x h i b i t one component of t h e f i b r e )
(x',E)
+
(xv,g)
+
p-1
(X
,o), n 2 . 2 ,
giving r i s e t o a diagram with exact rows
Here and l a t e r
where
h
[W,X]g
[
, Ih
s e r v e s a s basepoint f o r t h i s s e t .
6
i
and, by exactness, the o r b i t of
which a r e homotopic t o
that
denotes t h e s e t of (based) homotopy c l a s s e s of maps
g
Notice t h a t
i m $'
when r e s t r i c t e d t o
g'
i m $J
g
X
o p e r a t e s on
c o n s i s t s p r e c i s e l y of t h o s e maps
i s i n j e c t i v e , and we have t o show t h a t
i n j e c t i v e and s i n c e
71
V.
By induction we may assume
y-'(Yp)
=
g.
Since
a r e t h e i s o t r o p y subgroups d
r e s p e c t i v e l y , i t follows t h a t the s e t
Y
-1
(YE)
i,
6
is
Ispi)
i s i n one-one correspondence
Localization of homotopy types
84
with the set ker (coker $
localize their domain and
g
-+
so,
coker J, ) . The components of B clearly all
g
too, do those of a by Theorem 3.11. Therefore
the cokernel of J, splits into a product of p-local groups and the map
g
coker 0 + coker J, has components which p-localize. Hence
g
ker (coker 0
g
required.
+
coker J, ) = I01 by Theorem 1.3.6, and y
-1
g
(yp) =
9,
as
Notice that no finiteness conditions on X were needed for this
argument. But if the space X is of finite type then, by following the lines
of the proof of Theorem 5.3,we obtain the following corollary.
Corollary 5.4.
Suppose W is a connected f i n i t e CW-complex and X a
nilpotent CW-complex of f i n i t e type.
Let
S
5 T denote s e t s of primes.
Then:
a)
The canonical map
[W,XT]
b ) The canonical map
f i n i t e l y many primes
c)
map
+
[W,Xs] is finite-to-one.
[W,Xp] + [W,Xo] i s one-one f o r a l l but
p.
There e x i s t s a c o f i n i t e s e t of primes Q such that the canonicaZ
[W,XQl -+ [W,X
I
i s one-one.
Notice also that we may replace the partition of I7 into singleton
sets of primes, in the enunciation of Theorem 5.3, by any partition of lT.
We now illustrate, by means of an example, the fact that, even when
A
X is a sphere, the map X A n X is not a monomorphism in the homotopy
P
category of a l l CW-complexes
.
Proposition 5.5.
Let
W = (Si
V
S;)UAen+l
non-empty complementq s e t s of primes, and
Then there is an essential map
primes
p.
K:
w + sn+'
where n 1. 2, R and
A = (1,l) C nn(S;
such t h a t
K.
P
T are
v $j
= o for a l l
The main (pullback) theorem
Proof.
Let
W
K:
+
Sn+'
85
be the collapsing map and consider the
Puppe sequence
Then, for all primes p, E X
(CS;)p
or
wP "4. sP n
But, were
K
cA
has a left homotopy-inverse since either
P
is homotopy equivalent to S*l.
From the cofibration
P
= 0 for all p.
P
= 0, this would imply that ZX had a left homotopy-inverse and
+ 1 4 (CS;
V
Cs;Ip
we conclude that
K.
hence, by taking homology, Z would be a direct summand of ZR @ZT which
rISn+l i s not a monomorphism.
P
We now complete the proof of Theorem 5 . 1 .
i s absurd. Thus
Theorem 5 . 6 .
Let
Sn+'
-+
W be a connected f i n i t e CW-complex and X a nilpotent
CW-complex of f i n i t e type.
p C I'l U {O],
such that
i s the canonical map.
that e g
P
= g(p)
Proof.
Suppose given a f a m i l y of maps
g(p):
W
-+
xP'
n. g(0)
f o r a21 p f Il where r : X + X
P
P P
0
Then there i s a unique homotopy class g: W - + X such
r g(p)
for all
p.
Uniqueness has already been proved in Theorem 5.3,
have only to prove the existence of g.
If W
then the theorem follows from Theorem 1.3.6.
so
we
is a finite wedge of spheres,
Hence we proceed again by
induction. Let W = VUXen, n 1 2 , and assume that we have constructed g': V
such that epg'
= g(p)
IV for all p.
such an extension exists since
by Theorem 5.3.
(g'k)p
Let
i:W +X
be an extension of g';
0 for all p and hence g'A = 0
Now consider the diagram
+
X
Localization of homotopy types
86
7 U {O)
p C l
For each
a(p)
*
t h e r e i s a unique
*
epp = g ( p ) , t h e
on t h e set
0
-1
(epg').
d e n o t i n g t h e f a i t h f u l a c t i o n of
Note t h a t
used t o prove Theorem 5 . 3 .
action
x
a ( p ) C coker $ g l ( p )
coker $ , ( p ) e' (coker $
g
g
Further, since
eog(p)
is f a i t h f u l , i t f o l l o w s t h a t each
C=
g(0)
,p
such t h a t
coker $
(p)
by t h e argument
P
C
g'
n, and
the
n, r a t i o n a l i z e s
a(p), p C
to
coker $
i s f i n i t e l y g e n e r a t e d , i t f o l l o w s from Theorem 1.3.6
g
t h a t t h e family {o(p)} C n(coker $ )
d e t e r m i n e s a unique element
g' P
a C coker $
which p - l o c a l i z e s t o a ( p ) f o r a l l p . By n a t u r a l i t y w e
g'
a(O),
Since
conclude t h a t
a x
h a s t h e p r o p e r t i e s r e q u i r e d of
g.
P u t t i n g t o g e t h e r Theorems 5 . 3 and 5.6 w e o b t a i n our main r e s u l t ,
Theorem 5.1.
One can, of c o u r s e , g e n e r a l i z e Theorem 5.1, w i t h o u t changing a n y t h i n g
e s s e n t i a l i n i t s p r o o f , t o t h e case of a n a r b i t r a r y p a r t i t i o n of
mutually d i s j o i n t famil ies
Pi
Il i n t o
of primes.
I n o r d e r t o deduce t h e e x i s t e n c e of certain g l o b a l s t r u c t u r e s on
o u t of given s t r u c t u r e s on t h e
X Is, as w e w i l l w i s h t o do i n Chapter 111,
P
i t i s p a r t i c u l a r l y u s e f u l t o know how t o c o n s t r u c t
i n a " t o p o l o g i c a l " way.
We w i l l d e n o t e by
s i n g u l a r complex of
.
map, and
p:
Xo
w i l l assume t h a t
-r
l7X
P
by
Exp
r:
X
o u t of t h e maps
xp +. xo
t h e geometric r e a l i z a t i o n of the
EXP - + ~ X p ) ot,h e
rationalization
EX
l o c a l i z e d a t 0. We
P
are f i b r a t i o n s (without changing t h e n o t a t i o n ) ,
t h e c a n o n i c a l map
r
p
by a l t e r i n g t h e domains of
Theorem 5 . 7 .
There are maps
&p)o,
and
X
r
and
p
X
+
in t h e u s u a l way.
Suppose X i s a nilpotent CW-complex of f i n i t e type.
51 the topologiaal pullback
of
Xo
ex
)
PO
&EX
P
Denote
, Then t h e canonical
The main (pullback) theorem
map
X
+
x
a7
is a homotopy equivaZence.
Proof.
Consider t h e p u l l b a c k
-
square
The "Mayer-Vietoris" sequence i n homotopy g i v e s a n e x a c t sequence
... r n iTi
(5.8)
where
@
... - n p xiX
- -
<px,-r*>
nixo
nn.x
1 P
mnix)o
(n X ) x (rrn X )
l o
1 P
i s f i n i t e l y generated.
The maps
.1 -
n,rr
nn
are a l l p u l l b a c k diagrams.
a g a i n by Theorem 1.3.7.
x
mnlxp)o,
r*
<p*,-r*>,
i 2 2,
defined f o r
n
x
io
i l l
(Trn.x )
I P O
i P
But s o are t h e diagrams
The map
X
+
TI
which i s t h e i d e n t i t y on t h r e e c o r n e r s .
w X
i
P*
Hence i t f o l l o w s from (5.8) t h a t t h e
are a l l s u r j e c t i v e by Theorem 1 . 3 . 7 .
diagrams
-
...
i n d u c e s a map of p u l l b a c k
diagrams
I t t h u s i n d u c e s isomorphisms
&? TT
X, and s o is a homotopy e q u i v a l e n c e .
i
Of c o u r s e t h e r e i s a l s o a form of Theorem 5.7, mutatis mutandis, for
an a r b i t r a r y p a r t i t i o n of
II into m u t u a l l y d i s j o i n t f a m i l i e s of primes.
I f t h e p a r t i t i o n i s f i n i t e , i t i s e a s y t o see t h a t we need no l o n g e r i n s i s t
Localization of homotopy types
88
that
be of f i n i t e t y p e .
X
Theorem 1.3.7,
W
e may a l s o e x p l o i t Theorem 1.3.9 i n s t e a d of
ll
t h a t we a r e concerned w i t h t h e c a s e of a p a r t i t i o n of
so
i n t o two d i s j o i n t s u b s e t s .
Theorem 5.9.
Let
partition of
n.
n
be a nilpotent CW-complex and l e t
X
Denote by rp:
Xp
-+ Xo
rO: X
a d
.
canonical m a p s , which we assume t o be fibrations.
equivalent t o the topological pullback of
X
Q -+
= P U Q
0
be a
the
Then X i s homotopy
rp and
rQ'
The proof e x p l o i t s Theorem 1 . 3 . 9 j u s t a s t h e proof of Theorem 5.7
e x p l o i t e d Theorem 1.3.7.
W
e omit t h e d e t a i l s .
Often one reduces a problem i n v o l v i n g i n f i n i t e l y many primes t o one
i n v o l v i n g o n l y f i n i t e l y many by means of a c o r o l l a r y which is i n some s e n s e
T h i s c o r o l l a r y f o l l o w s a t once by u s i n g t h e diagram
d u a l t o C o r o l l a r y 5.4.
X
used i n t h e proof of Theorem 5.6, r e p l a c i n g
C o r o l l a r y 5.10.
by
Suppose W i s a connected f i n i t e
nilpotent CW-complex of f i n i t e type.
Given a map
a ) For a l l but a f i n i t e nwnber of primes
There e x i s t s a c o f i n i t e s e t of primes
b)
f €
QI
im([W,X
+
X
Q
and
X
P
by
CW-complex and
f: W
+
Xo,
X a
then:
p , f E im([W,X ]
P
Q
Xo.
-+
[W,Xo])
such t h a t
[W,Xol).
Combining t h i s w i t h C o r o l l a r y 5.4 w e g e t
and
f
be as i n Corollary 5.10.
e x i s t s a c o f i n i t e s e t of primes
Q
such that
C o r o l l a r y 5.11.
where
w
g:
-+
Let
xQ,
I n case
W, X
and rQ: xQ -+ X
W
f
Then there
factors uniquely as
f
-
i s the canonical map.
i t s e l f is n i l p o t e n t , we can r e f o r m u l a t e Theorems 5 . 3
and 5.6 u s i n g t h e u n i v e r s a l p r o p e r t y of l o c a l i z a t i o n , namely, t h e f a c t t h a t
e : W
P
-+
W
P
induces a b i j e c t i o n
e*: [W ,X ]
P
P P
+
[W,Xp].
W
e get
rQg
The main (pullback)theorem
Let W be a nilpotent f i n i t e CW-complex and X an arbitrary
Corollary 5.12.
nilpotent CW-compZex. Given t u o maps g, h: W
i f
gp
hp f o r aZZ primes
n.
89
+
X, then
g
n.
h i f and o n l y
p.
This is immediate from Theorem 5.3.
In case h = 0 this answers a
conjecture of Mimura-Nishida-Toda [ 5 3 ] affirmatively. From Theorem 5 . 6 we get
Let
Corollary 5.13.
W be a nilpotent f i n i t e CW-complex and
CW-compZex o f f i n i t e type.
such t h a t
cZass
g:
g(p),
w
e
x
g(p'),
Given m y f m i Z y o f maps
f o r aZI
p, p' c
n,
{g(p):
a niZpotent
X
Wp
-+
n)
Xplp €
there is a unique homotopy
g n. g(p) f o r a11 p.
P
However, we may further improve on Corollaries 5.12, 5.13 by exploiting
-+
Corollary 4 . 3 .
with
For, according to that result, if W
-
f*: [W,X]
2
[W,X], where f:
w -+ W
is quasifinite, then
is a map of a finite CW-complex W
Thus Theorems 5.3, 5.6 remain valid if the assumption that W
replaced by the assumption that W is quasifinite (nilpotent).
to W.
is finite be
Thus we
conclude
Theorem 5.14.
The conczusions of Corozlaries 5.12, 5.13 remain valid, i f
i s supposed q u a s i f i n i t e instead of f i n i t e .
W
Localization of homotopy types
90
6.
Localizing H-spaces
I n t h i s s e c t i o n we prove a theorem which w i l l be c r u c i a l i n our study
of t h e genus of an H-space i n 111.1, and which provides a n a t u r a l analog of
t h e b a s i c recognition p r i n c i p l e i n t h e l o c a l i z a t i o n theory of n i l p o t e n t groups.
X
Let
be a connected H-space.
so may be l o c a l i z e d .
-+
Xp
i s an H-map.
Then, f o r any CW-complex
For any monoid
M
and any element
x
in
M
x € M,
and we w i l l
f o r such an nth power, even though t h e r e i s , i n general, no unique
n t h power.
It i s thus c l e a r what we should understand by t h e claim t h a t a
homomorphism
$: M
Theorem 6 . 2 .
The map
-+
f: X
e,
let
P-local rmd
f,:
[W,X]
W.
Then
CW-complexes
Proof.
W
of monoids i s P - i n j e c t i v e (P-surjective,
N
Conversely,
true i f
property of
W, t h e induced map
we may, in an obvious way, speak of an n t h power of
xn
i s n i l p o t e n t and
may be endowed with an H-space s t r u c t u r e such t h a t
Xp
i s a homomorphism of monoids?
write
X
Moreover, i t i s p l a i n , from t h e u n i v e r s a l
l o c a l i z a t i o n , t h a t each
e: X
Then c e r t a i n l y
-+
(6.1)
i s f i n i t e connected.
be an H-map of connected spaces such that
Y
-+
i s P - b i j e c t i v e if W
P-bijective).
[W,Y]
f
We prove
i s P - b i j e c t i v e f o r a l l f i n i t e connected
P-localizes.
e,
(6.1) P-bijective.
This a s s e r t i o n i s c l e a r l y
is 1-dimensional, by t h e Fundamental Theorem of Chapter I.
t h e r e f o r e argue by induction on t h e number of c e l l s of
of Theorem 5 . 1 ) .
We assume
is
Y
W
We
(compare t h e proof
W = V U en, n 2 2 , and t h a t w e have a l r e a d y
proved t h a t
e,:
P,XI
* [V,%l
is P - b i j e c t i v e f o r a l l connected H-spaces
X.
W
e consider t h e diagram (of
monoid-homomorphisms)
*By monoid, we understand a s e t
endowed with a m u l t i p l i c a t i o n with two-sided unity
Localizing H-spaces
We prove
e*: [W,X] + [W,Xp]
e*ix = 1, s o
Then
$pexa = e*$a = 1.
ix" = 1, f o r some
f o r some
respect t o
flX
m
1
so t h a t
and
is P-injective.
$Jcm2
f o r some
e*:
Then
ipym = e*a
that
jam' = 1 f o r some
Thus
yml
with
e*x = 1.
Thus
We conclude t h a t
QXP.
e*a
ml
=
e*$c,
m m
=
W e now prove
ipe,x
m € P'.
x € [W,X]
h e r e we invoke t h e i n d u c t i v e h y p o t h e s i s w i t h
and i t s l o c a l i z a t i o n
m m
a 1 2
and
€ P';
Thus l e t
xm = $ a , a C TI X , and
m
e*a = Jipb, b € [ZV,$,], and b
= e*c,
It f o l l o w s t h a t
c f [CV,X]
e*
P-injective.
91
f o r some
=' : ,e
m
[w,x]
-+
[W,%]
P-surjective.
a € [V,X], m C P I .
1
= ipym'
= (e*x).($pb),
m2 6 P' , whence f i n a l l y
C P'.
am1
Now
= i x , x C [W,X],
mm 1-power of
( f o r a s u i t a b l y bracketed
b C vnXp.
bm2 = e*c, f o r some
Thus, by Lemma 6 . 4 below, ynrmlm2= ek(xm2.$c)
[W,%l.
y €
Let
= 1
e,ja = j Pe*a = 1, so
Thus
It f o l l o w s t h a t
(xm)m1m2 = $ a
c
<
y).
unX, m2 C P ' .
and t h e a s s e r t i o n i s proved.
(Note t h a t , i n t h e s e arguments, w e have w r i t t e n a l l monoid s t r u c t u r e s i n ( 6 . 3 )
multiplicatively. )
The converse i s t r i v i a l .
f,: II X
-+TI
n 11.
Thus
For i f we s e t
W = Sn
t h e n we know t h a t
i s a P-isomorphism t o a P - l o c a l group and hence P - l o c a l i z e s ,
Y
f
P - l o c a l i z e s by Theorem 3B.
The f o l l o w i n g lemma, t h e n , s u p p l i e s t h e one m i s s i n g s t e p .
Lemma 6 . 4 .
Let
X
attaching the cone
W
*
ZA
induces
be an H-space and l e t
CA
to
V
W = V
by means of a map
U
CA, t h e space obtained b y
A
-t
V.
The p r o j e c t i o n
Lacalization of homotopy types
92
4 i s central in the monoid
and the image of
Proof.
Let c: W
+
[w,x] in the strong sense that
W V CA be the cooperation map, In the terminology
of Eckmann-Hilton. Then, for x € [W,X], a
€
[ZA,X], we have
x.$a = c*<x.a>,
where <x,a> € [WVCA,X]
Thus
-
[W,X]
(xl.$al) (x,.4ia2)
c*<x1x2,a1a2> = x1x2. 4 (ala2>
[ZA,X].
X
=
(c*<xl,al>)(c*<x2,a2>) = c*(<xl,a1><x2,a2>) =
.
Remark. Note that we could have proved that
e*
the following stronger sense, namely, that if e,x
xn = yn for some nth power with n
€
P'.
in (6.1) is P-injective in
=
e*y, x , y € [W,X], then
In the presence of a nilpotency
condition, this sense of P-injectivity in fact coincides with the obvious one
(obtained by setting y = 1); and, indeed, it is true, with appropriate definitions, that
[W,X] is a nilpotent loop (non-associative group),
Corollary 4 . 3 enables us to deduce the following modification of
Theorem 6.2.
Corollary 6.5. Let
CW-complex.
X be a connected H-space mrd W a quasifinite
Then ZocaZization induces
and eg i s P-bijective.
The following consequence of this corollary will be very important
in the sequel.
Theorem 6.6.
Let W be a quasifinite CW-compZex and l e t
H-space such that Wp
9
Xp.
Then there e x i s t s a map
X be a connected
f: W * X such that
Localizing H-spaces
fp:
wp
eY
$.
Proof.
Let
g,
= ng*: R*(Wp)
g: Wp
-+
$
n C P' suchthat
6 . 5 there e x i s t s
n
93
-+
n,($)
gn = e l ( f ) , f : W
and t h a t , consequently,
isomorphism of homotopy groups
found the required map
b e a homotopy equivalence.
f
.
(n C Zt). Thus
-+
X.
n
g,
g": Wp
By Corollary
But i t i s c l e a r that
is,like
'\I
$,
g*,
an
so that we have
94
7.
Localization of homotopy types
Mixing of homotopy t y p e s
T h e i d e a o f m i x i n g h o m o t o p y t y p e s g o e s b a c k t o Zabrodsky [ 9 3 ] a n d h a s been
e x t e n s i v e l y used t o c o n s t r u c t examples and counterexamples i n homotopy t h e o r y ;
see, e . g .
[79,93,96] and Chapter 111 of t h i s monograph.
It seems t h a t l o c a l i z a t i o n
t h e o r y p r o v i d e s t h e r i g h t framework f o r d i s c u s s i n g t h i s i d e a and r e n d e r i n g i t
most e a s i l y u s a b l e i n a p p l i c a t i o n s .
We b e g i n by d i s c u s s i n g puZZbacks
i n homotopy t h e o r y , a t o p i c of some
independent i n t e r e s t , i n p a r t i c u l a r w i t h r e s p e c t t o l o c a l i z a t i o n .
Given a
diagram
X
If
(7.1)
Y-B
i n t h e category
g
T of based CW-complexes, we may r e p l a c e
and t a k e t h e ( s t r i c t ) p u l l b a c k
which we c a l l t h e weak
T.
by a f i b r e map
We o b t a i n a diagram
puZZback of (7.1) i n t h e homotopy c a t e g o r y H.
known t h a t t h e homotopy t y p e of
as a diagram i n
in
f
2
depends only on t h e diagram ( 7 . 1 ) , i n t e r p r e t e d
H, and i s symmetric w i t h r e s p e c t t o
we might j u s t as w e l l have r e p l a c e d
i n s t e a d of choosing t o r e p l a c e
f.
I t is
g
(or both
Of c o u r s e , i f
f
f , g, i n the sense t h a t
and
f
(or
g) by a f i b r e map,
g) were a l r e a d y a
f i b r e map, n o replacement would be n e c e s s a r y .
I f (7.1) were a diagram i n
To, t h e s u b c a t e g o r y of
T
c o n s i s t i n g of
connected CW-complexes, we would o b t a i n t h e weak p u l l b a c k i n the corresponding
homotopy c a t e g o r y
base point.
Ho by r e p l a c i n g
W
e would t h u s o b t a i n
2
i n (7.7.) by t h e component
Zo
of i t s
Mixing of homotopy types
U
x
zo
1. If
(7.3)
in Ho.
We are interested in the question of when we may infer that
zO
is, in fact, in NH.
Theorem 7 . 4 .
i f
W
e prove:
Suppose t h a t
X , Y C NH
i n (7.31.
Then
nlZo operates n i l p o t e n t l y on nnB, n 2 2 , v i a
Proof.
fu
Zo
E NH i f and onZy
.
The diagram ( 7 . 3 ) gives rise to a Mayer-Vietoris sequence
of groups with nlZo-action, where
Suppose that nlZo
G
i s the pullback of the diagram of groups
operates nilpotently on TInB, n
operates nilpotently (via u
vo)
nlZo operates nilpotently on
?
2. Then, since n lZ o
on the homotopy groups of X
it follows from Proposition 1 . 4 . 3 that nlZo
Now
95
TI
2
and Y,
operates nilpotently on nnZo, n ? 2.
B and hence on Im n2B C n l Z o .
However
here the operation is by conjugation and thus the operation of G on Im n2B
induced by the exact sequence
Im n2B
is also nilpotent.
-TI
Z
l o
-
Since G, as a subgroup of
we infer from Proposition 1.4.1 that r1Z0
Conversely, suppose that Zo
G
IT
X
1
x TI
Y, is nilpotent,
1
is nilpotent.
is nilpotent. Then an immediate
application of Proposition 1 . 4 . 3 to the Mayer-Vietoris sequence
Localization of homotopy types
96
shows that nlZo operates nilpotently on nnB, n 1 2 .
Of course, it is most useful to have a criterion for Z
to be
nilpotent which is independet of the maps uo, vo, but depends only on (7.1).
Thus we now enunciate
Corollary 7.6.
Let ( 7 . 1 ) be a diagram i n NH.
Then, i n the weak pullback
(7.3) i n Ho, Zo C NH.
The following immediate consequence of Theorem 7.4, generalizing
Theorem 2 . 2 , 1s also useful.
Corollarv 7.7.
Let ( 7 . 1 ) be a diagram i n Ho with X, Y C NH.
If X
OP
Y
i s 1-connected, then Zo C N-l.
We now suppose that (7.3) i s a weak pullback in N a n d we localize
at the family of primes P.
We obtain
yp
Diagram (7.8) i s a weak pullback in NH.
Proposition 7.9.
Proof.
Form the pullback in To
(7.9)
where we may
8:
Zop+ 2'
assume fp to be a fibration. There is then a map
yielding a commutative diagram
Mixing of homotopy types
in
91
NH, andhence a map of t h e P - l o c a l i z a t i o n of t h e Mayer-V
!to1
3
sequence of
(7.3) t o t h e Mayer-Vietoris sequence of (7.9); h e r e Theorem 1.2.10 p l a y s a
c r u c i a l r o l e i n e n s u r i n g t h a t , when we l o c a l i z e
G
i n (7.5) w e o b t a i n
t h e p u l l b a c k of t h e diagram
K1yP
-rB
rigp
1P
I n t h i s map of Mayer-Vietoris sequences a l l groups e x c e p t
mapped by t h e i d e n t i t y .
Thus
s
rnZoP
are
i n d u c e s an isomorphism of homotopy groups
and hence is a homotopy-equivalence.
Suppose, i n (7.31,
C o r o l l a r y 7.10.
Then Zo
€ Mi
and
Proof.
that
f
is a P-equlvalence and X, Y, B
is a P-equivalence.
vo
We a l r e a d y know t h a t
Z
€ NH
i s an e q u i v a l e n c e so t h a t , by P r o p o s i t i o n 7.9,
v
€
by C o r o l l a r y 7 . 6 .
vop
Now
fp
is an e q u i v a l e n c e and so
i s a P-equivalence.
Of c o u r s e , t h i s c o n c l u s i o n could more e a s i l y have been drawn w i t h o u t
e s t a b l i s h i n g P r o p o s i t i o n 7.9 i n f u l l g e n e r a l i t y .
We w i l l b e i n t e r e s t e d i n e s t a b l i s h i n g c o n d i t i o n s under which w e may
i n f e r t h a t t h e space
Z
i n (7.2)
is a l r e a d y connected, so t h a t
Z = Zo.
NH.
Localization of homotopy types
98
Obviously this holds if (7.1) is a diagram in To
in which f
(or g)
induces a surjection of fundamental groups. However, we will require a more
general criterion.
Proposition 7.11. Let ( 7 . 1 ) be a diagram i n
nlB i s of the form
17.21 in
H,
f*a.g,f?,
c1
E
Ti
To
i s which every eZement of
Then i n the weak pullback
X, f? E nlY.
1
i s connected.
Z
Proof.
Let us assume f a fibre map, so that ( 7 . 2 )
pullback in T. Given (x,y) E Z, x E X, y E Y, let k
o
to x, and m
a path in Y
from o
reverse of m, is a loop in B on
p
0 , so
is the strict
be a path in X
to y. Then fII *gi, where
i
from
is the
in X,
that there are loops h
in Y with
-
Thus f(x
* L)
-
fk*gm- fh*gp.
g(p *m), re1 endpoints,and, since f is a fibre map, we find
L' * h * II, re1 endpoints,
that
(II',m')
so
that
f&' = gm', where m' = p *m.
is a path in Z from o
to
It follows
(x,y).
We exploit Proposition 7.11 in the following way.
Let ( 7 . 2 ) be a diagram i n To i n which f,: nlX * TIIB is a
Corollary 7.12.
P-surjection and
of the primes.
g,:
TI
Y
1
-+TI
B i s a Q-surjection for some p a r t i t i o n
1
Then, i n (7.21,
Proof.
Z
yn = g,n
P
for m E Q, n € P.
are relatively prime we find integers k, II with km
II
and then y = f,Sk* g,n ,
+ an =
We are now ready to prove the mixing theorem which is the main
objective of this section.
=
i s connected.
Let y E alB. Then ym = f,S,
Since m, n
n
1
u
Q
Mixing of homotopy types
Let
Theorem 7.13.
with Xo
X , Y C NH
of the primes. Then there exists
2
0
X
and Let rI
Yo
with
Z C NH
There a r e c a n o n i c a l maps
Proof.
h: Y
2
99
s:
Zp
Xp
+
2
= P U Q
$,
Xo,
ZQ
t: Y
N
Q
Y
-f
be a partition
9'
Let
'0'
and c o n s i d e r t h e diagram
0
1
(7.14)
Y-%
Q
xO
Form t h e weak p u l l b a c k of ( 7 . 1 4 ) ,
Certainly
s
i s a Q-equivalence and
7.12 e n s u r e s t h a t
Corollary
Thus
u
7.10 g u a r a n t e e s t h a t
induces
up: Zp
2
Xp
u
and
i s a P-equivalence.
Thus C o r o l l a r y
Corollary 7.6 then ensures t h a t
i s connected.
Z
ht
i s a P-equivalence
v
induces
vs:
and
v
2 E NH and
i s a Q-equivalence.
ZQ r- YQ.
The following addendum i s important i n a p p l i c a t i o n s .
(i) Let X ,
P r o p o s i t i o n 7.15.
Z
Y
in Theorem 7.13 be quasifinite. Then
is quasifinite.
lii) Let X , Y in Theorem 7.13 be 1-connected. Then
Z
is 1-connected.
liiil Let X,
Y
in Theorem 7.13 have the homotopy type of
a finite 1-connected CW-complex. Then
Z
has the hornotopy type of a finite
1-connected CW-comp Zex.
Proof.
(i) Observe t h a t i f
generated L -module and
P
by Theorem 1.3.10,
is quasifinite.
A
A
Q
A = WiZ
then
is a finitely-generated
%
is a f i n i t e l y -
%-module.
Thue,
is a f i n i t e l y - g e n e r a t e d a b e l i a n group, so t h a t 2
100
Localization of homotopy types
(ii) Observe that
nlZ
is a nilpotent group which l o c a l i z e s t o
the t r i v i a l group a t every prime and hence is c e r t a i n l y t r i v i a l .
(iii) This follows from (i) and (ii),using the techniques of
homology decomposition.
Chapter 111
A p p l i c a t i o n s of l o c a l i z a t i o n t h e o r y
Introduction
I n t h i s c h a p t e r , we p r e s e n t some a p p l i c a t i o n s of t h e t h e o r y developed
i n the previous chapters.
I t would seem t h a t t h e r e are two main d i r e c t i o n s
a l o n g which t h e a p p l i c a t i o n s of l o c a l i z a t i o n t h e o r y should proceed.
First,
w e may w i s h t o s t u d y a problem concerning ' i n t e g r a l ' s p a c e s and maps by p a s s i n g
t o t h e corresponding l o c a l i z a t i o n s .
The l o c a l i z e d s p a c e s and maps o f t e n have
much s i m p l e r s t r u c t u r e , t h e r e b y making t h e l o c a l i z e d problem more t r a c t a b l e .
As a s i m p l e example, r e c a l l from Chapter I1 (Example 1.8) t h a t i f
t h e S t i e f e l manifold of 2-frames i n 7-space,
X
P
is homotopy e q u i v a l e n t t o
S",
P
t h e n f o r any odd prime
X = V
792
p,
t h e p - l o c a l i z a t i o n of t h e 11-sphere.
example arises i n o u r s t u d y i n S e c t i o n 3 of n o n - c a n c e l l a t i o n
is
phenomena.
This
As
a n o t h e r example, in connection w i t h o u r s t u d y of f i n i t e H-spaces i n S e c t i o n 2,
we are a b l e t o show, by p a s s i n g t o t h e l o c a l i z e d s i t u a t i o n , t h a t c e r t a i n
c a n d i d a t e s i n f a c t f a i l t o admit H-space s t r u c t u r e s .
L o c a l i z e d s p a c e s are n o t o n l y s i m p l e r t h a n t h e i r a n c e s t o r s , b u t are,
in a s e n s e , more ' f l e x i b l e ' .
The r i c h e r symmetry o f l o c a l s p a c e s stems from
t h e f a c t t h a t Z*, t h e group of u n i t s of
P
Z?
P'
which a c t s on t h e p - l o c a l
s p h e r e s , and on t h e homology and homotopy groups o f p - l o c a l s p a c e s , is v e r y
l a r g e , whereas
E* = {tl}. A v e r y s u b t l e example of t h i s symmetry, which
w i l l n o t b e d i s c u s s e d i n d e t a i l i n t h i s monograph, is S u l l i v a n ' s theorem t h a t
if
p
is a prime and
k
a number which d i v i d e s
p
-
1, t h e n t h e
S2k-1
admits a loop space s t r u c t u r e . Actually,
P
i2k-1
S u l l i v a n proves t h i s f i r s t f o r t h e p - p r o f i n i t e completion
of t h e
P
p-localized
(2k-l)-sphere,
(2k-l)-sphere
by making j u d i c i o u s u s e of t h e s t r u c t u r e of t h e group of u n i t s
of t h e p-adic i n t e g e r s , and t h e n u s e s t h e r e l a t i o n s h i p between l o c a l i z a t i o n
and p r o f i n i t e completion t o deduce t h e r e s u l t f o r
S2k-1
P
Applications of localizationtheory
102
The second type of application of localization theory, and one in
which we are particularly interested, derives from the fact (see, e.g.,
Example 1.2) that a space X
in NH is not determined uniquely in general by
the family of its p-localizations X although X can be reconstructed from
P)
the rationalizations Xp -+ Xo in case X is of finite type ( s e e Theorem
11.5.1 and subsequent remark).
We are thus able to construct many new
examples of spaces exhibiting various types of phenomena. Particularly
noteworthy in this regard is the construction of several sorts of exotic
finite H-spaces, a program pioneered by Zabrodsky.
The organization for the rest of the chapter is as follows. Section 1
discusses the concept of the genus of a space of finite type X in NH, which
by definition is the collection of all homotopy types Y
in NH which are
of finite type, such that the p-localizations of X are homotopy equivalent
to the corresponding p-localizations of Y. We illustrate and give some
general theorems concerning this notion and, in particular, study the possibility
of furnishing a space with a structure which is present in all of its localizations. In Section 2, we are concerned with the theory of finite H-spaces.
While we do not present an exhaustive study of the construction of new
finite H-spaces --we do not
enter into
some
of the more technical aspects
of the theory, such as An-structures--we do discuss in some detail the rank 2
case, where the classification problem is essentially solved, and give a
sampling of the sorts of strange behavior which finite H-spaces, in contrast
with Lie groups, may exhibit. In Section 3 , we discuss the relationship
between localization theory and the non-cancellation phenomena first discovered in
Hilton-Roitberg [451. The existenceof sucharelationship is not surprising inview of
the connection between the non-cancellation phenomena and rank 2 H-spaces.
Indeed, as we attempt to show, localization theory sheds considerable light on
Introduction
the examples of non-cancellation, and conversely our main theorem concerning
non-cancellation offers an excellent opportunity for application of the
fundamental Theorem 11.5.1.
103
Applications of localization theory
104
1.
Genus and H-spaces
We have seen in Chapter I1 that a space X C NH determines a
family {X Ip € l-l} of p-local spaces, its p-localizations, together with a
P
X Ip € II}, the rationalization maps. Moreover,
family of maps {rp: X
P
O
we have observed in case X is of finite type that X may be reconstructed
+
in a suitable way out of these two pieces of data (TheorenrsII, 5.1, 11.5.7).
However, if one is given a collection of rationally equivalent p-local
-
n}
such that H (Y 72 ) is a finitely generated
i P’ P
Z -module for all i and all p, one cannot deduce in general that there is
P
only one ‘integral‘ space X (i.e., a space X of finite type) whose
spaces
{Yplp C
are homotopy equivalent to Y
P
P
the following fundamental definition.
p-localizations X
Definition 1.1.
The genus G ( X )
for all p.
This prompts
of a space of finite type X € NH,
is the
collection of all objects of finite type Y ENH such that
X
P
%
Y
P’
for all p € II.
Further, a homotopy-theoretic property is said to be a generic property if it
is shared by all or none of the members of a genus.
Our definition of the genus G(X)
differs from the one originally
given in Mislin[59] in thatwe require afiniteness condition. The definition
chosen in this way in order that the genus sets should not be too big.
Actually all presently known examples of genus sets G(X)
complex are finite sets.
For instance, G(S1)
= IS1}
with X
a finite
whereas there exist
infinitely many different homotopy types X € NH with X “1 S1 for all
P
P
primes p, because there are infinitely many non-isomorphic abelian groups
A with A E Z for all primes p (cf. examples following Theorem I. 3.13).
P
P
We observe (using Theorem I. 3 . 1 4 ) that members of the same genus
Genus and H-spaces
105
have abstractly isomorphic homology and higher homotopy groups, so that
we may describe these groups as generic, or genus invariants.
However, their
fundamental groups are not necessarily isomorphic, unless they are abelian.
For instance, let X = K(N
1) and Y = K(N7/12,1) where
and
1/12'
N1 112
N 7/12 are the groups described following Theorem 1.3.13. Then Y C G(X)
but rr Y * n X.
1
1
From the results proved in Chapter I1 concerning the localization
of products, wedges, suspensions, loopspaces and mapping spaces, it is immediate
that for X, Y
and 2
spaces of finite type in NH
and W
a not necessarily
connected finite complex, the equality of genus sets G(X) = G(Y)
G(x
(i)
x
G(Xk)
(ii)
2) = G(Y x 2)
= G(Yk)
(iii)
G(ZX) = G(cY)
(iv)
G ( k ) = G(EY)
(v) G ( x W , o )
In case X, Y
v
k
2
implies that
= G(Yw,o)
and 2 are in addition 1-connected and if we denote by
the k-fold wedge of a space 2, then one can also conclude that
(vi) G(X
V 2) = G(Y V 2)
(61.0 G ( v k = G(v%
It is not known whether being of the homotopy type of a finite
complex is a generic property; but certainly quasifiniteness is a generic
property.
Example 1.2.
or if X
If X
is a sphere Sn, or more generally, a Moore space K'(A,n),
is an Eilenberg-MacLane space K(A,n)
abelian group and n
Theorem 1.3.14.
2
with A a finitely-generated
1, then we have G(X) = {X).
This follows from
106
Applications of localization theory
Let
Example 1.3.
element
€ n
LY
n-1
be t h e mapping cone
X
p.
Cka
if
p
r 2 2, of a homotopy
In t h i s c a s e
given by
that,
uLYe n ,
S r, which we suppose f o r s i m p l i c i t y t o be i n t h e s t a b l e range
and of prime o r d e r
so
Ca = Sr
-
k
denotes an odd prime, t h e c a r d i n a l i t y of t h e s e t
G(CLY)
i s (p-1)/2.
To prove t h e s e a s s e r t i o n s , we begin by showing t h a t i f
then
Y
has t h e r e q u i r e d form.
t h e obvious f a c t t h a t
Y
Y € G(CLY)
I t i s c l e a r from homology c o n s i d e r a t i o n s and
must be 1-connected, t h a t
Y may be put i n t h e
form
Y
for a suitable
rr
cB
=
sr uB en,
For every prime
B € T,-~S~.
h = h ( q ) : (C )
a q
Assuming
h
+
q, we have a homotopy equivalence
(CB)q.
c e l l u l a r , and invoking a c l a s s i c a l argument ( H i l t o n [ 3 3 1 ) , we
deduce a commutative diagram
with
u = u(q)
and
v = v(q)
of (1.4) r e a d i l y i m p l i e s
t h e desired conclusion
Bq
homotopy e q u i v a l e n c e s .
= 0
for
The l e f t hand s q u a r e
q # p; and, t a k i n g
q = p , we o b t a i n
Genus and H-spaces
since then u and v may be viewed as elements of
107
Z*
P
and we are in the
stable range. The same argument, applied in the unlocalized situation, proves
the assertion that
C =C
iff k
a
ka
3
51 (mod p ) .
It remains to show that any
G(Ca).
Now the composite
sn-l
of ka
-ska
r
-c
and the inclusion map is trivial,
Sn-l
sr
ka
kl
11
a
sn-l
Since
Cka, (k,p) = 1, actually belongs to
sr
~
so
there is a commutative diagram
@
I
k
a
,
ca
(k,p) = 1, it follows by the Five Lemma that
@*:H,(Cka;
ZP) H,(C,;
+
ZP)
is an isomorphism and hence that
:
(cka)p
-+
(CaIp
@P
is a homotopy equivalence. Since for q # p
(C )
aq
= sr v sn = (C
9
q
we have
)
kaq
our assertion is established.
For more results on the genus of complexes with two cells the reader
is referred to Molnar [631.
Remark. One gets similar examples in the dual situation, i.e. using 2-stage
Postnikov systems instead of spaces with two non-vanishing homology groups.
It is to be expected, of course, that many homotopy-theoretic
properties are In fact generic properties. We will illustrate this with a
few examples.
Recall the following definition.
Applications of localizationtheory
108
D e f i n i t i o n 1.5.
A space
n > 0
i s c a l l e d reducible provided t h e r e e x i s t s an
X
and a map
f : Sn
i s an isomorphism f o r a l l
i 2 n.
integer
A space
+
such t h a t
X
We s a y i n t h i s c a s e t h a t
i s c a l l e d S-reducible i f f o r some
X
Hi(Sn; Z) -+ Hi(X; Z)
f,:
k 1 0, C k x
reduces
f
X.
is reducible.
F o r i n s t a n c e , t h e Thom space of t h e normal bundle of a c l o s e d m a n i f o l d
embedded i n some Euclidean space i s r e d u c i b l e .
then
is S-reducible (Browder-Spanier
X
Let
Theorem 1 . 6 .
If
i s a f i n i t e H-complex,
X
[16]).
X € NH be of f i n i t e type and
Then
Y € G(X).
( i ) X i s reducible i f f Y i s reducible,
(iil
X i s S-reducible i f f
Proof.
Hi(X; Z) = 0
Further,
that
X
TI
Suppose
for
i > n.
can
n Y
f:
Sn
Hence
-f
X
reduces
Hn(Y; Z) E L
HI(Y; Z)
and
H (X; Z) is s u r j e c t i v e , and t h e r e f o r e
Hn(Y; Z) i s s u r j e c t i v e by Theorem 1 . 2 . 1 ( i i ) .
t o b e a counterimage of a g e n e r a t o r of
( i n t h e homological s e n s e ) and
m
Recall that
X
-
0
Y € G(X)
p.
k
C X
d u a l i t y w i t h (untwisted) i n t e g r a l c o e f f i c i e n t s .
i > n.
implies
Hence
Choosing
sn
g:
g
-f
Y
reduces
i s S-reducible i f f
X
is r e d u c i b l e f o r
is a Poincare? complex i f
and
for
Hn(Y;Z), i t i s c l e a r t h a t
To g e t t h e second a s s e r t i o n , w e observe t h a t
dim X <
H (X; Z) G Z
Then
X.
Hn(Y ; Z) i s s u r j e c t i v e f o r a l l primes
P
n P
nnY
Y.
i s S-reducible.
Y
X
k
-
dim X
s a t i s f i e s Poincar6
A Poincard complex n e c e s s a r i l y
h a s finitelygeneratedhomology groups (Browder [ 1 5 ] ) , a n d h e n c e i s q u a s i f i n i t e .
Theorem 1.7.
Y
Let
X E NH be
a Poincar6 complex and l e t Y
€ G(X).
Then
i s a Poincare? complex.
Proof.
cap product
Let
p € Hn(X;
+ 1.
Z) G Z be a fundamental c l a s s
so t h a t t h e
Genus and H-spaces
i
nu: H (X; Z)
is an isomorphism for all i.
tl
E H (Y; Z) Z Z
Hn-,(X;
Z)
If Y E G ( X ) , we pick a generator
and attempt to show that
is also an isomorphism for all i.
equivalence
-f
h ( p ) : X,
+
Y
0
.
For each prime p
choose a homotopy
From this we deduce a commutative diagram
!J
is a
*P
for some w(p) € Zt. This
which proves that nh(p),pp
is an isomorphism. Hence h(p)
generator of Hn(Yp; Z ) and v = o(p)h(p),pp
P
P
proves that nv
is an isomorphism for all primes p. Since nv = (flvIp,
P
P
this proves that nv is an isomorphism and hence Y is a Poincar6 complex.
Remark.
It may well be conjectured that the properties 'having the homotopy
type of a closed manifold' and 'having the homotopy type of a closed rr-manifold'
are generic properties. The latter conjecture may indeed be verified in certain
cases, e.g.when
X isl-connected and dim X is odd or of the form 4k with k > l ,
by using the Browder-Novikov Theorem in conjunction with Theorems 1.6 and 1.7.
We are going to give much attention to the question whether the
property of admitting an H-structure is generic. For aconnectedH-space X with
multiplication u: X
f o r which
e :X
P
from u .
x
X, there is a unique H-structure p(p) : X x X -+ X
P
P
P
is an H-map. We call p(p)
the H-structure induced
X
+
X
P
It follows from the universal property of p-localization,
+
on X
P
that ~ ( p ) is homotopy-associative (homotopy-commutative) if
p
is.
Applications of localization theory
110
N
Similarly, i f
is a c o n n e c t e d l o o p s p a c e , s a y f : X-QY,
X
loop s t r u c t u r e s f ( p ) : X
P
7ClYP
thenthere existunique
e :X
P
f o r which t h e maps
-+
X
are l o o p maps.
P
If c o n v e r s e l y one wants t o p r o d u c e a n H-space s t r u c t u r e o n X
H - s t r u c t u r e s on t h e p - l o c a l i z a t i o n s
one n e e d s a c e r t a i n ' r a t i o n a l c o h e r e n c e '
P'
More p r e c j s e l y we h a v e
condition.
Let
Theorem 1.8.
X C NH
Lie of f i n i t e type and suppose t h a t each X
equipped with an H-structure
H*(X;Q),
X
from g i v e n
induced from e : X
P
H-structure
f o r which
p
furthermore,
~ ( p ) such that the Hopf algebra structure on
-+
X
i s independent of
P'
ep: X
X
-+
(homotopy-commutative) then so i s
a Loop space, then the
for
Proof. By Theorem 1 1 . 5 . 7
If,
p.
~ ( p ) i s homotopy-associative
FinaZZy, i f i n addition each
p.
same i s t m e
Then X admits an
p.
i s an H-map f o r a l l primes
P
i s q u a s i f i n i t e and each
X
is
P
X
P
is
X.
X
is homotopy e q u i v a l e n t t o
-
X, t h e
(weak) p u l l b a c k of
nxP
liX
P
is a n H-space and
induced by
r.
r
(flxp)o
is a n H-map,
A xo;
i f we equip
b e induced by t h e p r o j e c t i o n
X
H - s t r u c t u r e on
P,O
liX
P
+
i n d u c e d from
-+
.
X .
P
X
P
(XpIo
-+
X
=
xP
Clearly
10
+(p)
is a n H-map f o r t h e
The c o n d i t i o n s t i p u l a t e d f o r
e q u i v a l e n t t o t h e e x i s t e n c e of a n H - s t r u c t u r e
r :X
w i t h the H - s t r u c t u r e
Let
+(p): o x p ) o
maps
fiXp)o
~ ( 0 )o n Xo
X
is
f o r which a l l t h e
are H-maps o r , e q u i v a l e n t l y , f o r w h i c h a l l t h e c a n o n i c a l
P
P
0
homotopy e q u i v a l e n c e s
X(p): X p,o
-+
Xo
are H-maps (for a more g e n e r a l
s t a t e m e n t c o n c e r n i n g t h e r e l a t i o n s h i p between H-maps and induced maps i n
homology compare Lemma 1.15).
Now t h e c o m p o s i t e map Xo
--t
@Xp),
can
-t
""P
,O
Genus and H-spaces
h a s components
{A(p)-'lp
€
n);
111
hence, by.assumption, it is an H-map.
Since, for a finitely generated abelian group
G,
is a split monomorphism (Theorem I.3.8), we deduce, using the fact that X
is of finite type, that the canonical map
are the maps
$(p)
above, is a (split) monomorphism in the homotopy category.
is an H-map, and therefore
Clearly $
flXp)o -+iiXp,,, whose components
$:
must be an H-map. The first
p
assertion of the theorem then follows.
is quasifinite and each ~ ( p ) is homotopy-associative
In case X
(homotopy-commutative), we obtain the corresponding property for v : X
by applying Theorem 11.5.1
(ps, p:
X
x
X
+
S
is the switching map).
In case one has in addition H-homotopy equivalences ~(p): Xp
J/
*
5
IIK(P),
Y
+
W(p)
a prime or 0, one gets a commutative diagram
rnY(p)
with
+
to
X respectively; here
for certain Y ( p ) C NH, p
X
x
induced by
- +
@
and
-c
A
3(i=iY(p))o
X =
fiY(0)
Yyp~(o)-~,
It remains to
prove that A
is a loop map; then the conclusion of the theorem is evident.
Certainly, A
is an H-map, since K ( o ) ,
k
if nY(o) = II K(Q, ri)
i=1
.
Hence, A
p
and
Y
being an H-map,
are. Notice that
X
Applications of localization theory
112
[A] = [A1]
+
... + [ A k ]
E H*(QY(o);A)
is a sum of p r i m i t i v e elements i n
hA(xl , . . . , ~ k )
%
H*(nY(o) ;A).
Thus
l i e s i n the
[A]
image of t h e cohomology s u s p e n s i o n , and i t f o l l o w s t h a t
is a loop map.
A
Remark.
I f we a r e only i n t e r e s t e d i n Theorem 1 . 8 i n t h e q u a s i f i n i t e c a s e ,
then we
can prove a l l b u t t h e f i n a l a s s e r t i o n of t h e theorem simply by
applying Theorem 11.5.1
t o get the H-structure
unique map which induces
p ( p ) : Xp
x
X
P
-+
X
P
1~: X x X
a t each prime
coherence c o n d i t i o n s t i p u l a t e d i n Theorem 11.5.1
maps
r : (XP,p(p))
P
+
(Xo,p(0))
+
X
as the
p.
The n e c e s s a r y
is f u l f i l l e d , s i n c e t h e
a r e H-maps.
I n t h e q u a s i f i n i t e c a s e Theorem 1.8 may b e g e n e r a l i z e d t o t h e e x t e n t
t h a t one r e q u i r e s t h a t t h e Hopfalgebra s t r u c t u r e s on
ep: X
+
Xp,
a r e isomorphic, r a t h e r t h a n e q u a l .
H*(X;Q), induced by
This g e n e r a l i z a t i o n is c r u c i a l
t o deduce t h a t a q u a s i f i n i t e H-complex admits an H-space s t r u c t u r e i f
admits a homotopy-associative H-space s t r u c t u r e f o r a l l primes
X
P
p ( s e e Theorem 1 . 7 ) .
It aeemsnot t o b e knownwhether thehomotopy-associativityconditioncould bedropped
i n t h i s assertion.
However one can prove t h a t being of t h e homotopy type of a
q u a s i f i n i t e H-complex is a g e n e r i c p r o p e r t y , and t h i s i s our n e x t o b j e c t i v e .
For t h i s , we w i l l f i r s t need some lemmas, which a r e of g e n e r a l use i n t h e s t u d y
of f i n i t e H-complexes.
R e c a l l t h a t f o r a n a r b i t r a r y connected s p a c e
is a coalgebra w i t h d i a g o n a l
F, H,(X;F)
w i t h c o u n i t induced by
that for
x E G*(X;F)
A,x
The elements
-
x 8 1
X
-+
0,
A*
X
induced by
and a r b i t r a r y f i e l d
A: X
-f
t h e projection onto t h e basepoint.
X x X , and
It f o l l o w s
one has
+ 1-3 x + Exi
x E H,(X;F)
63 yi;
such t h a t
A,x
deg xi, deg yi < deg x
= x 8 1
+ 1 8x
a r e , naturally, called
Genus and H-spaces
primitive elements.
a map
f: X
+
so t h a t , i f
They form a l i n e a r subspace
and an element
Y
x
x € H,(X;F),
is p r i m i t i v e ,
is a l s o p r i m i t i v e .
by
113
A,f,x
W
e d e n o t e by
=
Pf,:
f,x
0
wehave
1
PH,(X;F)
+
5 H,(X;F).
PH,(X;F)
= (f,@f,)A,x,
A,f,x
1 8 f,x
and hence
PH,(Y;F)
-+
For
f,x
t h e map induced
f,.
Let
Lemma 1.9.
a map.
X and
Then f,:
be arbitrary connected spaces and
Y
H,(X;F)
-+
i s one-one if Pf,:
H,(Y;F)
f: X
PH,(X;F)
+
Y
-+ PH,(Y;F)
i s one-one.
Proof.
Now every element i n
t h a t w e have e s t a b l i s h e d t h a t
let
w C Hi+l(X;F).
then
= 0
f,w
A,w
If
f,
is p r i m i t i v e .
Suppose, t h e n ,
is one-one i n dimensions 3,i P 1, and
= to 8 1
implies t h a t
H1(X;F)
A,f,w
+
1 8w
+
Zv
= (f,@f,)A,w
j
Qw
j'
8 1
= f,w
, deg w j 5 i,
j
1 Q f,w
deg v
+
+
Zf v 8 f w = 0. Thus Zf v 8 f w = 0 and t h e r e f o r e Zv a 9 w = 0 , s i n c e
"j
* j
* j
"j
j
j
f,
i s one-one i n dimensions 3 . T h i s means t h a t w is p r i m i t i v e and hence
w = 0.
If
Y
is a n H-space w i t h m u l t i p l i c a t i o n
u: Y
x
Y
-+
Y, then
H,(Y;F)
is a (not n e c e s s a r i l y a s s o c i a t i v e o r commutative) H o p f a l g e b r a , w i t h d i a g o n a l
A,,
u,
and m u l t i p l i c a t i o n
Lemma 1.10.
tmo maps.
Let
u.
be an arbitrary space,
X
Define
induced by
fg: X
+
Y
Y
an H-space, and
f ,g: X
by
( f g ) a = f ( a ) - g ( a ) 6 Y , a C X,
using the muZtipZication of
Then, i f
Y.
(fg)*x = f*(x)
Proof.
f g = u(fxg)A: X
(fg),x
-+
If
!J:
Y.
Hence f o r
= u,(f,@g,)A,x
Y x Y
-+
= u,(f,x@l
Y
x E PH,(X;F),
+ g,(x).
d e n o t e s t h e H-structure map, t h e n
x E PH,(X;F)
+
1
one h a s
= f,(x>
+ g,(x).
+
Y
1 I4
Applications of localization theory
A s an immediate consequence we have
Lemma 1.11. Let
inductively by
one has
Y be an H-space and
$(k)(y)
=
$(k): Y
y*$(k-1) (y), $‘(y)
-+
= y.
Y t h e k-power map, defined
Then, f o r
x E PH,(Y;F),
$ik)x = kx.
N
Define, by abuse of language,the k -power map
inductively by
~$(~”)(y)
= $ (k) ( p N - 1 ) (y)),
Y
$(k’N):
+
Y
$(kJ) (y) =. $(k) ( y) .
Let Y be a connected H-complex of f i n i t e homological dimension
Then the pN-power map $(pBN) = $(p) ,
Lemma 1.12.
N.
is a
{PI’-equivalence and induces
Proof.
The map
$(p)
induces, in homotopy, multiplication by pN
and is
therefore certainly a {el’-equivalence. We prove the second statement by
induction. If x 6 H1(Y; Z / p ) , then $ip)x = px = 0, since all elements of
H1 are primitive. Suppose $ ( p s i )
3 , where
i 2 1, and let y € Hi+l(Y; Z / p ) .
Then A*y = y
Hence A*$ipyi)y = $ie’i)y 8 1
deg Yj’ deg
$ip*i)(y)
induces zero on H*(Y; Z/p) in dimensions
= 0
1
+
1 8y
+ 1 8 $!psi)y
j
i*
is primitive. Lemma 1.11 shows that $(*psi+l)(y) =
p*$2yi)(y) = 0. Thus $*
€9
in dimension i
$iP)
€9
=
it is certainly
0 in dimensions 3 . This establishes the inductive hypothesis and the result
follows immediately.
From this lemma we can deduce the following consequence.
Lemma 1.13. Let
Denote hy
P
Y be a q u a s i f i n i t e connected H-compzex and l e t
a f i n i t e s e t of primes.
h(P):
X
+
X € G(Y).
Then there i s a P-equivalence,
Y.
z
and therefore
($*(p’i)(y))
+ 1; but
+ Cyj
j’
Genus and H-spaces
Proof.
p -equivalence
i
P = {pl, . . . , p m}
Let
f(i): X
g ( i ) = $(p,)
where
$(p )
1
0
+
and choose, f o r each
( s e e Theorem 11.6.6).
Y
$(p2)
... $A
( P J . ...
0
N
p -power map
denotes t h e
115
j
is a p -equivalence and, f o r
i
Y
p C P-{pij,
0
+
pi C P , a
Define
$(pm) o f ( i )
Y, a s i n Lemma 1.12.
Then
g(i)
one h a s
Define
where t h e product is performed in t h e monoid
We claim t h a t
h(P)
is a P-equivalence.
i t is enough t o prove t h a t , f o r
p
[X,Y], u s i n g some f i x e d b r a c k e t i n g .
Because of t h e f i n i t e n e s s assumptions
C P , h(P),:
j
h*(X; Z / p j ) +H,(Y;
B y L e m m a 1 . 9 i t i s e n o u g h t o check t h i s on p r i m i t i v e s .
be a p r i m i t i v e element.
u s i n g Lemma 1.10.
x = 0.
Hence
.i
x E H,(X; Z / p )
j
Then
h(P),x
= Pg(i)),x
Since
g(j)
h(P)
So l e t
Z / p ) i s one-one.
= zg(i),x
= g(j),x,
is a p -equivalence
j
h(P),x
= 0
implies t h a t
is a P-equivalence.
We a r e now ready t o prove t h a t b e i n g of t h e homotopy t y p e of a
q u a s i f i n i t e H-complex is a g e n e r i c p r o p e r t y .
Theorem 1.14.
Let
(i) X
(ii) If
Y
be a quasifinite H-complex and l e t
X € G(Y).
i s a quasifinite H-complex.
G(Y)
then there e x i s t s a
x
x
W".Y
x
z.
w
such that
Then
Applications of localization theory
116
Proof.
Choose a quasifinite H-complex V E G(Y), for instance V
and choose a rational equivalence B: X
integral homology B,
X
+
K = nK( Z,m,)
(see Theorem 11.6.6).
V
has a finite kernel and cokernel, B
for T a cofinite set of primes.
A:
-+
=
Y,
Since in
is a T-equivalence
Further choose a rational equivalence
such that A//:
nlX
+
II
1K is surjective; such a X
exists since H*(X;Q)
h(x l,...,x ) , an exterior algebra on odd-dimensional
n
generators. Clearly X is a nilpotent map, whose fibre has finite homotopy
as a composite of u and inducedfibrations A
groups. Therefore we can factor A
j’
where A
has fiber K( L/p n ) and a is (2N-1)-connected, N being the
j
j’ j
homological dimension of Y. Let S = T’ u {p 10 5 j < r} and let a = h(S) : X
j
Further let $j = $ ( p j ) : V
be an S-equivalence (see Lemma 1.13).
N
p -power map. Form
+
V be the
j
$ =
... o
$ r - l ~$ r - 2 ~
$o
and
X
Y
=
$j-lo$j-20...o
$oy
1 5 j < r.
We will prove that
{cr,B}:
+
V
x
has a homotopy left inverse. Notice that a
$6
is an S-equivalence and
is an S’-equivalence. Hence {a,$B}*: H*@ XV; L) -+ H*(X; L) is
surjective, since, for trivial reasons, H*(Y XV; Lp) H*(X; L ) is
P
surjective for all primes p . We conclude that there is a map p: Y x
-+
such that
-
!.I o ~ a , $ B ~ A:
X
+
To solve our lifting problem
X
I
X
,*l I
K.
v
+K
+
Y
Genus and H-spaces
I17
it is enough to prove that we can find a lifting in the following typical
diagram (i 11) :
Here ui = Xi-l o Xi-*
0..
.o
X ou
ui+l
and
may be assumed inductively to
exist such that
and
ur
= P.
We have an exact cohomology sequence for the map
(where we write the ‘relative’group in traditional notation), which breaks
up into short exact sequences
0
-
H*(YxV.,X; Z/pi)
-%
<a*’,
($i8)*>
H*(Y; Z/pi)
@
* H*(X;
H*(V; Z/pi)
z/pi) --t 0
ni+l
x
Tf
C H
(YxV,X;Z/pi)
then it follows from
(Di)
lies in
But
im(184Jt).
denotes the obstruction to the factorization pi,
and the naturality of the obstruction that 6x
$* = 0 on reduced cohomology with
by construction. Hence 6x = y
@J
is a pi-equivalence we infer that
From this we conclude that
Z/pi coefficients,
i
Pi
retract of the H-space Y x V.
1 and
<a*,($ B)*>6x = 0 = a*y.
i
Since a
y = 0 and, 6 being one-one, x
exists in
Thus X
(Di)
and hence that X
=
0.
is a
certainly admits an H-structure.
For part (ii) of the theorem observe that Z is an H-space by
(i).
Following up the proof of (i) with V = Z, we see that
of Y x Z, with retraction map
p,
then certainly X x W
p:
Y x Z + X.
Y x Z, since Y x 2
If W
X
is a retract
denotes the fibre of
is an H-space.
Applications of localization theory
118
Remark. It is not hard,to prove that the W constructed above actually
(See Mislin [60]; see also Wilkerson [921).
also belongs to the genus of Y.
To be able to construct specific maps between rational H-spaces
we will need the following lemma.
Lemma 1.15.
Let
X, Y C NH and assume that Yo i s an H-space whose rational,
homology i s finitely-generated over Q
i n each dimension.
Consider the
canonical map
Then
lil
(iil
e i s a bijection.
Xo i s an H-space whose rational homology
If, i n addition,
i s finitely-generated over Q i n each dimension, then
a b i j e c t i o n betueen H-maps Xo
H*(Yo;Q)
+
H*(Xo;Q)
+.
induces by r e s t r i c t i o n
6
Yo and Hopf-algebra homomorphisms
*
Proof. The assumptions on Y imply that H*(Yo;Q)
is a free
graded commutative and associative algebra, on free generators {yil i C I).
If deg yi = mi, then
Y -17K(Q,mi)
Hence the elements of
in NH.
[Xo,Yo] are in one-one correspondence with families
of homotopy classes {Xo+.K(Q,mi)}
which themselves are in one-one
correspondence with algebra maps H*(Yo;Q)
+
H*(Xo;Q)
by the freeness of
H*(Yo;Q).
To prove (ii), first observe that an H-map
induces a morphism of Hopf algebras H*(Yo;Q)
+
Xo
H*(Xo;Q)
+.
.
Yo certainly
Conversely, if
Genus and H-spaces
f*
i s a map of Hopf a l g e b r a s , then t h e f o l l o w i n g diagram commutes f o r every
K(Q,n):
r a t i o n a l Eilenberg-MacLane space
Therefore t h e same diagram commutes w i t h
t o t h e i d e n t i t y map
Xo x Xo
px:
f
119
+
X
Yo
and
-t
Yo
py:
K(Q,n)
this yields
Yo x Yo
+
r e p l a c e d by
Yo.
Applied
f o u x = p Y o ( f x f ) , where
denote t h e H - s t r u c t u r e maps.
Yo
Hence
is an H-map.
w e announced
We can now prove t h e e x t e n s i o n of Theorem 1 . 7
earlier.
Theorem 1.16.
p,
X
P
Let
X E NH be q u a s i f i n i t e and suppose t h a t , f o r every prime
i s equipped with an H-structure such t h a t the maps
e : X
P
-+
X
P
Then X admits an
induce isomorphic Hopf-aZgebra structures on H*(X;Q).
H-space structure.
Proof.
We w i l l produce an H-space
f o l l o w s from Theorem 1.14.
c a n o n i c a l map
r 2 : X2
+
Xo
corresponding s t r u c t u r e map
X
Equip
w i t h t h e H - s t r u c t u r e induced by t h e
p : Xo x Xo
e xe
xxx-
Q
-+
that
rQ:
XQ.
(+p(Q))
+ (Xo,p)
and d e n o t e by
u(p,6)
e ve
Xo.
X2; c a l l the
By C o r o l l a r y 11.5.11 t h e r e
such t h a t t h e two maps
xo
x
xo -bxo
xo
v
xo
and
have unique l i f t s t o
Then t h e r e s u l t
from t h e g i v e n H - s t r u c t u r e on
e x i s t s a c o f i n i t e s e t of primes
xvx-
E G(X).
Y
7
xo
Hence w e o b t a i n a n H-structure
is an H-map.
u(p)
Denote by
u(Q)
0
XQ
such
t h e €I-structure
t h e induced H-structure on (X ) = X
P
on
P,O
on X
. The assumption on
P
Applications of localization theory
120
the X ' s implies that there are abstract isomorphisms of Hopf algebras
P
with respect to the Hopf algebra structures induced by u
and p(p,o)
respectively. By the previous lemma we can realize these isomorphisms by
H-maps K(P) : (XP,,,u(p,o))
orp = K(p):
K(P)
(Xp,u(p))
H-may. Define Y
+
(Xo,u).
+
(Xo,u)
It follows that
is a rational equivalence which i s an
to be the weak pullback of the finite
family of
H-maps
{rQY;(p);p CQ').
It is now not difficult to prove that Y E G(X)
py:
Y
(Y,uy)
x
Y
.*
+
Y
such that the canonical maps
(Xp,p(p)),
and there is an H-structure
(Y,uy)
+
(XQ,u(Q))
p € Q', are H-maps. Then X € G(Y),
so
and
that X is an H-space.
An important special case is the following, for which the rational
coherence condition is automatically fulfilled.
Theorem 1.17. Let X be quasifinite in NH and suppose that, f o r each p,
is equipped with a homotopy-aesociative H-space structure. Then X admits
P
an H-space structure.
X
Proof. By the Hopf-Samelson Theorem there is, up to isomorphism,
only one coassociative Hopf algebra structure on H*(X;Q).
Therefore the
X * X induce isomorphic Hopf algebra structures on H*(X;Q)
P
P
the result follows from the previous theorem.
maps
e :
and
There are many ways to combine the mixing theorems of Chapter I1
with the results of this section. We will only mention one theorem, which
will be used later on.
Genus and H-spaces
Theorem 1.18.
by
P
and
Q
Let
X
and
Y
be q u a s i f i n i t e complexes i n
t u o complementary s e t s of primes.
equipped w i t h H-structures such t h a t
as Hopf algebras.
121
H*($;Q)
xp
Suppose
and
NH and denote
H*(Y ;I))
Q
Then there e x i s t s a q u a s i f i n i t e H-complex
and
Y
Q
are
are isomorphic
2
and H-homotopy
equivalences
zp “ x p ’ z
Q
“.Y
Q‘
Proof. Choose (by Lemma 1.15) an H-homotopy equivalence
Then the (weak) pullback
Z
in the diagram
Z-Y
1
S
f Q
T ‘Q,o
is quasifinite by Proposition 11.7.15.
Since r
and
prp
are H-maps,
admits an H-structure such that the canonical homotopy equivalences
Zp -%Xp
and
Z %- Y
Q
Q
are H-maps.
Z
122
2.
Applicationsof localizationtheory
Finite H-spaces. special results
We begin this section with a discussion of H-spaces of rank 2.
Recall that if X
is an H-space of the homotopy type of a quasifinite complex
(or, as we say, a quasifinite H-space), then
the exterior algebra on odd-dimensional generators xl,
deg(xi) = ni
and assuming n1 5
..., x .
Setting
... 5 nr, we define
rank(X) = r, type(X)
=
(nl,.
..,nr).
From the point of view of applications of localization theory, the only
interesting cases for rank 2 H-spaces occur when
so
we shall restrict attention to these cases.
In fact, we shall impose the
further condition that X be 1-connected;hence X will be a finite complex.
The classical examples of 1-connected H-spaces of type (3,7) are,
of course, S 3
x
S7 and the 2-dimensional symplectic group Sp(2).
homotopy structure of
Sp(2)
3
The
has been long known and can be described as
follows. The inclusion S = Sp(1)
+
Sp(2)
gives rise to a principal
SJ-bundle
s3 -+ Sp(2)
which is classified by an element in
as an element of
TI
+
s7
3
7(BS
) G
TI (S
6
3
) 2 2/12. 'When viewed
r 6 ( S J ) , this Blakers-Massey class u
with the class of the map induced by the commutator map
may be identified
Finite H-spaces,special results
123
and is known to be a generator. We now form the principal S3-bundle
S3
classified by
ko,
SO
+
Eku+ S 7 , 0
C
k
11,
C
that Sp(2) = Ew, and state our first theorem, which
summarizes results from various sources [18,45,46,75,79,93,941.
Theorem 2.1. The space
Ekw a h i t s an H-space structure i f f
Conversely, any 1-connected f i n i t e H-space
equivalent t o some
Proof.
Assuming k # 2, 6 , 10, we show that Eko
only with the cases k = 3, 4 , 5
=
o f type (3,71 i s homotopy
Eku.
H-space structure. This is clear for k
k
X
k # 2, 6, 10.
=
admits an
0 or 1 so we need concern ourselves
since, plainly, Ekw
9
E
(12-k)w'
For
3, we use the commutative diagram
where the vertical maps are the bundle projections, the lower horizontal
maps are maps of the indicated degrees and the upper horizontal maps are bundle
maps covering the maps on the bases. Localizing the right hand square of
(2.2) at any prime p # 3 gives a homotopy-commutative diagram
w i t h the columns fibrations by Theorem 11.3.12.
homotopy equivalence for p # 3
so,
But
3
is obviously a
P
by the Five Lemma, we conclude
(E3Jp = (Ew)p, P # 3.
Similarly, localizing the left hand square of (2.2) at any prime p # 2
124
Applications of localization theory
shows that
3 7
(E ) = (S X S )p, p # 2.
3w P
Thus, (E3w)p
admits an H-space structure for all p C P
and since the
condition stipulated in Theorem 1.16 on the various comultiplications on
is evidently satisfied for dimensional reasons, we may conclude
H*(E3w;Q)
our result for E3,
from Theorem 1.16.
Very similar arguments, using the diagrams
show that Ekw and E5,
also admit H-space structures.
The proof that EL
and
Eh
do not admit H-space structures is
quite complex and will not be reproduced here.
The result is due to
[94] and uses the theory of higher order mod 2 cohomology
Zabrodsky
operations.
See also Sigrist-Suter 1771 for a proof of this result which
i s based on K-theory.
(EZwIp and
(E6w)p
a8y we
~~t~all
, can see from the diagrams
admit H-space structures for all odd primes
problem of showing that E2w and
E6-
so
the
do not admit H-space structures is,
as indicated above, a purely mod 2 problem.
Also, the diagram
shows that E2w admits an H-space structure iff the same is true of E6w'
We now sketch a proof of the converse, emphasizing those points
relevant to localization theory. To begin with, we must invoke certain general
Finite H-spaces,
special results
results on the homology o f H-spaces [ 4 6 ]
12.5
to infer that an H-space X
satisfying the hypotheses of the theorem has no homological torsion. It
then follows that X possesses a cellular structure o f the form
AS f o r the second attaching map
and, of course, a = kw, 0 5 k C 6 .
various techniques " 4 6 , 621)
satisfies
1(B) = tu,'31;
(2.4)
here, j: ng(Ca)
and
B
may be used to show that
B,
l3 €
-+
3
ng(CapS )
T3(S 3 ) = L
Whitehead product of u
is the natural homomorphism,
are generators, and
and
i3.
[u,i3] E h9(C
,S
(J
3
6 a7(Ca,S ) = L
3)
is the relative
The condition ( 2 . 4 ) may be used to show
that X has the homotopy type of an S3-fibration over
7
S ;
indeed, close
examination of the set of homotopy types o f the form ( 2 . 3 ) with 6
as in
( 2 . 4 ) reveals that there are ten such in number and that each one has a
representative which is an orthogonal
S
3 -bundle over S7.
We compare X
with the corresponding principal S3-bundle
Ekw
Since
j(@) = j ( B ' ) ,
= S3
.
Ukwe7 UB, e10
the exact sequence
shows that
Now i f
( k , 3 ) = 1, then
i = 0 because S g ( S 3 )
Thus,
X %,EL i f
( k , 3 ) = 1.
is generated by
woZ
3
W.
Applications of localization theory
126
On the other hand, if k is a multiple of
3
element, (Ch)3
N-
7
and
S3 V S 3 ,
injective. Localizing X
where
t3
[ I ~ , I ~is
] the
~
3
E n3(S ),
i7
at
3 , then
kw i s a 2-primary
i, as well as its localization i3’ is
3 , we get
3-localization of the Whitehead product of generators
7
In particular, the fibration
C a7(S ) .
s3
3 + x 3 + s 37
admits a section S
7
3
+
Combining this section with the fibre inclusion
X3.
and utilizing the H-space structure on X3
induced from that on X, we obtain
a map
4 : s33
x
s37
+
x3
which induces a homotopy isomorphism and is hence a homotopy equivalence.
Assuming $
cellular and applying [ 3 3 ] , we obtain a commutative diagram
(see (2.5))
3 7
s3vs
I 3
with u
and v homotopy equivalences. Straightforward calculation using
the left-hand square of (2.6) now shows that i3(e3) = 0, hence
i(e)
= 0, so
that
X=E
kw
3,
if k is a multiple of
and the proof of the theorem is completed.
Remarks.
1. Our proof shows that E5w E G(sp(2))
so
that, by Theorem 1.8,
E g w is a loop space.
2.
The techniques used in the second part of the proof of Theorem 2.1
Finite H-spaces, special results
127
l e a d t o a d e t e r m i n a t i o n of t h e genus s e t s of t h e s p a c e s
G(Ew) = {Eu,E5w}, G ( E ~ J = {Ekw}
Indeed, f o r any
k, Ekw
we have
Ekw:
( k , l 2 ) # 1.
if
i s a n S-reducible Poincar6 complex
(it
i s i n f a c t e a s i l y checked t h a t t h e t o t a l s p a c e of a p r i n c i p a l G-bundle, where
G
is a L i e group, w i t h b a s e s p a c e a n-manifold,
thesame is t r u e f o r any
of t h e form (2.3) w i t h
by Theorems 1 . 6 , 1.7.
X € G(Eku)
B
is stably parallelizable),So
Hence
w i l l be
X
a s i n (2.4) and t h e proof of our a s s e r t i o n i s
e a s i l y completed.
We t u r n now t o H-spaces of t y p e (3,11) and r e c a l l t h a t t h e c l a s s i c a l
example h e r e i s t h e e x c e p t i o n a l L i e group
w h i l e n o t an H-space,
In fact, i f
Note a l s o t h a t
G2.
S3
x
Sll,
becomes an H-space when l o c a l i z e d a t any prime
p
# 2.
S2k+1 is any odd-dimensional s p h e r e , t h e n t h e s o l e o b s t r u c t i o n
S2k+1
t o p u t t i n g an H-space s t r u c t u r e on
[ I ~ ~ + ~ , Iof~ a~ g +
e n~
e r a] t o r
c l a s s i c a l l y t o have o r d e r
is t h e Whitehead s q u a r e
2k+l)
1 2k+l
€ T ~ ~ + ~ ( S
52, it follows t h a t , i f
.
S i n c e t h i s is known
p
i s odd,
S2 k+l
P
admits
an H-space s t r u c t u r e .
One may t h e n apply Theorem 1.18 t o t h e s p a c e s
G2
and
S3 x S l l
t o c r e a t e new examples of f i n i t e H-spaces of type (3,ll) and, i n f a c t , t h i s
was done i n [ & I , u s i n g some knowledge of t h e homotopy s t r u c t u r e of
A
G2.
c l o s e r a n a l y s i s of t h e s i t u a t i o n w a s c a r r i e d out i n Mbmura-Nishida-Toda
[54]
and can b e s u m a r i z e d as f o l l o w s ; we u s e t h e n o t a t i o n of [411.
G2
3
may b e d e s c r i b e d a s t h e t o t a l s p a c e of a p r i n c i p a l S -bundle
over t h e S t i e f e l manifold
792
= S0(7)/S0(5),
c:S3+G
(2.7)
Since
V
V
792
2
+ V
7,2'
has t h e c e l l u l a r s t r u c t u r e
v7,2 = S5
U2 e
6
U e
11
,
128
Applications of localization theory
it follows that
11
= sp ,
(v7,2)p
where P
(see Example 11.1.8)
denotes the family of odd primes. Hence if
classifies (2.7), we may view the localization ap
Moreover, since
all(BS 3
II lo(G2)
=
0, it follows that ap
Let B be the generator of nll(BS 3 )
ap
-
V
+ BS3
7Y2
as an element of
a:
is a generator of
corresponding to ap , i.e.,
B 8 1.
Now form the principal S 3 -bundle
Sk: S3
-f
\
+
V7,2, 0 5 k 5 14,
which I s classified by the composition
-
v7,2 c
'7,2
<a
V
S1'
,(k-1) B>
BS
,
where c is the cooperation map (In the terminology of Eckmann-Hilton)
arising from the attachment of el1 to form V7,2.
S1
=
6.
We may now state one of the main results of 1541.
Theorem 2.9.
(%=Xa
X
Note that X1 = G 2 ,
Each of the spaces Xk a h i t s an H-space structwle
iff k -&t(mod
15) .)
Conversely, any 1-connected f i n i t e H-space
of type (3,111 such that H*(X;22/2)
equivalent t o some
Proof.
i s primitively generated is homotopy-
5.
We sketch a proof of the fact that each
%
admits an
H-space structure. The proof of the converse Involves extensive homotopy
calculations and so I s omitted here.
Finite H-spaces,special results
129
The structure of the classifying map (2.8) for Sk
(XkI2
=
(G2j2,
(SIP
= (Eka)p
if P
where EkB is the principal S3-bundle over S1’
assert that each
P
=
shows that
{21’,
classified by
kB. We
(EkB)p, p € P, admits an H-space structure. In fact,
according to [54],
so
that certainly
statement for any
(Ea)p, p C P, admits an H-space structure. The corresponding
(EkB)p, p € P, now follows from the type of reasoning used
in the first part of the proof of Theorem 2.1.
structure for any p €
n,
Thus,
(X,)p
admits an H-space
and since the condition stipulated in Theorem 1.8
on the various comultiplications on H*(X,;Q)
is clearly satisfied for
dimensional reasons, we conclude from Theorem 1.8 that
%
does have an
H-space structure, as claimed.
Remarks.
1. Clearly
\
€
G(G2)
iff
(k, 15) = 1. In these cases,
%
is, by Theorem 1.8, a loop space.
2. The genus sets of the
X,
are computed as follows:
Theorems 2.1 and 2.9 suggest that a fruitful source for further
examples of finite H-spaces is the collection of total spaces of principal
G-bundles over spheres and related spaces. There are in fact a number of
papers in the literature (see, for example [18,32,96J ) describing
results along these lines, We content ourselves here with stating one of
the more comprehensive results in this direction, due to Zabrodsky [961.
Applications of localization theory
130
If
dn
-
ahits an H-space structure iff
1 z 7 , then Xm(n,d)
m <s odd.
In general, it is not reasonable to expect a homotopy classification
of finite H-spaces of a given type since the homotopy calculations involved
rapidly become prohibitively difficult. For a slight indication,of what
seems to occur for H-spaces of arbitrary rank, see Mislin-Roitberg [62].
Our next result
suggested by an old result in Lie group theory,
I s
according to which the integral homology of a Lie group has p-torsion, p
a prime, i f f it has q-torsion for all primes q
5 p.
The following result
of Mislin [58] shows that this result does not generalize to arbitrary finite
H-spaces
.
Theorem 2.11. There exists a finite H-space Z
such that H,(Z;
Z)
possesses 3-torsion but no 2-torsion.
Proof.
Y = Sp(6)
Let X = F
4
x
S7
x
S1',
F4
the exceptional Lie group,
and let P = {31, Q = {31'. As F4 has type (3,11,15,23), it
is clear that X
and Y are rationally equivalent. Moreover
Xp and Y
Q
evidently admit H-space structures and the resulting Hopf algebra structures
H*(ILp;Q)
and H*(Y
Q ;Q) have to be isomorphic for dimensional reasons. It
follows that the space Z guaranteed by Theorem 1.18
Since
Sp(6)
F4
Is
Is
a finite H-complex.
has both 2- and 3-torsion in its integral homology and since
homologically torsionfree, we conclude from
zp
%
$,
ZQ
that 2 has the advertised properties.
CX
YQ
131
Finite H-spaces, special results
5
S i n c e , by a cohomology argument,
Remark.
a s s o c i a t i v e s t r u c t u r e , n e i t h e r can
2.
cannot s u p p o r t a homotopy-
Thus, i t may s t i l l b e c o n j e c t u r e d
t h a t t h e pathology of Theorem 2.11 does n o t occur f o r q u a s i f i n i t e homotopya s s o c i a t i v e H-spaces.
Thus f a r , a l l of t h e examples of f i n i t e H-spaces which we have
c o n s t r u c t e d have t h e p r o p e r t y t h a t each of t h e i r l o c a l i z a t i o n s i s homotopy
e q u i v a l e n t t o t h e corresponding l o c a l i z a t i o n of a ' c l a s s i c a l '
example.
As
our f i n a l a p p l i c a t i o n i n t h i s s e c t i o n , we show, f o l l o w i n g Mimura-Toda [561
( s e e a l s o S t a s h e f f 1811 and Zabrodsky [96] f o r independent t r e a t m e n t s ) t h a t
t h i s need n o t b e t h e c a s e i n g e n e r a l .
Theorem 2.12.
There e x i s t s a f i n i t e H-space
such t h a t Z 3 is not
2
homotopy equivalent t o the 3-localization of the product of a Lie group and
odd-dimensional spheres.
Proof.
over
S9
Let
X'
5
be t h e t o t a l s p a c e of any o r t h o g o n a l S -bundle
5
(Such bundles c e r t a i n l y e x i s t s i n c e
3.
T ~ ( S) = Z / 2 4 h a s o r d e r
whose c h a r a c t e r i s t i c elqment i n
r 8 ( S 0 ( 6 ) ) 8 Z3
5
+
n8(S ) 8 Z 3
s u r j e c t i v e ; a l s o , i t may be v e r i f i e d , u s i n g t h e f a c t t h a t
n
is
5
13(S ) 8 Z 3 = 0,
t h a t t h e 3 - l o c a l i z a t i o n s of any two such bundles a r e fibre-homotopy e q u i v a l e n t . )
It f o l l o w s from a s t u d y of t h e a c t i o n of t h e mod 3 Steenrod a l g e b r a on t h e
Z/3-cohomology of
where
Eu
SU(5)
( s e e Oka [ 6 4 ] as w e l l a s [561) t h a t
h a s i t s p r e v i o u s meaning.
Hence
Xi,
and so a l s o
X3,
where w e
set
x
admit
H-structures.
=
x'
x
s3
x
SJ,
W
e may now apply Theorem 1.18 w i t h
Y = SU(5), P = { 3 1 , Q = {31' t o o b t a i n a f i n i t e H-space
which s a t i s f i e s i n p a r t i c u l a r
X
Z
a s above,
of t y p e (3,5,7,9)
Applications of localization theory
132
By comparing the effect of the Steenrod reduced power P i
(small) list of all the classical examples of type
on Z and on the
(3,5,7,9), we easily
conclude that Z does indeed satisfy the conclusion of the theorem.
The argument of Theorem 2.12 may be suitably generalized so as to
provide similar examples for other odd primes, but the prime 2 appears to
be exceptional
Non-cancellation phenomena
I33
3. Non-cancellation phenomena
Our first general theorem on non-cancellation phenomena concerns
total spaces of what we call quasiprincipal bundles (seeHilton-Mislin-Roitberg
We begin with the definition.
1401).
bundle with structural group G, and
g: X
principal G-bundle associated with 5 .
provided
g of
rr-
+
BG
---f
the classifying map of the
We say that
5
is quasipAn&pal
0.
Note that if 5
gof
f X a fibre
Let G be a topological group, 5 : F*E
Definition 3.1.
is a principal G-bundle, then the composition
is trivial since it factors through the contractible total space EG
of the universal bundle over BG; this justifies the terminology 'quasiprincipal'
In a similar way to that of our previous studies of spaces at
individual primes, it is plainly very useful to study fibrations (over a fixed
base space) by p-localizing the fibration (see Theorem 11.3.12).
We will
introduce the notion of the genus of a fibration which, in case the base
space consist of a point only, reduces to the definition of the genus of a
homotopy type, the total space of the fibration in question. The notion of
the genus of a fibration turns out to play a central role in the theory of
non-cancellation phenomena.
Definition 3 . 2 .
Two fibrations Si: Fi
+
E i + X, i = 1, 2,
with Fi, Ei, X C N H
are called p-fibre homotopy equivalent, if the localized bundles
(Si)p:
(Fi)p
We say that
(Ei)p
-r
c1
and
are fibre-homotopy equivalent for all primes p.
Xp
-r
belong to the same genus, if all the spaces
C2
Fi, Ei, X 6 NH are of finite type and if, in addition, C1
and
C2
are
p-fibre homotopy equivalent for all primes p.
Definition 3.3.
spaces F, E
If
c:
F
-f
E+X
is a fibration in M-l with all the
and X of finite type, then the genus of
6, G ( C ) , consists
Applications of localization theory
134
NH
of a l l f i b r e homotopy equivalence c l a s s e s of f i b r a t i o n s i n
over
X,
5.
which belong t o t h e same genus a s
We now prove t h e f o l l o w i n g , which e x t e n d s a r e s u l t of [40].
Let
Theorem 3.4.
Fi
NH which are p-fibre
El
and
E2
Ei
-t
-t
be quasiprincipaZ G-bundZes i n
X, i = 1, 2
homotopy equivaZent f o r a22 primes
are q u a s i f i n i t e rmd that
Assume
p.
(For example,
BG C NH.
G
that
may be a
connected topoZogicaZ group, or an arbitrary niZpotent topoZogical group [69 ]I.
Then El
F2
x
rmd
Proof.
with
6,:
Let
F +E
i
i
f
i
--t
gi:
p € Il
X
c l a s s i f y t h e p r i n c i p a l G-bundles a s s o c i a t e d
BG
-+
-
X; t h u s we a r e g i v e n t h a t
f i r s t aim is t o show t h a t
f o r each
are homeomorphic.
E2 x F1
g2 o f l
and
0
giofi
g of2
1
N
0.
= 0,
i = 1, 2.
To t h i s end, choose
a homotopy equivalence
O(P) : (Ellp
+
such t h a t
(f
Then, s i n c e
(g2)p o ( f 2 I p
(g*)p
0
5
1
1 P
= (f2)p OO(P).
0 , we o b t a i n
(fl)p
=
(g21p
0
(f21
P
0
@(PI =
and so,by Theorem 11.5.14,
g2 Ofl
Similarly
glof2
CI
0.
= 0.
Next, form t h e p u l l b a c k
i"
X
g2
* BG
Our
0,
p
n,
Non-cancellation phenomena
Since g2 ofl".O,
t h e product
135
i t f o l l o w s t h a t t h e i n d u c e d p r i n c i p a l G-bundle
over
Thus, t h e a s s o c i a t e d bundle w i t h f i b r e
E l x G.
F2
El
is
El
is
X
F
2'
S i n c e induced bundles commute w i t h passage t o a s s o c i a t e d bundles, we i n f e r
that
E12 = El
where e q u a l i t y means homeomorphism.
X
F2
Similarly, using t h e f a c t t h a t
glo
f2
=
0,
we f i n d
E12
= E2
X
F
1'
so t h e theorem i s proved.
I n o r d e r t o make use of Theorem 3.4 t o manufacture n o n - c a n c e l l a t i o n
examples i n v o l v i n g s p a c e s of f i n i t e t y p e , we w i l l b e e s p e c i a l l y i n t e r e s t e d i n
t h e case t h a t
F = F = F
1
2
is a s p a c e of f i n i t e t y p e . An immediate consequence
of t h e p r e v i o u s theorem is t h e n t h e following.
Theorem 3.5.
Suppose
F
+
Ei
+
X, i = 1, 2,
are quasiprincipal G-bundlesof the
same genus with BG € NH, and suppose t h a t the t o t a l spaces
quasifinite.
Then El
F
x
and
E2 x F
x
F
and E2
x
F
and
E2
me
are homeomorphic. Moreover, i f
a l l spaces involved are differentiable manifolds, and i f
then El
El
i s a Lie group,
G
are diffeomorphic.
W
e w i l l now develop a c r i t e r i o n , which w i l l e n a b l e u s , i n t h e c a s e
of p r i n c i p a l G-bundles over s p h e r e s , t o deduce t h a t two f i b r a t i o n s belong
t o t h e same genus.
More g e n e r a l r e s u l t s , i n v o l v i n g p r i n c i p a l bundles over
suspensions, may b e found i n [ 4 4 ] , Remark 2.
L e t then
(F,o)
be a p o i n t e d CW-complex,
(under composition) of self-homotopy e q u i v a l e n c e s of
e: H(F)
+
F
H(F)
F
t h e H-space
and
Applications of localization theory
I36
the evaluation map
-
h
e
It i s c l e a r t h a t
is a f i b r a t i o n with f i b r e
self-homotopy equivalences of
when
h ( o ) , h € H(F).
F.
Ho(F), the space of pointed
Moreover, i n t h e important s p e c i a l case
o
F = G , a topological group, and
i s t h e i d e n t i t y of
G, t h e r e i s a
canonical s e c t i o n
s: G
-+
H(G),
given by
which together with t h e f i b r e i n c l u s i o n
of
H(G)
Now t h e a c t i o n of
on
Ho(F)
ho.h.hil,
I-+
n n-1 (H(F)).
on
H(F), given by
ho € Ho(F), h € H(F),
Ho(F)
( i n f a c t , of
no(Ho(F))
Furthermore, t h e r e i s an a c t i o n of
n n- 1(F)
given by
(ho,a)
ho.a,
I-+
yields a representation
H(G)
H(G) = Ho(G) x G.
c l e a r l y induces an a c t i o n of
no(Ho(F)))
-+
as t h e t r i v i a l f i b r a t i o n
(ho,h)
on
Ho(G)
ho C Ho(F),
and, by d i r e c t c a l c u l a t i o n , we have:
C
H (F)
n n-1 (F),
( i n f a c t , of
Non-cancellationphenomena
Turning again to the case where F
we note that, i n general,
I37
is a topological group
is not a no(Ho(G))-module
s*
G,
map. However we
may prove (see [ 4 0 ] ) :
If the 'Scheerer' diagram
Lemma 3.7.
Sn-' x G
where
ho C Ho(G),
c1
€
TI
and v : G
(G),
n-1
map, i s homotopy-comtative,
lJ (ax11
____t
x
G
G
+
G i s the muZtipZication
then
Sx(ho.a) = ho*s*(a) y
in
V
n-1
@(GI).
Proof.
(t,x)
The hypothesis asserts that the maps
+
ho.(a(t)*x),
(t,x)
are homotopic. Now, s,(ho*a)
t E Sn-',
ho(cx(t))'ho(x),
+
is represented by the map
u:
x € G,
Sn-l
+
H(G)
given by
u(t) (x) = ho(a(t))
and ho.s,(a)
*x,
is represented by the map
v(t)(x>
=
v: S"-'
-t
H(G)
given by
ho(a(t)hol(x)),
where hi1 is a homotopy inverse of ho. The hypothesis therefore implies
that the adjoints of u and v
are homotopic, and
so
therefore are u
and v.
is connected; then, clearly, G €NH and we
We assume now that G
of the s&ace G. ( I n [ 4 1 ] it is
may speak of the p-localizations G
P
implicitly assumed that G is connected. That is why we spoke there of the
no(Ho(G))-actions.
Since H(G)
when G is connected.)
-
Ho(G)
x
G, no(Ho(G))
=
Localizing Lemma 3.7, we obtain:
no(H(G))
precisely
Applications of localization theory
138
I f the diagram
Lemma 3.8.
where
ho(p) C Ho(G )
P
is
(ho(p)
p-localization of an element
not assumed t o be homotopic t o the
ho € Ho(G)),
i s homotopy-comutative, then
are then fibre-homotopy equivalent.
Proof.
Only t h e l a s t s t a t e m e n t remains t o b e proved.
i t is e v i d e n t t h a t t h e elements
same o r b i t of
(sp)*(ho(p)*a )
P
T ~ - ~ ( H ( G ~ under
))
t h e a c t i o n of
But, by c l a s s i c a l t h e o r y , t h i s means t h a t
Scr
P
and
T~
(sp),(ap)
From ( 3 . 9 ) ,
l i e i n the
(Ho (Gp) 1 = r0 (H(Gp) 1
and
'ho(p) 'ap
.
are fibre-
homotopy e q u i v a l e n t .
We a r e now i n a p o s i t i o n t o e n u n c i a t e one of our c e n t r a l r e s u l t s .
Let
Theorem 3.10.
Suppose t h a t
k
G
be connected and l e t
i s an integer prime t o the order of
sn-l x G
sn-jx
K:
G
+
G
p(kax1);
1.
and that t h e diagram
,
the k t h power map, i s homotopy-commutative.
G-bundles
c l a s s i f i e d by
a
G
I X K
with
a € T ~ - ~ ( G be
) of f i n i t e order.
a , ka
belong t o the same genus.
Then the principal
Non-cancellationphenomena
139
Remark. Under the hypothesis of Theorem 3.10, it is in fact possible to
prove directly that
ka’ of
(a’
the adjoint of
a)
3
0
and this was the manner in which non-cancellation
examples were first constructed.
(See [45] for the case G = S3 and [441
for the general case. See also Sieradski [75] for related results using
yet a different approach.)
Thus, in case the base spaces of the fibrations
involved are spheres, it is possible (though not necessarily desirable) to
bypass the theory of localization altogether. However, Theorem 3.10 has an
obvious local version and, as we show later, there are fibrations Si, i = 1, 2,
over a space X which is not a sphere but such that the local version of
Theorem 3.10 applies to
Xp
Sp”
for some family of primes P, while the
for p f P
(cl)p, (C,)
equivalence of
3
follows from other, more obvious,
considerations.
Proof of Theorem 3.10.
then
(La),
and
(€,ka)p
If p € Il
is prime to the order of
are fibre-homotopy equivalent since they are both
(p,k) = 1, then clearly
fibre-homotopy trivial. On the other hand, if
G + G
KP: P
P
is a homotopy equivalence and we may apply Lemma 3.8 with ho(p) =
K
P’
Applying Theorems 3.5, 3.10 in the case G = S3, we get:
Theorem 3.11.
Let
3
a C T~-~(S
)
have order m, l e t
k be an i n t e g e r
prime t o m, and suppose
k(2k-l)w o z3a
(3.12)
Then Ea
x S
3 and
Eka
a,
x
S3
Furthermore, Ea & Eka
=
o
3
6 +n+2(~ )
.
are diffeomorphic.
unless k
!
+1
(mod m).
140
Applications of localization theory
Proof.
A s t a n d a r d homotopy c a l c u l a t i o n shows t h a t t h e diagram
3
of Theorem 3.10 (with
is homotopy-commutative i f f (3.12) h o l d s .
G = S )
Thus t h e f i r s t p a r t of t h e theorem f o l l o w s from Theorems 3.5 and 3.10.
To prove t h e second p a r t , n o t e t h a t
so t h a t a homotopy equivalence
t i o n , a homotopy equivalence
h: E
6:
Ca
a
+
+
E
i n d u c e s , by c e l l u l a r approxima-
ka
Cka.
By a s l i g h t m o d i f i c a t i o n of t h e argument used i n Example 1 . 3
(a need not b e s t a b l e , b u t
S3
is an H-space),
it follows t h a t
'
k
5
+1
(mod m),
a s claimed.
A s a c o n c r e t e example i l l u s t r a t i n g Theorem 3.11, l e t
a = w , k = 7 .
3
Then c o n d i t i o n (3.12) is m e t because
3
i n t h e course of proving Theorem 2 . 1 ,
Eo & E70, Ew
has o r d e r 3; indeed, a s noted
w o C w
w OC w
S3 = E7w
x
generates
x
S
3
3
ng(S ) = 1213.
Thus,
.
A second a p p l i c a t i o n of Theorem 3.5 and ( t h e l o c a l v e r s i o n o f )
Theorem 3.10 is provided by t h e s p a c e s
$
s t u d i e d i n S e c t i o n 2 ( s e e Theorem
2.9).
Theorem 3.13.
x
We have diffeomorphisms
s3 =
x1
x4
x
s3 = x7
x
s3 =
X13
x
s3,
x3
x
s3 = X6
x
s3
.
Proof.
C,,
Consider, i n t h e n o t a t i o n used i n Theorem 2.9, C 1
and
3
P. = 4 , 7 , 13. Since .rrll(BS ) = 12/15, i t is c l e a r from (2.8) t h a t
(El),
and
and
(C2lp
p = 5.
map of
5
have, f o r
a
are fibre-homotopy e q u i v a l e n t e x c e p t p o s s i b l y f o r
p = 3
For t h e s e primes, we may i d e n t i f y t h e l o c a l i z e d c l a s s i f y i n g
with
!Lap =&Bp.
P. = 4 , 7, 13,
Now, i f
6' € II (S 3 )
10
is a d j o i n t t o 6 , w e
Non-cancellation phenomena
141
31 (a-1) , 12w = 0, 156' = 0.
since
Thus, we may apply Theorem 3.10 and t h e succeeding remark w i t h
P = 13,5>
t o conclude t h a t
equivalent f o r
p = 3
(c,),
c3
a r e a l s o fibre-homotopy
P
p = 5.
and
Similarly,
(5,)
and
c6
and
belong t o t h e same genus and t h e proof
i s completed by a p p e a l i n g t o Theorem 3 . 5 .
Thus f a r , a l l o u r examples have been p r i n c i p a l G-bundles f o r
G
A s a f i n a l a p p l i c a t i o n of
a L i e group and hence a t most 2-connected.
Theorem 3 . 2 , we p r e s e n t examples of n o n - c a n c e l l a t i o n w i t h t h e s p a c e s involved
being a r b i t r a r i l y h i g h l y connected manifolds ( s e e 1 4 0 1 ) .
It is t o o b t a i n
t h e s e examples t h a t we have made t h e g e n e r a l i z a t i o n ( D e f i n i t i o n 3.1) from
p r i n c i p a l t o q u a s i p r i n c i p a l bundles.
Theorem 3 . 1 4 .
p > q
es
and l e t
C IT
n-1
be a f k e d odd number,
q
+
Denote by
1 mod p, where
e : SO(q+l)
q
=
2p
e x i s t s , since
s
(Sq)
-
2.
n
has degree
such an
a
Let
q 1 3, let
s: Sq
+
Sq
+
be an element of order p and l e t
a map such that
SO(q+l)
denotes the evaZuation map;
i s a regular prime f o r
p
be a prime,
p
sO(q+l).
a = s a E IT
n-1
Let
(SO(q+l)).
Consider
nka:
the orthogonal
Sq-bundle over
i s c l a s s i f i e d by
(il
liil
(iiil
Sq
IG(na)
If
Sn
3
{nkal(k,p) =
I
2 (p-1) / 2
(k,p) = 1 then
f i v ) If k 2 +1 (mod p)
Proof.
Bka
Sn
+
whose associated principal
SO (q+l) -bundle
Then
ka.
G(na)
+
Since
11
Ba x Sq
and Bka
x
Sq
are diffeomorphic
then Ba $ Bka.
a = sa, and a ,
a
a r e of o r d e r
p, i t f o l l o w s t h a t
142
Applications of localization theory
-
a = e a , and t h e n
= sq uka-en u
Bka
en+q,
Thus ( i v ) is proved i n t h e same way a s t h e l a s t s t a t e m e n t of Theorem 3.11,
a
using t h e f a c t t h a t
has order
p.
Next, we want t o show t h a t
# 1 then
if
(k,p)
if
(k,p) = 1, t h e n
equivalent, i f
case.
q
qka
I
(qa)q
qka
E G(n,)
iff
s i n c e t h e n even
G(qa)
and
(nka)q
(k,p) = 1.
(Bkalp
+ (Balp.
Clearly,
Conversely,
a r e c e r t a i n l y fibre-homotopy
# p , s i n c e they a r e both fibre-homotopy t r i v i a l i n t h i s
It remains t o s t u d y t h e s i t u a t i o n f o r
q = p.
To t h i s end, c o n s i d e r
t h e diagram
Here,
1,
is induced by t h e obvious i n c l u s i o n ,
r e s p e c t i v e evaluationmaps,
(ep)*
E*
e* by t h e
and
by l o c a l i z a t i o n and
(gp)
*
l o c a l i z a t i o n ' (see t h e d i s c u s s i o n f o l l o w i n g Theorem 11.3.11).
by ' f i b r e w i s e
Setting
a' = G p ) * i * ( a )
we w i l l show t h a t
under t h e
a'
and
ka'
l i e i n t h e same o r b i t of
ao(Ho(Sq))-action, thereby proving t h a t
P
a r e fibre-homotopy e q u i v a l e n t .
n-1 (Ho (Sq))
p
'TI
i t follows from t h e c h o i c e s of
Since
nn-l+q(Sq)
p, q
and
0 Ep
n
that
n n-1 (Ho (Sq))
= 0
p
so t h a t
?'*
is injective.
Hence, i f w e now s e t
a" = ? * ( a ' )
(qJP
IT
n-1 (H(Sjf))
and
(nkcr)p
Non-cancellation phenomena
we a r e reduced, by Lemma 3 . 6 , t o showing t h a t
same o r b i t of
IT
n-1
under t h e
(Sq)
p
a"
143
and
ka"
l i e i n the
(H ( S q ) ) - a c t i o n induced by l e f t -
IT
O
O
P
T h i s l a t t e r a s s e r t i o n i s c l e a r because
composition.
k o a" = ka"
as
i s an H-space;
Sq
P
and
k C E* = no(Ho(Sq))
P
P
(k,p) = 1. T h i s completes t h e proof of ( i ) .
as
The a s s e r t i o n ( i i ) f o l l o w s
(Bka)p $ (BRa)p i f
now from ( i ) by observing t h a t
k
$kR
(mod p ) .
We
g e t (iii) from (i) by v i r t u e of Theorem 3 . 5 , once w e have v e r i f i e d t h a t t h e
bundles
nka
nka: Sq
gk: Sn
and l e t
-+
p r i n c i p a l bundle.
is a
A:
-
are a l l quasiprincipal.
Sq+l
+
BSO(q+l)
jk
Sn
B
kcr
be t h e c l a s s i f y i n g map f o r t h e a s s o c i a t e d
To prove t h a t
BSO(q+l)
Let
gkofk
N
0
i t s u f f i c e s t o show t h a t
there
making t h e f o l l o w i n g diagram commutative
uZkti,Jka>- $q+l
/
//A
(3.15)
/
i!
i
where t h e t o p l i n e i s t h e Puppe sequence of
X
t h e a d j o i n t of
is adjoint t o
X oJka? 0
and
s: Sq
order
SO(q+l).
Now
gk
jk
( s e e [ 4 0 ] ) . We t a k e f o r
is a d j o i n t t o
ka
and
AoZka
Hence (3.15) w i l l b e commutative, provided t h a t
s o k a = ka.
g k o x rr 0.
Our c h o i c e of
n
(S")
n+q
+
BSO(q+l)
n, q
i s z e r o , and t h a t
p, we i n f e r t h a t
g u a r a n t e e s t h a t t h e p-primary component of
x
g ox
k
i s a suspension.
=
Thus, s i n c e
gk
i s of
0.
On t h e o t h e r hand, w r i t i n g
'n
f o r t h e p-primary component of
'TI,
Applications of localization theory
144
By our choice of
n , q , t h e second summand is zero; f o r
2p
-
3 < n
-
1 < 4p
Thus
(S2q+1)
Jka = [i,i] o y , y C
and
Thus,
i s a suspension.
y
A o
Now
’TI
n+q
C
[I ,I]
TI
2q+l
(BSO(q+l))
is an element of f i n i t e o r d e r prime t o
h o[I,I]
It follows t h a t A o J k a =
X
o[I,I]
oy
N
2
azq(SO(q+l)).
p y since
p > q.
This completes t h e proof of t h e
0.
theorem.
We consider now another type of non-cancellation phenomenon, which
i n v o l v e s H-spaces
.
Let
Theorem 3.16.
Ekw and
XI1
have themeanings given i n Section 2 .
Consider the bijections
G(G2) = IX1,X2,X4,X71
%
Then one has, for spacesin
E~~ x E~~
Xk
x
=
Xk =
Proof.
G(Sp(2))
-
( 2/15)*/I&l)
-i;
and
G(GZ) respectivezy,
E ~ , x E~~
iff
IZ
=
iii i n (z/’iz)*/
+I
xm
iff
iz
=
i;
x
Xn
in
(2/15)*/
+1
We w i l l o n l y consider t h e c a s e of t h e spaces i n
o t h e r c a s e i s similar and a c t u a l l y s i m p l e r .
f:
3 x?, a x m x xn
Suppose given
G ( G ~ ) ;t h e
k, a , m y n
and
-
145
Non-cancellation phenomena
Denote by P
so
the set of odd primes. Now
we can assume a homotopy commutative diagram
where A , B
are localizations ofhomotopy equivalences. Since
we may regard C and D as 2x2-matrices with entries in
The map B
TI
10(S3) = 22/15
22/15. Thus
is represented by the P-localization ofa unimodular integral
matrix. The axes of A
are P-localizations of integral vectors
(al,a2),
(a3,a4), such that
is
is unimodular. The.homotopy commutativity of the left-hand square in (*)
expressed by the matrix equation
KC = DB
over L/15. Taking determinants yields W E +mn (mod 15).
given k, E, m, n
-
certainly Xm, Xn
with W x Xi
of type
(units mod 15) with c F =
< G(X,)
Xm x Xn.
in
Conversely,
(22/15)*/{+1},
then
and by Theorem 1.14 there exists an H-space W
It follows that W
(3,11), and hence W
c"
is a 1-connected finite H-complex
Xi for some i relatively prime to 15.
--
--
From the first part of the proof we conclude that ill = mn and hence
i
=
Applications of localization theory
146
in
( Z/15)*/{+1}.
I t follows that
-
Xi
$
and hence
3 x x t = x
m
x x .
n
A s our f i n a l r e s u l t , we have the following corollary.
Let
Theorem 3 . 1 6 .
Eko and
Then the powers of
Ekw, XQ are related by:
2
fi)
E
w
(ii) X:
Proof.
4
Hence X
1
c= X4
-
4 -
have t h e meanings given them in Section 2.
Xt
'u
E
c1
X:
2
50
+ X:
c1
2
x7,
X;
Theorem 3 . 1 5 g i v e s
X2
2
X
X2
- -
7 -
X4
2 -
X4
7'
2
X;
E
'u
2
X:
2
c1
E50
-
x47 '
and
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Index
Abelianization 20
F-monomorphism 83
Action (of nilpotent group) 34
Function space 48, 63, 77
Fundamental Theorem 7
Basic commutators 5
Bousfield-Kan 1, 47
Generalized Serre class 43, 69
Browder-Novikov Theorem 109
Generic property 104, 115
Genus 32, 102, 104, 133
Class (of abelian groups) 43
Genus invariant 105
Cofibre sequence 48, 58, 77
Cofinite (set of primes) 26,84
Half-exact functor 41
Cohomology of groups 1, 15
€!all 5, 25, 26
Commensurable 24
Hilton 1, 40, 67
1-Connected 47, 52
H-map 109
Cooperation map 92, 128
Homological dimension
(of quasifinite space) 79, 114
CW-complex 47
Homology decomposition 100
Homology of groups 20, 39
Decomposition (= partition) 29, 31
Homotopy-associative H-structure 120
Dror 48, 76
Homotopy category 47, 94
Hopf algebra structure 110, 112, 119
0-Equivalence 61
H-space 50, 90, 104, 119, 128
Evaluation map 136
Induced H-structure 109
Fibre sequence 48, 58, 77
0-Isomorphism 61
Finite H-space 122, 130, 131
Finite type 59, 79
Lazard 1
Five-term exact sequence 20
0-Local 4
Index
Local cell 48
155
Orthogonal bundles 141
Localization 1
functor 2
p-fibre homotopy equivalent 133
of connected CW-complexes 52
p-local 4
of homotopy types 47
p-universal 4
of nilpotent complexes 72
P-bijective 5, 92
of nilpotent groups 3, 19
P-equivalence 59
Lower central series 3, 20, 67
w-series 34, 67
Lyndon-Hochschild-Serre spectral
sequence 14, 41, 44
P-injective 5, 23
P-isomorphism 1, 5, 24
P-local abelian group 7
cell 57
group 1, 4
Malcev 1
space 41, 52, 72
Mayer-Vietoris sequence (in homotopy) 87, 95
sphere 57
Milnor 26, 32
P-localization 1, 47
Mislin 33, 104, 130
theory 4, 7
Mixing homotopy types 94
P-localizing functor 4
Moore-Postnikov system 68
map 4, 41, 52, 1 2
P-surjective 5, 23
Nilpotency class 3, 34
P-universal 4
Nilpotent action 2 , 34
Partition (of the set of primes) 28, 51
complex 62
Pickel 33
homotopy types 1
Poincar6 complex 108
group 1, 3
Postnikov decomposition 65
Lie group 62
map 67
k-Power map 114
N
k -Power map 114
space 48, 62
Primitive elements (of coalgebra) 112
Non-cancellation phenomena 102, 133, 144
Index
156
P r i n c i p a l refinement 65
Tensorial c l a s s 43
P r o f i n i t e completion 101
Thom s p a c e 108
Pullback 21, 2 6 , 28, 3 0 , 8 6 , 8 8
Type (of H-space)
122
i n homotopy t h e o r y 94
Theorem 79, 82
0-Universal 4
Pushout 28
Upper c e n t r a l s e r i e s 3 , 20
Q u a s i f i n i t e (space) 4 9 , 79, 89
Weak pullback 9 4 , 96
Q u a s i p r i n c i p a l (bundle) 133
Zabrodsky 50, 9 4 , 1 0 2 , 129
Rank 122
Rank 2 H-space 1 0 2 , 122
R a t i o n a l i z a t i o n 2 , 4 , 2 4 , 26
Reducible (space) 1 0 8
Roitberg 2 6 , 6 2 , 77
Scheerer diagram 137
S e r r e c l a s s 2 , 43
Simple (space) 62
S p l i t e x t e n s i o n 37
S-reducible (space) 1 0 8
Stallings-Stammbach Theorem 2 1
S t m b a c h 43
S t i e f e l manifold 57
Symplectic group 122