WITT SPACES:
A GEOMETRIC CYCLE THEORY FOR KO-HOMOLOGY
AT ODD PRIMES
by
Paul Howard Siegel
B.S., Massachusetts Ind'itute of Teclnology
(1975)
SUBMITTED IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE
DEGREE OF
DOCTOR OF PHILOSOPHY
at the
INSTITUTE OF TECHNOLOGY
MASSACHUSETTS
May 4, 1979
Signature of Author........
Department
of_ Mathematics, May 4, 1979
Certified by.........
Thesis Supervisor
..
.
Accepted by..
Chairman, Departmental Committee on Graduate Students
ARCHIVES
OF %&Y:'LcOY
JUN
.i
1979
LIBRARIES
2
WITT
SPACES:
A GEOMETRIC
CYCLE THEORY FOR
KO-HOMOLOGY AT ODD PRIMES
by
Paul Howard Siegel
Submitted to the Department of Mathematics on May 4, 1979
in partial fulfillment of the requirements for the
Degree of Doctor of Philosophy.
ABSTRACT
We solve a problem posed by Dennis Sullivan:
to
construct a class of piecewise-linear cycles with signature
which represent the connective version of KO homology at odd
primes,
ko
0 2[-].
The construction is one of the first
applications of the recent intersection homology theory of
Goresky and MacPherson.
It demonstrates the power of the
theory in the study of the geometry of spaces with singularities.
We also present what appears to be the first direct
adaptation of classical surgery (spherical modification) to
the setting of singular spaces.
Specifically, we consider a class of stratified p.l.
pseudomanifolds which we call Witt spaces, whose rational
intersection homology groups satisfy a Poincare duality
theorem.
To each Witt space
X, we associate a p.l. invariant
w(X)
taking values in W(Q), the Witt ring of the rationals.
This invariant generalizes the signature of manifolds, and
it possesses many of the same beautiful properties:
cobordism invariance, additivity, and a product formula.
Sullivan has shown how to use such an invariant to define a
canonical orgientation in
ko* ® 2[
],
The orientations
induce a natural transformation p
of homology theories,
from Witt space bordism to
ko, 0 Z[\].
To complete the
construction, we use rational surgery to prove that the
cobordism class of a Witt space X
is determined by w(X),
and we obtain an explicit description of the cobordism
groups of Witt spaces.
The only nontrivial groups occur in
dimensions 4k, and for k > 0, they are precisely
W(Q).
The structure of W(Q)
is known, and we conclude that
y
is an equivalence at odd primes.
Thesis Supervisor:
Title:
Edward Y. Miller
Associate Professor of Mathematics
3
TABLE OF CONTENTS
Abstract.............
..
.......
.
2
.. ..
Introduction.........
Chapter I:
Chapter II:
Chapter III:
4
Geometric Preliminaries.............
Algebraic Preliminaries............
12
33
Witt Spaces and Generalized
Poincar6 Duality.................. 43
Chapter V:
Chapter VI:
Appendix...
References.
Properties of the Witt Class w(X)..
Rational Surgery on Witt Spaces.....
65
85
Witt Space Bordism Theory.......... 112
.. .
. ...0................ ......... 121
... . .........................
.
-
Chapter IV:
131
Figures...........
134
Acknowledgements..
.137
Biographical Note.
139
4
WITT SPACES:
A GEOMETRIC CYCLE THEORY FOR
KO-HOMOLOGY AT ODD PRIMES
Introduction
In this thesis, we construct a geometric cycle theory
with signature,
primes,
representing connected KO-homology at odd
.
ko,* 0[]
The construction is one of the first
applications of the intersection homology theory of Goresky
and MacPherson.
We also present what appears to be the
first direct adaptation of classical surqery
modification)
(spherical
to the setting of "manifolds with singularity."
To put the result in perspective, we give some of the
historical
background to the problem before describing the
construction.
Dennis Sullivan
Floyd type theorem
the signature,
Theorem
[9]
relating
[22]
discovtred
the Conner-
smooth oriented bordism,
and connected KO-theory at odd primes:
(Sullivan):
For compact p.l. pairs,
there is a
canonical isomorphism (equivalence) of homology theories:
(XA)
® S
Q,
Z[2]
ko, (X,A) 02Z []
,
So
Q,
(pt)
where the theories are regarded as
2Z/42Z-graded.
5
He found
(1)
two interesting applications:
A geometric description of connected
KO horhology
obtained by geometrically
singularities
at
as a bordism theory with join-
ko* 0 Z [3],
odd primes,
like
(at least)
[See
generators of smooth bordism.
"killing"
[8],
[6],
[23]].
There
is a commutative diagram:
so
SSO
i
J
* (X, A)
p
0s
Q * (X,.0A)
Q*
in which
J,
k*(X,A) 0 2Z[ 1]
Z
(PO)
is the bordism theory with singularities,
is the natural inclusion
(gotten by regarding a smooth
manifold as a cycle in the bordism theory),
natural projection, and
i
p
the
is
gives an equiv1,ence of theories
p
[8].
(2)
A construction of a canonical
KO* o 2Z [%]
block bundle over a finite
for a p.l.
a canonical
particular,
KO*
manifolds via Alexander duality
0
[%]
[22].
complex,
orientation
orientation
and,
in
for p.l.
The techniques used
for this construction generalize to give a more general
result.
If
the collection
of cycles of a geometric homology
6
theory is closed under the operations of taking cartesian
product and transversal
intersections with p.l. manifolds,
and if it possesses a "nice" signature invariant extending
then the cycles have canonical
the classical signature,
KO* 0 2[k]
orientations.
Note that among the properties
of a "nice" signature are cobordism invariance,
additivity,
and a product formula with respect to closed p.l. manifolds.
For further details see Chapter VI and the Appendix.
A natural synthesis of the themes of these two applications
would be achieved by solving the following
Problem:
Construct a geometric cycle theory for
built from a class of p.l. cycles
invariant,
induced by the orientations
-
ko, *
]
with "nice" signature
(at odd primes)
such that the equivalence
:
is
6'
M[
ko* 0
2Z
described
in
(2)
above.
Goresky and MacPherson were also interested in finding
a class of spaces with cobordism invariant signature for the
purpose of extending the Hirzebruch L-class to the setting
of "manifolds with singularity."
Their investigation of
cycle intersection phenomena in stratified spaces resulted
in their beautiful intersection homology theory
proved that for a stratified p.l. pseudomanifold
[11].
X
They
of
7
with only even codimension
4k
dimension
intersection homology group
H k (XA)
strata, the
and
is self-dual,
they proceeded to define a cobordism invariant signature
and L-class for the collection of spaces with only even
strata.
codimension
This paper begins where
[11]
left off, and makes use of
intersection homology theory to solve the problem stated
above.
Techniques for computing the cobordism groups of
stratified spaces with even codimension strata are not
currently available,
so we enlarge the class of spaces to
permit certain odd codimension strata.
Specifically,
impose the following local
homological
If
condition:
L
0.
There are three observations which confirm
definition
is
link
is the link of an odd codimension
H m (L;V) =
stratum, then
intersection
we
that this
reasonable.
Atiyah in
Observation 1:
[4]
studied examples of smooth
fibrations
M 29
X
N2k
with sign
(X) /
signature of
X.
0, where
sign
(X)
denotes the classical
Using a theorem of Putz, we may regard
8
fibre bundle as being p.l., and the mapping cylinder
the
of the fibration provides a stratified p.l. cobordism of
X
The cobordism has odd codimension stratum
to zero.
with link
If the extended signature is to be cobordism
M.
invariant, we need a general condition on
sign
that
H
N
(X)
(M;Q) = 0.
= 0.
Atiyah's
For a manifold
M
which ensures
analysis provides
M,
coincides with ordinary homology,
one:
intersection homology
so we get the same link
condition as above.
Observation 2:
additive.
We want the extended signature to be
An appealing geometric proof of additivity
follows immediately from cobordism invariance if the
"pinch cobordism"
1.).
is a permissible cobordism
is the only odd codimension
v
The pinch point
Its link
singularity introduced.
carried out in Chapter IV,
H
Observation 3:
(L;) =
is a suspension,
L
A geoiLctric calculation,
with suspension points identified.
a fortiori,
(see Figure
shows that
H
(L;2Z) = 0;
0.
An intersection homology spectral sequence
argument proves that
H
(X2k;Q)
condition is satisfied in
X.
is self-dual if the link
This result is proved in
Chapter III.
Using Observation 3, we associate to each permissible
9
stratified p.l. pseudomanifold
satisfying
X4k
space
Hm
the
(X;V).
X4k
link condition),
(that is,
to each
the inner product
The inner product space is determined up
to permissible cobordism by its equivalence class in
the Witt group of the rationals.
class
dim
the Witt class of
(X)
/
0
incorporates
(mod 4),
X,
we set
= 0.
a "nice" signature in the
Applying Sullivan's result,
in
We call this equivalence
and denote it
w(X)
ko* 0 Z[1]
W(O),
by
w(X) .
This p.l.
If
invariant
sense used above.
we obtain canonical orientations
for permissible cycles.
It remains to compute the cobordism groups.
applying a technique of rational surgery,
By
adapted from
classical spherical modification to the setting of
stratified spaces and intersection homology, we show that
the Witt class
w(X)
determines the cobordism class
of
X.
We also construct geometrical representatives in positive
dimensions
q E 0
Therefore,
(mod 4)
for each class
Q
Witt
.
and denote the bordism theory
These results can be rephrased
as:
Theorem:
W().
we give the collection of permissible
spaces the name Witt spaces,
based on them by
in
There are isomorphisms
10
QWitt(pt)
=Z
W(),
Q Witt(pt)
q
QWitt(Pt)
q
q > 0
q /
= 0
0
0 (mod 4)
q-
and
(mod 4)
determined by the Witt class invariant.
ko, 0 2Z [%]
of homology theories:
p
transformation
induce a natural
orientations
Witt
Witt
which reduces to the signature
groups when
sign
q
0
:
@ ko
>
7 2Z[%]
.
The
homomorphism on coefficient
(mod 4):
Witt
q
(pt)
q
The group structure of
c
-Z
W(V)
is
Z[%]
classical
ko
(pt)
0 2[%]
[13].
There
q
is
a canonical isomorphism
W(2)
where
4.
T
@ T,
is a direct sum of cyclic 2-groups of order 2 or
The composition
cyclic summand
gotten by projection onto the infinite
11
Witt
Q
q
(pt)
W(Q)
-1
Z,
coincides with the signature homomorphism above.
Tensoring with
Theorem:
2[%2],
we conclude:
The natural transformation
Witt
Witt 0
2Z [12--
ko* 0 2 [%;
is an equivalence of homology theories,
signature homomorphism
coefficient
(tensored with
reducing to the
Z[6])
on
groups.
Finally, we note here that John Morgan has also
extended the signature to a class of stratified
whose cobordism groups he has computed.
spaces
These spaces give
a cycle theory for the homology theory ass.ciated to
G/PL.
He uses different techniques from those in this thesis, and
thoroughly studies the 2-torsion in his theory.
We gratefully acknowledge some helpful remarks regarding 2-torsion which he provided early on during the
preparation of this thesis.
12
CHAPTER I:
homology theory and the theory of bordism
from intersection
All
with singularities.
Stratified
p.l.
E
space
6 are taken for the most
[11].
Definition 1.1.:
manifold
are piecewise
pseudomanifolds.
The contents of sections 1 part from
spaces
topological
stated.
unless otherwise
Section 1:
and results
definitions
chapter we present
In this
linear
Preliminaries
Geometric
X
A
(closed)
n-dimensional
(p.l.)
pseudo-
is a compact space containing a closed subsatisfying:
(1)
dim
(E
(2)
X -
E
) < n - 2
xis
an n-dimensional oriented
(p.l.)
manifold
(3)
C(X -E)
Definition 1.2.:
=
X.
A stratification of a pseudomanifold
is a filtration by closed subspaces
X=xX n
such that
xXn-1
for each point
n-2
p E X
x
n-3
-X
i
, ....2X 1 D X 0
Xi,
there is a
X
13
filtered compact p.l. space:
V = Vn D Vn-
and a mapping
V.
J
x B
i
in
X..
to
V
V
(p.1.)
x
which,
B
i
is the p.1.
(interior point of
X
If
-*X
for each
j, takes
homeomorphically to a neighborhood of
Here,
I
B
x
= a point
i
D'..D
is
not empty,
i-ball
and
p
p
corresponds
B ).
it
is
a manifold of dimension
i,
called the i-dimensional stratum of the stratification.
We also consider relative objects:
Definition 1.3.:
An n-dimensional
boundary is a compact pair
(1)
A
is
singular set
(2)
dim E
an
(n-1)-dimensional
A
neighborhood
[0,1) = N.
(p.l.)
pseudomanifold with
There is a closed subspace
X -
such that
n-manifold which is dense
ing
satisfying the properties:
ZA*
< n - 2
(3)
(X,A)
pseudomrnifold with
in
of
A
homeomorphism
S(EA
in
U A)
with
is an oriented
X.
is collared in
N
GE
E
X;
X
i.e.,
there
is
and an orientation
0 : A
x
x
[0,1]
-N
a closed
preservsuch that
14
Here is the relative definition of stratification.
A stratification of an n-dimensional
Definition 1.4.:
(X,A)
pseudomanifold with boundary
is a filtration by
closed subspaces:
n
stratifies
A,
and the
obvious sense),
(A
A
[0,1])
_x
Section 2:
=
0
1
n-3
A
given by
stratify
=
X - A
X
(
A
(in the
respect the collaring
filtrations
By the last
X.
in
A
- A. 1
X
the
x
2
of
the filtration
such that
of
n-
n-l
we mean
condition,
X. r N.
Piecewise-linear chains and perversity
Assume throughout that
X
is
an n-dimensional
pseudomanifold with fixed stratification.
chains
is
complexes
of
X
the complex of
C* (X),
Definition 2.1.:
the direct
T
C,(X),
T
C e C
(X),
triangulations
ranging over all
compatible with the p.1.
For
structure.
the support of
C,
denoted
the union of the closures of the i-simplices
coefficient
in
c
is
geometric
under refinement of the simplicial
limit
with
..
not zero.
As a p.l.
a
JC,
is
whose
subspace
of
X,
15
D E
A perversity
Definition 2.2.:
=
p
(P2 '''''n)
+ 1.
If
Y C X
such that
is
i
(p,i)
) < i
n-c
2)
IDCI
C
(X)
is (p,i)
is
We will
the condition
in 2.2.
< i
and
a subspace
dim
(Y) < i
and
c > 2.
is the subgroup of
C
C
(X)
such that
allowable
(p,i-l) allowable.
Remark 2.4.:
(Y)
or
on occasion.
a perversity,
p
for all
consisting of those chains
IC
p,
pc+l =c
c-
-
Definition 2.3.:
1)
and
allowable if
c + p
-
of integers
a sequence
for
an integer and
is called
is
p2 = 0
p(c)
We also write
dim (Y 0 X
dim
|DI.
has a well defined support,
C*(X)
a chain
Therefore,
is invariant under refinement.
JCJ
frequently use an ec-uivalent
Y
Namely,
dim (Y r) Xn
c)
is
< i
(p,i)
-
form of
allowable if
c + p
for all
c > 2.
Proof of equivalence:
dim (Y
so the second condition
Since
X n-c C
Xn-c, we have
nx
)n-c
< dim (Y () X n-c)
implies the first.
On the other
16
hand,
(Y r
dim
Xn
max
=
j=O,...,n-c
j,
For each
Pc+j
c +
{dim (Y nX
.n-c-j
j, so the second condition
dim (Y
) Xnc)
< i
+ p c+
(c+j)
-
< i
+ c + p
.
implies,
Therefore,
dim
The
H(X) ,
(Y C
i th intersection
Xn-c
< i -
c + pc
homology group of perversity
is the i th homology group of the chain
pP
complex
CP (X).
There are homomorphisms for each perversity
a- : H*(X)
p
-+
HP
W- : HP (X)
p
-+.
H,(X)
n-*
p:
(X)
0 [X]
: H*(X)
-+Hn-*
.
which factor the classical cap product homomorphisms:
These are roughly defined as follows.
The map
w-
p
is
17
T
is
p
by sending the elementary cochain
a
where
,
an
is
a
chain supported on the
T, to the p.1. n-i
i-simplex in
assume
p
on the cochain level
a-
Define
a triangulation.
a-,
For the map
induced by chain level inclusion.
i
a
dual to
in
class in
fundamental
Section
X, corresponding to the appropriate
Transversality and Intersection products.
3:
Definition 3.1.:
Suppose
dimensionally transverse
is
[11],
Lemma 3.2.:
C A @D,
D E
and
j
-
n = 2.
are said to be
CW(X)
J
C fi D)
(written
i +
ICI () IDI
if
allowable.
(r2,k)
In
p + q = r
and a chain
C E C (X)
A chain
H n-i(D(a),DD(a)).
the following lemma is
Let
be as in 3.1.
C, D
then the intersection
chain
proved:
If
C Ax D,
C f) D E
Cr
DC A D,
is defined
and
a(C 0 D)
=
3C t) D +
(-1)n-iC 0
D.
is
The transversality lemma, which we use frequently,
proved in
[16].
Lemma 3.3.:
Suppose
p + q =
r,
A e C?(X)
1
and
B E C9(X).
J
18
Suppose
K
:
Then there is a p.2 . isotopy
DA A B.
X x [0,1]
-- +X
such that
(1)
K(x,0) = x,
(2)
K(x,t) = x
for all
(3)
K(x,t) e X.
-
(4)
K(IAIl) = K
for all
X
(JAI)
dimensionally
x
X
E
xE
all
t E
whenever
x E X
-
supports
K1
j|Aj,
transverse
to
B.
denotes the induced map on
Corol lary 3. 4.:
p + q = r.
and
3C'
Suppose
Suppose
C e
DC A D.
such that
E e C i+l (X)
C
Here
K
C, (X).
(X),
D E C
1
=
C'
X
(A), which is
J
(X), and
Then there exists
DE
[0,1]
-
C,
C' e Cw(X)
C' A D, and
C.
=
Now we come to the
homology theory its
Theorem 3.5.:
intersection
(1)
result which gives intersection
name.
Suppose
p + q < r.
There
pairing:
S:HP (X)
i
x
H0 (X)
3
-- + H
i+j-n
defined by:
[C] f)
[D]
=
[C r) D]
(X)
is
a unique
19
for every dimensionally transverse pair of cycles
C e CP(X)
I
(2)
D E
and
If
C
(X).
I
f E
H9(X), and
3
(Ol,2,...,n-2), then
E EH
p + c + s < t
=
C 0 T1
r)
(3)
The
(T,
)
p + q = r,
That
p p
=
A n W-(C)
p
=
<A,w- (C)>=
p
and
c
: H
Section
4:
w
fn
is,
if
A,
= A 0 [X]
a-(A U B)
p
<,)
T)
B
E
H* (X),
C
H
E
is
-- + 2Z
x-(A) 0
p
(cap nroduct)
a-(B)
q
O(a-(A) n C)
q
r
E(w-(a -(A)
r
p
() C)).
the Kronecker pairing
H*
H*
x
---
2Z
is the usual augmentation.
The basic sets
QP
and the intersection
pairing
theorem.
For any compatible triangulation
perversity
(X)
then
w-a-(A)
Here
(c r
=
T
(n-j)
(n-i)
(1)
and if
intersection pairings are compatible with the
cup and cap product.
and
r
=
HS(X),
k
WE
T
of
X
and
p, Goresky and MacPherson define the basic
20
subsets
lQ}.
and
Q
1
C.
1+1
is a full i-dimensional
Q.
T,
the first barycentric subdivision of
T',
subcomplex of
0 < i < n,
For
for all
(modulo
i
n).
The relation
of
these sets to intersection homology is given by the lemmas
below.
Lemma
4.1.:
is an isomorphism
There
S:Im(H 1 (Q1 )
--
>H
HP (X)
(Q?+)
111
induced by inclusion of cycles.
Lemma
4.2.:
If
> 1
i
and
p + q = t
=
(0,1,2, ...,n-2),
then there are canonical simplex preserving deformation
retractions:
X i+1
where
Tn-2
Lemma 4.3.:
is the
If
n-2
i +
H(X)
i
-
QP
n-i+l
-
n-2
skeleton of
j=n,
~Im(Hj(Q
and
)
T.
p + q=t,
H3 (Q)),
5
21
N
canonically.
Lemma
4.1 implies
that
the groups
H
(X)
are finitely
X, since
generated and independent of the stratification of
any two stratifications have a common subordinate triangulation.
Lemmas 4.2 and 4.3,
algebra,
yield the
Theorem 4.4.:
Intersection
E
Let
augmentation, where
p + q = t,
and
combined with some elementary
: H
t
0
(X)
-+
Pairing Theorem.
Z
be the usual
(0,l,2, ...,n-2).
=
j
i +
If
=
n
then the augmented intersection pairing:
HP(X)
1
x
H9(X)
3
+Ht
0
'E
X)
is nondegenerate if these groups are tensored with the
Section 5:
0.
sets
The basic
Whereas the basic sets
proving the intersection
different
basic
applications.
if
p < q,
R
Q
are perfectly
pairing theorem,
sets which are better
suited to
there are
adapted
The main drawback of the sets
then it is
The basis sets
on the other hand;
generally not true that
R
W(.
and complexes
to other
Q
is that
QC
QC
.
rationals
behave well under inclusion relations,
they play an important role in Chapter III.
22
They are defined as follows.
T
be a triangulation of
centric subdivision
X
R
Define
Tn I D
=
T,
IT n-2 I
=
IT n -3
simplexes which are
(p,i)
' .Z.jT
T'
to be the subcomp]ex of
1
with first bary-
Consider the stratification of
T'.
skeleta of
X given by the
X
0
.
Let
consisting
of all
allowable with respect to this
stratification.
W
Let
those
be the subgroup of
simplicial
properties
Property 5.1.:
The inclusion
i
of the complexes
: W
Im (H.1 (R?)
1
If
--
p <
H.(RP))~
1
1+1
~,
W
R
are:
C 1 (X)
i
an isomorphism on homology,
Property 5.2.:
consisting of
i-chains with boundary supported on
Two fundamental
induces
C T(R )
so
H?
(X).
1
there is an inclusion of chain
complexes:
WPC0
W *.
23
Intersection homology with coefficients and
Section 6:
the
G
Let
Q/Z
intersection pairing.
be an abelian group.
the subgroup of chains C e C
(p,i)
allowable and
(X) 0 G
JaCj
is
This complex may differ from
CE(X;G)
Define
1
such that
(p,i-1)
to be
jCj
allowable.
is
(Note:
0 G).
CP(X)
One can check that:
1
~Im(H
1
(Q?; G)
-+Hi
+?; G)
1+
)
HP(X;G)
and
HP(X;G)
1
Suppose there is
and
p+q=r
D
E
C
J
a pairing
i+j
in
C ( D
holds.
[11]
E
C A D,
H.i (RP
1+1 ;G))
--
G 0
G'
If
-n9=.
(X,G'), with
arguments
chain
~Im(H.i (R?;G)
1
DC A D,
-+G",
and suppose
CY(X;G)
CE
and
C
can be modified to yield
A aD,
an
intersection
we get
intersection products with coefficients.
If
p + q < r,
there
is
an intersection
pairing:
0
:
then the
Cr (X;G"), and the Lefschetz boundary formula
By applying the transversality lemma,
Theorem 6.1.:
and
H1 (X; G)
1
x
H9 (X;G')
3
--
H +
1
j-n
(X;G")
24
such that
[C] 0
[D]
=
[C 0
transverse pair of cycles
D]
for every dimensionally
C E CP(X;G)
D e C
and
1
let
In particular,
G' = 2Z,
G = Q/2Z,
J
(X;G').
G" = Q/2Z.
and
Consider the pairing induced by the abelian group structure
q/zZ:
of
'2/2Z ® 2Z -+/2Z.
The group
Q/2Z
is divisible,
so the universal
coefficient theorem for cohomology implies,
finite
for
A
a
complex,
H
i
(A; Q/2Z)
Hom
(H (A) , /2Z),
with the isomorphism induced by chain level evaluation.
Some elementary algebra,
and an argument very much like
that used to prove Theorem 4.4 yields the
Q/2Z
intersection
pairing theorem:
Theorem 6.2.:
is
Let
i
+
j = n
a nonsingular augmented
H ?(X;J/2Z)
x
and let
intersection
H.(X;2Z)
--
+
p + q
= t.
There
pairing:
H
(X;Q/2Z)
:,.Q/2Z.
I
25
Section
7:
In
Bordism with
singularities.
Sullivan introduced the concept of a bordism
[23],
theory based upon a class of cycles and homologies with
Let
or "bordism with singularities."
singularities,
specified
rn
Akin
pairs.
be a sequence of classes of compact p.l.
n e 2Z,
,
gave a list
[2]
of axioms which ensure that
is rich enough to support a homology theory
{yn}
defined for compact p.l.
(T,T0 ),
built upon cobordisms of continuous maps
where
(X,X0 )
necessities,
E
tnL,
some
and produces
n.
and
f : (X,X0 ) --
We present slightly
criteria for
{Xn}
(T,T O)
The list represents the bare
Z/2Z-bordism theories
unoriented).
oriented).
(T,T ),
pairs
(that
is,
stronger axioms providing
to support a Z-bordism
theory
(that
is,
In chapter VI, we give an interesting example
of such a theory.
First,
recall the following standard definitions:
Definition 7.1.:
X
C X
X
C N
that
(X,X )
is collared if
C X,
and a p.I.
h(X 0 x 0)
special
be a compact p.l. pair.
there is a closed neighborhood
N,
homeomorphism
such
= X0 C N.
h
(Note definition
: X
x
1.3
I -* N
(3)
is
a
case).
Definition 7.2.:
totally
Let
A locally compact polyhedron
n-dimensional
if
X = C9(In X)
X
_ In-1(X)).
is
Here
26
Remark
7.3.:
sional
iff
Akin
some
[2]
shows that
X
is
(not necessarily proper)
Now,
let
of
X
Every simplex of
X
n
Assume
.
K.
n > 0.
pairs,
,
E,
is
all
n.
satisfies the following axioms.
{yn}
Axiom 1:
If
(XX
0)
If
(XX0
and
YY)
0(
(X,X0 )
,n, then
E
En
Axiom 2:
and
K
face of an n-simplex of
(X,#)
for
Suppose
(Y,Y0 )
n-dimen-
be a sequence of compact p.1.
X E yn
Write
k), then
{fn}
[1]
totally
triangulation
(and hence any)
satisfies the following condition:
a
as in
.
is
X,
k-skeleton of
the intrinsic
Ik (X)
X0 C X
is
E.n,
then
is totally n-dimensional
X
collared.
n
Axiom 3:
If
(X,X0 )
E
J'
and
'
(X,X 0 )
J
~
Hn(XX;)
i=j~
(X,X0 ), denoted
Axiom 4:
If
(X,X0) Ey n
an orientation for
(X,X0 ),
is an orientation for
by
[X,X 0 ].
Here
for
is a sum of generators of the
[X,X 0 ]
cyclic summands of
infinite
An orientation
J.
some finite
Z,
Hn (XX0'
then
X0
then
If
E .n-l.
3[X,X 0 ]
E
[X,X 0 ]
Hn-(X ;')
X 0 , called the orientation induced
D* : Hn(XIX0;)
b. Hn- 1 (X
0
;2Z)
is the
is
27
connecting homomorphism
Axiom 5:
Axiom 6
(X,X 0 ) E Y
If
(X x I,X x I UX0 xIn+
then
(V,V )
A
If
Y = CS(X-V)
n,
(V,Bd V U V0 )
in
(X ,X 0 ) .
Axiom 7
Vr1Y )
A C. X
,E
(X,X0
subpolyhedron,
(Bd VBd
be a closed
a regular neighborhood pair for
= CZ(X0
Y
and
n
0
), then
and
n-1
Let
(Pasting).
X
:
be p.1. triples withZ X and 0Y disjoint.
(X , 0 )
Suppose
Let
[Xtx 0
i
Z
[Y y 0 ],
X0
-+
n
3
(YY0
[Z,Z 0 ]
and
iy :
and
,
6n
Z
(ZZ0 )
and
n-1
Let
be orientations.
be p.1.
--
embeddings which
are orientation preserving and reversing,
respectively, with
respect to the restrictions of the induced orientations on
Y
Furthermore,
suppose
X0
Y
=
orientation
XC
W
and
(X U
[W,W0 ]
Y - W
'
1
(Z) U
Y
(X,i,,(Z )),
(W,W0 ) =
Then,
Y (Z) UY
i
X
and
X
,
and
,
X0
'
Z
,
Let
(Cutting):
(Y,Bd VUY 0 ) C
X0 X1, Y D Y
in ordinary homology.
(Yl
Z
1i
= ZO
1)
(Z)
n
E n-1
and there
is
an
such that the natural embeddings
are orientation preserving.
Now we define the notions of oriented cobordism and
28
oriented map.
Definition 7.4.:
[X,X 0 ]
orientations
(X,X0 )
Let
[Y,Y 0 ],
and
respectively.
(Y,Y )
and
(X,X0 )
An oriented cobordism between
an oriented pair
yn, with
be in
(Y,Y )
and
n+ , with orientation
(W,W0 )
[W,W 0]
satisfying
1)
i
are p.1.
There
W
: Y -+
i
embeddings
W
: X -
and
preserving and
which are orientation
reversing respectively with respect to the induced
orientation
on
*
W0
2)
i
3)
There is a compact subspace
r) i
a)
and
(Y)
=
i
(X)
W
i
(Y) n
W
WO
- i
=
i(X) U
(W ,i (X 0) U
b)
An oriented map of
such that
W
x)
(Y
W1 U
i
C
iy (Y)
Y ))
EIn
(X,X0 ), with its orientation
[X,X ],
is a continuous map
(T,T0 )
is
denote this
i
W
'
(X)
f : (XX
a compact p.l. pair with
0)
--
T0 C T
{f,[X,X 0 ]1, and refer to
'
0 )T, where
closed.
{f,[X,X 0 ]}
singular 3-cycle.
Given oriented maps
(TT0),an oriented cobordism
g between
: (Y,Y
f and
0)
'
f : (X,X0
(T,T 0 )
and
We
as a
is
29
(W,W )
is an oriented cobordism
(Y,Y ), together with a continuous map
F o
such that
and
f
=
x
F o
i=
F
as is
easily
: (W,W)
(T,T )
(T,T0 ).
checked.
Under disjoint union,
group structure:
and
*(T,TOr
is an equival-
Denote by
the set of equivalence classes of oriented maps
into
and
g.
y
Cobordism of oriented maps into
ence relation,
(X,X0 )
between
Q n T,T 0
)
g
{f,[X,X 0]}
this has an abelian
$
the zero element is
(4,$) ---. (T,T0 ),
:
{f, [X,X 0 ]} + {f,-[X,X 0 ]} = 0, using the product
~I,
xIUx0 X]}.
{f oa 7
cobordism
Define the connecting homomorphism:
n
Q
0
(T, T0
n-l(T
0
by
9n fi[XiX 0
where
define
Finally,
F*
if
: Qn (S'S0
F
Q*
* (T,T )
: (S,S0 )
Qn(T,T )
F*({f,[XX0]})
Let
I
ffIX0
is the induced orientation
D*[X,X 0 ]
before.
=
denote the functor
=
from
on
X 0 , as
is continuous,
by
{F o f,[X,X 0]'
compact p.1. pairs to
graded abelian groups, along with the connecting morphisms
30
n
Theorem 7.5.:
Q,
is a homology theory.
The proof proceeds as in
Proof:
simply keep track of
[2];
orientations.
3
called an oriented bordism
Such a homology theory is
theory with singularities.
Definition 7.6.:
of compact p.l.
n >
for
spaces,
be a sequence of classes
{&n}
a&=
Let
We say
0.
d&
is a class
of singularities if:
(1)
4
(2)
If
L
(3)
If
L E C"
n, all
E
E
n
,n , then
L
is totall r n-dimensional
J
some
J > 1.
L 74 $, then
and
An orientation for
n (L;2Z)
is denoted
L
=
7Z,
[L], as
before.
(4)
So
For
n > 0,
is the 0-sphere.
L
E
iff
.6n
S
*
L
E
n+ ,
where
This is closure unier suspension and
desuspensi n.
L
n = x,X0) ' (X0) is a totally
Let n-dime]=sional compact
pair,
x C
orientable,
X - X0 '
with
x0
k (x; X) E .6} .
X
Here
collared,
kk (x; X)
such that for
denotes
the link
31
x
of
in
Then
X.
is readily verified
satisfies axioms 1 -
=P.&
[2].
7,
as
The associated homology theory
called the oriented bordism theory based on the
is
*
J,
class
of singularities.
o7
Bordism theory with
8:
Section
2/kZ
coefficients.
QT.
Suppose we have an oriented bordism theory
in an abelian
Coefficients
*
;G).
The
[8], yield-
as described in
geometrically into the theory,
ing the theory
can be introduced
G
group
G = 2Z/k2Z
case
admits a
particularly simple and convenient description first
;Z/kZ)
;,O
[18]
in the setting
of
.
by Morgan and Sullivan in
realized
In general,
a
Z/kE-cycle
of dimension
n
is
obtained
, equipped with an oriented
(X,X0 ) En
from an oriented pair
k
embedding
Y0
X 0 of k
copies of
Y
onto
XO'
1=
Y
where
n-1,
ori:entation
by identifying the copies of
preserving manner.
is defined similarly,
into
A
,n-.
by embedding
k
copies of
(Y0 Y 1 ).
The boundary consists of
k
-
-{interior
1=1
along the images of
of image of
Y1
.
X0
in an
2Z/kE-cycle with boundary
X0 , and identifying the copies of
(YOY1 ) 1
Y
Y01'
with identifications
(Y0 Y 1)
Here
32
(
The homology theory
built
from maps of
;2Z/k2Z)
Z/k2Z-cycles,
is
the bordism theory
modulo -naps of
2Z/kZ-
cycles with boundary.
>f bordism with
We need the following properties
2Z/k2Z
coefficients,
play a role
in the discussion
Property 8 .1.:
is
both easily
verified.
in the Appendix.
The homomorphiEm
multiplication
by
k,
These facts
i
:
2Z/k2Z
induces a natural
-2Z/k-.ZZ,
which
transformation
of theories:
Q*(
This is
Property
reduction
;2Z/k2Z)
;2Z/k2Z4)
,(
--
Sullivan's "replication" homomorphism.
8.2.:
The projection
mod k,
Q*(
ZZ/k'-9I
-.
2Z/k2Z, which is
induces a natural transfurmation
;2Z/k-9,2Z)
-- +Q*,(
; 2Z/k2Z)
I
33
Chapter
Algebraic Preliminaries.
II:
This chapter contains a description of the Witt theory
of inner product spaces.
of classification
Section 1:
Witt rings.
We employ the terminology of
is found in
Let
finitely
R
be a commutative ring with
generated projective
1 : X
Another treatment
[3].
Definition 1.1.:
form
[13].
x
module over
An inner product
X --
R
unit.
3
on
Let
X
to
HomR(X,R)
be a
R.
X
which is non-degenerate,
two homomorphisms from
X
is a bilinear
i.e.,
the
given by:
x i (x00,-)
y 0 I-'P(',y
0
)
and
are bijective.
We call
(X,1)
an
inner product space; there is the
obvious notion of isomorphism of inner product spaces over
R.
The inner product is
according as
symmetric or skew-symmetric
34
r(x,y) = f(y,x)
or
In this
R = 2Z/p2Z
We can associate to
(X,?)
p
el,...,e
B =
=
where
1.2.:
has matrix
a)
B
b)
If
B'
(X,r')
prime,
and
R =
2Z.
if
p.
( . .)
3(e.,e.).
:ji
J
invertible
(X,
)
has a matrix
and only if
(e1 , . .
= ABAt
B
B
and
,e )),
(XI')
then
, foi some
A.
3]
We sometimes write
I
(XS)
=
Given inner product spaces
with the form:
<B>.
(X ,
) , ... ,
(X 1 . .EDXn')
.
define the orthogonal sum
X 1 1....Xn
X.
the mratric:
(with respect to
matrix
[13,
is
be a basis for
.
invertible
Proof:
for
is necessarily a free R-module (a vector space or
Let
(X,3)~
-1(y,x).
we will be concerned primarily with
free abelian group).
Lemma
=
.
X
thesis,
R = C,
the cases
Then
(x,y)
to be
n,'
we
35
f(x e.. .x
Clearly
(X1 6...
'ye. . .ey n)
is
6Xn)
(iFyi)
an inner product space.
may also construct the tensor product
X 1. . .QXn
is
with the unique form
X...
Xn')
(X1
We
which
satisfying
'
n
'(x 1
Again,
(X1 Q...Xn,')
[13,
10].
p.
N C X
its
R.
Let
X
such that
orthogonal
Oxn'
is
Definition 1.3.:
space over
.-
N
split
be a symmetric inner product
if
there exists
is a direct
complement.
a submodule
summand of
X,
N = NL,
and
(NL= {x e XJr (x,n) = 0
for
n e N}).
all
Remark
1.4.:
A split
"metabolic" in
inner product space
space
(X,)
is
called
to us, we have the following
convenient description of split
Lemma 1.5.:
X
[3].
In the cases of interest
it
T i (xi'01 yi
i=1
n
an inner product space
(X,r)
is
'
1
Let
over
R
spaces.
be a ring such that
R
is free.
Then
every inner product
(X)
is split iff
has a basis with respect to which the matrix of
6
has
36
0
I
n
the form:
,where
is
I
the
identity
matrix of
I nAn
rank
n.
If
2
is
a unit,
then with respect to some
0
the matrix of
basis,
a
(In
Proof:
[13, p.
1
is
0)
U
13].
In the case of skew-symmetric
X
(X, )
is symplectic if
6 (x,x) =
0,
all
E X.
Parallel
Lemma 1.7.:
to Lemma 1.5
If
R
a field),
inner product space over
symplectic basis;
0
is:
is either a Dedekind domain
or a local ring (e.g.
S
spaces, we
concept of being split.
have an analogue of the
Definition 1.6.:
inner product
that
R
is,
(e.g.,
Z)
then every symplectic
is free,
and possesses a
a basis such that
the matrix of
1
n
is
(n
Proof:
[13, p.
7].
Finally, we distill the collection of symmetric inner
product spaces over
R
into the Witt ring
W(R)
by means
37
of an equivalence relation.
Two symmetric inner product spaces
Definition 1.8.:
X'
X
belong to the same Witt class, denoted
there exist split spaces
isomorphic
to
X'
,
equivalent to the form
a useful
S'
such that
if
X @ S
is
we omit explicit
s').
By abuse of terminology,
This is
and
(For convenience,
S'.
mention of the forms
S
~ X',
X,
we say that
a split
space
is
<0).
relation,
it
satisfies
the following
properties.
Lemma 1.9.:
X - X'
If
1)
X
2)
X E Y
3)
(X , ) E
(X,- )
4)
(1>
- X
Proof:
) Y
~X'
9 Y'
~
E Y'
0 X
Y
-
Y',
then:
~ <0)
3
[13, p. 14].
Consequently,
W(R)
and
the collection
has attractive structure,
of Witt classes,
denoted
suitable for study in terms
of standard algebraic concepts.
Theorem 1.10.:
W(R)
is a commutative ring with unit,
using orthogonal sum as addition operation, and tensor
38
product as multiplication operation.
U
Immediate from 1.9.
Proof:
Structure of
Section 2:
We derive
the additive
In the process,
W().
a prime,
and
W(V).
structure
of the Witt ring
we determine
W ZZ/p7Z),
for
p
W(2Z).
We begin with the statement of a lemma concerning the
"diagonalization" of the matrix of an inner product space.
Lemma 2.1.:
If
R
is
then every symmetric
a local
ring in which
inner product space
possesses an orthogonal basis.
basis
e 1 ... ,e
so that
X = (u 1 )G.. .$(un>,
Proof:
[13, p.
That is,
(e.,e.) = 0
where the
u
Now,
p (X,)
given
in
(Xr)
X
a unit,
over
R
possesses a
for
i
are units in
j.
-
Thus
R.
a nonsingular
rational
on Lemma 2.1.
(X,3)
in
W(Z/pZ),
for
following result,
is
6].
Anyone who has diagonalized
matrix has relied
2
W(V) , we associate elements
p
prime, by means of the
due to Springer
and Knebusch.
First,
39
if
we fix a prime
rational number
n
E
2Z
of rationals
E
2Z,
and
u
is
any non-zero
n
p
=
*u,
a unit
in
2Z ()
the subring
which, when written in reduced
denominators prime to
Lemma 2.2.:
additive
p.
Fix a prime
(That is,
p
E
2Z.
2Z
form, have
(2Z-(p)) -I
)
There is a unique
homomorphism
-+
ap : W(
W(2Z/P2Z)
which maps each generator
<pnu>
as
0 (mod 2).
n E 1
(mod 2) or
residue class of
Proof:
then
can be written uniquely as:
q
q
where
p
u
n
in
2Z
to
(u >
/(P) .Z ()
or
(Here
0,
u
according
is the
= 2Z/p&).
U
[13, p. 85].
Remark 2.3.:
This homomorphism is
the second
residue class
form homomorphism.
We can collect
p
.
note that
the homomorphisms
P
for all primes
p
and thereby obtain an additive homomorphism
40
S:W(M
(@)+W
(2Z/pZ)
p prime
Note that
we also have
i
a homomorphism
: W (2Z)
inner product space as
an integral
gotten by regarding
-W(W
a
rational one.
The next, remarkable theorem is proved in
90]
.
recall
First
Definition
space
is
d+
the integer
- d-
d+
where
inner product
d~)
(resp.
diagonal of a diagonalized matrix for the form on
0
1)
The sequence
.W(2Z)
-*W(')
-()W(Z/p2Z)
-p0
is split exact.
W(2Z)
~Z
canonically,
and the
isomorphism is
:
W (2Z )
-+.
Z
,
given by the signature homomorphism,
sign z
is
the space.
p
2)
88-
entries on the
the number of positive (resp. negative)
Theorem 2.5.:
p.
a standard definition.
The signature of a rational
2.4.:
[13,
41
the
signature of
3)
sign 1
to its
(XI) e W(2Z)
i(X,3)
e W(Q))
The exact sequence
o Sign
is
split
(that
is,
by the homomorphism
, where
--+W(Z)
W(Q)
signature
.
which takes
sign
: W(0)
2Z
-
is the signature homomorphism.
Therefore
there is
a canonical
e D W(2Z/p2z)
W(2Z)
W()
isomorphism:
p
2Z ED
W (2Z/p2Z)
p
It
remains to compute
Lemma 2.6.:
The
additive structure
1)
W(2Z/22Z)
Z/22Z,
2)
W(2Z/p2Z)
2Z/42Z,
p = 3
3)
Proof:
for all primes
of
W(2Z/p2Z)
p.
is:
generated by (1>.
generated by
<)
for
(mod 4)
W(2Z/p2Z)
<d>
for
p = 1
2Z/22Z E
and
[19, p.
Remark 2.7.:
W(2Z/pZZ)
where
2Z/27Z, with generators
d
is
a non-square
in
<1>
2Z/p2Z,
(mod 4).
3
47].
[19,
p.
47]
structure is described by:
actually
shows that
the ring
42
1)
<d> 0 (d>
2)
O>) 0 ) =<'
3)
(1> 0 <1)
=
(d) 0 <d)
=<(l>
(1>
=
for
p = 2
for
p E 3
(mod 4)
for
p =1
(mod 4).
1>
We isolate the following obvious corollary to Theorem
2.5 and Lemma 2.6,
as it
has important ramifications in
Chapter VI.
Corollary 2 .8.:
W(Q)
~
Z
T
(canonically), where
T
is
an infinite direct sum of cyclic 2-groups of order 2 or 4.
2
It is also useful to state the following fact about
.
W(O)
Proposition
2.9.:
If
(Xr3)
is
a symmetric rational
product space such that its equivalence class in
the zero element,
Proof:
Witt's
then
(X,f)
[13,
p.
W(V)
is
is split.
This follows from the definition of
theorem
inner
W(Q)
and
8].
M
43
Chapter III:
Witt spaces and generalized Poincar6 duality.
Introduction.
Section 1:
In section 2, we define the class of Witt spaces.
Roughly,
a Witt space is a stratified p.l. pseudomanifold
link
homological
the following intersection
satisfying
condition:
is the link of an odd cocimension stratum
L2'
at a point
x
X, then
in
H
where
M
=
= 0,
(0,0,11,2,2,...); that
In section
pseudomanifold
Wn(X;r)
(L;Q)
3,
for an arbitrary
is
mn(c)
=
triangulated
c-2
[-2
*
If
p.l.
X, we define an increasing filtration on
corresponding to the increasing sequence of
perversities:
M
dim X-2
(0,0,ll,2,2, ... , [
2
dim X-1
(0,0,1,2,2,3,...,[dim Xn =
(0,1,1,2,2,3,
dim X-1
2
44
Recall
=
n(c)
[
Notice that
c-1
-I.
the perversity
index of each odd co-
dimension stratum gets incremented by 1,
in order of
decreasing codimension.
Associated to the filtration
converging to
collapses
Wn (X;e).
is an E1 -spectral sequence,
We prove that
for a Witt space.
That is,
the
E
spectral
= 0
sequence
for
s > 0.
The generalized Poincare duality theorem for closed Witt
spaces X,
stating
m
that
H*(X;Q)
is
augmented intersection product,
section pairing theorem
of a Witt space
values
in
W(T),
dim X = 4k,
X,
a p.l.
in
W (Q).
invariant taking its
the Witt group of the rationals.
k > 0,
w(X)
then follows from the inter-
we introduce the Witt class
w(X)
is
Otherwise,
If
is the equivalence class of the
rational inner product space
dim X = 0,
under
(1.4.4).
Finally in section 4,
w(X)
self-dual
H k (X; ()
merely the rank of
we set
w (X)
in
W(Q).
H 0 (X;Q)
If
times
6]>
= 0.
For discussion and applications of a special case of
the Witt class,
Section 2:
Let
see
[3],
[10]
Witt spaces.
X
be a q-dimensional p.l. pseudomanifold with
45
Then there is a p.l. homeomorphism:
Z(x)
[1,
= q p.
X
~
at
x.
Sd(x)-l * L(x).
is a pseudomanifold of dimension
L(x)
The space
kk(x,X)
DX.
homeomorphism
dimension of
be the intrinsic
d(x)
Let
[20].
up to p.1.
unique
kk (x,X), is
x,
of
The link
X E X -
Let
X.
collared boundary
(possibly empty)
-
d(x)
homeomorphism
unique up to p.l.
1,
420 and 424].
as above.
manifold
X
Let
Definition 2.1.:
We say
be a q-dimensional p.l. pseudo-
Z(x)/2
for all
X
E X
Remark 2.2.:
-
DX
is
X
a Witt space
(L(x);
such that
k(x)
) =
if
0,
E 0
(mod 2).
The definition places intersection homological
restrictions on the links of odd codimensi*n intrinsic
strata only.
section
The discussion of the spectral sequence in
3 will provide
justification.
For applications, we must translate the Witt condition
in 2.1 into a statement about arbitrary stratifications
of
X.
To this end, we prove a few lemmas which culminate
in Proposition 2.6.
with m(c)
=
c-2
Recall that
m
refers
to the perversity
46
Let
dimension 22.
manifold
K
L
be a closed p.l. pseudomanifold of
L = S0 * K
Suppose
of dimension
2k-l.
for some
Then
H
p.l.
(L;E)
pseudo-
=M
.
Lemma 2.3.:
0k
Proof:
T
The 0-sphere
is the discrete set
be a triangulation of
ci
stratification:
S
S0
=
K, and
T
T2 Z-1 D ZT
be the triangulation of
let
ef
be the
c7
L
{a,b}.
Let
the associated
T 2Z.-30
23
'T
0
).
Let
induced by suspension,
stratification
on
L
similarly
and
induced.
X
(S
Note that
S
*T 2
29i-l
.4
*Z=
* T 2k-3...
T
* T0 D
22-3~
0
is not the stratification asbociated to
z e Cm (L) , where perversity
Consider a cycle
4.
defined with respect to
a triangulation
S
k0
Then,
of
L
Izi
) Sj = $.
subordinate to
S
z
is a sum of k-simplices with coefficients.
v
of
L.
If
If
v
Izi
in the
Int(a) C K,
E Int(a*T)
define
of
lies
Izi.
f(a) = T.
By
interior
define
or
v
E
of a unique
f(a)
[20, p. 17],
chain which we denote
One checks easily that
f
f(z)
f(z)
is
Choose
such that
Each vertex
a
of
= C, the barycenter of
a.
Int(b*T),
Extend the map
S.
where
f
simplex
Int(T) C.
linearly
K, then
to simplices
determines a p.1. geometric
in
C ,(L),
supported in
is a cycle and,
in fact,
IKI.
)
That is:
47
If(z)I
the
same homology class as
of
Izi
IzI
v
structure on
Izi
x
[0,1].
chain
H(z
DH(z x
[0,1])
= f(z)
- z.
[0,l])j
is
|H(z
x
for instance,
See,
-* L
[0,1]
x
a p.l.
that
The triangulation
and subordinate to the product cell
x
[0,1])
= v
H (v,0)
by
[11].
and
over simplices of
[20, p.
Appealing again to
[0,1].
x
(L).
the set
and extend linearly
f(v),
=
IzI)
E
IZI
H :
Define
Izi
H
vertices
having as its
{(v,0),(v,1)I
H(v,1)
in
can be used to construct a triangulation of
[0,1]
x
z
represents
f(z)
We claim that
(rn,Z)-allowable.
is
in
we obtain
17],
CP+1(L), with
it is easy to see
Moreover,
(mi, Z+l) -allowable.
This proves
the claim.
Let
IWI = a
W
*
be the obvious p.1.
If(z)I
chain in
DW = f(z).
and
trivial.
Therefore
was arbitrary,
Corollary 2.4.:
dimension 2k.
manifold
Proof:
represents
f(z)
Suppose
L
in
be a closed p.l.
L
Let
0
L = Si
0 <
j
as
S0
< 2k.
r0
in
H
C
with
(L),
H m(L)
(L).
is
Since
z
(L) = 0.
H
we conclude
K, where
Rewrite
z
W E
Then
implying that the homology class of
Ck+1 (L)
* K
for some p.l. pseudo-
Then,
~1
pseudomanifold of
(Sj1* K) ,
H
(L) = 0.
and apply 2. 3.
I
48
Corollary 2.5.:
Proof:
Let
H' (L;Q)
= C*(L)
C*(L;V)
Hm(L)
Then
Hm(L;Q)
= 0.
manifold,
with
X
Let
.4 =
Xi,
0
Xi
Xi /
and
boundary
Xn.n-2
Xn
(Xn
be the link of
(X-DX)
be a stratified p.l. pseudo-
(possibly empty)
stratification:
L(Xi,x)
induced by the definition:
0 Q,
0 (.
Proposition 2.6.:
E
be as in 2.4.
The result follows from 2.4 and the isomorphism of
groups:
x
L
X.1t
=
$.
-
I
DX
Then,
0).
''
at
_1
X
and
Let
x,
xherwhere
is a Witt space if
and only if
H (L(Xix);Q) =
for
i
= q -
Proof:
k > 1.
(2k+l), whenever
Suppose
There are p.l. homeomorphisms:
is Witt.
X
Sd(x)
k(x,X)
9k(x,X)
Let
j
=
d(x)
-
i.
L(Xi;x)
Then
z S
0
~
* L(Xi;x)
S
j
* L(x)
>
0, and
* L(x)
by
[1, p.
423].
49
Case 1:
j -
If
follows
result
Case 2:
1 = -1,
The converse
of
stratification
X
from the fact
j - 1 >
If
0,
Pk(xi;x)
then
a Witt space.
is
the result
follows
follows from the fact that
is
X
so the
~L(x),
from 2.5.
any p.l.
to the intrinsic
subordinate
stratification.
I
Section
3:
homology
The intersection
spectral
sequence
and duality.
triangulation
r
T, and associated stratification
2r+1 < q
be the largest integer satisfying
q > 2) .
pseudomanifold with
is a q-dimensional p.l.
X
Assume
U7.
Let
(assume
Assume rational coefficients unless otherwise
specified.
Definition
3.1.:
Let
be the perversity defined by:
>k
m(c) =
Pkc(c)
1 < k < q
for
c < k
[-2
for
f
c-1
c > k
=
n(c)
where
c-2
[2]
and
=
2 < c < q.
50
Since
odd.
= n(c)
m(c)
Note that
for
and
m
P2r+l
k
= n.
p
on the chain complex
F
There is a filtration
[see
even, we may assume
c
Wn(X)
1.5]:
(0
(X)
F
={W(2r+1)2s(X)
W(X)
for
s < 0
for
0 < s < r
for
s > r
The filtration is induced by theinclusions of complexes:
0c
.
=
w*
P2r-1
2r+l
Following
E 1 -spectral
c W*
[21,
C-
W
p. 469],
3
C
there is
p1
a convergent
sequence with
st = H s~t (FSWr* (X)/Fs-1i W
Est
and
E
nO
isomorphic
(X))
to the bigraded module
G (H
(X)),
associated graded to the filtration
F
(H (X)) = Im(H
Theorem 3.2.:
(2 r+l) -2s (X)
Assume
(1)
E
(2)
E St = 0
Proof:
0
t
ott
0(
Part
(1)
Im(H
X
is a Witt space.
(X) --. H
for
H n(X)).
Then
t (X))
s > 0.
is immediate, since
F
(H (X) ) = 0.
the
51
Part
(2)
follows from the equality
2s3
Im(H,
(3)
0 < s < r
We now prove
(3).
Fix
W 2s+3
J
E
W.
t
_
s + 1 <
j
s
2s+1
(X).
with
< q -
0 < s <
for
2s+l
J
2s+l
(X), for
r -
1.
j
< s + 1
j
> q -
Note that
and
s,
s.
be represented by the cycle
H 2s+1(X)
Let
z
~
(X)) = H
1.
-
so assume
2s+l
H
(X) -
.
We construct a chain
w E W
P2s+1
(X)
j+a
such
that
@w = z - y,
where
y e W.
P2s+3
(3)
This will prove
(X).
and therefore
prove the theorem.
The chain
stratum of
dim( IzI
is constructed locally to reduce the
I z I K X q-(2s+3), where
dimension of
q-(2s+3)
w
n
(f.
Xq-(2s+3)
Xq-(2s+3)
To begin with,
- j =
j
is the
we have:
(2s+3) + p2s+1 (2s+3)
- s -
2.
52
If,
in
fact,
(4)
then
z
the following stronger inequality holds:
dim(IzI 0 Xq-(2s+3))
92s+3
W.
E
(X).
<
j-
(2s+3)
=
j-
s
-
+ p2s+3 (2s+3)
3,
This follows from the fact that
P2s+p(c)
= P2s+3(c)
y = z
In this case, we may set
/ 2s+3.
c
if
and
w
0, and we're
=
done.
Suppose
(4)
does not hold.
The open stratum
is the disjoint union of interiors of simplexes
dim a = q -
where
(2s+3).
subset of these simplices
j
-
s -
(where
Let
2.
I
is
a e C,
a finite
Let
C
denote
for which
index
set)
the set
j
of
in the first barycentric subdivision of
Izi n
Orient each
simplices
Int(T
)
in
is:
Int(a)
The subchain of
T.
T'
whose
closures
-
in
\ Int(a'))
{T
s - 2
a
a
intersect
=
i i}iEI
simplices
satisfying
D Int(T).
z
T,
the non-empty
dim(jzJ
and consider
Xq-(2s+3)
consisting of
Iny(a)
jin
m
53
Z.
T.
1
1
where
v.
an d
e C s+1(k(u,T')),
k(c,T')
boundary of the classical dual to
Lemma 3.3.:
Dv. = 0, for all
Proof:
i
I.
E
Dz.
=
3z
-
Now,
-a
(-
*
1
Z., I
The re fore,
/
*
v
v. +
Int (c)
z
as
contains
By definition of
0.
)
Int (
does not contain
(5),
i.
z = z. +
1
(z-z.).
1
(-)s-1. * 3v.
1
I|a(Z-z.i)|
in
dual(a,T').
(z-z.).
=T.
Dv.
T',
is a cycle, we have:
z
(5)
if
Decompose
in
)
Since
Fix
a
denotes the
z.,
Int(T.)
if and only
however,
10 Int (a)
It follows, by taking supports
v. = 0.
that
2
The canonical simplicial isomorphism of
s+l
k(a,T)'
maps
the complex
v.
to a cycle
WT'(9,k(a,T))
V.
1
e C
tk(a,T')
(kk(a,T)).
and
We have
associated to the restriction of
54
kk(a,T).
to
(k(cr,T)),
v. e W
s+1
I
Lemma 3.4.:
(s+l)-simplices in the
is a sum of
v.
Clearly,
Proof:
first barycentric subdivision of
for each
Fix
(6)
Qk(a,T).
Iv |
it remains to show only that
i.
for all
av.
Since
0,
=
(m,s+)-allowable,
is
i.
i
v
The cycle
I.
E
can be written uniquely as
v
=
a(<p0''..
'
s+1-i
'
T
G<y04'''.'ps+1
Here
the simplex in
d(p)
T'
(7)
'
spanned by
k
to be the maximum integer
d(p)
For
such that
p> a,
^ =
>'Os+l>
with non-zero coefficient in
simplex
Set
is the rational coefficient of
a(y 0 '..'s+l)
= -1
if
p
does not appear.
dim(Iz.I (I Int())
On the other hand,
since
z
j
E
P2s+1
C.
(X), we get the
s -
2 + d(y)
(6).
+ 1.
J
following inequality, valid when codimension of
is greater than or equal to 2
in a
Then
=
-
define
(codim(p,X) > 2):
p
in
X
55
n
dim(lzl
(8)
<
P2s+l (codim(-p,X))
Now,
implying
codim(y,X)
d(p)
(9)
y > a
Each
<
(s+1)
r) Int(y))
< dim(jzj
Int(p))
j - codim(p,X) + p 2s+(codim(p,X)).
= m(codim(y,X)),
< 2s+l.
Combining
since
(7)
and
P > a,
(8),
we get:
+ m(codim(P,X)).
- codim(p,X)
has a unique expression as:
P = a * V,
where
V
e 9k (a,T).
The cycle
v
(10)
v.
that
P
=
a * v,
v = Vk
then
d(P)
in a simplex
0
is
the maximum integer
,. '
(v 0 ''
ps+
<' 0
s+1 >
d(y)
= dim(|v'
Int(v)).
Observe that
(12)
codim(vZk(a,T))
k
such
with non-zero
coefficient in (10). Therefore,
(11)
.vS+1 >.
a(
a<p 0 <..'Ps+1
If
then has the expression:
= codim(p,X).
56
Then
(9),
and
(11),
dim( V-
I
(12)
yield the inequality:
<
) Int(v))
-
(s+l)
codim(v,Yk(o,T))
+ r(codim(v,Zk(a,T)))
This
codim(v,Pk(a,T)) >
inequality implies
that
I.
2
is
.
when
(ms+l)-allowable.
I
By hypothesis,
m
that
Hs+1 (9k(a,T))
x.
i
s+2
E WM
(k(a,T))
X
is Witt.
= 0.
Therefore,
wit h
Dx.
corresponding chain in
= v..
1
w
implies
there is
a chain
Let
be the
x.
i
WJe define:
9,k(cr,T').
w. =
and
Proposition 2.6
(-l) j-sT
*
X
w
=
iE I
Note that:
Dw. =
Lemma 3.5.:
Proof:
It
W
a
(-1) j
D.
1
*
x.
1
+
T.
1
*
ax..
1
EW Wp2s+1M
(X).
j+1
suffices to prove
W.
1
1 2s+
E W.
3+1
(X
,
1
f or all
i.
57
The argument proceeds as in 3.4.
is
Iw I
(p2s+l, j+1)-allowable.
codim(p,X)
1:
Case
P E
Let
star(cy,T),
with
A counting argument analogous
p > a.
3.4 proves:
dim(|wj |
<
) Int(I))
codim(p,X)
Since
we show that
> 2.
Suppose
to that in
First
(j+1)
< 2s+1
- codim(p,X)
+ m(codim(p,X)).
= p 2s+l(codim(p,X))
m(codim(p,X))
so
Int(p))
dim(jwl 1)
Case 2:
Suppose
dim(|w
| n
P
<
Int())
j+l - codim(p,X)
<
+ p2s+l(codim(p,X)).
Then
a.
= dim(|T.
* xi|
= dim(JT.
* v
< dim(Iz I
Int
int(1))
fl
I
) Int(W))
()).
(p2s+1,j )-allowable,
Since
Izi
is
dim(Iwil
n
Int(p))
<
j
- codim(p,X) + p2s+l (codim(p,X))
58
p
Suppose
star
lw. l|n
Then
(0,T)
.
Case 3:
Int(P)
These three cases prove the assertion that
0.
is
j+l)-allowable.
|3w
J
is
* Xii
U
ITi * v
Next, we must verify that
able.
Note
be as
above.
Case 1:
(13)
that
I|wji
=
aTi
three cases
We treat
a.
p
dim(
* xj sInt(y))
(
Ti
11
again.
= dim(IT
= dim(Iwi 1A
<
Let
Then
Suppose
>
(p2s+ 1 , j)-allow-
.
(p2s+l,
Iw |
=
j
-
Int(y))
(codim(yp,X)
i* xi(A
-
Int (p) )-1
1
+ 62s+l (codim(p,X)).
(14)
dim(IT
S*
Int
xj I r)
(p)
)
Also,
= dim(I-u
<dim(|zi n
j
Together,
dim(IwI
(13)
and
r
(14)
Int(p))
-
*
vi I
Int(p))
Int(p))
codim(p,X)
+ P2s+ 1 (codim(y,X)).
imply that
<
j
-
codim(p,X),X)
+ 2s+l (codim(y,,X)).
59
n
dim(0|Dwi
3:
Case
Then
P < a.
Suppose
(p)
Int
Suppose
y
)
Case 2:
'
<
dim(IT1
<
dim( zi
<
j
Int(y))
) Int(p))
- codim(p,X)
star(cyT).
These three cases prove that
0
Then
I|w I
+ 62s+l(codim(y,X))
Dwi
is
.,)
Int(y)
= 0.
(P2s+ 1 ,j)-allowable,
completing the proof of the lemma.
a
Let
Y
CY
of
E
y
(X).
W.
J
w
ar
z -
=
w.
Then
Moreover,
y
is
a cycle and
it is clear from the construction
that
dim(IyI
n
Int(cy))
<
j - s - 3.
Now repeat the argument for each of the simplices remaining
Denote the
in
C.
by
{w }
.
j+l-chains obtained at all of the steps
If we set
w
=
and
w
CreE
60
y =
z -
Dw
we have:
dim(IyI1
for all
G E Tq-(s+3.
Int(a))
<
j
-
Therefore,
yE
s -
(4)
3
holds and
W 2s+3 (X).
W
W.
J
The theorem is proved.
2
Remark 3.6.:
The proof of 3.2
condition on links is
satisfied
shows that
if
the Witt
only for odd codimension
strata of codimension less than or equal to
2k+1, then
the spectral sequence collapses partially.
Esit = 0
We now use
for
s >
(r-k).
3.2 to prove a proposition which,
when
combined with the intersection pairing theorem, yields
Theorem 3.8,
generalized Poincare duality for Witt spaces.
We include the rational coefficients for emphasis.
61
Proposition 3.7.:
X
Let
q > 0.
be a Witt space,
The
homomorphism of graded groups,
H 1,(X;Q)
i,
-
H
(X
induced by inclusion of chain complexes,
Proof:
For
q
For
q = 0
or
1, the result claimed is obvious.
> 2, Theorem 3.2
(15)
Im(H*(X;Q)
is an isomorphism.
implies:
-,H*(X;Q))
=
H
(X;O)
Therefore:
(16)
(X;Q) > dim H(X;Q)
dimH
pairing theorem
The intersection
(17)
dimH
Combined,
(18)
(X;C)
0 i
(16)
and
dim H M (X;Q)
i
= dimZH
q j
(17)
=
0
for
.(X;qD)
(I.4.4)
<
j
< q.
implies:
for
0 <
j
for
0 <
j < q.
< q.
yield:
dim H'(X;D)
MO 3
The proposition follows from (15) and
(18).
I
62
Finally, we get generalized Poincar6 duality for Witt
spaces.
Let
Theorem 3.8.:
X
There is a
be a Witt space.
nondegenerate rational pairing:
H m(X;O)
21~
for
i +
j
= q;
i,j >
x
Hm(X;
J
)
Q
0.
The pairing is given by augmented intersection product.
Proof:
Theorem 1.4.4 provides a nondegenerate rational
pairing
O)x
Proposition
on the
H n(X; ()
for
-+
3. 7 gives an isomorphism
i
+
j
= q;
ij
> 0
.
H (X;
(ind iced by inclusion
chain level):
i.
J
J
(X;
J
(X;
for
0 < j < q.
Intersection product of cycles is compatible with this
homomorphism by
[11].
The theorem follows.
a
63
Section 4:
Let
The Witt class
X
w(X).
be a Witt space of dimension
Theorem 3.8 gives a nondegenerate
on the vector space
H
(X;V).
4k,
k >
0.
symmetric bilinear
That is,
form
is
H k(X;Q)
a
symmetric inner product space.
Definition 4.1.:
q,
w(X),
2k
X
Let
be a Witt space of dimension
q > 0.
q = 4k,
If
Hm
(1)
is
k > 0,
the Witt class of
the equivalence
(X;Q)
in
denoted
of the inner product space
W(), the Witt group of the rationals.
0,
w(X)
If
q
=
If
q
/ 4k, set
(2)
If
(X,DX)
set
w(X)
=
rank(H0 (X; C)).(l>
Remark 4.2.:
sign
= 0
w(X)
in
in
Clearly
: W(T)
W(D).
W(().
is a Witt space of dimension
= w(R), where
Let
class
X,
q
0
(mod 4),
X = X U cone(X).
w(X)
--
Z
is a p.l.
invariant.
be the signature homomorphism
of 11.2.5.
Definition 4.3.:
integer
sign
The signature of
(w(X)).
X,
sign(X),
is the
64
Remark 4.4.:
This signature extends that for pseudo-
manifolds which can be stratified with strata of only even
codimension and, therefore,
p.l. manifolds
[111.
the classical signature for
65
Chapter IV:
Section 1:
w(X).
Properties of the Witt class
Introduction.
In this chapter, we study properties of the Witt class,
For the most part, these are analogues of properties
w(X).
of the signature of manifolds.
(1)
Cobordism invariance:
X
If
(2)
(3)
=
4k+1
k,
then
= 0.
w(X)
Additivity:
Z 4k-1 = aX4k
If
then
4k
w(X UZ Y)
=
w(X)
-Z 4k-1 = 3Y4k
and
+ w(Y).
Multiplicativity with respect to signature of
closed manifolds:
X
If
is a Witt space, and
M
is a closed
manifold,
(*)
w(M x X) = sign(M)
- w(X).
These properties are crucial in the application of
Sullivan's
ko, ® ZZ [
bordism theory.
]
orientation argument to Witt
See Chapter VI and the Appendix.
In section 2, we give a proof of Witt cobordism
invariance of the Witt class.
The proof is modeled on the
classical proof of cobordism invariance of the signature for
66
manifolds
[12].
In section 3, we give a geometric proof of additivity.
"pinch cobordism," we deduce additivity
By means of the
from cobordism invariance. Incidentally
easily
this
result
implies the Novikov additivity theorem for the signature of
[cf.
manifolds.
5,
588]
p.
Section 4 contains a Kunneth formula for intersection
homology of a product
and
X
M
X,
x
where
M
is a closed manifold
Specifically, we prove that
a closed Witt space.
chain level cross-product induces an isomorphism:
H*(M;Q)
0 H(X;
for all perversities
formula
[10]
(*),
)
H
(MxX;
From this, we deduce the product
p.
which generalizes a similar formula stated in
for the case where
X
is the quotient space of a
prime order diffeomorphism on a smooth manifold.
Section 2:
Cobordism invariance.
Theorem 2.1.:
Let
space with boundary.
(Y,DY)
Then
be a
w(DY)
Proof:
The argument resembles
follows
[12].
that
(4k+l)-dimensional Witt
= 0.
in
[11],
which in turn
67
Y = Y U cone(DY)
Let
Y
be the space obtained from
We will construct
by adjoining the cone on the boundary.
a commutative diagram of rational vector spaces
(assume
rational coefficients):
i
k+()
=
(c)
i +
2k,
n-(4k+1)
= m(c)
is surjective, and
algebra
in
m +
=
n(4k+l)
4k,
in
The vertical map
a
3
(over
Q).
is the isomorphism induced by
implies that
A standard
i(H
(Y) )
k(aY).
=dimnH
Here are the details.
in linear
arguent
is self-annihilating
Hmk (3Y), under the pairing of III.3.3,
dimQ i(H k+1 (Y))
(4k+l)
1.
as in 111.3.7.
[12]
c <
2 <
(0y)
the row sequences are exact and,
dual to each other
inclusion,
for
= 2k -
= m(4k+1)
In the diagram,
fact,
2 < c < 4k,
for
= n(c)
n-(c)
and
H
H2k (0Y)
H
H2k+1 (Y
Here
H k Y
-H k(0Y)
and
It follows that
w(DY)
The top exact row fits into the
long exact sequences on homology corresponding to the
inclusion of chain complexes:
(1)
C'2(Y)
Similarly,
C C
= 0.
(Y).
the bottom row is a segment of the long
68
exact sequence gotten from:
(2)
C
(Y) -- C
(Y)
Allowability is defined with respect to the stratifications
on
and
3Y
Let
be the set of point singularities in the
{v }
By reasoning just
and
C
k
2k+t (Y) '2k+l
C
+2(Y)/Cm
(here
(1)
I
a finite
is
index set).
the quotient
we find that
[11],
as in
from
complex arising
I2
links
of 4k-dimensional
be the corresponding
{L }
Y, and let
stratification of
set
Y.
induced by that of
Y
has only two non-zero terms:
Z
.
(D
) s Z2
k (L
k
B k(L ) (
PY)
B k
iE I
The differential on the quotient complex identifies
with the inclusion homomorphism.
H2k+1(
(Y) /C
(Y))
The Witt condition on
Y
H k (Li)
Thereforo
(L
EHk
)
H k
ensures that
0,
all
i c I,
so
m
m+"
H 2k+l1(C * (Y)/C (Y)
m
H2 k (3y).
69
This proves the existence of the top row of the diagram.
An entirely analogous argument yields the bottom row.
The duality of the row sequences follows from the intersection pairing theorem
a
remains to show that
is
(M+)+(n-)= t.
Note that
(1.4.4).
For thlis,
surjective.
refer
It
to
Remark 111.3.6, noting that the Witt condition on links is
v,
satisfied for all odd codimension strata except
the cone
I
point.
Section 3:
Additivity.
w(X)
The cobordism invariance of the Witt class
suggests a simple geometric proof of the additivity of
w(X).
"Additivity" refers to the property described in the
following proposition.
Let
Proposition 3.1.:
Z,
Witt space, and
(1)
Y = X
(2)
X1
(3)
(XVZ)
be an oriented 4k-dimensional
Y
X , X2
subspaces of
and
Z
(X2 -Z)
= w(X ) + w(X2
is biTollared in
Y
are oriented 4k-dimensional
[Y]
=
[X 1 1 +
[X2
.
X2 = Z, and
'
w(Y)
such that:
U X2
Witt spaces with boundary, so
Then,
Y
70
Proof:
Let
K1 , K 2
respectively.
be triangulations of
Denote by
K ,
K2
X1 , X 2
the complexes with
isomorphic simplicial collars added on the outside
The induced triangulation of
Z
the collars)
L.
Then
as in
[20,
will be denoted
is p.l. homeomorphic to
"pinched space"
(on the outside edge of
IKI
p.
IK
=
24].
the polyhedron underlying
UL K2I
The
the complex
defined by:
J =
where
and
Li,
v
L2
[K
U
cone(L
1
1
)] v
ILI
induced map
p,
to
in
X1 , X 2
v
I|J.
in
p :
Orient
Ii
Figure 2. Attach a collar
x
C
with collared boundary.
To check
check that
homeomorphic
so that the
IKI
I
to
P
is
H k(Zk(v,P);Q)
Z
x
CP,
P
See
forming the space
is a pseudomanifold
It can be oriented so that
We call
that
to
by
is triangulable.
It is not difficult to see that
lJI.
JlI
IKj
IJ
By constructing a triangulation explicitly,
we see that the mapping cylinder
-
--
on homology carries the orientation of
IJ|.
to that of
IKI
Z
is the common cone vertex.
collapsing
aP =
[K 2 Ui cone(L 2)]
are the triangulations of
Now, define a continuous map:
P.
[n
'
J
is
Y,
[20].
P
the pinch cobordism.
a Witt space,
= 0.
But
[-1,1] U cone(3(Z
it
is
ik(v;P)
x
enough to
is p.l.
[-1,1])).
This is
71
Z, with suspension
homeomorphic to the suspension of
normalization as
S
Therefore,
points identified.
has the same
Z
*
Zk(v,P), implying
H k(S0
H k(kk(a,P);O)
*
Z;Q)
0,
with the latter isomorphism from 111.2.5.
By cobordism invariance of the Witt class, we have
w(Y)
= w([J )
= w(X1
=
Section 4:
+ w( 2)
w(X1 ) + w(X
2
)
111.4.1)
(by definition
Multiplicativity with respect to signature of
manifolds.
Suppose
space,
M
a p.1.
is
manifold,
both without boundary.
without boundary.
Then
X
and
M
x
X
is
is
a Witt
a Witt space
In this section, we prove the product
formula
(1)
where
w(M x X) = sign(M)-w(X)
sign(M)-w(X)
in the abelian group
is
sign(M)
W(Q).
times the element
w(X)
72
The proof proceeds as follows.
M
For closed manifold
X, we construct an iso-
and arbitrary pseudomanifold
morphism
ip : H * (M; Q)
for every perversity
Assume
X
0 HP (X; Q)
i+j=2k
X; Q),
x
This is done in Theorem 3.1.
p.
is a Witt space,
specialize to the case
-H+PH(M
p
=
m.
dim(M
x
X)= 4k,
and
We have an isomorphism
H . (M;Q) 0 HJ (X;Q)
i
1
H MXX;Q)
-2k
which can be regarded as an isomorphism of rational inner
product spaces.
The inner product on the right hand side
is that given by Theorem 111.3.8, while that on the left
side is uniquely determined by:
(a b ,cd>=
(-1 )dim b dim c<a, cM -(b , d)X
if
dim a + dim c = dim M
dim b + dim d = dim X
otherwise
0
where
<,>M
H* (M;Q)
if
and
and
dim M E
H
1
<,>X
(X;
(mod 2)
are the intersection pairings on
It follows at once that
or
dim M
= 2
(mod 4)
w(MxX)
and
= 0
1313
73
dim M = 0
if
= sign(M)-w(X)
w(MxX)
(mod 4).
First,
Now we formulate and prove Theorem 3.1.
define
a chain homomorphism
C*(M) 0 C
(X)
-b. C
(MxX)
Given
(resp.
d.)
be the homology class in
(resp.
H.(Id.I,I3d.i)) corresponding to
J
Denote by
c.
H.
J1J
J
d.
x
c.
(M) 0 C*(X) , let
c . 0 d. e C
1
J1
as follows.
the chain in
, |3ci
)
i p:
d.).
(resp.
c.
x ( 'c.x d1)C C (MxX)
C.
corresponding to the ordinary exterior cross product
6.
x
d.
J
1
in
H.
j
(Ic.IxdIrIcixld.I U
respect to the product stratification on
lies
in
i'j
is a chain map.
CP
(MxX).
1+j
Set
i
(c. 0 d.)
1
Let
ip
= c.
Ic.xIDdi).
c.
X,
M
x
x
d..
With
x
1
d.
J
Clearly
be the homomorphism induced on
rational homology:
i
i
Theorem 3.1.:
Proof:
: H*(C*(M;V) 0 C
is
(X;Q)) -+H
(MXX;0).
an isomorphism.
The proof breaks down into two main lemmas
(injectivity and surjectivity)
and two technical lemmas.
We assume rational coefficients throughout.
Lemma 3.2.:
ip
is injective.
74
there is a canonical
By the classical Kunneth theorem,
isomorphism
Let
q
(X;
= H*(M;Q)
)
0 C
H* (C* (M;Q)
0
H (X;0).
p + q = t,
be the perversity such that
the homomorphism:
iq
H*(M)
(X)
0 H
-0
H*q (MxX).
The dual homomorphism is:
(i q)*
:
(H (MxX))*
There are canonical
V,
0 H (X))
(H,(M)
isomorphisms
(2)
(H (MxX))*
~ H
(3)
(H* (M)
(X))*
0 H
(MXX)
(H*(M) )*
H,(M)
0
(H (X))*
0 HP (X)
by 111.3.8.
So, we regard
(i
)*
: H
(ii')
(MxX)
as a homomorphism
H(M)
+
0 H
(X).
and consider
75
Injectivity of
follows from the following claim.
ip
0 i
(ia)*
The composition
Claim:
is the identity
homomorphism.
observe
To verify the claim,
(iP (~
where
<,
H*(M)
0 H
0 6)
E 6,E
iq(
intersection numbers
(MxX), with
above.
and such that
and
0
m+x
0 < k < m+x.
are obvious.
Given a representa-
z'
= Yc.
x d..
SJ
homologous to
z'
a, we produce a cycle
for
z
(3)
(2) and
The cases of dimension
tive cycle
with
is surjective.
i*
a E H
r-Q
(M) ® HP(X)
(X) , and that, by 111.3.8,
Lemma 3.3.:
z
H
is the pairing of
determine the isomorphisms
Assume
that
In the course of the
construction, we use some technical lemmas whose proofs
Let
skeleton
Let
skeleton
S
be a triangulation of
Si, and
T
T.,
perversities
M, with
i-dimensional
(m-j)-dimensional coskeleton
be a triangulation of
and basic sets
p
(3.4 and 3.5).
and
with respect to perversity
q.
p
{Qk}
X
and
Sm
.
are deferred to the end of the section
with i-dimensional
{Q }
for
(We will refer to a chain as allowable
if its support is so.)
76
(p,Z)-allowable with respect to the
is
z
The cycle
(product) intrinsic stratification on
stratification refines it.
k > m
We treat the case
since any other
MxX
Z < m.
initially that
Assume
later.
The first step is to apply stratified general position
[16]
so as to be disjoint from
z
to shift
SS(in- t+1)q
[Qq nT
X
z .
Denote the shifted chain by
is
a strong deformation
r
:
M x X -
]
>MZ~)
retraction:
S(m-Z+l)
X[Q
intrinsic skeleta of
1zl1
follows from 3.4 that
v
the map of vertices
1zlJ,
simplices of
support in
M
x Q
U
imply that
S
x X.
12
is
and
Moreover,
T1
2
and,
it also
has a linear extension to
-. r(v)
z 2 = r*(z )
Also,
(M x QP) () (S
z2
S
U S
can be triangulated so that
yielding a cycle
and the fact that
Z-1,
MxX.
x Q
-+ M
2
which preserves products of simplices in
therefore,
there
By lemma 3.4,
Z-2
with
the delfinition of
x X)
r,
has dimension
(p,t)-allowable.
111.2.3 proves that
Decompose
z2
z2
is homologous
to
z
.
An argument entirely analogous to that in the proof of
into the sum of two chains:
XX
77
z2
Jul C MXQ
where
l|ul
=
Q
Note that
M x Qp
is a regular cell complex with 0-skeleton
w
+ y1 , where
lylp
,1 C
Z-1
M x X,
lies in the nonsingular part of
homologous to u
XQP
13CSXQP IU S
1
X
S
y1C
3w = 3u
with
|y 0
'_
C
QP,
and
and
-Say Q
By
lemma 3.5,
J
-
.
for chains
C
c
E C
i
Since
lvi
keeping
n
3v
Tx 2
fixed.
x X,
X-2
YJC-2S Int (aP
stratified
(M),
E C
(X)
d
E C
z 3 = w + v, with
levI
d
and
(X).
w
E
Image(i ).
is disjoint from
using stratified general position
so as to be disjoint from this subspace,
v
chain in
a
d
..
c
= iP(
y
and
is (p,t)-allowable and
S (m-Z+2) X [Qq
we shift
(M),
E C
So we now have a cycle
K
.-
c 1x dj)
ip
y0
)
.
Q1-l
Z 0 9-2
= y0
x X.
k-2
X
we can find a chain
w
IVI C
XQP
_1
l3vI C S
Since
and
and
1v
= u + V.
2
xX
The shifts can be chosen to keep the
since they take place in
which is the interior of
pseudo manifold of dimension
a
x + k - 2
and
78
(S
X)
x
,-2
k,
of dimension
x
q_ 0
x-1
x
x-2
)
U
a -2
x-
Denote the shifted chain
x-3.
of dimension
and
v1
.
by
\ (S (m-t+2)
Tx
x-2
[Qq_
x-1
|vj
subsets
involve closed
Now,
r :
S
use the deformation retraction,
X -
x
k-2
S(m-Z+2)
[Qq_
x
0T
x-l
-3
S
with support in
as
y 2 + v,
Sk-3
where
x
c,-2
e C
Z-X22 (M)
fashion,
The chain
x d2)
and
shifting
r*(v{)
v'
L
C
to
Now,
2 (X)
v
where
c
e C
(M)
proceed
and deforminq
y3 E
in the same
v'
to
Image(i ), and so on.
We will then have
Continue until the process stops.
y, homologous to
y
and
v
r*(V
Decompose
will have a decomposition
r*(v!) = y 3 + v", where
constructed a cycle
homologous to
by 3.5, where
E
d
Q
x
x X U St - 2
is the subchain with support in
-
Z-2 x QP. Then y2
ii
S)-2
X U
y2
-
c
r,(v )
(p,Z)-allowable chain
x-2
.
to get a
given by 3.4,
=
d
j+k=E.
k
E
i
(y c i0 d)
i
Ck(X).
k
z, with
k
r,(v!).
79
k < m.
This completes the proof for the case
Z >
If
m,
the procedure is very similar,
in indices being the primary modifications.
first
shift
z
with changes
Specifically,
so as to be disjoint from
(0)- +1
T 1, yielding a chain z.
-
X
Use the deformation retraction:
X -
: M x
r
S()
x
) Tx
[Qx-(-m)+
2
Mx QP
US
k-m
in-i
to get a chain
r*(z )
M x QP -
S
x X,
Decompose
r*(z )
homologous
and which is
to
lies
y-1
Now,
-S xXUS
xQ
.
v,
in
with support in
into a sum of two chains
and
C M x QP
e
+ V,
|vi C Sm-l x X.
By
3.5,
Image (ip )
.
1yi-mn
1y1
repeat the shift,
deform,
decompose procedure on
using the deformation retraction:
r :
Sr
M-1
xX-S
X
(p,k) -allowable.
r,(z1 ) = y 1
where
z,
x
M x [Q q_-
x-(k-M)
n T(xm-2
P
M-1
x-M+1
80
k <
as in the case where
And so on,
m.
5
The
first
retractions
lemma provides the deformation
technical
used
in the proof of
3.3.
We thank Javier
Bracho
for his assistance on this argument.
Given integers
Lemma 3.4.:
0 <
j
< m,
M Xx
there
M
is
S
x
i,
j
1 < i < x
with
and
an embedding:
[Q
1
n T
x-2
] *[S
m-j-l
X
X U M X Q]x-i+1
which is the identity on
S
[Q9 r) T
]
1
x-2
x
and
S
and which takes
M
X
x
x-i+1
to a union of join lines.
It suffices to define compatible embeddings on
products of simplices in
APE
S'
and
T':
bxAq Ap xAd
(A
A(AA
S',
a
_P
5
Ab
(j)
i
) r) S
= A
AP
-
.-
-
A(x
where
x Q
(
Proof:
.
x XUM
M-3-1
81
AC = A
T ',
and
A
Let
{v 1,
{v 1 },
' d- A q f-QiP 1
T
{v 1 },
in these decompositions:
x e A
Any points
be vertices of the simplices
{v.}
1
and
A
=
Aa
=
AC * A
y 6 Aq
Ab
have unique expressions in
terms of barycentric coordinates:
and
y =
a
Zc.v
Define
and
Both are p.1.
b V
a +
+
,
x =
with
Ya.
+
b
d Vi,
with
Yc.
+
d.
1
1
:
A
-+
I
by
t(x)
=
b
s :
A
-
I
by
s(y)
=
Yd.
t
=
1,
a.,b.
> 0
=
1,
c.,d.
> 0.
1
1-
maps.
X =
{(x,y)Jt(x)
< s(y)}
Y =
{(x,y)|t(x)
> s(y)}
Z =
{(x,y)|t(x)
=
s(y)}
Ap x A :
.
Consider the closed convex subspaces of
See Figure 3.
We define the embedding separately on these subspaces.
It
is easy tb check compatibility on intersections.
Define
by
iZ
:
Z-Aa x Ac *Ab xAdc
aa x Ac
(Ab
q
p
d
82
a
1t,
Ea.v.
(J
c.vc
11)
Ea.
Ec.
1
(x, y)
b
+
(1-t)
Ed.v.
Eb.
Ed.
(
)
1
1
H-0
(x,y)
Define
di
Eb.v.
Ac P
a x
hX
if
t
= t(x)
if
t
= t(x)
0,1.
=
Ap d
q
Aa xAC *(Ab
d
0,1
by
Ea.v
( -t
+ ( 1-t
S
if
(x,y)
Define
(-s)
+
(I3-
t
t(x)
=
(-),3
S
t
# 1
S
)s
s = s(y)
and
0 0
-
(x, y)
1-S
Ed.vd
Zb.v.
Za v
Ec.v
hy
:
~a
if
c
s(y) = 0
qe
b
t(x) = 1.
or
a
x
p~x ad
ql
cb
similarly:
Zava
( l-t
i
(x,y)
-+
(x,y)
EC.V
I)(1-)
1-s
+ (
zc.v
Eb.vb
t
if
t
if
t(x) = 0
=
t(x)
#
0
or
and
Ed.vd
+
(IS
1t
tS
s
s
s(y)
=
# 1
s(y) = 1.
Continuity is checked using barycentric coordinates,
and the definitions agree on overlaps.
seen to be a union of join lines.
Since
The image is easily
Ap x Aq
is compact
Hausdorff and the map is injective, we have an embedding as
83
claimed.
I
Thus,
the strong deformation retractions along join
lines in the join induce the strong deformation retractions
in
Ap
x
Aq
which we used in 3.3.
Specifically, we obtain
the deformation retraction
r
:
-
Ap x Aq
which preserves the convex sets
A
them to the convex sets
b
x Ac
a
X,
Y,
Ad
x
U A
x A
and
Z,
Ab x Aq,
and
x Ad
and carries
Ab x Ad
respectively.
The next lemma was used to conclude that the chains
in a product space
Lemma 3.5.:
with
Let
dim X = x,
S
(X,X0 )
c E Zx+y(X
chains
(X,X0),
with
a] E Z
J
and
(Y,Y )
dim X0 = x-l
Given a chain
are in the image of
Q
x
X YX
bj e Z
and
dim Y = y,
x
U
Y
x
Y)
(Y,Y0), for
j
X
dim Y=
there are
= l,...,J,
finite, such that
J
c =
.
aj x bi.
j=1
Proof:
ip.
be tinite complexes
.
y.
By the Kunneth formula [21, p. 235]
dimensional chains we have the isomorphism:
for top-
y-.
84
H
x
(X,X ) ® H
0
y
(YY )
0
=
Z
x
(x,X ) 0 Z
0
y
z +y(X x Y,X
(YY
x
0
YO U X 0 xY)
H +y(X x Y,XxY
C E Z +y(X x YX x Y 0 U
element of the tensor product
under the isomorphism.
X0 x Y), we can find an
a
Therefore,
0 b3
c =
which maps to
a
x
b
c
.
Since
x Y).
This completes the proof of Theorem 3.1.
2
85
Chapter V:
Rational surgery on Witt spaces.
Section 1:
Description of program.
In this chapter, we develop a technique for performing
We use it to prove that
rational surgery on Witt spaces.
the Witt class
In particular,
4k,
and
determines the cobordism class of
w(X)
if
X.
is a closed Witt space of dimension
X
w(X)
= 0, we find a
.
n'n
}I
"surgery basis"
with respect to which
H k (X;Q)
for
01
the inner product has matrix
1
diag[( 1 0''
0
0 1
By
collapsing a regular neighborhood of an irreducible
representative for
(see Section 2), we obtain
X
with
(1)
w(X1 )
(2)
rank H' k(X1,)
space
and
al
0
=
=
rank H
(X;Q) -
that
X
X
and
X
.
We say
by an elementary surgery,
X
is obtained from
are killed.
(with a collar
the mapping cylinder of the collapse
gives a Witt cobordism between
added)
2.
{ctL31 }
Roughly speaking, the homology classes
Moreover,
a Witt
and call the cobordism the trace of the surgery.
Iterating this procedure
space
between
X
with
n
X
and
Hm
2k Xn ;Q)
X
n
.
=
n
0
times,
we construct a Witt
and a Witt cobordism
Coning off the copy of
produces a Witt cobordism of
X
to zero.
X
n
in
Y
DY
;Z) = 2Z
A
similar procedure works when the dimension of
4k+2.
In that case, we find a
{ (ca
),.,(Ct
is
symplectic basis
with respect to which the skew-
n)}
symmetric form on
H k
diag[(diag[( 0
0)].
0
2k+1
1
X
has matrix
(X;Q)
Then proceed as above.
In section 2 we
The chapter is organized as follows.
define and prove existence of irreducible representative
cycles.
In section 3,
we study the regidlar neighborhood of
an irreducible cycle and show that the "mapping cylinder" of
a collapse as described above is a Witt cobordism.
in section 4,
Finally,
we construct a surgery exact
diagram very much in the classical vein
[17]
to compute the
effect of surgery on intersection homology, verifying
and
(2)
above.
Irreducible p.l.
Section 2:
Definition 2.1.:
G =
2Z
(1)
A representative cycle
H .
(2)
the generator
|z
of
H.(I z;Z)
J
on every j-simplex of
of
Remark 2.2.:
for
a e Hk(X;G),
J
Z
(1)
J
z
is irreducible if
Q,
or
geometric cycl-s.
Izi,
has coefficient
1
in some triangulation
|z1.
Condition
(2)
implies that the generator of
87
H.(Izl;E)
has coefficient
1 on every j-simplex of
J
Izi.
in any triangulation of
be a p.l. pseudomanifold of dimension
X2k
Let
we say
X
2k
By abuse of
[X].
with irreducible fundamental class
language,
Izi
Thissection is devoted
is irreducible.
to a proof of the following lemma.
Let
Lemma 2.3.:
a E H
Given
z
which is
Proof:
J
X2k
be an irreducible p.l. pseudomanifold.
(X;q), there exists a representative cycle
irreducible,
if
0 _ j < 2k-1.
by a general position argument,
lvi
such that
p.
67]
E
v
Step 2 uses the technique of piping
to produce a cycle
satisfies the condition
j-cycle,
a representative cycle
supports a simplicial cycle with coefficient
1 on every j-simplex.
[20,
Step 1 constructs,
The proof proceeds in 2 steps.
z
Hi.(Izl, Z)
which, additionally,
=
If
Z.
J
is a
denotes the complement of the intrinsic open
j-stratum.
Step 1:
In some triangulation
T
representative simplicial cycle for
j-simplices if necessary,
choose a
X,
of
a.
By reorienting
we may assume all non-zero
Clear denominators
coefficients of j-simplices are positive.
so as to obtain an integral cycle
y =
Y
GET
n(a)a
G
such that
88
all distinct non-zero coefficients share no common factor.
If
= 1
n(a)
whenever
/
n1(a)
0,
v = y
then set
and
Step 1 will be completed.
If not,
let
be the connected components of
E 1 ,...,E
the intrinsic open j-stratum of
Y
lyl.
=
we can associate a unique coefficient
of simplices in that component.
union of
n.
copies of
the obvious p.l. map.
Let
obtained by identifying
f(x)
=
f(x').
Then .W
having coefficient +1
f*(w)
=
y,
where
with respect to
X -
g
: W -+
shifts
HO =
to
x'
ZX
H1
if
x,x'
[1,
W
f
S(wx
[0,1])
g,(w)
-
f*(w)
p.
418],
-
W;
= g.
homotopy
Then
(Z )
and
y
j-cycle
w
[20,
p.
We move
51].
H
X.
Let
The linear trace of the
: V
x
[0,1]
-+
X
with
is a p.l. geometric cycle
on each j-simplex.
(p,j)-allowable.
[0,1]
=
wx 1- wx 0.
g,(w)
sequence
that is, only if
g*(w)
x
=
W
into general position
the nonsingular part of
Its support is also
W
-l
E f
Y
W -+
and such that
which, when triangulated, has coefficient +1
Give
:
f
Perform a finite
f, bringing
a p.1.
and
and
be the quotient space of
be the reulting map.
X
yields
f
= 1,...,n,
on all j-simplices,
ZW
Zx,
be the disjoint
supports a simplicial
Int(T) C
only if
W
the coefficient
denotes the induced map on chains.
f*
of shifts of the map
f(Int(T)) C
x
f(W-E W) C: X -
Note that
f(T)
W
n.,
Let
i
E.,
To each component
the orientation such that
- y.
Then
Since
H*(w x
H(x,t)
[0,1])
= H(x,0)
=
=
f(x)
89
when
x e
x e W
-
is
ZW,
EW
for all
t, and
for all
t,
y
such that
in
H
[v]
=
g,(w)
(X;V).
Let
an orientation respecting pipe
E6 H.(X;Q).
lvI.
P
jvj
of
(X-EXv l r) (X-E X))
complete.
F
to
Fi+1
[20, p.
67).
is a subcomplex of some triangulation
is 3 cally unknotted
at all points in the interiors of j-dimensional
lvI.
v
We now construct
from
Note that since
of
be the
Then
J
1 < i < Z-1, with distinct pipes disjoint.
X, the pair
v
be the ccnnected components
{F . }
ofthe intrinsic open j-stratum of
for
[0,1])|
represents
Let
is the desired representative, and Step 1 ir
Step 2:
when
IH,(w x
it follows that
the same homology class as
g*(w)
c X - EX
We conclude that
(p,j+l)-allowable.
multiple of
H(x,t)
simplices
Therefore the piping construction can always be
carried out.
Let
Z
be the p.l.
the piping is done.
Then
subspace of
Z
X
we obtain after
supports a simplicial cycle
which has coefficient 1 on every simplex in
Z -
E Z
is connected,
this cycle generates
Z.
H
Since
(Z;2Z).
(p,j)-allowable and, by "filling
Clearly, its support is
in the pipes," we see that it represents the same homology
class as
g,(w)
in
this cycle such that
H
(X;Q).
[z]
satisfies the conditions
Let
[v]
(1) and
E
z
be the multiple of
Hk (X;Q).
J
Then
z
(2) of definition 2.1.
I
90
Remark 2.4.:
If
X
is an irreducible p.l.
manifold of dimension 2,
class
a
e H
1
(X;Q)
pseudo-
it is easy to see that any
is represented by a cycle whose
support is a p.1. embedded
S
1
91
Regular neighborhoods.
Section 3:
Let
X2k
U
e H
Iz|
in
be an irreducible representative for
a
(closed) regular neighborhood of
X
U = U U
X.
The stratification of
-
Since
Izj
is
and
Let
437].
adjoined.
DU
U:
induces a stratification on
U
U
in
X
corresponding to a stratum
X U cone(X r) DU)
p.
[1,
U
Then
aU,
with the cone on
U
be
cone(DU)
v0
{v 0 }, where
(r,k)-allowable,
U
(X;), with
is a pseudomanifold with collared boundary
stratification induced from that of
z
Let
be an irreducible pseudomanifold.
is the stratum
is the cone point.
the cone point
v0
constitutes the entire 0-stratum of the stratification of
U.
The main result of this section is:
Proposition 3.1.:
If
<a,a> = 0,
is a Witt space and
X
then:
H
;(U.)
=
0.
To prove this, we prove a series of preliminary lemmas.
Let
i
:
C
(U;2Z)
complexes, where
for
2 < c
< 2k-1
-+Cm(U;
mand
)
be the inclusion of chain
is the perversity:
m -
(2k)
= m(2k) -
m
(c)
-
1
=
k
=
M(c)
2.
-
An argument identical to that given in the proof of IV.2.1
proves that
the quotient complex
only two non-trivial terms,
C
(O;2)/C ~(0;
)
has
and there are canonical iso-
92
morphisms of these terms:
k+2 /C mk+2
B m( U)
Cm
Clm /C k_
k+1
k+l
k
=
Z (U).
identified
The boundary homomorphism on the left is
We therehy obtain:
with inclusion on the right.
Lemma 3.2.:
There is an exact sequence of integral inter-
section homology groups
H + 1 (U) -
(integer coefficients assumed):
H k(DU)
.
H
()
-.
F(U)
k0
-k
a
of
We made no use of the irreducibility
proof of 3.2,
Lemma 3.3.:
Then
H
but it
is
u
Let
in the nexL
be a generator
= 2Z
(U;)
crucial
of
z
in the
lemma.
Hk(lzl;2)
and the homology class
of
u
= Zk(lzI).
is
a
generator.
Remark
3.4.:
(1)
Since
Jul
is
(m,k)-allowable, it is also
(m-,k)-allowable.
(2)
[u]
Since
e Hk(U;eZ)
[z]
E
Hm(X;C)
is non-trivia 1.
is non-trivial,
If
u =
w,
for some
93
W E
Cm
k+l
|wl
projection, we may assume
also,
this would imply
Therefore
Proof of 3.3.:
Let
C
[u]
E
cone(9U)
*
= v0
E Hk~ (U; 2Z) ,
assume
|yI
*
$
y
jl A
then
Int(a),
Izi
f(v)
=
with
*
--
f,(y)
in
jz
Jul () Wu
so that each simplex lies in
Izi
ITI =
U.
3
If a vertex
lul
let
and we
c,
0
v
T
E
IyI
denote
Iz|.
Extend this map on vertices of
T.
linearly over simplices,
:yj
(U; 2Z)
* A = A, by the usual convention.
=
the simplex in
f
y E C
satisfies:
a e U
Then every
a unique simplex with interior in
Define
with
By pseudoradial projection, we can
Triangulate the cycle
lies in
U.
the structure of a derived neighborhood
33].
A
a
Suppose
cycle
Jl= Jlt () lI
where
(U;2)
U0 0
U
Give
p.
C
.
k+l
in
DU
cone(DU)
UO = Cl(U-N).
Let
with representative
lyI A V 0 =v
Then
[20,
U.
C
is non-trivial.
of
N U
There is a p.l. homeomorphism of
E
(X;2Z)
H
(U;2Z)
H
denote a collar
N
w
Since
U.
is trivial in
[u]
contradiction.
so by pseudoradial
'
Iwi r) v 0
then
(U; 2Z)
obtaining a p.l.
|y|
map
U C . Then f determines a geometric cycle
Ck (U; Z) .
Since the support of
f*(y)
lies in
94
Iz i,
E Ck
(Y)
(U;2Z)
to define
Proceed as in 111.2.3.
.
kA
a p.l.
lyt
H
map
DH*(yxI)
satisfying
implies that
represents
and
H*(yxI)
(.
Hk (Izl;2Z)
j
for some
E
and geometric chain
x I -U
= f
-(y)
y.
Therefore
e Ck+l (U; 2Z).
f*(y)
= 2Z
by hypothesis.
and so
(yxI)
f
The definition of
But,
7Z,
H
f*(y)
Izi,
is a k-cycle supported in
(
=
Therefcre
f*(y)
=
j-[u].
By the intersection pairing theorem
(1.4.4), there is
a nondegenerate pairing induced by intersection product:
Hk
That
n +
is,
(c) = n(c)
S
E
n+
Hk (U;Q)
2 < c
for
n+(2k)
Choose
OwQ.
(U;D)
is the perversity satisfying
n+
Here,
(U;Q) x H k
(i-)
< 2k-1
= n(2k)+l = k
a generator
<UaI> =
Lemma 3.5.:
j-u
such that
1.
The following diagram commutes:
+
and
(n+)
=
t.
95
p
H
(U;0)
k
n+
H k(U;)
All
The map
Proof:
C
n+
k
4-
Hk
Ii
(U;Q)
H
-
n
Hk(U;q~
j
0
Hk
is given by:
g
(U;Q),
rn~
Finally,
the map
m--
= Ck
(U;Q)
j
Since
implies that
of
g
immediate
from the
3.1.
Refer to the diagram of 3.5.
hypothesis of the proposition,
a
surje-tive.
3.
We may now prove Proposition
Proof of 3.1.:
is
h
is
Ck+6(U;)
Similarly,
is injective.
the description
definition of
that
of chain complexes.
maps are induced by inclusion
Commutativity is clear.
C k(U;)
0
-+
generates
m-^
Hk (U;E)).
commutativity imply that
ioh
g
Under the
(Reca l
is the zero map.
Injectivity of
j
and
is the zero map also.
inherits a Witt space structure from
X,
so 111.3.7 implies
that
i
map.
The proposition now follows from surjectivity of
is an isomorphism,
so
h
U
must in fact be the zero
h.
I
96
One corollary will be useful when we analyze the
surgery exact diagram in Section 4.
Refer to 3.2 for
notation.
Corollary 3.6.:
Assume rational coefficients.
Under the
condition of 3.1,
H
3,(Y)
(3U)
~{y}
Image (q*) ,
where
E
H
(3U)
with
I
= a.
Section 4:
Let
The surgery exact diagram.
X
be an irreducible Witt space of dimension 2k.
Assume there is a decomposition of
H
where
y
VO
=
(X;)
=
1 =
(Yl''O.ryd'
H
V
E
(a
,6
_(X;,):
)
and with respect
this basis the intersection pairing is given by:
,3
0a ,Yi) = 01 lY
Oilal
)=
=
<
,1>
1
.
(Xlal>
=
=
0,
0
i
= 1,...,F
to
97
Let
U
be a regular neighborhood of an irreducible
z
representative cycle
is
in
bicollared
X
(X -
X=
The passage from
Note
X
[1,
X
for
p.
437].
Int(U)) U
X
to
U
We may assume that
TU
Form the space
cone(
U).
is called an elementary surgery.
is homeomorphic to
by collapsing
a1.
X,
X
the space obtained from
to a point.
The following sequence of propositions, whose proofs
will be deferred briefly, partially describe the effect of
the elementary surgery.
is an irreducible Witt space.
Proposition 4.1.:
X
Proposition 4.2.:
Let
collapse
X
-+
Y
be the mapping cylinder of the
X, with a collar
X x I
attached.
can be given the structure of a Witt cobordism between
and
on
X .
(We call
Y
There is an isomorphism
Hk (X
which preserves the
X
the trace of the elementary surgery
X).
Proposition 4.3.:
Y
Then
;)
~V0
intersection form.
98
Repeated application of these propositions yields the
result alluded to in the introduction.
Theorem 4.4.:
Let
dimension 2k,
k > 1.
X
be an irreducible Witt space of
If
= 0,
w(X)
then
X
Then
w(X)
is Witt
cobordant to zero.
Proof:
dim X 2 0
Suppose
(mod 4).
that the inner product space
Choose a basis
{a,
1
,a
2
H
'3 2 ,..
(X;O)
which the matrix of the form is
Let
4.1 H
VO
(
4.3.
2
4'2 '..,'c'n'n)
We obtain
(X,)
X
isomorphic to
rational inner product space.
V;
(al,Y
), and apply
Xl, Witt cobordant to
-.
setting
01)( )].
-
V
=
(a ,,...
Now decompose
,a.',')
and
H
cobordant to
Hk(X;Q)
0.
as
(X ,q),
and
V=
Continue until we obtain
=
with
V
repeat the procedure.
X, with
implies
with respect to
diag[(
and
0
is split, by 11.2.9.
4an' }
.,
=
Xn,
The cobordism to
X
is obtained by pasting the traces of the elementary
surgeries along their boundaries.
Now attach
cone(X )
to
the cobordism.
If
H m(X;V)
{a
1
dim X
is skew-symmetric.
1 1 ...
form is
= 2(mod 4), then the intersection form on
,n'
diag-
6
n}
Choose a symplectic basis
with respect to which the matrix of the
0 1
1
0 1
0'***''l0)].
above shows that a sequence of
The same reasoning as
n
surgeries will yield a
99
cobordism of
X
to zero.
a
We now supply the proofs of 4.1 -
Proof of 4.1.:
The fact that
is bicollared in
Witt space.
X
X
k = 1,
Therefore
Six
is a
Int(U)
is path connected.
There-
is irreducible.
If
to
X
aU
general position proves that the
X -
nonsingular part of
fore
is a Witt space and
immediately implies that
k > 1,
If
X
4.3.
z
then
X
has at most point singularities.
is an embedded
[-1,1].
S1
and
The assumption that
X -
trivial implies
Int(U)
U
is homeomorphic
E Hm(X;Q)
[z]
is connected,
so
X
is nonis
irreducible.
U
Proof of 4.2.:
The trace of the elementary surgery is
homeomorphic to the following polyhedron, defined analogously to the trace of a spherical modification
(X -
Int(U))
cone
(OU)
x
U
[0,1], attach
along
DU x 1.
the resulting space as
attached as above.
aU x 0
along
[17].
and
There is an embedding of
aU x I
Now attach
with
U
cone(U)
and
To
U
in
cone(DU)
= v*U
along
U
via this embedding, yielding the trace of the surgery
Y.
Y
and
is
clearly a pseudomanifold with collared boundary,
it can be oriented
so that
aY = X
-
X.
is a Witt space, it suffices to check that
To prove that
kk(a,Y)
Y
satisfies
100
H
But
Zk(v,Y)
U,
Proof of 4.3.:
Steps 1 -
4
(Pk(v,Y);q)
=
0.
so Proposition 3.1 completes the proof.
The proof is broken into a number of steps.
provide the ingredients for the surgery exact
diagram, which constitutes step 5.
Step 6 completes the
proof using the diagram.
Step 1:
Transverse chains.
Let
U
denote
cycle of
X
restricted to the regular neighborhood
aU
(by abuse of notation)
its boundary chain.
C*(X)
Let
consisting of chains
TC*(X)
c
dimensionally transverse to
c e TC
and,
1
c
and
by the Lefschetz boundary formula:
=
DU n
U +
(-1)2
c (icU.
This defines a homomorphism
t
by
t(c)
m
: TC(X)
= c r\ U.
1
-
m
C(U)
1
Dc
are
To each element
(X), we can associate the intersection chain
(C 0 U)
U, and
be the subcomplex of
for which
U.
the fundamental
for each
i
c
r) U,
101
U C U).
(We use the inclusion
m
C
Define
(U,3U)
Give the graded group
to be the image of
C
t
in
C
(U).
the structure of a chain
(U,9U)
complex, defining
6
by
6(t(c))
Note that
6
C
(U, 3U)
=
3c C
-+
3-
(U, 3U)
CM
1-1
U = t(cc).
if
is well-defined:
3(t(c))
=
3(c r) U)
U +
c
=
t(c)
= 0,
(-1)
2
then
k-ic r
3U
=
0.
Dimensional transversality implies that
dim(1
ac n U
so
| c f )U|)
cP) UI A
< i-2
0
=
and
c n
In particular,
6 (t (c) ) =
aU =
Dc n
It is easy to check that
t
is a chain homomorphism.
0.
U = 0.
6o6
Let
=
K
0.
We conclude that
denote the kernel of
t, with the induced chain complex structure.
short exact sequence of complexes:
There is a
102
0
-
K
M~
--
m
TC* (XW
m
C* (UF
-
U)
-~-V
0,
-aHk(K
( M(UDU))
Hk
In steps 2 -
)
H (CM (Ua U))
H+
-- Hk(TCm(X))
4 we analyze the homology groups occurring in
this sequence.
Step 2:
= H
H,(TC M(X))
The inclusion
(X).
of chain complexes
i : TC(X)
InCo(X)
induces an isomorphism on homology.
In other words,
Hm (X)
.
H,*(TC *(X))
Proof of surjectivity:
Apply the stratified general
position lemma(I.3.3) (or its corollary 1.3.4)
z E C
to a cycle
(X).
Proof of injectivity:
m
a chain
w E
Step 3:
The complex
Let
m+
C +(X),
Apply the relative part of 1.3.4 to
where
Dw
=
z
and
C +(U).
denote the sequence with
z
m
E
TC (X).
+
and an associated long exact sequence in homology:
103
m+(c)
m+(2k)
Although
m+
m(c)
=
=
2 < c < 2k-i
for
m(2k)+1 = k
is not a perversity, we may define
(m+,j)-
allowability of the support of a geometric chain with
respect to a stratification.
consisting of chains
is
c
Consider the complex
of dimension
IOcl
(m+,j)-allowable and
is
respect to the stratification on
j
C,
such that
(U)
Icl
(m+,j-l)-allowable with
U.
Recall that the cone
point is the 0-stratum.
The inclusion
C
mm+
(U)
sequence of complexes.
Cp
'-.
(U)
induces a short exact
We can analyze the homology of the
quotient as in Lemma 3.1 and obtain an isomorphism:
m
ATfl+
H k+l(U)
A
Hk+l(U)
and an exact sequence
0-*Hk (U
PHk
()
Hk- 1 O)
Hk- 1(
We get another long exact sequence
from the short
sequence induced by the inclusion
C>(U)
n+
is the perversity:
--+ C,
exact
(U), where
104
n+(2k)
=
2
for
n (c)
n+ (c)=
n(2k +
2k- 1
< c <
k.
)=
There is a commutative ladder:
0 --
0
H
-- +
(k- lH (U) -~+Hk _ ($ --
--0 HHk ())
0
k- 1
k-l1
U)
Hn
kl(DU)
~H
-+0+H
Hn
kl(U)
k
k
-+
H'
Hk
(^)
(U)
All vertical maps are induced by chain complex inclusions.
is Witt,
X
Since
and
U
DU
Proposi-
are also Witt.
$ 0 0
tion 111.3.7 and the 5-lemma imply that
is an
isomorphism.
We now identify
j
> k.
Let
N
m
J
H . (U, U)
m
(U)
J
H.
with
DU
be a simplicial collar of
cone( U) U
is a p.l. homeomorphism
cone lines in the cone on the left.
N
In
cone( U) U
will
N
cone(DU) U
N.
denote the chain map induced by pseudoradial
projection along cone lines from the cone vertex
boundary of
p,(v)
preserving
U, we can triangulate
be subordinate to the simplicial structure on
p*
There
U.
in
~ cone(aU)
any chain so that its intersection with
Let
in dimensions
= v,
Cl(U-N),
and
pop,
p:
= p,.
Define homomorphisms:
C, (U) -+C*(U).
v
to the
Note that
105
(U),
--. C
: C .(U,DU)
J
J
J
j
> k + 1
and
-- w
(U,3U)
: Z
(cycles)
Z(U)
by
j+lv1
(c (C U))
is
'P
in Step 1 shows that
t
The discussion of
*
$k (t(c)) = p,(t(c)
well-
defined.
For
j
> k + 1,
1P _ (6t(c))
=
$ _ (t(0c))
=
P*(t(0c) +
=
p,(0(t(c) +
=
Dp,(t(c)
(-l)
n\ DU))
(x
V
(-1)vj+V*(c n\ DU)))
j+1V*(C
i (t(c)).
=
$.
J
induces a homomorphism
(
H. (UaU) -aH
J
(U)
j >k
-
on homology for
In fact,
),
*
Therefore
na eU))
($'
J
.*
Proof of surjectivity:
is an isomorphism for
Let
m+
().
J
z E Z.
j
> k.
The cycle
p*(z)
106
M+
H . (U), by an
in
z
as
the same class
represents
J
argument just like that used in previous pseudoradial
C m(U,
U)
=
3w,
C M+fUU)
where
z E
Let
Z
w E C +1 (U).
(U,U)
C.
J
(z) = P, (z +
$
and
(z)
p,(w) (\ U
Then
=
*p(w) t'\ U
=
p*(3w) Cn U
=
P*
z +
(-l)
(-l) j+lv* (z n
DU)
p~op*
since
U,
j+1 v* (z A
(U), via a chain we denote by
U)),
are homooqous
y.
Therefore
H
Cm
j+l (U,
yAUe
.
U
and
The complex
D
(X -
Int
Let
X-U
=
be the subgroup of
such that:
6(yAU)
($P)
in
By the relative
=
y j
DU.
($n(z)AU) j
represents the same class as
We conclude that
(U, U).
Step 4:
(z) n
U)
=p.
the chains
stratified general position lemma, we can assume
Then
lies in
\ U
(Z)
i (z) /
i
and suppose
and
6 (p* (w) t'\ U)
Since
lies in
U
= p*(z).
$p(p,(z) n U)
and satisfies:
Proof of injectivity:
i (z)
p*(z) /
The chain
projection arguments.
z
is injective.
(X-U).
(U) ) U cone ( U) .
C. (X-U)
J
Let
DT
(X-U)
consisting of those chains
z.
in
107
dim(Ic.I
dim(Iac.
J
where
v
v)
(
J
j
<
r) v)
<
-
2k +
(j-1) -
is the cone vertex of
X-U.
(k-2)
2k +
(k-2),
Allowability of chain
supports is with respect to the stratification on
induced by that on
X-Int(U).
Thus the cone point is
contained in, but may not be all of,
K.
(by means of the inclusion
$
(c)
= c
(X
D
-
X -
i (X-Int(U)), where
the 0-stratum.
j:
Define homomorphisms for all
S
X-U
Int(U)
C X-U)
by:
by abuse of notation,
X-Int (U)
represents the restriction of the fundamental
cycle of
X
(1)
to
X-Int(U).
Then
c = c C\ (X-Int(U))
$.
J
+ c,1 U
is
for
injective, because
c e TCm(X)
J
and
(2)
= c (\ U =
t(c)
Clearly the
c.
J
0
for
c E K..
J
are chain maps and we regard them as an
inclusion of chain complexes.
We will show that the induced maps on homology
are isomorphisms
Proof of surjectivity:
for
j
Let
N
< k.
be a collar of
3U
in
108
X -
p*
Int(U), and
be the chain map on
induced
C*(X^U)
by pseudoradial projection along cone lines to the inner
boundary of
Izi
v
A
$.
=
m(X-U)
J
If
N.
If
z E D.
H . (DM (X-U)).
Proof of injectivity:
a,
then
p,(z)
=
aw, where
Iw
v
=
l(
$,
and
a
p,(z)
p,(w)
E
then
represents
p,(z)
the
e K.,
X-U.
Let
w e Dj+
< k,
But
JJ
since its support lies in
j
p*(z)
is a cycle,
z
same homology class in
represents
for
e H.(K*).
represents
1
(X-U).
Kj+
ap*(w)
=
If the cycle
J
1.
a
j
If
also.
< k,
z
Assume
then
Moreover,
P*(Dw)
= p*p,(z)
= p*(z)
This proves the claim about
($4),,
j < k.
Finally, analysis of the quotient complex
Am-A
m
C,(X-U)/D*
H m(aU)
Step 5:
-
(X-U)
yields an exact sequence:
Hk (Dm ~(X-U))
-
H m(X-U)
-+0.
The surgery exact diagram.
Recall that
X
is a Witt space of dimension 2k.
There
is a commutative diagram of rational homology groups, with
exact row and column
(assume rational coefficients):
109
H (J)
k
q
1H
3, Hk (DM*~ (X-*U) ) -- H(X)
k+1 (U)
+1~l(U)
(U)
i
III
Hk (U)
H k (XU)
0
appears in Lemma 3.2.
q*
The homomorphism
The isomorphism
on the left was proved in Step 3.
The horizontal sequence is a segment of the exact
sequence in Step 1.
We have used the isomoiphisms proved
in Steps 2, 3, and 4 to relabel some of the groups.
The vertical exact sequence is a segment of the
long exact homology sequence obtained from the inclusion of
complexes
D'*(X-U)
mA
-- + C*(X-U),
as discussed in Step 4.
The commutativity of the upper left
follows
triangle
from the chain-level definitions of the connecting homomorphism of the horizontal and vertical sequences and the map
q,.
The homomorphism
triangle commute.
Step 3, and
t
of
t,
Step 1.
Here
is defined to make the lower right
g
$
is the rational isomorphism of
is the homomorphism induced by the chain map
In fact,
g
is given by:
g(y)
=
(,ay)6,
110
S
generates
inner product on
Step 6:
n+
Hk (U)
Conclusion.
{a ,
Recall the basis
and the decomposition
of H (X)
,,yl,...,Y,}
(X)
is the
H (X).
Assume rational coefficients.
H
(,>
as in 3.5, and
1.
~V0
By remarks in Step 5,
t*(y ) = 0,
i
= 1,.
..
,
where
and
t(a)=
0
Similarly,
$(t,(
1
))
-
3.
By exactness,
@ (a!) @ Image
Hk(D*(X2U)) =
as a rational vector space, and
to
V 00
product.
(a )
V'0 0 (a')
1
(3*)
is isomorphic
by an isomorphism compatible with intersection
Now, corollary 3.6 and commutativity of the upper
left triangle imply:
Image
(j)
=
(a!)
@ Image(3*)
111
Finally,
since the vertical sequence is exact, we conclude
that:
=V
H X(X-U)
H; V0k
V
The isomorphism is compatible with intersectiun product.
This completes the proof of 4.3.
0
112
Witt space bordism theory.
Chapter VI:
Section 1:
Introduction.
In this
a
chapter,
show that
we
Witt spaces provide
KO
geometric cycle theory for connected
odd primes.
Specifically,
in section 2,
Witt
based on the class of Witt spaces,
QW
homology at
the bordism theory
is
,
introduced
and using results from Chapter V and a simple construction
given in
section 3 of this chapter,
it follows that the
coefficient groups are:
WiA-4-
"
Q
z
(pt)
Witt
Qq
(pt)
q
~0
Witt
Qq
(pt)
W(0)
q
The
isomorphisms are
Finally,
[24]
ko*
if
q
if
> 0
and
q > 0
q
and
$
0
q
(mod 4)
= 0
(mod 4)
induced by the Witt class.
in section 4,
the technique of D.
Sullivan
is used to construct canonical orientations in
0
2[%Z
for Witt spaces,
transformation of
P
Witt
:
and the induced natural
theories:
Witt
-,
A ko, 0
Z [%],
is shown to be an equivalence at odd primes.
113
Section 2:
Let
The bordism theory
Xd
be the class
QWitt
of oriented
pseudomanifolds
L
satisfying:
It
(1)
L
(2)
H
is
is a Witt space
(L;Q) =
0
not difficult
if
to check
(oriented)singularities
bordism sequence
spaces.
Let
dim L
{,
Witt
Q.
n}
Witt
Q Wit(pt)
The
Z
is
a class
of
The associated
is the class of all oriented Witt
eo.
The coefficient groups
Wi tt
(pt)
are:
= 2
Witt
(pt) = 0,
q
Witt
(pt)
q
that
denote the bordism theory based on
QW
Qq
22.
(use 111.2.3).
the class of singularities
Proposition 2.1.
=
=
if
W(Q),
q
if
isomorphisms in dimension
t 0
(mod 4)
q >
0
q
> 0
and
qg=
0
(mod 4).
are induced by the
Witt class.
Proof:
The case
q = 0
is evident.
When
q E 1
(mod 2),
observe that any odd dimensional Witt space bounds the cone
on itself.
114
The case
invariance
V.4.4,
is
q
: 2
(IV.2.1)
(mod 4)
follows from cobordism
and surgery
In applying
we can use the fact that any Witt space
not irreducible
space
X'.
is
Witt cobordant to
Simply take connected sum
2k-stratum of
X,
say
2k-stratum of
X'
is
Finally,
show that
in
E 1 , ...
the case
0
Witt
of components of the
so that the
(mod
4),
IV.2.1
and V.4.4
the homomorphism
Witt
4k
(t
is
.
.
+W0
injective.
construct explicit generators
Section
which
an irreducible
E ,MIin order,
q
induced by the Witt class
w
X2k
# E 2 #...#E
E
w
that
(V.4.4).
In section 2,
Witt
k
(pt)
4k
for
we
and prove
is also surjective.
3:
____
Generators for
__
Q Witt
*
(pt).
We use the method of plumbing disk bundles over spheres,
due to Milnor
[7
]
to construct representative Witt
spaces for each bordism class in
and
q
= 0
(mod 4).
QWitt (pt), when
q
In particular,
> 0
The representatives can be stratified
in such a way that the singularity subset
point.
q
they have stratifications
even dimensional strata.
E
is a single
with only
This yields information on
ev
115
[11].
Theorem 3.1.
entries,
[7]:
Let
symmetric,
Then,
(M,3M)
1,
M
is
H 2 k(M)
(2)
be an
nxn
and with even
there is
of dimension
(1)
and
k >
for
B
4k
matrix with integer
entries
on the diagonal.
an manifold with boundary
such that:
(2k-l)-connected,
DM
is
(2k-2)-connected,
is free abelian;
The matrix of intersections
is given by
H 2 k (M) ® H 2 k(M)
-ZZ
B.
I
Given an element
a basis
the
{v }
for
inner product,
with even entries
E W(Q),
it is possible to find
V
with respect to which the matrix of
B,
is an integral symmetric matrix
on the diagonal.
manifold with boundary
represents
[V,3]
(M,3M)
the Witt class
Theorem 2.1 produces a
such that
[V,3]
in
W(Q).
exact sequence of rational homology groups
H 2 k(M;)
In the
long
(rational
coefficients assumed):
2
-
k(3M)
the transformation
bases
the
{v.}
intersection
H 2k(M)
i,
-
H 2 k (MDM) -- + H 2 k-l(aM)
has matrix
and {v*}, where
pairing of
{v}
H 2 k(M)
B
-
i*
-H
with respect to the
is dual to
with
{v.}
H2k(MaM):
under
116
H 2 k(M)
Therefore
H 2 k(MrM)
the Novikov group of
m
H2k (M;Q), where
isomorphic group
represent
x
the class
of
(V,3).
w(M)
This proves the result
Proposition 2.2.
(M,
M
=
V.
M)
and the
M U cone(DM),
That is:
[V,f]
=
+
E W(Q).
used in section
2:
The homomorphisms
w
Witt
4k
(t
k >
+W0,
which are induced by the Witt class,
0,
are surjective.
2
There
is
Corollary 3.3.
and
where
q = 0
also the corollary:
Every cobordism class in
(mod 4),
(M,TM)
is
has a representative
Q
M
Witt
, for
q
= M U
cone
q >
0
(TM),
a q-dimensional manifold with boundary,
obtained by plumbing disk bundles over spheres.
I
In particular, since
M
has a natural
with only even dimensional strata,
concrete result about
ev
stratification
we obtain the following
117
The homomorphism of graded cobordism groups
Corollary 3.4.
Qev (t
QWitt (t
is surjective.
Remark 3.5.:
Simple examples
show that the homomorphism
in 3.4 is not injective.
Section 4:
ko*
Suppose that
a, of
p.1.
(1)
a geometric cycle theory for
Witt spaces:
2Z[% ].
Q*
is a bordism theory based on a class
spaces satisfying
the following properties:
Or is closed under the operations of taking
cartesian product with a p.l.
manifold,
and intersecting
transversely with a closed p.l. manifold in Euclidean space.
(2)
Y
has a signature invariant defined on the cycle
level which is cobordism invariant:
There is a homomorphism
sign
:
Q* (pt)
2Z
which extends the identification
Q*(pt)
-+
2Z.
118
relative cycles
The signature can be extended to
(3)
(X,3X), so that it is additive:
If
X,Y
sign (X UZ
then
z
Y)
= sign (X, Z)
The signature
(4)
Z =
are relative cycles such that
is
3X,-Z =
Y,
+ sign (Y, Z).
multiplicative
with respect
to
closed manifolds:
If
X
is
a cycle
in
t1,
and
M
is
a closed manifold,
then:
sign (M
sign
Then,
x
X)
(M)
coefficient
denotes
a natural
theories:
is
[2Z[].
Z/4 2Z -periodic.
The
groups are:
ko q(pt)
0 2Z[%]
~
Z [4]
0
Sullivan shows that
q = 0
signature
the classical
-- + ko, @
ko* 0 2Z [2]
where
has defined
[24]
of homology
p:
that
sign (M) -sign (X) ,
Dennis Sullivan
transformation
Recall
=
(mod 4),
for
if
q :
0
(mod 4)
otherwise.
[X]
e Q
q
(pt), where
of M.
119
E
2Z[]
In chapters III and IV,
(1)
through
(4)
hold
koq (pt)
7Z[ ]
.
Ppt([X]) = sign(X)
we demonstrated that properties
for the class
of Witt spaces,
so we
Witt
Witt
ko,*
-*
In Chapter V and Chapter VI,
computed
QWitt (pt).
Recall
2 Z [%]12
.
receive a natural transformation:
sections 1 and 3,
there
are canonical
we
iso-
morphisms:
Witt
pt)~
Witt
Qq
(pt)
q
Witt
Qqit(pt)
q
and
Recall
from
(11.2.8)
W(0)
if
0
otherwise.
T
The projection
is
a direct
Witt
,
(mod 4)
and
4) T,
summand:
ZZ
~
q >
isomorphism
sum of 2-groups of order 2
onto the
q
- 0
the canonical
W(0)
where
q
and 4.
0
120
when
q >
Therefore,
0,
(mod 4),
is
homomorphism.
the signature
the natural transformation
Witt
(5)
q - 0
® 2[ ]
:
Witt
1t O 2Z[ ]
-~+ko, 0 2Z[ ]
induces an isomorphism on coefficient groups.
We have
proved:
Theorem 4.1:
The natural transformation
(5)
is an
equivalence of homology theories.
I
For details of Sullivan's construction,
Appendix.
see the
121
Appendix:
Sullivan's
construction
of
ko*
0
Z[
]
orientations.
In chapter VI,
p
we defined a natural transformation
Witt
:
Witt
-, ako, 0 2Z[%]
(X
by assigning to every Witt space
orientation class in
koq(XX)
,
X)
0 Z[%].
a canonical
What is needed to
carry out this assignment is the alchemy of Dennis Sullivan:
Given a geometric homology theory
cycles
Y,
properties
with signature invariant,
in section
listed
data into
signature
*
for 1-cycles.
canonical
Specifically,
VI.4,
based on a class of,
and satisfying the
the
Sullivan translates
2Z[4-]
ko*
orientations
he constructs a natural
transformation of homology theories:
11Y:
Q
--
ko*
[%
such that the induced homomorphism on coefficient groups is
the signature homomorphism.
The construction is found in
may be unavailable to the reader,
[22]
and
and
[24]
[24].
Since
[22]
will almost
certainly be so, we sketch the arguments here.
The first ingredient is the following algebraic result,
which Sullivan calls Pontrjagin-duality for KO-theory.
Theorem:
[Sullivan]
group
(X)
KO
0
2[z]
Let
X
be a finite complex.
The
is naturally isomorphic to the
group
122
of commutative diagrams:
a
KO . (X; 0)
1I
KO.(X;Q/
where
])
-
b C/
[
of coefficient groups,
i
be specified.
diagrams
[
is the map on homology induced by the homo-
i*
morphism
and
a
and
T
are to
(The group structure on the collection of
is the obvious one).
Sullivan constructs an exact sequence:
Sketch of proof:
(2)
ji
,
(1)
0 -+KO
2Z[4] 3--KO^i (X) ) KO (X;Q) -+KO^ (X) 0 -+0
(X)
from the arithmetic square:
2Z
Here
2Z
-
=
lim
Ikt
[%2]^
zz 2[%]^,
2Z [%] /n2Z[],
0
.
2[ ] -' Q
where the
inverse
limit
n odd
taken over the direct system of odd positive integers,
is
123
partially
ordered by the relation:
if
m/n
.
m<n
The corresponding homomorphism:
2Z[ k]/n2Z [ .- ]
Similarly,
KO
2Z [
]I/m2Z[
]i
m.
KO i(X; 2Z [
(X)
]/nZ
[
] )
.
is reduction modulo
-!+
n odd
The inclusion:
2Z [%]antu/nZ
[%map] ohmo o/Z
v+
)
KO . (X; 2Z [%]/n2Z [%-
KO . (X; (Q/2Z [ %]l
By composition with the homomorphism
T
)
induces a natural map on homology:
in the diagram,
we get a collection of homomorphisms:
T
n
:
KO
L
(X; 2Z [-1-/n2Z [%2-])
-+p
2Z[
]121/n2Z [ -2] C
/2Z[%]
124
Now,
there is a canonical isomorphism of cohomology theories
induced by evaluation
KG
(X;Z[%]/nZ[%])
Tn
n
odd:
P-Hom(KO. (X;2Z[%I/n2Z[3]),2Z[
This isomorphism is discussed in
regard each
for
(cap product)
[22].
as an element of
KO
T = 1iM T
4
thereby obtain an element
ni
]/n2Z[%])
We can therefore
(X;ZZ [/nZZ%])
of
and
KO(X).
Using the isomorphism obtained from the universal
coefficient theorem for KO-theory, namely:
KO
we identify
KO
^
(X;Q).
a
(X;Q)
-Hom(KO.(X;
Q),Q),
in the diagram with a unique element of
The commutativity of the diagram implies that
0 a e KO
(X)
@ KO
(X;0)
in the short exact sequence
associate a unique element
is sent to
(2).
v E KO
0
in
KO
(X)
0
Therefor-, we can
(X)
0 Z[Z]
with the
diagram (1).
The.next result, also Sullivan's,
enables us to
translate the above isomorphism into more geometric terms.
It is a Conner-Floyd type theorem
(cf.
cobordism to KO-theory at odd primes.
[9])
relating smooth
125
[Sullivan]:
Theorem
periodic theories:
so
Q*
where
(XA)
Sketch of proof:
KO
f
:
L
MSO(4q)
A
Q 2Z[
],;21
and where
pair
has
2Z[k]
MSO(4r)
of
A4k
with the property
=
the universal
is
KO* (XIA)
Sullivan constructs an element
-1 ch(A 4 k 0 C)
where
-+
structure induced by the signature.
[]
Q
(MSO(4k))
2Z [12]
a compact p.1.
is
(X,A)
(p t
'FQ
Q*(pt)-module
the
is a canonical isomorphism of
)
Z/42Z
There
L- 1 e H*(BSO(4k);q)),
Moreover, if
L-class.
is
-*MSO(4(q+r))
the natural
map,
then
f*A 4 (q+r)
">x"
A4q
x A4 r,
where
denotes the external product in KO-cohomology.
Proceeding as in
[9],
one uses
A
to define a natural
transformation of cohomology theories
Q*O(XA)
KO* (XA)
G) 2Z[%1-1,
126
or,
a natural
by Alexander duality,
transformation of
homology theories:
so (XA)
KO* (X, A)
A
E
2Z [%]
such that
A
so
KO 4k (Pt)
Q4k (pt)
the
is
0
2Z [
]2 ~ 2Z[]
homomorphism:
signature
[M]
F-
sign(M).
(That is,
Q4k(pt)
is
so
=~
-4k
A*
-4k
SO
-1,
(pt)
(pt)
KO
ph
Q 2Z [-]
2Z[%
homomorphism).
the signature
By the multiplicativity of the elements
deduces that,
A 4 k' one
in fact, there is a natural transformation of
2Z/42Z-periodic theories:
so
Q*
C
(XA)
Q*,(pt)
Z[
--
+ KO*(XA)
0
Z[
Since this transformation induces an isomorphism on
127
coefficient groups,
we conclude that it is an equivalence
of theories.
I
From the two theorems above, we see that elements of
KO
(X)
are commutative diagrams:
0 2Z[]
y
Q i+4*()0
ji*
where
are homomorphisms satisfying the
cy,T
"periodicity
relations":
f
T((V
--
X)
x
(M -+pt))
= sign(M).T(V
G((V
f
-.
X)
x
(M -+pt))
f
= sign (M) -cy(V -.
X)
and
The elements of
Now,
suppose
homology theory
be a p.l.
of
(XA)
Q 2Z [-]
(that is,
in the relative
are analogously described.
case)
X
KO
X).
X
that
Q*
a cycle in the geometric
mentioned earlier.
Let
h
: X
v- ]RN
embedding into a large Euclidean space such that
has codimension 4k,
X.
is
and let
A canonical orientation
be a regular neighborhood
U
y
X
in
ko
n
(X)
G 2Z [%]
corresponds by Alexander duality to a canonical element of
128
ko
0 2Z
(U, aU)
That
is,
[4]
it
corresponds
0 4*(U,9U)
0
4 *(U,3 U;Q/ 2Z[
2])
to a commutative
"diagram":
(3)
[2]
V/2Z
which satisfies the periodicity relation.
Sullivan
described how to obtain such a diagram from the signature
invariant
(in
Given
and transversality.
cl)
[(M,DM),f]
transversality theorem
assume
f
(X) C
M
in
(U ,U),
Q
([8],
[15])
the relative block
implies that we can
is a cycle with the same local
as
X,
with dimension 4k,
in
Y,
and has
say.
a signature
That is,
sign(f
1
(X))
signature, this procedure associates to
a ([(M, M),f]).
well-defined integer
.
a
For
T,
it
n
Z
r
Define
bordism of
recall
that
2Z/n2Z-manifolds.
assign to a singular
By relative
T
: Q(U, 3u;2Z/n2Z)
Q*(-;2Z/n2Z)
[M,g]
in
(3)
to
-+Z/nZ
2Z[%].
is,
Using transversality
2Z/nZ-cycle
a
a
a compatible
odd, which we can then tensor with
To do this,
is a cycle
[(MM),f]
to define
suffices
collection of homomorphisms:
for
(X)
and by cobordism invariance of the
transversality again,
be
f-
structure
geometrically,
as above,
a 2Z/n2Z-cycle
129
g
(X)
Z/n2Z-coefficients
in the bordism theory with
associated to
If
Q
1.8).
(see
obtained by identifications on the boundary,
sign(g
as
signature invariant of
we define:
of the
Additivity
guarantees that the correspondence:
Y
-+
Q*(u , DU; 2Z/n2Z)
[R,g] I-
Z/n2Z
sign(g
is a well-defined homomorphism.
Let
(X))
T
=
lim
4-
T
n odd
of the signature
By multiplicAtivity
manifolds,
it is clear that
relations.
T
commutes.
Finally,
is
sign(Y),
Z/nZ-manifolds.
in the case of
Tn :
-
(X))
(x)
g
is the relative cycle from which
Y
a,
T
the diagram
The definition
(3)
n
0 a;['].
with respect
to
satisfy the periodicity
corresponding to
of orientations
a,
for relative
cycles proceeds similarly.
c*(pX)
where
c
: X -*Ppt
Specializing
=
sign(X)
satisfies:
p
Note that the orientation
E
kon (pt) ® Z[
]
is the collapse map.
now to the bordism theory associated
to
130
the class
of Witt spaces,
Theorem:
Every Witt space
a canonical
X=
the theorem:
(X, aX)
orientation class
P
of dimension
E
ko
q >
(X, X) 0 2Z1.%].
0
has
If
then the homomorphism
C* : ko (X)
carries
we have
px
to
0 Z[%] --
ko
(pt) 0 Z [%]
sign (X).
a
131
References.
1.
E.
Akin, Manifold phenomena in the theory of polyhedra,
(1969), p. 413-473.
Trans. Amer. Math. Soc. 143
2.
E. Akin,
Stiefel-Whitney homology classes and bordism,
Tr'ans. Amer. Math. Soc.
3.
J. Alexander,
G.
Hamrick,
(1975), p. 341-359.
205
and J. Vick,
and maps of odd prime order,
Soc.
4.
221
M.F. Atiyah,
Analysis
(1976),
1969,
M.F. Atiyah and I.M.
p.
73-84.
Singer,
The index of elliptic
87
(1968), p. 546-604.
N.A. Baas, On bordism theory of manifolds with
W.
Browder,
Math.
Surgery
Springer-Verlag,
8.
Global
(eds. Spencer and Iyanaga), University of
singularity,
7.
Math.
169-185.
operators III, Ann. of Math.
6.
Trans. Amer.
The signature of fibre bundles,
Tokyo Press,
5.
p.
forms
Linking
S.
Scan.
33
(1973),
p. 279-302.
on Simply Connected Manifolds,
New York and Berlin,
Buoncristiano, C.P.
Rourke,
and B.J.
1972.
geometric approach to homology theory,
Soc. Lect. Note Series 18,
A
Sanderson,
London Math.
Cambridge Univ. Press,
1976.
9.
P.E.
Conner and E.E.
Floyd, The relation between co-
bordism and K-theory,
No. 28,
10.
P.E.
Springer-Verlag
Lecture
Notes
1966.
Conner and F. Raymond, A
quadratic form on the
132
quotient
of a periodic
map,
Semigroup
Formum 7
(1974), p. 310-333.
11.
R.M.
Goresky and R. MacPherson,
theory,
Intersection homology
to appear.
(Announcement of results appears in:
La dualitd
de Poincar4 pour les espaces singuliers, C.R.
Sci. Paris,
12.
F.
s4rie A, 284
New York and Berlin,
D. Husemoller and J. Milnor,
Springer-Verlag,
14.
K.
Janich,
McCrory,
Trans.
16.
C.
Mccrory,
Symmetric Bilinear Forms,
der Signatur von Mannig-
(1968) ,
6
p.
Eigenschaft,
35-40.
Cone complexes and p.1.
Amer.
Math.
Soc.
207
Stratified general
Symposium on Algebraic
transversality,
(1975),
position,
J.
Milnor,
664, p. 141-145.
A procedure for killing homotopy groups of
differentiable manifolds, Proc. Symp.
v.
18.
3
Proc. of
and Geometric Topology,
Springer-Verlag Lecture Notes No.
17.
1973.
durcheine Additivitats
Inventiones Math.
C.
1966.
New York and Berlin,
Characterisierung
faltigkeiten
15.
1549-1551.
Hirzebruch, Topological Methods in Algebraic Geometry,
Springer-Verlag,
13.
(1977), p.
(1961),
J. Morgan and D.
in Pure Math.
p. 39-55.
Sullivan, The transversality
characteristic class and linking cycles
theory,
19.
Acad.
A. Pfister,
Ann. of Math.
99
(1974),
in surgery
p. 463-544.
Quadratic Forms, Lecture notes by A.D.
133
McGettrick,
20.
C.P.
University of Cambridge,
Rourke and B.J.
Sanderson,
Piecewise Linear Topology,
York and Berlin,
21.
E.
Spanier,
1967.
Introduction to
Springer-Verlag, New
1972.
New York,
Algebraic Topology, McGraw-Hill,
1966.
22.
D.
Sullivan,
Geometric Topology Notes,
Cambridge,
23.
D.
Sullivan,
Singularities
MA,
Part
I,
1970.
Singularities in spaces,
Symposium
II,
Proc.
of Liverpool
Springer-Verlag
Notes No. 209, p. 196-206.
24.
D.
Sullivan,
MIT,
unpublished
(handwritten)
notes.
Lecture
134
Figure
1.
135
(
z ~
Y-
V
X, U
1I
x.1
U
7u x[-I A
z
v,v
Figure
2.
4-i-
136
12
i
/
CIL~
/
-,
--
/
/Z
A
Figure 3.
y
137
ACKNOWLEDGEMENTS
It gives me great pleasure to thank Mark Goresky for
all of the support he provided during the preparation of
this thesis.
His most obvious mathematical contribution
is the remarkable intersection homology theory,
my primary
which he developed jointly with Bob MacPherson at Brown
tool,
University.
In addition,
he first described for me the
and our many conversations
problem which this thesis solves,
Mark later showed
were crucial in leading me to a solution.
me Atiyah's example, which has become my favorite "motivation"
for the definition of Witt spaces.
His remarks also clarified
the connection between additivity and cobordism invariance
of the signature.
I could go on.
Finally, he commented
helpfully on a swarm of preliminary drafts.
On the less technical side,
Mark generously shared with
me his personal and mathematical wisdom,
and gave unfailing
encouragement at some critical junctures.
I am also happy to express my gratitude to Ed Miller
for the active interest he took in my work and mathematical.
career.
ideas,
His broad understanding of topology, wealth of
and willingness to discuss mathematics have been
valuable and inspiring.
And I warmly thank Professor James Munkres,
teacher of mathematics I have encountered,
the finest
for introducing
138
me in such beautiful fashion to much of the topology I know.
The Department of Mathematics has kindly permitted me
to finance my graduate education with a teaching assistant-
ship;
I thoroughly enjoyed the experience of working with
students.
Finally,
of its forms)
I thank my parents for their support
and Eugenia Parnassa,
(in all
without whom I would
have finished this thesis six monthsearlier.
139
BIOGRAPHICAL NOTE
The author was born September 12, 1953 in Berkeley,
California.
He grew up in Framingham, Massachusetts during
its quaint pre-"Golden Mile" era,
and attended Framingham
North High School.
In September 1971,
he began studies at M.I.T., where he
was elected to Phi Beta Kappa in 1974.
B.S.
in Mathematics in June 1975,
his graduate work in topology.
He graduated with a
and stayed on at M.I.T.
for
He held a four year teaching
assistantship in the Department of Mathematics.
Lest he overstay his welcome at M.I.T.,
the author will
spend next year as a Visiting Member at the Courant Institute
of Mathematical Sciences in New York.