A geometric approach to homology theory

London Mathematical

Society Lecture

Note Series.

18

A Geometric Approach to Homology Theory by S.BUONCRISTIANO, C.P.ROURI(E, and B.J.SANDERSON·

CAMBRIDGE UNIVERSITY PRESS CAMBRIDGE LONDON . NEW YORK . MELBOURNE

Published by the Syndics of the Cambridge University Press The Pitt Building, Trumpington Street, Cambridge CB2 1RP Bentley House, 200 Euston Road, London NW1 2DB

Contents

32 East 57th Street, New York, N. Y. 10022, USA 296 Beaconsfield Parade, Middle Park, Melbourne 3206, Australia

©

Cambridge University Press

1976 Page

Library of Congress Catalogue Card Number: 75-22980

Introduction

1

I

Homotopy functors

4

II

Mock bundles

19

Printed in Great Britain

III

Coefficients

41

at the University Printing House, Cambridge

IV

Geometric theories

81

(Euan Phillips, University Printer)

V

Equivariant theories and operations

98

VI

Sheaves

113

The geometry of CW complexes

131

ISBN: 0521 209404

VII

Introduction

The purpose of these notes is to give a geometrical treatment of generalised homology and cohomology theories.

The central idea is that

of a 'mock bundle', which is the geometric cocycle of a general cobordism theory, and the main new result is that any homology theory is a generalised bordism theory.

Thus every theory has both cycles and cocycles;

the cycles are manifolds, with a pattern of singularities

depending on the

theory, and the cocycles are mock bundles with the same 'manifolds' as fibres. The geometric treatment, pl

which we give in detail for the case of

bordism and cobordism, has many good features.

easy to set up and to see as a cohomology theory. transparent

Mock bundles are

Duality theorems are

(the Poincare duality map is the identity on representatives).

Thorn isomorphism and the cohomology transfer are obvious geometrically while cup product is just 'Whitney sum' on the bundle level and cap product is the induced bundle glued up. Transversality the geometric interpretations

is built into the theory -

of cup and cap products are extensions of

those familiar in classical homology. Coefficients have a beautiful geometrical interpretation and the universal coefficient sequence is absorbed into the more general 'killing' exact sequence.

Equivariant cohomology

is easy to set up and operations are defined in a general setting.

Finally

there is the new concept of a generalised cohomology with a sheaf of coefficients (which unfortunately does not have all the nicest properties). The material is organised as follows.

In Chapter I the transition

from functor on cell complexes to homotopy functor on polyhedra is axiomatised, the mock bundles of Chapter II being the principal example. In Chapter II, the simplest case of mock bundles, corresponding to pl cobordism, is treated, but the definitions and proofs all generalise to the more complicated setting of later chapters.

In Chapter III is the geo-

metric treatment of coefficients, where again only the simplest case,

1

pl bordism, is treated. is given in this case.

A geometric proof of functoriality for coefficients

NOTE ON INDEXING CONVENTIONS

Chapter IV extends the previous work to a generalThroughout this set of notes we will use the opposite of the usual

ised bordism theory and includes the 'killing' process and a discussion of functoriality for coefficients in general (similar results to Hilton's treatment being obtained). In Chapter V we extend to the equivariant case and discuss the Z 2 operations on pl cobordism in detail, linking with work of tom Dieck and Quillen.

Chapter VI discusses sheaves, which work

nicely in the cases when coefficients are functorial (for 'good' theories or for 2-torsion free abelian groups) and finally in Chapter VII we prove that a general theory is geometric.

The principal result is that a theory

has cycles unique up to the equivalence generated by 'resolution of singularities'.

The result is proved by extending transversality

to the

convention for indexing cohomology groups.

This fits with our geometric

description of cocycles as mock bundles - the dimension of the class then being the same as the fibre dimension of the bundle.

It also means that

coboundaries reduce dimension (like boundaries), that both cup and cap products add dimensions and that, for a generalised theory, n

h (pt.) ~ h (pt.). However the convention has the disadvantage that n ordinary cohomology appears only in negative dimensions. If the reader wishes to convert our convention to the usual one he has merely to change the sign of the index of all cohomology classes.

category of CW complexes, which can now be regarded as geometrical objects as well as homotopy objects.

Any CW spectrum can then be seen

as the Thorn spectrum of a suitable bordism theory.

The intrinsic geo-

metry of CW complexes, which has strong connections with stratified sets and the later work of Thorn, is touched on only lightly in these notes, and we intend to develop these ideas further in a paper. Each chapter is self-contained and carries its own references and it is not necessary to read them in the given order.

The main pattern of

dependence is illustrated below. I

t

7~T IV

V

VI

~ VII The germs of many of the ideas contained in the present notes come from ideas of Dennis Sullivan, who is himself a tireless

cam-

paigner for the geometric approach in homology theory, and we would like to dedicate this work to him.

2

3

It then follows that

I·Homotopy functors

(3)

if

a,

T E K, then

Notice that we do not assume A subset and we write

L

C

E K.

K is a subcomplex if L is itself a ball complex,

(K, L) for such a pair.

If (K , L ) is another pair, and o 0 then there is the inclusion

K

The main purpose of this chapter is to axiomatise the passage from functors defined on pl on polyhedra.

cell complexes to homotopy functors defined

Principal examples are simplicial homology and mock

bundles (see Chapter II). Our main result,

is a union of balls of K.

anT anT

C K, L cLare subcomplexes, o 0 ---(K , L ) C (K, L). An isomorphism f: (K, L)'" (K , L ) is a pl o 0 1 1 homeomorphism f: IKI •.• fK I suchthat fiLl = IL I, and a EK 1

implies

f(a)

E K.

1

1

In the case where K and K

plexes, there are simplicial maps f: (K, L)'" (K , L). 1

3.2, states that the homotopy category is iso-

morphic to the category of fractions of pl formally inverting expansions.

cell complexes defined by

K x L of ball complexes The categories

K, L is defined by K x L =

----

{axT

I uEK,

TEL}.

Bi and Bs

Now define the category

Bi to have for objects pairs

Analogous results for categories of simplicial complexes have been

morphisms generated by isomorphisms

proved by Siebenmann, [3].

morphism is an isomorphism onto a subpair).

In §4, similar results are proved for A-sets.

This gives an

alternative approach to the homotopy theory of A-sets (compare [2]). In §6 and §7, we axiomatise the construction of homotopy functors Here we are motivated by the coming applica-

tion to mock bundles in Chapter II, where the point of studying cell complexes, rather than simplicial complexes, becomes plain as the Thom isomorphism and duality theorems fall out.

(K, L) and

and inclusions (i. e., a general The category

Bs has the

same objects but the generating set for the morphisms is enlarged to include simplicial maps between pairs of simplicial complexes. Subdivisions If L', L are ball complexes with each ball of L' contained in some ball of L and L' (K, L)). The inverse isomor-

Z(K x ~ , L x ~ ). 2

phism is called a subdivision isomorphism and we write

To see that z - z, consider

2

Now if f : (K , L ) ....• (K , L ) is a morphism in Bi, then

sd : T(K) - T(L).

1

1

2

2

T(f) : T(K , L ) - T(K , L ) is clearly well-defined by 2

2

1

1

Now let Ph be the category of compact polyhedra and homotopy classes of continuous maps.

Let T : Bi ..• S* be a functor satisfying

Axiom C, where S* is the category of based sets.

We define T: Ph -S*

as follows. An element of T(P) is an equivalence class of elements of T(K), where

IKI = P.

Suppose IKil = P, i = 0, 1. Then

Uo

E

T(Ko)

T(f)[z] = [Z(f)z]. Proposition 6. 1. Proof.

The functor

T satisfies Axiom C.

Let e : (K , L ) - (K, L) be an expansion. o 0

We have to

is 15

14

show T(e) is an isomorphism. Suppose then that T(e)z

= T(e)z. Construct o I in two steps. First, find z

that z : z _ z o I using Axiom G twice; then find z using E.

z E:

2

E:

Z(K x I, L x I) such

Z(KU(K XI), L U(L xI» 0 a

K=>K,K: I

T satisfies the following axioms for any

2

Half exact:

T(K, Kl UK)"'" T(K, KI)"" T(K2,

Kl n K)

is exact. Excision:

Compatibility of extension to Bs Now suppose that Z is in fact defined on Bs.

Then we have T

defined using Z I Bi, and T extends uniquely to Bs by Theorem 5. 1. Proposition 6. 2.

Lemma 7. 1.

But by Axiom E we have T(e) is onto.

T(K UK, I 2 is an isomorphism. Proof.

Order 2 is obvious; to see exactness,

a concordance on K

The extension of T to Bs is given by

K ) ...•T(K , K n K ) I 2 I 2

2

use E to extend

to one on K. For excision, use the definition of

Z(K, L) and G. T(f)[z] Proof.

=

[Z(f)z].

Now suppose given functors

By uniqueness,

it suffices to show that T(f) is well-

defined on Bs by the above formula, case f simplicial.

and it is sufficient to consider the

zq for q

E

Z' defined on Bi and

extended to Bi as above, and suppose that in addition we have Axiom S (suspension). *

There are given natural isomorphisms

Let K be any cell complex, and define (K x I)'

by deriving each cell on the half-way level. so is (K x 1)'; and if f: KI - K

2

Then if K is simplicial,

is simplicial,

then the deriveds may

be chosen so that L(f x id) : (K x I)' ....•(K x I)' is simplicial. I

2

Then we can define

The

result therefore follows from Lemma 6.3. that z. = Z I K x {i l.

z o - z I if and only if there is z

E:

Z (K x I)' such

1

Proof.

Consider

Q = (K X ~2),

to be the composition sq Tq(L) ...• Tq-1(L x I, Lxi)

0

1

E:

Z(K

X ~2)"

it is easy to see that Q col-

{o} )

TO) L)

2

lapses to both these subsets. "i

Moreover,

Tq-1(W, KULX

Tq-1(K,

I

and (K x (~l U~l»,.

I

+-

obtained by deriving each cell

of K x {v }. Then Q contains isomorphic copies of (K x 1)'; namely, K x ~l

T(i)

and hence

"il

Therefore, E

given a z

E:

Z(K x I), we get

Z(K x I)' and vice versa.

where W = K U x {1 }L x I, and i is an excision, j extends the identiL fication L - L x {O}, and as a map K"'" W is homotopic to the inclusion by an extension of the obvious homotopy on L.

7. COHOMOLOGYTHEORIES Now let Bi with L

=

C

Then easy arguments show that the long sequence is exact, and

Bi be the subcategory consisting of pairs

¢, and suppose that

Z : Bi - S* is a functor.

extend Z to Bi by defining Z(K, L)

=

(K, L)

we have shown

Then we can

Ker {Z(K) - Z(L) l. Suppose Z

now satisfies axioms E and G. Let T denote the associated homotopy

* See the note on indexing cohomology groups at the end of the introduction.

functor.

16

17

Theorem

7.2.

A sequence

sat~fying

E, G, and

S defines

~~~ll1pact

polyhe~al

pair~

Remarks

7.3.

sq~, sq."

E

sq~ + sqTj suspension

Zq-I(K E

Zq-I(K

to return

2,

i),

and use

to

In fact,

homology theory

Given

x I)'.

Finally,

~,

I g I "" n I g

1/ E

Z,

E

II-Mock bundles

on the category

Zq(K), form

G to construct

excision,

by formal

with

We describe and inverse

gk

=

example.

and suspension

of

imply co-

using Puppe sequences.

&I-spectrum

for

zq(~k)

11. q We will ~XPlain

qII q §5 for a specific

S*, q

theory pi

half exactness,

g

-+

Zq(K).

A classifying

taking ail-set

theory

use amalgamation

argument, q Axiom S need hold only for T (,). 3,

Bi

is in fact an abelian group functor.

addition':

x I, K x

zq:

a cohomology

Tq()

1.

This is seen by 'track

of functors

q T ()

Thus

can be constructed

by

in detail in Chapter 'f" Inl't e comthe theory t 0 In

REFERENCES

FOR CHAPTER

proofs.

proof of the

Quinn's

Another

~-sets

[2]

Homotopy theory and calculus

fractions.

Springer-Verlag,

Berlin

C. P. Rourke

and B. J. Sanderson.

Oxford Ser-, 2 (1971),

p]

L. Siebenmann. Akad.

J. Stallings.

[51

J. H. C. Whitehead.

[61

are constructed.

notes on polyhedral

Combinatorial

Seminar

Categories

Verlag,

C. P. Rourke linear

homotopy types.

Proc.

Kon. Ned.

topology,

homotopy 1.

Bull.

1969. Amer.

+

topology.

a

it the block over

of a pi

topology.

I. H. E. S.

-1

p ~ (u).

tive,

and then

Springer-Verlag,

~(a)

=

¢ if dim a

empty set for total space, Figure

Berlin

is similar

to

a


can be regarded as an extension of the cap

product with [M]. 2.

(3.3) generalises both Lefshetz duality and Spanier-Whitehead

duality; e. g. , for the latter, take X = M = Sn (d. Whitehead [13]).

31

q By Spanier-Whitehead duality, T ( ,) is indeed the dual

3. theory to pi

submanifold.

Then we extend to give relative theorems,

embeddings and for general subpolyhedra.

bordism.

with block transversality Proof of 3.3.

Let N(X), N(Y) be derived neighbourhoods of

X and Y, and define

Ifi

to be the composition

I.

n

i*

We also give the connection

[12].

Let f : W ..• M be a map between compact pi closed, and suppose that N C M is a submanifold.

Tq(X, Y) ;:. Tq(N(X), N(Y)) ~ T +q(N(X) - N(Y), N(X) - N(Y))

theorems for

manifolds with W

Then we can regard

f : W ..• M* as the projection of a mock bundle ~, as in §3. Now ~ can be subdivided so that N is a subcomplex of the base (this involves a 1

homotopy of f); then C (N)

1*

is the restriction

manifold by 1. 2, and f is now transverse

of ~ to N, and hence a

(in some sense) to N!!

Notice that the proof is easily adapted to give an E-version by where a is amalgamation;

making the diameters of cells of M* < E. Further,

see Fig. 3.

by a whole family of manifolds. is a general subpolyhedron. verslity in this setting.

N can be replaced

In fact, the natural setting is where

Let X C M be a subpolyhedron.

f : W ..• M is mock transverse

In X, or f In

We say that

to X if f is the projection of a mock

bundle in which X underlies some subcomplex of the base. W

N

We now show how to treat relative trans-

We write

X.

= X n aM to a get a relative theorem (there are counterexamples otherwise; see 4.2 For technical reasons,

we need a condition on X

and 4. 3 below). We say that X that at each point x

E

X

is locally collared in (M, X) provided a ------there is a neighbourhood in (M, X) which is

a the product of a neighbourhood in (aM, X ) with the unit interval. a collaring is equivalent to collaring [8; p. 321].

Local

Fig. 3 'To see that

Let M be a compact mani-

and suppose

Then restrict

f : M ..• y

theorem 4. 1.

by Cohen's theorem. Ifi

regard

Relative transversality

fold with boundary and X C M a polyhedron with X = X n aM locally ---------a -1 collared in (M, X). Let f: W - M be a map such that f aM = oW,

To see

is surjective,

C

as the projection c of a mock bundle (in which the blocks might have extra boundary over X) Ifi

to X to get a genuine mock bundle.

is injective, combine this proof with the second half of the

flow

In Xo;

making f mock transverse

then there is an E-homotopy of f reI oW to X.

proof of 3. 2. Proof. 4. APPLICATION TO TRANSVERSALITY We observe that the mock bundle subdivision theorem (together with Cohen [3; 5. 6]) implies various transversality with the simplest case first,

Suppose f low

is the projection of the mock bundle UK,

and choose a ball complex L with

theorems.

We deal

the case of making a map transverse

to a

IL I = M

extending K, and so that X

is a subcomplex of L. This is done by first extending to a collar via the product ball complex K x I, and then choosing any suitable ball structure on M -

IK I x

[0, 1) and adjoining the two.

Following the proof of 3. 2,

we can suppose that f is the projection of a mock bundle ~ such that 32

33

~ 10M

is a subdivision of ~. Choose a further subdivision

so that L' <J L. with

~1 I K 3;~.

Theorem 4. 4.

~';1.' of ~

X

1

W if and only if W

In x.

Then amalgamating over L, we have ~ , say, over L 1

Thus Theorem 4. 1 (the version for embeddings) recovers

It only remains to observe that the homotopy of p ~

takes place within cells of K, and hence can be shrunk to the identity,

strengthened form of [12; 1. 2].

and this extends to give a modified homotopy of f by the HEP. Remark 4.2.

Proof.

In fact, the proof of 4. 1 used only that K extends

to L with X a subcomplex of L. on Xa than local collaring.

This needs a much weaker condition

A necessary and sufficient condition is that

the ambient intrinsic dimension [1] of X at X

is constant on the ina teriors of balls of K. This is always true if the ambient intrinsic dimension of X at x

E

Suppose X

II/W

so that X n E(v)

with

IK I = M

E(v)

1- W;

then there is a normal block bundle

is a union of blocks.

Choose a ball complex K

so that the blocks of v are balls of K, and so that X

underlies a subcomplex

(1.e., simply triangulate the complement of

and throw in the blocks of v!).

we

Then the inclusion

IKI

is

the projection of an (embedded) mock bundle with X a subcomplex of I the base.

Xa equals that of Xa at x.

a

Notice that the restriction

of this mock bundle to

gives

E(v)

the Thorn class of v. Example 4. 3. K n X is a I-cell.

Let X = 'the letter

Then, if f:

point of K n X, there is no homotopy of f reI aw making f mock trans-

suppose Wjp X by ~/K; then we construct a normal

block bundle v on W in M by induction over the skeleta of K so that it restricts

to a normal bundle for

~(a) in

(J

for each

(J

E

K. This

is an easy consequence of the relative existence theorem for block

verse to X. Transversality

Conversely,

T' with the top in aM so that

oW ....• aM is not transverse to the mid-

bundles [11; 4. 3]. Then X

for embeddings

Now suppose that f: W ....•M is a locally flat embedding (the condition on local-flatness

will be removed later).

E

locally-flat embedding (1.e., f-1(a) inclusion R~_C R:

K, we have f I

= aHa),

:

v.

Extension to polyhedra This subsection anticipates

We say that

f : W ...•.M is an embedded mock bundle if f is the projection of a mock bundle ~/K, and for each ball a

-.h

~(a) ...•.a is a proper

and f looks locally like the

for some k, n). We then observe that the sub-

gularities).

§3 of Chapter III (manifolds with sin-

Observe that if f: Y ....•M is a map where Y is a polyhedron,

and we apply the process of Cohen's theorem [3; 5.6] and regard

f as

a 'bundle' over M*, then Cohen's proof shows that the blocks, although not manifolds, are polyhedra with collarable 'boundaries'; M*, then f

-1 •

(a) is collarable

in f

-1

(a).

i. e., if

So we define f : y ...•. K to

division theorem for mock bundles applies to embedded mock bundles to

(J E

yield an embedded mock bundle, and that the homotopy which takes place

be a polyhedral mock bundle if for each

in the proof can be replaced by an ambient isotopy (by uniqueness of

in C1{(J).

collars [5]). Thus Theorem 4. 1 applies to give a relative transverality

versality theorem for two polyhedra in a manifold and similar relative

theorem for embeddings via an E-ambient isotopy (fixed on aM).

versions and versions for embeddings.

E

K we have f

-1 • ((J)

collarable

Then the subdivision theorem works and we thus get a trans-

lransversality

Connection with block transversality

(J

implies transversality

In the case of embeddings,

mock

in the sense of Armstrong [2].

This is proved by using the collars to construct neighbourhoods of the Suppose X C M is a compact polyhedron and W is a locally flat submanifold of M. Recall [12] that X is block transverse

to W in M

if there is a normal block bundle v /W in M such that X n E{V) E(vIX

34

n W). We write

X

-.h

W or

xlv.

=

form cone x transverse

star;

compare with the proof of 1. 2. McCrory

[15]has shown that mock transversality equivalent to both block transversality

for polyhedra is symmetric

and

in the sense of Stone [16] and to

35

i

transimpliciality

in the sense of Armstrong

The transversality

definition

Suppose given

I inn {}0 ...• An+l. gIven by r:.. x II A x

[2].

of the cup product

~q, r{;K

then for some large

m we may assume

(t , .•. , t , s) ~ o non

«1 - s)t , ... , (1 - s)t , s).

m This determines a natural isomorphism E W, E (1]) elK I x 1 so that PeP are restrictions of the projection m IKI x 1 ...• IKI. Now consider E(~)1]X1m E(1]) x 1m c /KI x 1m x 1m also based homotopy equivalences .. m 0' 1 0 l' By mductIvely making ~(a) x I transverse to 17(a) x 1m in ax 1m x lID o 1 0 1 I/J(Y): ISY/ ...•Y and 1/I(X):x we get a mock bundle E(~) x I~ n E(1]) x I~ ..• K It can easily be seen that this gives the cup product,

using the alternative

definition.

we sketch a proof below connecting the transversality restriction

to the diagonal definition.

generalising

s2m(~ x 1]) =

denote q-fold suspension.

considered

(see [10; p. 334]). l/J(nlxl)

This proof has the virtue of

to the more complex situation

Let s

However

Consider

definition with the lelxll

0

e(Y) : nsy

...•sny.

There are

...•slxl

the composition

° In1/l(x) I

:

Inxl

..• lnslxll

...•lsnlxll

...•nlxl.

in Chapter V. We have now proved:

From definitions

q m m smW x t(K x I ) u t(K x I ) x sm(1]) and the total spaces

on the right are transverse without being moved. Let i: IK I x 12m ...• 2m IKI x IKI x 1 be given by i(x, y) = (x, x, y). Then applying i* we get

Lemma 5. 1.

Suppose

X is a Kan based A-set.

There

is a based

(weak) homotoEY eq~ival~nce Inxl-nlxl. Now define an n-spectrum a Kan A-set

Desuspending

both sides reveals

the coincidence

S

of definitions. em :

5. THE CLASSIFYING SPECTRUM

For each

and A-sets

the category

We give a simple- minded definition A-sets.

Basic facts about A-sets

Given a based Kan A-set

contained

of spectra

in the category

of pairs

nx

i "

=

of S (PL)

0 1, ... , n, are just restrictions

of the

for

cobordism

pl

S (PL)

as follows. m

Let R co = URn.

Then a k-simplex

kook

denote the space of loops on the based complex of Y.

CW complex

Y,

There is an identification

Base simplexes face operators

*k

€

S(PL)m

are defined by taking

are defined by restriction.

1. 1 and general position that

36

of maps by

projection.

I

and let SY denote the singular

and homotopy classes

is a compact polyhedron X c A x R such that 71 I : X - A m k is the projection of an m-mock bundle over A , where 71 is the natural

o~+1a = * o' The operators

0. in X,

Let ny

h* is defined on

as follows, Define

I

of CW complexes

theory

of

The n-spectra

O. : (nX)(n) ..• (nX)(n-l)

Z is given

in [10] will be assumed.

X, we define a based Kan A-set

if and only if 0n+l a = *nand

€

-ns m- l'

It follows from 5. 1 and [13] that a cohomology

n-spectra

m

and a homotopy equivalence

m

Sm

as follows.

S (PL)

m

X

=

¢, and

It follows from the proof of is a Kan complex (compare

[11;

37

k k k+ 1 k() 1 1 Dehne e : 6. -+ 6. byes = 2S + 2vk+l' ek : !j(PL)(k) -+ (Ug(PL) )(k) defined by o

2.3]). m

m-1

k

00

L\. x ReI

X c

e

m

gives a mock bundle 11 /CK such that s ~/K x I is isomorphic to the # amalgamation of the pull- back 1f 11 by the pinching map 1f: (Kx I)' - K,

Then we have

: g(PL)

m

where (K x I)' [8M] = 0 in I·

nn- 1 ® F 0 because

it is the bordism class of the boundary of the complement of a regular

-

-

neighbourhood of 8M in M. The next lemma is important in what follows. Lemma 2.8. dary

Suppose that M is a

(p,

n+1)-manifold with boun-

V and label it by b

;

with singularities V, representing

an element in H (p,

W,

58

N has singularities

then

V,

I

cp 2'

up to codimension p + h + 1 at most, at most.

N:

of

with

W

to each manifold

nn- 1) we are able to associate a nn(p), such that

representing an element in In fact, if W'

with boundary, W, OW,between M, oM and N, oN such that

(c)

E BI;

nn-1).'

identify each copy with V ® bl•

(p,__ n)-manifold

exists a bordism

OW has singularities

I

in codimension one only. Therefore,

that W' has singularities

(b)

H (p,

the above identification on its boundary, becomes a (p, n)-manifold

has singularities up to codimension p + h, there is a (P, n)-bordism W has singularities

E

n n- 1 ® F 0, i. e. V ® w(bI) = aV. Take a copy aV consists of a number of copies of V (non

constantly labelled in general);

oM. Then, if oM has singularities up to codimension p and M s[W] ~ [V] + im

(a)

2

O. 8uppose V constantly labelled by b l

is not necessarily

Take [V] + im?>

l

l

0

n n- 1)

[8M] + im ~ .

It is straightforward

the sequence

Thus there is a well defined

map: s : nn (P)

Fig. 11

1m ~2'

E

E

[W] we can assume, by Lemma 2. 8,

8W' in codimension at most one and that there

W' - VI with singularities

8N in codimension two

Then [sW] - [8W'] = ¢2[8W] and so s is an epimorphism.

up to codimension p, up to codimension p + 1.

59

running:

Description of the map l Define a map I: defined homomorphism; I

l

fiJ

1

n

® F

..• n 1

= 0 because let

by bI E BI.

[M]; l

is a well

®

non

F

[W] =

..• n

mula when

(p).

lli> 1[M] and suppose M constantly labelled p)

and observe that

of W to the empty set by means of a

(p,

n+1)-manifold with codimension

n*(-;p).

Pick a representative

O.

N, of V to

l/J

is exact.

p

3. FUNCTORIALITY The classes of links constructed in Section 2 summarize the whole structure of the resolution we refer to

one singularities. there exists a bordism,

(-))=}

a(Mx bIL(bJ,P

is bordant to W. So stick the two bordisms together and get a bordism

=

q

This spectral sequence collapses to the universal coefficient for-

Then take M x bIL(bI,

Assume now l([M])

n

H (p, p

l

l/J

n

=

® F ..• n (p) by l[M] non so we have the sequence

n

If

V of l([M]):

p

P,

p'

map f : p .•• p'

such that N has singularities

p

geometrically.

Therefore,

from now on,

as a linked resolution. are linked resolutions of G, G' respectively, is said to be a map of linked resolutions

a chain

(or simply

linked map) if f is based and link preserving, i. e. :(a) f(bP) E B'P for each bP E BP. remove from N a regular neighbourhood of SN in N to get the required (b) Let bP E BP. If we relabel each stratum of the link bordism between M and Ii> (SN). Thus we have proved that the sequence 1 L(tP, p) according to f and if f(L(bP, p)) denotes the resulting object, above is exact; which is enough to ensure the existence of a monomorthen f(L(bP, p)) = L(fbP, p'). phism l : H (p, n ) ..• n (P) induced by l. ann So there is a category, e, whose objects are linked resolutions Now it only remains to prove exactness at n (P). ---n E P ••• G and whose morphisms are linked maps. If Cib* is the category sl = 0 : sl[M] = 0, because [M] has no singularities; hence We claim that ~ [SN] = [M]. In fact

SN in codimension one at most.

sl

=

1

of graded abelian groups, we have the following

O.

Ker s

C

im l:

let [M] E n (P) and assume, without loss of n

generality, that M has codimension one singularities means that [SM] E im dism; therefore

l/J. 2

C

im

p

[M] = [M'] where M' is without singularities

l = im t.

=

n

® F

a

whose image through

l

Proposition 3.1.

0

But then SM can be re-solved up to bor-

determines an element of n Thus Ker s

SM; s[M]

sake of simplicity wedisregard

and hence

is [M].

Proof.

[M] ..• [T(M)] p

We have seen how the exactness of p is used in

the proof of the universal-coefficient

theorem.

As pointed out before, if p is any based ordered chain complex augmented over G, then the theory way. But now the singularities necessarily

60

solvable;

the topological component of n*(-; -).)

Let T: p .•• p' be a morphism of

we associate a (p', n)-manifold,

the

e.

If [M] En (-; p), p

T(M), to M by relabelling

n

all the strata

of M according to the based map T. The correspondence

The proof of the proposition is now complete. Remark 2. 9.

e ..•CLb*. (For

n*(X, A; p) is a functor

p

T* : n*(-;

p) .••

gives a well defined natural transformation n*(-;

p')

Corollary 3. 2.

and the functorial properties If the linked map T: p .•• p'

of theories

are clear. is a homotopy equi-

valence, then T* is an isomorphism.

n*(-, p) can be defined in the same

in codimension greater than one are not 2

Proof.

they give rise to the E -term of a spectral sequenC11 theorem.

This is an easy consequence of the universal-coefficient

There is a commutative diagram

61

0-

H

° (p, ° (p',

j

AI)j

H,(T, 0n(X, 0'" H

Obviously the above definition of

0 (X, A))'" 0 (X, A; p) - H (p, 0 l(X, A)) - 0 n n 1 n-

0 (X, A)) n

T.

H, (T, 0n_1 (X,

lX, A;

~i

p) •••

H

j)

0

(p, 1

n-

A)I

I

p'

E

e.

nn (X, A; p)

is independent of the chosen

Now we fix our attention on a particular the canonical resolution of G and written

l(X, A)) - 0

y.

t. 1. resolution,

called

It is defined as follows:

£

in which the side-morphisms equivalence.

Therefore

T

are isomorphisms,

*

because

T

y:r

is a homoto

is also an isomorphism.

If G E }, y' = {r', lfi' L We proceed by p p p P induction on p. Write (~) = (~o' ~I' ~/ For p = 0, put ~o(bo)=I/>E(bo) Proof.

°

Let

°

p

~

Define -(M, f) = (-M, f). Two singular (p, n)- for each b E B. Inductively, let E B.p ~hen -I/>p_1I/>p(bP) EKerl/>~_l Ofth dO " "t " and therefore it determines a basis element, b , in F Ker I/>p' 1; b'P has cyc 1es (M , f) , (M , f) are b or dan t 1 e 1Sl01l1umon 2 " (X A) B d" ° ° 1 1 t" a canonical word w(b'P) and cancellation rule c(b'P) induced from those (M1 U - M2'1 f 1 IUf 2 ) ~ or ds 111 , . or Ism IS an equ1va ence re a lOn .p __ of 0- through the map (I/> lfi) Therefore the assignment in the set of singular (p, n)-cycles of (X, A). Denote the bordism class of p-l' p-£' tfl- (b'P " w(b'P) c(b'P)) defines lfip with the required properties. (M, f) by [M, f] and the set of all such bordism classes by On(X, A; p), ,

W is called a bordism.

An abelian-group structure

62

is given in Un(X, A; p) by disjoint union.

I

63

Lemma 3.4. Proof. If

n*(x, A;

Functoriality

gives a functor

p)

on

: G ....•G', cJ>

~* : n*(X, A; y) ....•n*(X, A; y'), where

y

tG

O.<X,

"b - db.,

t1

,

2

p,)

is the canonical

Q/ ~

n*(X, A; p )

Cib, assign the homo-

E

¢

is the canonical

3

where t ..

1, J

and Q/ are the natural transformations

obtained in the usual

extension of cJ> described in Lemma 3. 3 and ¢* is the induced homo-

way by relabelling the cycles according to the canonicallifting's

morphism described in Lemma 3.4.

id : G .• G. By Corollary 3.2, t. . is an isomorphism 1,

In view of the previous corollary we shall write n* (X, A; G)

Therefore,

instead of O(X, A; y) and cJ>* instead of ¢*. A (y, n)-cycle [bordism]

in order to prove the theorem,


(M')

M" - 0 by the proof of the universaln

1m lJl*. Let M

(G", n+l)-manifold with

be a G-manifold and W" a

aw"

= lJI(M). We show how to modify W" in order to get a G-bordism between Mn and a G'-manifold M,n.

3

Ck;J g

If M' is a (G', n)-manifold,

Therefore

coefficient theorem.

g'

(b) g

o.

Relabel each component of the (n+I)-stratum

g' 2

3

of G, obtained from the G"-labels through a lifting

2

(a)

The relabelled n-stratum stratum of Mn.

Fig. 14

(b) If the singular set, S(M), of M has more than one component, the relabelling construction can be performed componentwise and one

of

aw"

of W" by elements G t G" such that

..•.-

-

coincides with the n-

If two components are labelled by the same element of G",

the corresponding liftings coincide. Let V be a component of the n-stratum V

of W" and g , ... , g 1

V

gets a (G', n-l)-manifold, !3(M), whose bordism class is independent of the new G-labels around V; g' = ~ g. is an element of G'. Attach a the various choices. i=l 1 Next we show that !3(M) depends only on the bordism class of new sheet (V x I) ® g' to V iff g' '" 0 and label V by the G-relation M, i. e. if M

')(1 ® D )(V)

o

Summing up, we have the following:

(a)

n*(-; G) is a functor on the category of R-modules

(b)

n*(-; G) is additive

(c)

For every short exact sequence of R-modules,

there is an

associated functorial Bockstein sequence. so that V' may be borded to 9f by a (G', n+1)-manifold with singularities given by (1 ® D )(V). o We now turn to the main object of this section, i. e. putting coefficients in an R-module.

Properties

(a), (b), (c) form the hypothesis of Dold's Universal-

coefficient theorem [1]. Therefore we deduce that there is a spectral sequence running

In the following R will be a commutative 2

E p,q

ring with unit.

=

Tor (n (-'R) G) =} p q' , p

n (-'G) *'

If G is an R-module, let n*(-; G) be bordism with coefficients This completes the discussion of the case of R- modules as co-

in the underlying abelian group G; n*(-; G) has a natural R-module structure.

In fact, we must exhibit a ring homomorphism

a: R ..• HomZ(n*(-; G), n*(-; G)). The above additivity lemma, together with functoriality,

tells us that there is a ring homomorphism

efficients.

In later chapters we shall only deal with abelian groups;

but

it is understood that everything we say continues to work in the category of R- modules. REFERENCES FOR CHAPTER ill

defined by a'(f) = f*.

Therefore we can define a by the composition

[1]

Math. Zeitschrift,

80

(1962/3).

a

R ..• HO~(n*(-;

G), n*(-; G))

a~ la' Horn (G, G) Z 78

A. Dold. Universelle Koeffizienten.

[2]

P. J. Hilton. Putting coefficients into a cohomology theory. Konikl. Nederl. Akademie van Weterschappen (Amsterdam), Proceedings,

Series A, 73 No. 3 and Indag. Math. 30 No.3,

(1970), 196-216. 79

P. J. Hilton and A. Deleanu.

[3]

efficient sequences. (1970),

[4]

On the splitting of universal co-

Aarhus Univ., Algebraic topology Vol. I,

180-201.

J. Morgan and P. Sullivan.

The transversality

class and linking cycles in surgery theory. (1974),

IV· Geometric theories

characteristic

Ann. of Math. 99

463-544.

In this chapter we extend the notion of a geometric homology and cohomology (mock bundle) theory by allowing (1)

singularities

(2)

labellings

(3)

restrictions

on normal bundles.

The final notion of a 'geometric theory' is in fact sufficiently general to include all theories (this being the main result of Chapter Vn). A further extension, to equivariant theories,

will be covered in Chapter V.

In the present chapter, we also deal with coefficients in an arbitrary geometric theory.

A geometric theory with coefficients is itself

an example of a geometric theory and it is thus possible to introduce coefficients repeatedly! The chapter is organised as follows.

In

§1

we extend the treat-

ment of coefficients in the last chapter to cover oriented mock bundles and in

§§

2 and 3 we deal with singularities

and restrictions

on the normal

bundle. In §§4 and 5 we give interesting examples of geometric theories, including Sullivan's description of K-theory [11] and some theories which represent (ordinary) Z -homology. p

Finally

§6

deals with coefficients

in the general theory. 1. COBORDISMWITH COEFFICIENTS We now combine Chapters n and III to give a geometric description of cobordism with coefficients.

It is first necessary to introduce oriented

mock bundles (the theory dual to oriented bordism).

We give here the

simplest definition of orientation, an alternative definition will be given in §2.

n n Suppose M , V -1 are oriented manifolds with V

C

aM. Then

we define the incidence number £(V, M) = ±1 by comparing the orienta-

80

81

2.

tion of V with that induced on V from M (the induced orientation of aM is defined by taking the inward normal last);

orientations agree and -1 if they disagree.

E(V, M) = + 1 if these

An priented cell complex K


st(a2, K», a, aI' a2 EK. ~ver a; (T, q)-cocyc1es [cobordisms] ~, T/ over K are isomorphic, K l 2 We briefly recall the notion of simplicial cohomology with coefwritten ~ 2" T/, if there is a (Pi) homeomorphism h: E ~ - E T/ such ficients in a stack and its relation with Cech cohomology. Let T/K be chat h I ~(a) is an isomorphism of T(a)-manifolds [bordisms] between a stack over the oriented simplicial complex K. A (-p)-cochain, fP, ((a) and 1] (a). Suppose given ~q/K and L C K, then the restriction

X. If K is a cell complex, FK(a) = F(st(a,

K», F (a

with coefficients in ~

E

T,

F induces a stack, FK/K, by:

K, an element of T(uP); (-p)-cochains form an abelian group by

coordinate addition, C-P(K, T). There is a coboundary homomorphism: oPfP(JJ+1) =

L:

+1[a:

~(a) =

< ( ) = F(st(a1,

is a map which assigns to each p-simplex

o-P : C-P(K, T)- C-p-l(K,

The manifold [bordism]

T), given by

uP+1]T(a
X +1

n

n

=

¢}

There is a category TCW consisting of TCW --------~-~ complexes _'"

N.. } ~ N..

1,1-

is the cone on L. with -1

The cone flag C. is a fibre or model for the framified set X Lemma 2. 5.

f: M -+ X, g : X -+ Y are both transverse

where M is a compact pl Then g

136

0

manifold and X, Yare

maps

TCW complexes.

-1

--

---

-'

and the framified set L. is a link for X. There are obvious notions of -1

restriction

--

-

of a framified set to a suitable subpolyhedron and of product

f : M -+ Y is transverse.

137

of a framified set with a manifold.

A pl

homeomorphism is an iso-

Moreover, by uniqueness we can choose the structure

T.

on aT to be the

morphism of framified sets if it commutes with all the extra structure.

product of the framification of S. 1 with

In particular two isomorphic framified sets have the same (or identi-

agree and extend the strata to M by adjoining the 'cones' on their

fiable) system of links.

intersection with IlT. L e.

The final condition is:

(5) h. restricts

to an isomorphism of framified sets

1

{N.1,1:J N.1,2 :J Remark 3.2.

...

1

N..1,1-I}

3!

N ..

x

1,1-1

Choose the indeXing to

M. == (M ). U T x (C(S. 1)' - cone pt. ).

a

J-

1

1

L ..

Finally the isomorphism

A regular neighbourhood system is constructed

inductively by defining N == X and N . is a simultaneous system n,n n n,J of second derived neighbourhoods of X in X .. Then define n J X: == X - int(N .) and proceed with the construction for 1

J-

ht is provided by the chosen product structure

on T. Induction now gives all the structure of a framified set to M. Uniqueness is clear.

n,1

X'1 :J X'2 :J •••

:J X'n-l :J ¢.

Existence and uniqueness of regular neighbourhood systems thus follows from the usual regular neighbourhood theorem. Notice that a framified set is a building with bricks the strata {X. - X. I} 1

and plan defined by using the cone structures

1-

(see §l

Example 4). We are not interested in the specific ordering of the strata of ~ but only in the partial ordering given by the geometric structure of X, and we will allow an isomorphism of framified sets to change the order.

The importance of framified sets lies in the following theorem,

which is essentially an observation. Theorem 3. 3.

Let f: M ...•X be a transverse

map to a TCW, Fig. 20

then M has the, structure of a framified set determined up to isomorphism by the map f. Proof.

We can also describe this framification (at least as a building) Since the image of f is contained in a finite subcomplex

quickly as follows.

of X we can assume without loss that X is finite and proceed by induction on the number of cells of X. Mt == f-1(e) and Nt 1 == cl(f-1(e»

,

X -- X*1 :J X*2 ::>•••

In fact we will produce a framifica-

tion of the same length as the number of cells. (i. e. Mt ==

Let X == X

a

u e.

Define

of and Nt, 1 == T in the

Now let M == cl (M - T) then M a a x S. 1 have the structure of framified sets by induction.

138

T

J-

::>X* t-- e

A

be the corresponding filtration of X.

Then Mt == f

-1

(X;).

In other words

the building is the pull-back by f of the dual building to X.

notation of the previous section). and IlT 3!

Take the dual complex to X and let

Notice that the 'models'

C. are just the closures of the cone

-1

flags

139

xn .

e. n (X'" ~ X'" ::J ••• 1

1

2

But all the product structures

1

required product structure

are coherent in the Dk factor and the

on T. is seen. 1

Since the models depend only on X, we say that the framification of M is modelled_on X. It is clear that there is a 1 - 1 correspondence

4. MANIFOLDSAND MOCK BUNDLESMODELLED ON X

between transverse maps f: M - X and framifications of M modelled on X.

We now rephrase the results of §3 by omitting the first stratum

This observation will be developed in the next section when we

'classify' homotopy classes of maps from one TCW to another.

To end

throughout.

suspension and thus carries

this section we will prove the lemma left at the end of the last section.

1,

l

we can fwd an isomorphic system with smaller closed strata

over to stable maps.

as the first cell in X) and let

and larger cone flags (d. Stone on 'minidivision' [6]); this has the effect,

of the duals to cells other than

for the framification given by a transverse

associated stratification

•

map f; M ...•X, of replacing

A.

+

A

f

the nelghbourhoods T. x D. by nelghbourhoods of the form T~ x D. A

1

A

1

1

where T: ~ T. and D:t-= D. u collar, 1

1

1

1

1

and it is not hard to see that we

A

o

-1

M:)

Let e. be a typical cell of X.

(X (X)).

and write

E

1

= cl(x(X) 1

C.

X be the basepoint (regarded

x (X) be the subcomplex of X* *. If f : M ...•X is transverse

of M starts

X(M, f) where

1

.,

1

1

Using the fact that L.

1

of a framed X manifold.

.

a framed Xo manifold.

We call this framification an extension of the original one.

11

=

1

1

1

aD 1.. The set of basic links {L.) 1 C

(see IV §3) called free

S., we have an intuitive notion 1

The precise formulation is in terms of

killing as in IV §4. Suppose X is finite and X inductively that framed

X(M, f)

Collapse the notation of §3

defines a theory of manifolds with singularities

X

then the

and we label the cone point bye .. L. is in

fact a framified set embedded in SI'= X manifolds.

consisting

n e.), then C. is the cone e.L., where L, is

the link associated to e

can choose all the collars so that there are diagrams

*

Let X be a based TCW. Let

First observe that, by inductively choosing collars on the frontiers of the N..

The idea is to obtain a formulation which is invariant under

= Xo

U

e.. 1

Suppose

Xo manifolds have been defined so that L

is i The theory of framed X manifolds is the theory

obtained from this theory by killing L. and labelling the new stratum of Proof of Lemma 2. 5.

Let e

be a cell in Y and Tk = (g f) Choose an extended

This means that we can regard

made of generalised handles of the form T ~x D:. 1

is transverse

+ n Tk i

D

aDt

in a similar subtube.

as included in M as pt. x D:

1

6:1 x D:'-) n T. = (Q 1 1

C

T~ x D:. 1

x T~) x Dk 1

= Q' x Dk say.

140

1

(from the definition of transversality

M as

Since go f I : D: -Y 1

1

Here we are regarding Thus

1

singularities framed

bye .. In general define a framed 1

X' manifold where X'

C

X manifold to be a

X is a finite subcomplex.

From the

definitions of killing and framified sets, it is easy to see that a framification of M modelled on X is equivalent to a framed

for TCW's) we have

= Q x Dk say where Q is a manifold with boundary and

Q x Dk meets

ek.

k is a product Tk x Dk of codimension zero

We have to show that cl(Tk) in M which meets aM in a similar subproduct. framification for f: M -. X.

-1

0

X manifold

embedded in M. From Theorem 3. 3 and the transversality

theorem we

have:

Dt

Proposition 4. 1.

There is a 1 - 1 correspondence

between the

set of homotopy classes of maps [M, X] and the set of cobordism classes of framed X manifolds embedded in M. In particular n

UX means the group of cobordism classes of framed

1T

(X) ~ Un where n X--

X manifolds em-

bedded in Sn.

141

We can extend the proposition to maps [Y, X] where Y is an unbased TCW using an extension of mock bundles to TCW's. be a TCW then a subset

E

C

Let Y

enlarged by adding a trivial I-disc bundle.

We thus have the stable

version of 4. 1 and 4. 3 (for details of the stable category see Adams [1]):

Y is the total space of an embedded mock

bundle (of dimension -q) provided that for each cell e

i

E

Proposition 4. 4.

Y there is a

diagram

There is a 1 - 1 correspondence

between stable

homotopy classes of (based) maps Sn -+ X and cobordism classes of stably framed

X manifolds.

There is a 1 - 1 correspondence

stable homotopy classes of stable maps

{Y, X)

and framed

between

X mock

00

bundles over Y - * (i. e. embedded in S Y - *). Remarks.

1.

direct generalisation

= 1

where M.

The above notion of a mock bundle over Y is a

of the definition for cell complexes in Chapter

n.

h~1(E) and is a proper submanifold of D. of codimension 1

1

2.

q. The following proposition is a generalisation of Lemma 1. 2 of Chapter II and is proved by a similar argument to Lemma 2. 5. We omit the

We have been deliberately careless

about dimensions; 3.

details.

this will be remedied in the next section.

See Example 2 at the end of the next section for a clarifica-

tion of the relation between this representation Proposition 4. 2.

Let E

C

and 'killing'.

Y be an embedded mock bundle and

f : M -+ Y '!-.i!'ansy~!,~emap, then f-1(E) is a proper submanifold of M

5. THE CYCLES OF A HOMOLOGYTHEORY