Notes on Cobordism Theory 9781400879977

Table of contents :
Preface
Contents
I. Introduction-Cobordism Categories
II. Manifolds with Structure-The Pontrjagin-Thom Theorem
III. Characteristic Classes and Numbers
IV. The Interesting Examples-A Survey of the Literature
V. Cohomology of Classifying Spaces
VI. Unoriented Cobordism
VII. Complex Cobordism
VIII. ϭ₁ - Restricted Cobordism
IX. Oriented Cobordism
X. Special Unitary Cobordism
XI. Spin, Spin c and Similar Nonsense
Appendix I. Advanced Calculus
Appendix II. Differentiable Manifolds
Bibliography

Citation preview

NOTES ON COBOPDISM THEORY

BY ROBERT E. STONG

PRINCETON UNIVERSITY PRESS AND THE UNIVERSITY OP TOKYO PRESS

PRINCETON, NEW JERSEY

1968

Copyright @ 1968, by Princeton University Press All Rights Reserved

Published in Japan exclusively by the University of Tokyo Press; in other parts of the world by Princeton University Press

Printed in the United States of America

Preface

These notes represent the outgrowth of an offer by Princeton University to let me teach a graduate level course in cobordism theory.

Despite the fact

that cobordism notions appear in the earliest literature of algebraic topology, it has only been since the work of Thorn in 195^ that more than isolated results have been available.

Since that time the growth of this area has been

phenomonal, but has largely taken the form of individual research papers. To a certain extent, the nature of cobordism as a classificational tool has led to the study of many individual applications rather than the development of a central theory. In particular, there is no complete exposition of the fundamental results of cobordism theory, and it is hoped that these notes may help to fill this gap. Being intended for graduate and research level work, no attempt is made here to use only elementary ideas.

In particular, it is assumed that the

reader knows algebraic topology fairly thoroughly, with cobordism being treated here as an application of topology.

In many cases this is not the

fashion in which development took place, for ideas from cobordism have frequently led to new methods in topology itself. An attempt has been made to provide references to the sources of most of the ideas used.

Although the main ideas of these sources are followed closely,

the details have frequently been modified considerably.

Thus the reader may

find it helpful to refer to the original papers to find other methods which are useful.

For example, the Adams spectral sequence gives a powerful computational

tool which has been used in determining some theories and which facilitates low dimensional calculations, but is never used here. Many of the ideas which appear are of the "well known to workers in the field - but totally unavailaole" type and a few ideas are my own.

The pattern of exposition follows my own prejudices, and may be roughly described as follows.

There are three central ideas in cobordism theory:

1)

Definition of the cobordism groups,

2)

Reduction to a homotopy problem, and

3)

Establishing cobordism invariants.

This material is covered in the first three chapters.

Beyond that point, one

must become involved with the peculiarities of the individual cobordism problem. This is begun in the fourth chapter with a survey of the literature, followed by detailed discussion of specific cobordism theories in the later chapters. Finally, two appendices covering advanced calculus and differential topology are added, this material being central to the 'reduction to a homotopy problem' but being of such a nature as to overly delay any attempt to get rapidly to the ideas of cobordism. I am indebted to many people for leading me to this work and developing ny ideas in this direction.

Especially, I am indebted to Greg Brumfiel,

Peter Landweber, and Larry Smith for numerous suggestions in preparing these notes, and to Mrs. Barbara Duld for typing. I am indebted to Princeton University and the National Science Foundation for financial support.

Finally,

I am indebted to my wife for putting up with the foul moods I developed during this work.

CONTENTS Chapter I.

Introduction-Cobordism Categories

Chapter II.

Manifolds With Structure-The Pontrjagin-Thom Theorem

14

Chapter III.

Characteristic ClaGses and Numbers

27

Chapter IV.

The Interesting Examples-A Survey of the Literature

40

Chapter V.

Cohomology of Classifying Spaces

59

Chapter VI.

Unoriented Cobordism

90

Chapter VII.

Complex Cobordism

110

Chapter VIII.

~l - Restricted Cobordism

147

Chapter IX.

Oriented Cobordism

176

Chapter X.

Special Unitary Cobordism

237

c

Chapter XI.

Spin, Spin

Appendix I.

Advanced Calculus

Appendix II.

Differentiable Manifolds

Bibliography

and Similar Nonsense

1

283

CHAPTER I

Introduction - Cobordism Categories

In order to place the general notion of cobordism theory in mathematical perspective recall that differential topology is the study of the category of differentiable manifolds and differentiate maps, primarily in relation to the category of topological spaces and continuous maps.

From a slightly

less theoretical point of view, it is the study of differentiate manifolds by topologists using any methods they can find.

The guiding principle is

that one does not study imposed structures such as Biemannian metrics or connections and this distinguishes differential topology from differential geometry. As in any subject, the primary problem is classification of the objects within isomorphism and determination of effective and computable invariants to distinguish the isomorphism classes.

In the case of differ-

entiable manifolds this problem is not solvable, since for any finitely presented group

S

one can construct a four dimensional manifold

with fundamental group

S

in such a way that

hoiaeomorphic if and only if

S

and

T

M(S)

and

M(T)

M(S)

will be

are isomorphic, but one cannot

solve the word problem to determine whether two finitely presented groups are isomorphic (Markov [7fc]).

In special cases one can solve the problem,

but cobordism theory works in another way - by introducing an equivalence relation much weaker than isomorphism. Briefly, two manifolds without boundary are called 'cobordant' if their disjoint union is the boundary of some manifold. to note that every manifold M χ [0,·»).

M

It is worthwhile

with empty boundary is the boundary of

To get a nontrivial theory it is standard to restrict attention

to compact manifolds.

The first description of this equivalence relation was by H. Poincare: Analysis Situs, Journal de l'Ecole Polytechnique, 1 (1895), 1-121 (section 5, Homologies).

His concept of homology is basically the same as the

concept of cobordism used today. The next development of cobordism theory was by L. S. Pontrjagin: Characteristic cycles on different!able manifolds, Math. Sbor. (U.S.), 21 (63) (19^7), 233-284 (Amer. Math. Soc. translations, series 1, no. 32). This paper shows that the characteristic numbers of a closed manifold vanish if the manifold is a boundary (providing the invariants for classification). The cobordism classification of manifolds is reasonably elementary in dimensions 0,1, and dimensions. in dimension

2, since manifolds are themselves classified in these

Using geometric methods the cobordism classification problem 3 was solved by V. A. Rohlin:

the boundaiy of a

A

3-dimensional manifold is

U-dimensional manifold, Doklady Akad. Bauk. S.8.S.R.,

81 (1951), 355. The first application of cobordism was by L. S. Pontrjagin:

Smooth

manifolds and their applications in homotopy theory, Trody Mat. Inst, im Steklov no. !)5, Izdat. Akad. Nauk. S.S.S.R. Moscow, 1955 (Amer. Math. Soc. translations, series 2, vol. 11, 1959)·

Pontrjagin attempted to study

the stable homotopy groups of spheres as the groups of cobordism classes of 'framed' manifolds. This amounts to the equivalence of a homotopy problem and a cobordism problem.

The lack of knowledge of manifolds has

prevented this from being of use in solving the homotopy problem. The major development of cobordism theory is the paper of R. Thorn: Quelques propri&tgs globales des vari6t£s differentiables, Comm. Math. Helv., 28 (1951+), 17-86. This paper showed that the problem of cobordism is

equivalent to a homotopy problem.

For many of the interesting manifold

classification questions the resulting homotopy problem turns out to be solvable.

Thus, Thom brought the Pontrjagin technique to the study of

manifolds, largely reversing the original idea. For a brief sketch of cobordism theory there are three survey articles of considerable interest..

For an insight into the early development

of the theory (up through Thorn's work) see V. A. Rohlin:

Intrinsic homology

theories, Uspekhi Mat. Hauk., I^ (1959), 3-20 (Amer. Math. Soc. translations, series 2, 30 (1963), 255-271).

A short article which covers many of the

examples of cobordism classification problems is J. iiilnor:

A survey of

cobordism theory, Enseignement Mathematique, 8 (1962), 16-23. in the survey of differential topology by C. T. C. Wall:

Contained

Topology of

smooth manifolds, Journal London Math. Soc., ^O (1965), 1-20, is a discussion of representative cobordism theories, with outlines of the methods by which these problems are solved.

Cobordism Categories In order to formalize the notion of cobordism theory, it seens useful to set up a 'general nonsense' situation.

As motivation, one may consider

the properties of differentiable manifolds. Let

denote the category whose objects are compact differentiable

manifolds with boundary (of class maps (again

C )

C )

and whose maps are the differentiable

which take boundary into boundary.

This category has finite

sums given by the disjoint union and has an initial object given by the empty manifold.

For each object of

object of

and for each map the restriction of it to the boundary.

,

one has its boundary, again an

-

k.

Further, the boundary of the boundary is always empty. additive functor 3 : M

—*-

This defines an

. For any manifold M, the boundary of

is a subset whose inclusion is a differentiable map i(M) : 3M —>·M.

This inclusion gives a natural transformation i : 3 —> I of additive functors, I :

Kr'—*•

KJ~ being the identity functor.

Finally, the Whitney

imbedding theorem shows that each differentiable manifold is isomorphic to a submanifold of countable dimensional Euclidean space. Thus a small subcategory

has

Rf (submanifolds of R ) such that each object

of fi" is isomorphic to an object of

/fc^.

Abstracting these properties, one has:

Definition: A cobordism category (£,3,i) is a triple in which: 1) C- is a category having finite sums and an initial object; 2) 9

: C

—s- C is an additive functor such that for each object

X

of (ί , 83(X) is an initial object; 3) i : 3 —*• I is a natural transformation of additive functors from 3 to the identity functor I; and Ό

There is a small subcategory is isomorphic to an object of

of

C. such that each object of

C-Q.

As noted in motivating this definition, (/θ^,3,χ) is a cobordism category. There are many more examples, and in fact the purpose of cobordism theory is to study the interesting examples. The precise choice of this formulation is based, somewhat vaguely, on the definition of 1

adjoint functors'. The purpose of this definition is not to establish a general nonsense

structure; rather the definition will be used to follow the framework of previously developed theory and to try to unify the ideas. To begin, one has in any cobordism category the idea of a 'cobordism relation'.

Definition: the objects and

T

sum of

of r

X

If and

C

is a cobordism category, one says that I

of

C

are cobordant if there exist objects

such that the sum of

and

3V.

X

and

U

is isomorphic to the

This will be written

One has easily:

Proposition: a)

=

is an equivalence relation on the objects of

b)

X = Y

c)

For all

d)

If

C- .

implies X,

3X

where

.

and

and

is an initial object. Z

and

Z'

are sums of the pairs

respectively, then

Proof:

BOTE:

In all of the above for

Mote:

A

and

A + B

denotes an object which is a sum

B. **

If one is unhappy with equivalence relations on a category, one

may reduce to considering

s

as an equivalence relation on the set of

isomorphism classes of objects of about existence of

This is the reason for the assumption

Definition: object.

An object

An object

X

X

of

of

C. is closed if

bounds if

X 5 0

3X

where

is an initial

0

is an initial

object.

Proposition: a) X

closed and

b)

and

X

X'

J EI

Y

closed.

closed implies their sum is closed.

c) X

bounds implies

d) X

and

e) X

bounds and

Y

implies

X

is closed.

bound implies their sum bounds. Y = X

implies

Y

bounds.

Proof: a) follows directly from

b)

e)

is an equivalence relation.

is immediate since

Proposition: (under

=

above.

**

The set of equivalence classes of closed objects of

= ) has an operation induced by the sum in

C.

This operation

is associative, commutative, and has a unit (the class of any object which bounds).

Proof: form a set.

The existence of That the sum in

implies that the equivalence classes gives rise to an operation on this set

follows immediately from the propositions.

Associativity and ccmmutativity

hold for isomorphism classes of objects, hence also here. **

- 7Definition:

( C, a ,i)

The

coborQi"~ semi~roup

of the coboraism category

is the set of eq1.;.iv"lence classes of closed objects of

C:.

the operation induced by the sum in

C

with

This semigroup will be denoted

(l(C, a,i).

~:

(l(C,a,i)

1)

may also be described as the semigroup of

c:

isomorphism classes of closed objects of

modulo the sub-semigroup of

isomorphism classes of objects which bound. 2)

The semigroup l'l( $-;3,i)

)tt*

cobordism group

is quite easily identifiable as Thom's

of unoriented cobordisru classes of closed manifolds.

In order to clarify this slightly, in the usual expression for equivalence one has

X equivalent to

X v 3V ~ Y v a(XxI) X

v

Y

= 3T

Y if there is a

giving

X " Y.

V with

3V

=X v

Y.

Then

The implication X,, au ;; Y v 3V

implies

is an easy geometric argument by looking at components and

piecing together manifolds with boundary by means of tubular neighborhoods of their boundary components. Within the literature of cobordism performed.

the~e

are a few standard

const~uctions

These may be generalized to the categorical situation as will

no'f be sho'fn. Construction I:

Let

(L, a ,i) be a cobordism category, }f

with finite sums and an initial functor. are pairs

For any Object (C,f)

with

X of C an

object, and F ;

y_,

L:

form a category

object of

Map(C,C')

such that the diagram

)E

an additive

elX whose objects

C, and f c Map(F(C),X) a11d

whose maps are given by letting Map«C,f),(C',f')) ~ E

~

a category

be the set of maps

commutes. If

is an initial object of

map, then

and

C/X

and

is a sum for

D+D'

F(D)

If

is a sum for

and

F(D' ) in

give a well defined map sum of

(D,g) and

(D',g')

in

and

D'

in

The maps

(B',g') . then

g

and

(D+D'.g+g')

g'

is the

to define the functor

Define the natural transformation

Then

Remarks:

(D,g) and

C/X. and

C/X.

D

and

Let 5 : C/X

is the unique

is an initial object of

are objects of F(D+D')

£

by

is a cobordism category.

1) This is the algebraic-geometric (Grothendieck style)

notion of the category of objects over a given object. 2)

If one begins with the category

and takes

to be the forgetful functor to the category of topological spaces and continuous maps, then

is the unoriented bordism group

as originally formulated by M. F. Atiyah:

Bordism and cobordism, Proc. Camb.

Phil. Soc., 57 (1961), 200-208.

Construction II: category, and let

Fun

Let

be a small category,

1 a cobordism

be the category whose objects are functors

and whose maps are the natural transformations.

If

0

is an initial object of

Ci, the constant functor

is an initial object of F.G : for

are functors, let

F(A)

and

G(A)

and let

: Then

jy, and

sum for

F

I

by letting

H(A) be a sum

:

and

be the maps exhibiting

jQ

and

H :

Fur

as the sum.

are natural transformations which exhibit

H

as a

G. and let

be the natural transformation given by the map whose evaluation at any object Then

A

of

is a cobordism category.

Remark:

Many standard examples fit this construction.

is the category with one object A functor

whose maps are a finite group is given by selecting a manifold

and a homomorphism induced map

A

Suppose

Since

G

is a differentiable action of

is finite, the G

on

X.

Thus

is the unoriented cobordism group of (unrestricted) G-e.etions as defined by P. E. Conner and E. E. Floyd;

"Differentiable

Periodic Maps", Springer, Berlin, 196k (section Si).

Relative Cobordism In order to study the relationship between two cobordism categories it is convenient to have available a 'relative cobordism' semigroup.

In

the geometric case this is made possible by joining together two manifolds with the same boundary to form a closed manifold.

In the categorical

situation, the idea is to replace a pair of objects having the same boundary

by a pair of closed objects.

For this one needs the idea of the

Grothendieck group construction. Recall that for any category with finite sums for which the isomorphism classes of objects form a set, group of

, one defines

1

the Grothendieck

to be the set of equivalence classes of pairs

of objects of is an object

where A

of

(X,X')

is equivalent to

(X,X')

(Y,Y') if there

such that

is

an abelian group under the operation induced by the sum in Let

and

i be two cobordism categories,

an additive functor, and

a natural equivalence

of additive functors such that the diagram

commutes.

Let

_

with and with

be the category whose objects are triples closed, and

Map

(X,Y,f)

an isomorphism

the set of

such that

commutes.

Then

C an object of

.

has finite sums and a small subcategory such that each object of

is isomorphic to

Let of and

.

be the collection of pairs

of objects

for which v

of _

if there are objects

such that

and

set of equivalence classes

u

Then the

forms an abelian group under the operation

induced by the sum. One has a homomorphism

where

subcategory of closed objects of where

is the

by sending

into

is an initial object of

(2. and

are the unique isomorphisms of initial objects. If one has a homomorphism

such that the composition with

3

is the quotient homomorphism of

then one can define a relative cobordism semigroup as follows: For objects

and

one writes

if there exist objects

U

and

U'

of

with

and for which Using the fact that

a

equivalence relation.

is a homomorphism one easily sees that The relative cobordism semigroup

set of equivalence classes under induced by the sum in One has homomorphisms

=

of elements of

5

is an

i2(F,t,a) is the with the sum

and the triangle

is easily seen to have period 2 In order to clarify the relationship between the homomorphism

a

and

the joining of two manifolds along their common boundary, consider elements

of

_

as a manifold with boundary together with additional

structure on its boundary. isomorphism where

-X'

For

choose an

and let is

X'

the boundaries of

a(x,x') be the class of

with its opposite structure (e.g. orientation), and X

and

X'

are identified via

class does not depend on the choice of

g,

This

for if

g'

is another isomorphism,

one may attach

I

so

that the difference of two representatives is cobordant to Identifying a cobordism of

3X*0 with

and

this is isomorpnic via identified using

f

to the image under

a,

may then find a cobordism of

using

a

F

of

Y

x

I

Y

and

Y',

gives

k" - but

with ends

suppose one has

g. One

say

Thus one may find a cobordism of so that

in X«I

does not depend on the choice of

so that

is cobordant to a closed manifold structure.

k"

with ends identified by

. Thus

With this choice of

3X*1

where

Y

bounds.

D

with additional

and This is the usual geometric

description for cobordism of manifolds with boundary.

Remark:

One may let

initial objects, with determining

a.

F

C.

be the subcategory of

the inclusion.

Then

β

QJ

consisting of

is epic, uniquely

The relative cobordism semigroup in this case is then

identifiable with the cobordism semigroup of

(2,'.

- Ik Chapter XI

Manifolds With Structure - The Pontrjagin-Thom Theorem

The standard cobordism theories are based on manifolds with additional structure on the tangent or normal bundle. taken from the paper:

R. K. Lashof:

The exposition given here is

Poincare duality and cobordism,

Trans. Amer. Math. Soc., 109 (1963), 257-277. Denote by

the Grassmann manifold of unoriented

r-planes in

r+n R an

and let

be the

r-plane in

r-plane bundle over

and a point in that

with universal

r-plane. Then

r-plane bundle

Definition:

Let

be a fibration.

n-dimensional vector bundle over the space then a liftings to

consisting of pairs:

B^

X

structure on

of the map

with

If

is an

classified by the map £

is a homotopy class of

C,; i.e. an equivalence class of maps where

£

are homotopic by a homotopy

and

£

are equivalent if they

such that

for all

Note:

A

structure depends on the specific map into

There is no way to make

BO^.

structures correspond for equivalent

bundles, since the correspondence is dependent upon the choice of the equivalence. Let

be a compact differentiable

boundary) and let i

(C ) manifold (with or without

be an imbedding.

The normal bundle of

is the quotient of the pullback of the tangent bundle of

by the subbundle

T(M).

(living

the Riemannian

metric obtained from the usual inner product in Euclidean space, the total.

space

N

of the normal bundle may be identified with the orthogonal

complement of

T(M)

in

or the fiber of

identified with the subspace of with a

x

The normal map of

M

may be

i

(m,x)

is given by sending

covered by the bundle map yr

Composing with the inclusion into

provides a map

which classifies the normal bundle of the imbedding

Lemma:

at

consisting of vectors

orthogonal to

into

H

If

r

i.

is sufficiently large (depending only on

is a one-to-one correspondence between the

n),

there

structures for the

normal bundles of any two imbeddings

Proof:

For

in

r

sufficiently large, any two imbeddings

of

are regularly homotopic and any two such regular homotopies

are homotopic through regular homotopies leaving endpoints fixed. regular homotopy is a homotopy

such that each

(A H( ,t)

is an immersion and such that the differentials define a homotopy).

See M. Hirsch:

Math. Soc., 93 (1959), 21*2-276. gives a homotopy from

Immersions of manifolds, Trans. Amer.

Then a regular homotopy from

and two homotopies defined in

this way are themselves homotopic relative to endpoints. a well-defined equivalence for the two bundles. lifting property for the map quite easily.

to

f

Thus one has

Applying the homotopy

then establishes the correspondence

**

Definition:

Suppose one is given a sequence and maps

(B,f) of fibrations

such that the diagram

commutes,

j

being the usual inclusion. 11

normal bundle of

M

in

defines a unique

via the inclusion

A

class of sequences of of

M,

structure on the

(B,f)

structure 11

structure on

structures

M

is an equivalence

on the normal bundle

two such sequences being equivalent if they agree for sufficiently

large

r.

and a

(B,f) structure on

If

A

(B,f) manifold is a pair consisting of a manifold Mn.

¥ is a manifold and

normal bundle, one may imbed

is a submanifold of

K

in

r

W

large, and extend by means

of the trivialization to an imbedding of a neighborhood of into

with trivialized

M

in

so that the neighborhood meets

along

M.

This may then be extended to an imbedding of

normal planes to

M

in

normal planes to

W

in

map, then

v|

structure on

iv

Remarks:

induces a well-defined 1)

The induced

W

in M

The of the

is a lifting of the normal

is a lifting for the normal map of W

orthogonally

are then the restriction to If

W

(B,f)

M.

Thus a

structure on

(B,f)

M.

(B,f) structure depends only on the

equivalence class of the trivialization, not on the specific choice of trivialization. 2)

If

f

' is an isomorphism of manifolds, the normal bundle

is trivialized, being zero dimensional.

If

i :-

is the inclusion

of the "boundary, there are two choices of trivialization, via the choice of inner or outer normal.

If

j :

is the inclusion of a direct

summand, then the normal bundle is again zero dimensional, so trivialized.

Definition:

The cobordism category of

(B,f) manifolds is the category

whose objects are compact differentiable manifolds with

(B,f) structure

and whose maps are the boundary preserving differentiable imbeddings with trivialized .normal bundle such that the map coincides with the functor

3

(3,f)

applied to a

(B,f) structure induced by the

structure on the domain manifold.

(B,f) manifold

W

is the manifold

3W with

(B,f) structure induced by the inner normal trivialization, and maps is restriction.

The natural transformation

i

The

8

on

is the inclusion

of the boundary with inner normal trivialization. The cobordism semigroup of this category will be denoted The sub-semigroup of equivalence classes of will be denoted

Clearly

Proposition:

Proof: large

r

Let

M

R.

Extend to an

by the usual inclusion of

The normal map for

M x I

inner normal along

M * I

M

defines a

which induces the given structure on M * o. M x 1

Thus the structure on

I

is the composition of the projection on

Thus the lifting for

(B,f) The

gives rise to an induced structure on M * 1,

and with these structures one has

««

for some

a lifting of the normal map.

and the normal map of M.

n(B,f).

is the direct sum of the

IT1 be a closed manifold imbedded in

with

structure on

n-dimensional closed manifolds

The cobordism semigroup ft(B,f) is an abelian group.

imbedding of in

£!(B,f).

in the category.

is an inverse for the structure on

M

in

- 18 Considering

BO

as the space of

r

product on the subspace of

of R'"

RS oo

finite dimensional subspace

r-planes contained in some

R

and taking the usual inner

consisting of vectors with only finitely

many non-zero components, one obtains. a Riemannian metric on the universal bundle the map

r y •

If

~:

is an

X ~ BO

r

r-plane bundle over a space

x

classified by

one has induced a Riemannian metric on

,

~.

(Note:

For the normal bundle of a manifold this coincides with the metric obtained from the splitting)·.

obtained from the total space of

~

at least one to a point, denoted

co.

bundle

g: X

~

= g*n

n: Y ---+ BO , r ---+

BOr

by a map

jr: BOr

---+

BOr+l

T~,

If

~

~

is the bundle induced from a Y,

then the usual bundle map

Tn.

induces a vector bundle

which may be identified as the Whitney sum of yr

bundle.

Then Tj*{yr+l)

is the space

by collapsing all vectors of length

T~ ~

n induces a map Tg :

The map

~,

The Thom space of

j;{yr+l)

over

and a trivial line

may be identified as the suspension of Tyr.

r

One then has a commutative diagram Tg

~TB

ETB

[~r>l

,.,,1' Tj

ETBO -4TBO r

groups, where spaces

r+l

E denotes suspension, and TBO , r

TB

r

denote the Thom

Tyr

The main theorem is the generalized Pontrjagin-Thom theorem: ~:

The cobordism group of

is isomorphic to

n-dimensional

lim ~ + (TB .oo).

r-

n r

r

(B,f)

manifolds

- 19 ~:

A)

0: n (B,f) ~ lim ~ + (TB ,~). n z-"O' n r r

Definition of the homomorphism

n £ nn(B,f) be represented by a (B,f) manifold M , Let r i : M~ Rn+ be an imbedding with a lifting v; M ~ B which defines Let

0

r

the given

structure on M,

(B,f)

the nOI'!!lal bundle of M,

Let

N denote the total space of

thought of as a subspace of Rn+r x Rn+r ,

e: Rn+r )( Rn+r ~ Rn+r : (a,b)

the eValuation map

°

N is mapped differentiably and on M = M x the imbedding

i.

C

For some sufficiently small

-+

a+b,

To define a map Rn+r

el , N Sn+r ~

Tf; (yr) ,

c: Sn+r ~ N /aN

u., and let

E

outside or on the boundary of

NE

£ > 0,

the subspace of

~

0

is the composition of

e

= T(o x

(~OlT)}

Replacing of

£

0

E-loc

-1

The map

r

r Yn

r

in Y

f*(yr} r

and

and

The composition

is a map of pairs

£-loc

will be homotopic,

not change the homotopy class of

lim lIn+r(TBr'~)

Sn+r

n x (V011) ; N -+oyr x B ,

is a bundle map into

Replacing

equivalent lifting simply gives a homotopy of T(0

r-

as

£ by a smaller value does not change the homotopy class

e since the maps

gives rise to

is

Multiplication by 1/£

n with the inclusion of

T(;;- x (VOlT)} : TN ~ TBr'

induces the map

N£,

by collapsing all points of

E

to a point,

N on M,

is the projection of

E,

begin by considering Sn+r

denoted by where

the subspace

N this map restricts to

N consisting of vectors of length less than or equal to imbedded by this map

Under

Tgro~e

e.

x

("01l))

Clearly , the map

e.

and so does

n r M -4 R +

and thus one has defined an element of

represented by the map

v by an C Rn+r+1

To show that this element depends only on the cobordism class of the manifold

M

and not on the choice of the imbedding, let

manifold and

W

be a

(B,f)

an imbedding with a lift giving the same

is assumed large).

Let

imbeddings

j|

i

and

(B ,f ) structure on M r r

chosen so that

is

and let j x 1

orthogonally along

r

be a regular homotopy of the

and is agreeing with

(here

on

3W

be a map

and imbedding a tubular neighborhood of

j(3W) x l.

3W

The map

is sin imbedding on a closed neighborhood of the boundary and may be homotaped to an imbedding

by a homotopy fixed

on that neighborhood of the boundary.

is a regular homotopy and

corresponding to its normal map one may find a covering map agreeing with M x 1,

v

on

W

Since the normal map is constant near

one may modify the lift to agree with

(B,f) structure on of

M x 0.

3W

agreeing with

v

~

is induced from that of on

3W.

is a neighborhood of (image

and the map to give a map i

imbedding of

one may find a lift

one has a collapse where

Taking

W

Since the

Following the previous construction with

the imbedding

defined by

on M x l.

W

F),

a map

which compose

. This provides a homotopy for the maps and

j.

empty shows that the class of M.

9

Further, if M = M'

0

is independent of the

with

so that the class of the map only on the cobordism class of

M.

then 6

depends

B)

6

If

M^

is a homomorphism. and

M2

represent two classes in

choose imbeddings

for which the last coordinate is positive for negative for

i^.

i^

and

If tubular neighborhoods are chosen small enough

to lie in the same half spaces then

is represented by n+r

where and

8a

represents

homotopy classes, C)

S

d

collapses the equator of

S

Since this map represents the sum of the 0

is a homomorphism.

is epic.

Let

r

large, represent a class in and since

compact, s.

The map 1) h^

be deformed to a map

for some h^

so that

is differentiable on the preimage of some open set of

containing

and is transverse regular on

is a differentiable manifold. 2)

If

3)

The map

is a bundle map on a normal tube of

in agrees with

h^

on the preimage

V

of a closed

neighborhood of =>. Since

classifies the normal bundle of

(by a further homotopy if necessary) that and that

h

r

M,

one may assume

is the normal map

is given by the usual translation

of vectors zo the origin on a normal tubular neighborhood of

M,

Now

00

is a fibration except at the point

since

does not contain

theorem applies so that the deformation n+r S - interior V

o

to be constant on V, homotopy of

6

the covering homotopy f

may be covered by a homotopy of

which is pointwise fixed on the boundary of

o 0

BO

r

6^.

under

n n+r S - interior V

on

Taking the homotopy Tf 06 r

one may cover the homotopy of

to a new map

the inverse image of

V.

, and

The inverse image of h , which is r

M.

to

h

by a

r

B^ under

8^ is

F u r t h e r i s

a lift of the normal map Thus one has a

structure on

(B,f) structure on the lift N^

of

and since

agree with

large

0

r

TB^-B^

8^

deforms to

M

be a

Rn+r

with

8.^ on a neighborhood

one may homotope

(B,f) manifold such that

the standard map

to

8^

Then

(relative to

0([M]) = 0. defined by

to

0([M])

Thus for some M

is homotopic

One may choose the homotopy for

for some

H^

in

is monic.

so that

map

M

8.

to the trivial map

Tf^oL

and hence a

agrees with

by pushing the complement of

is the class of

Let

in

Using the given imbedding of

the resulting map

M

D)

M.

M

t e

By compactness

As above, one may homotope

N £ (M) X [0,p])

in a neighborhood of

Gr

g

to a

which is differentiable near and transverse on is a submanifold of orthogonally along

normal map and

H^

M.

Rn+r x I

with

3W = M

One may also assume

meeting is

the

agrees with the usual translation of vectors map on

a neighborhood of

W.

Applying covering homotopy, one may deform

to a map with

8|w

with covering the normal map

structure on

W

for small H^

of

W.

which induces the original

Thus

and

L

This defines a

(B,f)

(B,f) structure on

M.

[M] is the zero class of

Tangential Structures

It is frequently desirable to define

(B,f) structures on manifolds

by means of structures on the stable tangent bundle. and

Let

with

The map

obtained by assigning to each orthogonal

I

B = lim

plane induces a map

n

plane its

with

2

I

the identity.

is a fibration and one has the induced fibration .

Since

is again

B.

The induced

bundle maps give a diagram

with

1*1' If m"

and

I'I*

is imbedded in

and vectors are related by one has maps

both being identity maps. P

, H

large,

the maps

obtained by translation of normal and tangent x„ = H

v„. H,n N and \

Following these by the inclusions with

T = Iv.

A

(B,f)

structure on

M

as previously defined is precisely a

homotopy class (through liftings) of liftings to The maps

I'

and

of liftings of to

B*

v

I*

B

of the map

define an obvious equivalence between these classes

and the homotopy classes (through liftings) of liftings

of

T

Such a class of liftings of

structure on the stable tangent bundle of

is a

M.

Structures For Sequences Of Maps

If instead of fibrations one is given only spaces such that 2r^r

one may replace the maps

The resulting maps

C^

and maps

is homotopic to

by homotopy equivalent fibrations.

g^ may be deformed inductively to give commutative

diagrams by means of covering homotopy.

structure is then a

structure for some such fibration sequence (chosen).

Since the cobordism

group is given by homotopy of the Thorn complex, which depends only on the homotopy type of the fibrations, the resulting cobordism group does not depend on the choice of equivalent fibration sequence.

Rinj> Structure

If one has an

r

space, they span an defines a map

plane in r+r'

r+s

space and an

r'

plane in

plane in

r'+s' This

and induces a map corresponding to the Whitney sum of vector bundles.

GQ

q

is a point and provides a base point in each so that

g

(the usual

is mapped via standard inclusions.

The twisted map

is homotopic to

the usual map

by a rotation of

to interchange factors. structure on

This gives the usual homotopy commutative

BO.

Having similar multiplications f

H-space

so that the maps

preserve products up to homotopy, one may define a ring structure in

(B,f) cobordism, for the multiplication defines a

(B,f) structure on

the product manifold The map

induces a map

giving a product in the stable homotopy, making it into a ring.

This ring

structure is the same as that of the cobordism groups.

Relative Groups

If one has commutative diagrams

in which

hr

maps, a

and

d^

are fibrations and

(B,f) structure induces a

'reduction'

h.

This gives a functor

(B,f) manifolds to that of manifold via

h

W

(C,d) structure on

and

kr

are fibre preserving

(C,d) structure by means of the h

from the cobordism category of

(C,d) manifolds.

with boundary having a

the same

gf

An

(n+l) dimensional

(B,f) structure on its boundary inducing 3W

as is induced by a

(C,d)

structure on

W

is a relative manifold.

Using the standard 'piecing

together' homomorphism one has defined a relative cobordism semigroup. If one imbeds map to

B^,

3W

in

(r large) with lifting of the normal

extends to an imbedding of

W

(orthogonally along the boundary

using a tubular neighborhood) in lifting of the normal map of lifting on

W

selecting a to

Cr

which agrees with the

h-induced

3W, then one may apply the tubular neighborhood map to this

imbedding-lifting situation to construct a map

If U

W

is cobordant to

V

(relatively) one may find a

giving a cobordism of

3W

identifications the closed One may imbed 3W x 0

U

3W'

so that with proper boundary

(C,d) manifold

in

and

and

(B,f) manifold

bounds.

;o give the proper identifications at with a lifting of the normal map to

in the manifold with

(C,d) structure in

B^

and fill

along its boundary Ignoring corners

(which don't affect the homotopy situation, but rather involve the identification of

the normal maps and their liftings define

a homotopy of the maps for

W

and

V.

Ignoring lots of details one sees easily that the relative cobordism group

(-W being constructed from

to the stable homotopy group

(n+l) W

x

I)

dimensional is isomorphic

Further, the cobordism

triangle is identifiable as the exact homotopy sequence of the 'pair'

(TC,TB).

Chapter III Characteristic Classes and Numbers As mentioned in the introduction, the determination of invariants which distinguish manifolds in one of the principal aims of differential topology. In the framework of cobordism theory, the use of characteristic classes provides invariants called characteristic numbers which are cobordism invariants.

In order to set up the machinery for these invariants, the

ideas of generalized cohomology theory play a central role, and for this the basic reference is G. W. Whitehead:

Generalized homology theories, Trans.

Amer. Math. Soc., 102 (1962), 227-283.

Definition:

A spectrum

E

is a sequence

(Enln E

of

spaces with

base point together with a sequence of maps the suspension.

is another spectrum, a map

^F is a sequence of maps

Examples:

£ being

l)

h

from

to

with

The sphere spectrum

where

is the identity map. 2)

If

(B,f) is a sequence of fibrations

maps

with

as in Chapter II, then

is

a spectrum, known as the Thorn spectrum of the family the maps

(B,f).

In particular,

define a map of spectra

Tf : If one chooses base points bundle

induces a trivial

a map Tb :

such that r-plane bundle

Since Note:

defining

, this gives a map of spectra

The identification of

choice of framing of the fiber over

then the

with

requires a

Definition: the spectrum

The homology and cohomology groups with coefficients in

E^ are defined by

and

where

X/A

is the space obtained from

(the base point), and

[ , ]

»

X

by collapsing

A

to a point

is the smash product

V),

denotes homotopy classes of maps.

H*( ;E) and

are functors satisfying all the axioms of

Eilenberg-Steenrod as cohomology and homology theories with the exception of the dimension axiom. One defines and

X/0

base point

to be

is the disjoint union of p,

Definition:

one writes

where X

and a point.

is the empty set If

is a space with

for

A ring spectrum is a spectrum

and a pairing

Y

with a map

i.e. a collection of maps

such that the maps of the diagram

represent classes of the group

related by

such that the diagram a -1 S P -A a

commutes, where

a

l-a

2

a

A -A •< p q

— A -S*1 P

is the multiple composition of the suspensions of maps

a^

is a map whose class in the group

and

times the class of

[Note:

If

this is not a group, but

so that one does not need the group structure to find a map.]

Example:

Let

R

be a ring with unit,

K(R,n)

space (the only non-zero homotopy group being

R

an Eilenberg-MacLane

in dimension

n)

and

be a map corresponding to the identification

let

The spectrum and

is a ring spectrum

is the usual cohomology with coefficients in

R.

With a ring spectrum one has the usual sorts of products, such as cup products in cohomology making

Definition:

Let

into a commutative ring with unit.

be a sequence of fibrations

A Thorn class is a map of spectra How let with

3M

be a

imbedded in

where A

(B,f) manifold,

N

an imbedding

with the usual orthogonal framing along a

tubular neighborhood of Let

is a ring spectrum.

be the. normal bundle of

(denoting the neighborhood by M

and

be a lifting defining the

N'

the normal bundle of

(B,f) structure on M

the induced map on the Thom complex.

3M.

with

Let

Consider the map

where

A

is the diagonal map,

it the bundle projection,

p

is the collapse

onto the Thorn complex, and the last map is the obvious collapse.

Under

this map the vectors of norm at least one are sent to the base point, as are all vectors over

3M,

i.e.

Letting

B'.

Thus one induces a map

be the standard (scaled) collapse, the

projection into

TN/TH'

sends

into the base point also and hence

defines a collapse Letting

be a Thorn class, one has a composite map

which represents an element of

Letting

defines a class only on the

go to infinity

This element is easily seen to depend

(B,f)

Definition:

r

If

structure of if

is a

M. (B,f) manifold and

Thom class, then the fundamental class of If

3M

is a

(M,3M) is the class

is empty, this class will be denoted

If one collapses the complement of the tubular neighborhood of the boundary by

one has the map which defines the boundary homomorphism in homology

If one composes the map defining

[M,3M] with the map

d»l,

it is immediate

that the resulting map is the suspension of the one which defines

Proposit ion i Under "the toundaxy liomomorphism in fundamental class of

Definition:

A universal characteristic class with

then the

X

structure given by a lifting

x-characteristic class of the

bundle is

where

map into the limit 3pace. M

If

x-normal

defined by

If

Mn

M.

is a closed

x-characteristic number of M

by evaluating

(B,f) manifold, the

is a lift of the normal map (of some imbedding)

(B,f) structure on

Definition: — — — — —

—

is a

is the usual

is the class

where

then the

coefficients for is an

with a

the class

characteristic class of

3M.

where

r-plane bundle over a space

Thus:

A—homology j the

(M,3M) is sent into the fundamental class of

(B,f) bundles is a class

defining the

[3M].

(B,f) manifold and *

represented by a map

E

HP(B;A), ^

is the class in

x(M) on the fundamental class of

is represented by a map

x

M.

obtained

Thus :

and then is represented by the map

s

The usefulness of characteristic numbers in cobordism theory arises from the result of L. S. Pontrjagin:

Characteristic cycles on differentiable

manifolds, Math. Sbor. (N.S.), 21 (63) (19VT), 233-281|; Amer. Math. Soc. translation 32.

Theorem: the

x e HP(B;A) and

If

x-characteristic number of

class of

Since

to show that for

(B,f) manifold, then

depends only on the

(B,f) cobordism

x-characteristic numbers are clearly additive, it suffices

M = 3W

one has

inclusion one has

6

is a closed

M.

Proof:

where

M

M°

x[M] = 0.

Letting

be the

1, so

is the cohomology coboundary homomorphism induced by the collapse Since the cohomology sequence

is exact,

hence

Remark:

Being given fibration sequences

of manifold

one may think

as a relative characteristic class. M

with

(B,f°h) structure on

characteristic number

3M,

Being given a

(B,f)

one has defined a relative

since the normal map gives

To see that such numbers are relative cobordism invariants, one may suppose by additivity that there is a 3W = M V) (_u) joined along

3M = 3U, with

(B,f) manifold U

a

W

with

(B,foh) manifold.

One

- 33 -

then has (w,a'l) -~ E(aW/rf;) ~ l:(aw/u) ~ E(M/aM) giving

p*[M,aM)

decomposition of

= j*a[w,aw) ~~d

aw

by the orientation assumption in the

= p*q*y(W,U)

y(M,aM)

(M,3M) -ll>- (aw,u) so y[M,aM) = " q*y,j*a[w,aw) sequence of the triple

(w,aw,u)

H*(W,U) is zero, so y[M,aM)

> =
.

the composition

-3! H*(aw,U) §.4 H*(W,aW)

Note:

Taking

B empty, this reduces precisely to

the closed case. In addition to the manifold theoretic treatment of characteristic numbers by using the Thom class to construct fundamental classes, one may also give homology and cohomology theoretic descriptions of characteristic numbers which are frequently useful.

In particular, these will be needed

later. As in the construction of the map y

over a space

giving a map

rf;,

one has for any

r-plane bundle

X the composition

rf;: Ty

~

(X/rf;).Ty.

Applying this to the bundle Thom class and inclusion of

Br

f~(yr)

in

over

B gives

Br'

and composing with the

-

inducing on homotopy the homomorphism

If

is a closed

(B,f) manifold, so that

maps into

under

the normal map, one has the commutative diagram

Thus the homotopy homomorphism is the homomorphism sending the cobordism class of a manifold class of

11

M .

into the image under the normal map of the fundamental

Thus the pairing of homology and cohomology of

B

into the

cohomology of a point gives

which coincides with the evaluation of characteristic numbers. In addition, the composition

gives rise to the Thom homomorphism in homology

determined by the Thorn class the map

U.

is the composition of

(Note:

This is given by

since

ir„l with the map used in defining cap

products.) The homomorphism

may then be interpreted

as the composition of the Hurewicz homomorphism in ^-homology given by Thorn homomorphism determined by

U

and the

(at least after letting

r

go to

infinity). It is more common to consider the Thorn homomorphism in cohomology theory.

If one begins with the map

and chooses a map

representing a class

then the composition

represents a class in

This defines the cohomology Thorn

homomorphi sm

The construction of

shows that is the projection and

where is interpreted

as a class x e HP(B;A) one has the sequence of elements

In particular, for

lifting to elements If

is a closed

(B,f) manifold with cobordism class represented by

the map Hp-n(pt;A).

, then (Bote:

It is immediate that this agrees with the previous

homology interpretation. naturality of the map

All interpretations are really based on the ).

Definition: — — — —

The Thorn class

—

A-orientation — — — — .

f w

if for each point

of

determined by

and

Remarks:

is said to be an

there is a framing of

so that the classes

are the same in

1) This is the assertion that the bundles

(uniformly) Ar-oriented in the sense of A. Dold:

are

Relations between ordinary

and extraordinary cohomology, Notes, Aarhus Colloquium on Algebraic Topology, Aarhus, 1962. 2)

If the class of

as an element of

does not

have order 2, this gives a preferred orientation to the fiber

r

6 .

If this

class has order 2, then all A^ cohomology is of order 2 and orientation doesn't enter the situation.

Proposition:

Let

be an

bundle over a finite complex on

5.

X

A-orientation,

5

an r-plane

with

structure

Then the Thom homomorphism

is an isomorphism.

Proof: of

The composition

Over any cell

Dn

of

: X

defines an

the bundle

5

is trivial, and

being path connected and simply connected the class over

Dn.

Over

Dn

A-orientation

Ur°T5

Dn

orients the

bundle

£

one then has the Thom space equivalent to

DnASr,

with the Thom homomorphism being just the suspension isomorphism.

Thus the Thom homomorphism defines a homomorphism of the sequence of

X

into the reduced cohomology

which is an isomorphism on

k

hence also on

A, spectral

spectral sequence of **

T£

- 31 Corollary:

If

U: :!',§-----!';

!'; orientation and

is an

is a

(B,f)

~U

manifold, then the Thom homomorphisms

HP(3M;!,;) - -

tf

Hp+r(TN';~)'

and

HP (M,3M;A) __ lip+r(TN,TN' ;~)

are isomorphisms.

tf

Let 3M

C

be a manifold, and let

n r l

R + -

M be imbedded in

in the usual way, and let

normal bundle of

a)

(M/3M)

b)

(M/~)

and and

are dual in

Sn+r+l.

[B

C in

and

be the normal bundle of

M,

v'

3M.

(Atiyah [14])

Theorem:

v

~+r with

The pairs

or

Tv,

(Tv/Tv')

Sk

are dual i f

B

and

C are diSjoint and each is a

strong deformation retract of the complement of the other.) Proof: as

Let

N

be a tubular neighborhood of

M in

If+r x R with

a base point Consider

~+r.

Consider collapsed to

00.

(Tv/Tv')

as

(N x 0

U

aN x [0,00)

U 00)

and then one may

collapse the complement which is

retraction.

·pt

the

- 38 -

Considering

(M/dM)

as

(M

1)

x

U '" one

may

collapse the complement by

a strong deformation retract onto the subset

iT

x

on

° U aN

x [0,2J

aN

with vertex some point with last

2

x

IJ c, where c is the cone

coordinate larger than

N '"

- N

N

just

n

Tv.

Rn+r - l •

Z---

where

2,

_-- • t

This subset is clearly

Similarly, removal of this set

gives a set collapsing onto Note:

~ r~ T\I

Tv

M

M/aM.

All deformations are obtained by radial deformations toward points

in question, and scalar multiplication expansions in the fibers of the normal bundles of M. If

**

B,C C Sk

are disjoint sets as above, let

stereographic projection to map f : B x C ~ sk-l of and

by

B, C respectively! f

B, C into

Rk

f(b,c) = (b-c)/llb-cll • feB

x

c) \) feb

factors homotopic ally through a map

as disjoint subsets. Letting

f

One then has defined a duality as follows: representative map

a: SP+i ~ B ,Ai

defines a class in

Hk-l-P(C;~),

~:

H*(Rk_b,Rk_B;~) Proof: Sk_B

b, c be base points

B , C ~ Sk-l. For

a c

i p (B;A) -

choose a

and then

denoted

Let b c B C Rk with

Let

is a proper subset of sk-l

C)

x

Po c Sk_B V C and apply

Da.

B an imbedded disc.

Then

= 0.

H*(Rk_b,Rk_B)

are contractible,

=H*(Sk_b,Sk_B)

H*(Sk_b )

the pair gives the result.

**

by excision.

= H*(sk_B) = 0

Since

Sk_b

and

and the exact sequence of

Corollary:

If

is an imbedded disc then is an isomorphism.

Proof:

One has the diagram

in which the end maps are both isomorphisms since the groups are zero, and to complete the proof one need only check

Theorem:

D", which is clearly an isomorphism.

(Alexander Duality) For any polyhedral pair

**

in

s an isomorphism.

Proof:

By naturality it suffices to consider the case

and then

one may apply a Mayer-Vietoris argument using induction on the number of cells of

B,

the corollary and the five lemma.

Theorem:

**

(Spanier-Whithead duality) For any pair

as above

is an isomorphism.

Proof:

Mote:

D

is given by the composition of isomorphisms

See Spanier [110], pages 295 and h62.

Theorem:

(Hsiang and Wall [58]) A class

orientation if and only if the class

is an

in the sense of Whitehead; i.e. for each point Hn(M,3M;S)

obtained by collapsing

is a Dold

onto

orientation

the class in where

is a disc

- 39(a) neighborhood of

q

and the class

defined by the unit in Proof: q

The map

n TD

n D ~ ~ - 3M

given by taking the disc neighborhood of

M/3M ~ Dn /3D n which is patently dual to the Tv,

but

n TD

is homotopy equivalent to the Thom space

**

of a fiber. Note:

in

have Kronecker product the class

HO(pt;A).

defines the collapse

inclusion of

D-ICCL)

Similarly, the collapse

Sn+r ~ Tv /Tv

I

is dual to the map of

M

to a point. Corollary: for any an

~

(B,f)

If the Thom class manifold ~f,

U:

~ ~ ~

is an

~

orientation, then

the fundamental class [M,aM]

E

H (M,3M;!l) n

is

orientation in the sense of Whitehead.

~:

(Poincare-Lefschetz duality)

:it-orientation, then for any

(B,f)

manifold

If

U: TB

~

"-'

r.f

-A

is an

one has isomorphisms

given by the cap-product with the fundamental class

[M,aMj.

The cap-product relation is given by

where

t; is the map given from the diagonal.

a class in

Hq(M;~),

If E i(M/CP)

4

Aq+i

represents

the cap product is represented by

A similar formula defines the other homomorphism. ~:

These isomorphisms are just the composite of Spanier-Whitehead

duali ty and the Them isomorphisms.

**

- U o -

Chapter IV The Interesting Examples - A Survey of the Literature Since cobordism theory is a classificational tool, the interest really lies in the investigation of specific classification problems.

Numerous

examples have been considered and hence a vast literature exists, with few really central theoretical tools, largely due to the idiosyncrasies inherent in the examples.

The purpose of this chapter is to list many of these

examples and indicate briefly what is known and where to find it in the literature.

Example 1:

Framed cobordism:

Historically:

fl".

First application of cobordism theory, intended to

study the homotopy of spheres. Ob.jects:

Framed manifolds, i.e. manifolds with an equivalence class

of trivializations of the normal bundle. Determination: (B,f) cobordism with each space for the identity subgroup

1

B r

of 0 ), so

r

r

contractible (classifying

ST^r = Iim n r+®

.(3Γ) is the

π η+Ι

stable homotopy of spheres (Pontrjagin I/O/]). Results:

A vast literature exists but is largely unrelated to cobordism.

Use of surgery (Milnor [63], Wallace [/37]) to construct framed cobordisms shows that representatives frequently may be taken to be homotopy spheres (Kervaire-Milnor [fc/]). Recent work of Conner and Floyd [fI] has placed the

e-invariant of Adams [4] in a cobordism framework.

Example 2:

Unoriented cobordism:

·

Historically: The turning point for cobordism theory. Objects:

All compact manifolds, i.e. the category (/^^ji)·

Determination: fr

Equivalent to

the identity map. Calculation:

of dimension

i

(B,f)

cobordism with

and

(Thorn [127]). is the polynomial ring over

for each integer

i

not of the form

Z^

on classes

2S-1.

Even dimensional

generators may be taken to be the classes of real projective spaces. [127]).

x^

(Thorn

Odd dimensional generators were constructed by Dold [U3].

Characteristic numbers:

Z^

cohomology characteristic numbers give

complete invariants (Thorn [127]).

All relations among these numbers

(expressed tangentially) are given by Wu's formulae

relating

to the action of the Steenrod algebra (Dold t^]).

Example 3 Objects:

Complex cobordism: Stably almost complex manifolds - manifolds with an equivalence

class of complex vector bundle structures on the normal bundle. Determination:

(B,f) cobordism with

space for the unitary group Calculation: dimension

2i

U^

the classifying

(limit of complex Grassmann manifolds).

is the integral polynomial ring on classes

for each integer

i, with

x^

of

represented by a projective

complex algebraic variety (Milnor [82 ], Novikov

In fact, every

class is represented by such a variety (Milnor; see Hirzebruch [54] or Thom [129]). Characteristic numbers:

Cobordism is determined by integral cohomology

characteristic numbers (Milnor [82]).

All relations among these numbers

are given by the Atiyah-Hirzebruch [17] form of Riemann-Roch theorem relating complex

K-theory to rational cohomology (Stong [117], Hattori [52]).

Relation to [«7]).

maps onto the squares of classes of "J*"^ (Milnor

- 42 Relation to

n;r:

Every framed manifold bounds a complex manifold.

Todd class homomorphism

Q*«BU,f),(Bl,f»

group induces the Adams

e

Example

--7

The

Q of the relative cobordism

homomorphism (Conner and Floyd [41 ]).

4: Oriented cobordism: n~O.

Objects:

Oriented manifolds.

Determination:

(B,f)

cobordism "ith

space for the special orthogonal group

SO

Br '" BSO

r

the classifying

r

(limit of G~assmannians of

oriented planes) (Thom [12,J).

Q~O 6 Q is the rational polynomial ring on classes x4i

Calculation:

of the complex projective spaces

n~O

Z on

4i

dimensional generators (Milnor [SI

has only torsion of order two and the quotient

descri~ed as follows:

x 2k '

for

k

SO SOQ* /2l'l* = kernel

d

d l

x 2k_l '

Let -~*

be a

not a power of

l

so

n*

has no odd

Q~O/Torsion is a polynomial ring

torsion (Milnor [8/], Averbuh [21]) and over

(Thom [127]).

CP(2i)

Z2 2,

J,

Novikov [Q2, 93]).

n~O/2n~0

may be

~olynomial ring on classes

and X2j2

Let

d

l

:

~It;;

--71,y;,

and the image of the torsion will be the image of

(Wall [130]). Characteristic numbers:

Cobordism is determined by

cohomology, all relations among the

Z2

Z and

Z2

numbers being given by the relations

of Wu together with the vanishing of the first Stiefel-Whitney class (Wall

[130]).

All relations among the

Z numbers are given by the Riemann-Roch

theorem (Stong [117]). U

Relation to Relation to kernel

d

l

Q* maps onto

I

1*:

(Milnor [8Z]).

Q~0/2Q~0 is mapped isomorphical1y to the subring

described above, the

(Wall [130]).

Q~O/TorSion

Xi

being (well-chosen) generators of ~

Example 5:

w^

Historically:

spherical cobordism: This cobordism theory arises in Wall's determination

of oriented cobordism, and was completely determined by Wall [130]. Objects:

Manifolds for which the first Stiefel-Whitney class

w^

is the reduction of an integral cohomology class; is induced by a map into the sphere

S1.

Determination: —

fibration over

(B,f) •i

B0r x s

cobordism with

Br

the total space of the

induced from the path fibration over

K(Z2,l)

by the map realizing the cohomology class being the generator. Calculation:

Given by the polynomial ring

Characterist ic numbers:

Z

described above.

cohomology determines cobordism, all

relations being given by those of Wu together with the vanishing of Relation of

and

describing the image of Exanple 6: Objects:

Maps monomorphically into

, with

as above.

Bordism:

Let

cobordism category of

be the forgetful functor from the (B,f) manifolds to the category of topological

spaces which takes the underlying topological space. cobordism category of

(B,f) manifolds 'over' a space

a subspace one has a functor

One then has the X.

If

is

induced by the

inclusion. Determination:

(B,f)/X

cobordism is just the cobordism theory based •p

on the fibration

ir-being the projection.

relative bordism group of the pair a

is the piecing together previously described, and is given by

The where

-

Historically:

kh -

These groups were originally defined by Atiyah [13],

who called them the

(B,f) bordism groups of the pair

(X,A).

He

reserved the name cobordism for the dual cohomology theory with coefficients in the spectrum

TB. The unoriented bordism of a pair

(X,A)

is essentially

trivial, being isomorphic as 7 T *

module to

is determined by

(See Conner and Floyd [3fc]).

Zg

cohomology.

, Cobordism

in this theory were determined by Landweber [fc3, £4],

Operations

Used by Brown and

Peterson [2