Surgery with Coefficients

Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann

591 G. A. Anderson

Surgery with Coefficients I

I

Springer-Verlag Berlin. Heidelberq • New York 1977

Author Gerald A. Anderson Department of Mathematics Pennsylvania State University University Park PA 1 6 8 0 2 / U S A

AMS Subject Classifications (1970): 57 B10, 57 C10, 57 D 65 ISBN 3-540-08250-6 Springer-Verlag Berlin • Heidelberg • New York 1SBN 0-387-08250-6 Springer-Verlag New York • Heidelberg • Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin - Heidelberg t977 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210

INTRODUCTION

This

set of notes

is d e r i v e d

at the U n i v e r s i t y

of M i c h i g a n

author's

doctoral

thesis.

complete

and

in 1973,

from a seminar and p o r t i o n s

It is i n t e n d e d

self-contained

account

to give

of surgery

given

of the

a reasonably

theory

modulo

a set of primes. The material

first

necessary

definitions exception

of relative Included

theorem

of H i r s e h

but

which

in a ring,

2 contains

Gp/H

differs

theorem.

satisfy

Normal and the

i is m a i n l y w i t h the

colocalization

immersion

the theory

is j u s t i f i e d

including

fibration.

Chapter

no new ideas,

and

of the

the b a c k g r o u n d

of

classification

and H a e f l i g e r - P o e n a r u .

collapse-expansion of spaces

the theory.

localization

The d e f i n i t i o n

and Shaneson,

contain

and contains

is a sketch

Chapter torsion.

chapters

to d e s c r i b e

and n o t a t i o n

spaces.

described

three

Chapter

3 discusses

duality

construction

invariants homotopy

modulo

groups

Whitehead

from the one given

by a W h i t e h e a d - t y p e

Poincare

the

of local

with

by Cappell local

the theory

coefficients

of a local

Spivak

a set of primes

of the

classifying

normal

are space

are computed. Chapter

theorem.

Briefly,

obstruction

4 contains groups

to f i n d i n g

the m a i n

surgery

are c o n s t r u c t e d

a homotopy

obstruction

to m e a s u r e

equivalence

the

(over a ring

and

with given torsion) dimension,

cobordant to a given map.

Below the middle

the technique is due to Milnor and Wallace.

homotopy equivalences over the integers,

~he simply connected

case is essentially done by Kervaire and Milnor, by Browder and Novikov;

Considering

and globalized

the general case is due to Wall.

We

show that the obstruction lies in a Wall group of a localized group ring. Surgery over a field was first considered by Petrie and Passman,

and Miscenko noticed that Wall's groups behaved

nicely away from the prime 2.

More recently,

and Pardon have considered rational surgery case),

Connelly,

(in the non-simple

and the methods of Cappell and Shaneson

rings with a local epimorphism

~

+ R)

general case, with rings of the form

Giest

(which uses

also apply.

The

R~, is due to the author

in his thesis. Chapter 5 gives the geometric definition of surgery groups, and the generalization to manifold n-ads. approach is also briefly discussed.

Finally,

Quinn's

the periodicity

theorem, in the non-simple case, is proved. Chapter 6 describes

the result of changing rings

in surgery groups by means of a long exact sequence. include a Rothenberg-type

sequence, the general

Corollaries

periodicity

isomorphism and determination of the kernel of s

L2k_l(

~)

s

÷ L2k_l(~)

~

finite, by simple linking forms,

generalizing the original odd-dimensional due to Wall and clarified by Connelly.

surgery obstructions

Finally,

five appendicies are included:

torsion notions for n-ads,

the algebraic construction of

Ln(W÷w';R) , the computation of manifolds,

Ln(~;R),

surgery on embedded

and homotopy and homology spheres.

arranged into categories.

Whitehead

Undoubtedly,

The reference has been

some errors and

omissions have occurred in this arrangement,

but I hope the

general drift is helpful to the reader. A number of people have been of greaD help in writing these notes.

I am indebted to my thesis advisor

C.N. Lee for many helpful suggestions and discussions.

I

would also llke to thank Dennis Barden, Allan Edmonds, Steve Ferry, and Steve Wilson, who participated

in the seminar,

Frank Raymond, Jack Mac Laughlin and W. Holstztynski.

Massachusetts

Institute of Technology

TABLE

Chapter

i.

OF C O N T E N T S

Preliminaries

i.i M o d u l e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 H o m o l o g y and C o h o m o l o g y w i t h T w i s t e d Coefficients ................................. 1.3 A-Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 M i c r o b u n d l e s , Block B u n d l e s and S p h e r i c a l Fibrations ................................... 1.5 The I m m e r s i o n C l a s s i f i c a t i o n T h e o r e m ......... 1.6 I n t e r s e c t i o n Numbers . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 A l g e b r a i c K-Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8 L o c a l i z a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

i

7 ll 14 19 23

Chapter

2.

Whitehead

Torsion

28

Chapter

3.

Poincare

Complexes

39

2 4

3.1 P o i n c a r e D u a l i t y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.2 S p h e r i c a l F i b r a t i o n s and Normal Maps ......... 45 Chapter

Chapter

Chapter

with

54

4

Surgery

Coefficients

4.1 4.2 4.3 4.4 4.5

Surgery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The P r o b l e m of Surgery w i t h C o e f f i c i e n t s ..... Surgery O b s t r u c t i o n Groups ................... The Simply C o n n e c t e d Case . . . . . . . . . . . . . . . . . . . . The Exact Sequence of Surgery ................

54 57 60 74 80

5.

Relative

82

5.1 5.2 5.3 5.4

Handle S u b t r a c t i o n and A p p l i c a t i o n s .......... G e o m e t r i c D e f i n i t i o n of Surgery Groups ....... C l a s s i f y i n g Spaces for S u r g e r y ............... The P e r i o d i c i t y Theorem, Part I ..............

6.

Relations

Surgery

Between

Surgery

I01

Theories

6.1 The Long Exact Sequence of Surgery w i t h Coefficients ................................ 6.2 The R o t h e n b e r g Sequence ..................... 6.3 The P e r i o d i c i t y Theorem, Part II ............ 6.4 Simple L i n k i n g Numbers . . . . . . . . . . . . . . . . . . . . . . Appendix

A.

Torsion

Appendix

B.

Higher

Appendix

C.

L Groups

Appendix

D.

Ambient Surgery and Surgery S u b m a n i f o l d Fixed

82 85 94 96

i01 105 109 Ii0

for n-ads

122

L-Theories

124

of Free

Abelian

Groups Leaving

127 a 129

Appendix

E:

References Symbol

Homotopy

and

Homology

Spheres

.................

135

.................................................

138

Index ...............................................

154

Index ......................................................

156

Chapter

i. P r e l i m i n a r i e s

i.i. Modules. Let

A

be a ring

with involution,

i.e.

(not n e c e s s a r i l y

a map

commutative)

A ÷ A, w r i t t e n

k~

~*, so

that

(a)

(~i+12)* = ll* + ~2'

(b)

(Ii~2)* = ~2'~i*

(c)

~** = ~.

We will u s u a l l y units in

A.

be finitely Then

M

assume

i E A.

A

Unless otherwise generated

inherits

denotes the group of

stated,

and right.

all

Let

a left A-module

M

A-modules

will

be a A-module.

structure

by d e f i n i n g

l.m=m.l*. The dual of

M

with A-module

structure

f~ M*,

If

l ~ A.

by giving case

N

M

and

N=A, M @ A A

sum of copies of so that

M @ N

we may choose k.

If

for some

M

given by N

N

M@A

k.

with

= l*f(m), we define

as above.

M

is p r o j e c t i v e

to be free,

M

if there

is s t a b l [ free i.e. M ~ A k

a stable basis

M ~N

In the

(x @ l)W = x @

is free if it is isomorphic

is free.

is s-free,

M* = HomA(M,A)

(f'~)(m)

structure

is a A-module

A.

by

are A-modules,

a left A-module

A A-module

N

is defined

~*l.

to a direct is a A-module

(s-free)

if

is free for some

(s-basis)

is a basis

The m a i n example of A = Rw

for some

~ng.g,

Rw

will be a group ring,

(usually commutative)

a multiplicative The r i n g

A

ring

R

group w i t h a h o m o m o r p h i s m

with

w:w ~ {±l}.

is d e f i n e d to be the set of all finite

ngg R,

g e w.

The i n v o l u t i o n

i,

sums

is given by

(~ng.g)* = ~w(g)ngg -1

1.2. Homology

and Cohomology

Let

X

a homomorphism.

w i t h Twi.sted Coefficients.

be a finite Let

CW-complex,

A = H~

and

M

~ = ~l(X) and

a A-module.

w:~ ÷ {±i}

Define

~{i(x;M) = HI(C,(X) @,,~.~) }Ii(X;M) = Hi(HomA(c , ( X ) , M ) ) , where

C,(X)

the u n i v e r s a l

is the chain complex of cellular cover

X; C,(X)

and based A-modules. with compact HI(X;A),

If

supports.

X

is not compact,

We write

we use c o h o m o l o g y

Hi(X) , Hi(x)

for

Hi(X;A).

determines Let

an element Z/2H

in

HI(x;~/2H)

act on

H

to be the bundle a s s o c i a t e d ÷ X

in

is a chain complex of free

We can define this a l t e r n a t e l y

E ÷ X.

chains

bundle w i t h fiber

and so a double

non-trlvially to

is a principal x - b u n d l e

associated

as follows:

E

w i t h fiber

Let

~t

cover

and define

and so define M.

w

= ~

H. ~

Ht

Now to be the

~ zt.

Then

Hi(X;M)

= H i ( X ; ~ t)

Hi(x;M)

= Hi(x;~)

where we use bundle If

c

is an n-cell in

en:cq(x) where

(or sheaf) homology

defines

linearly to chains since we

supports.

This defines M

This extends

Cn(X) and, in fact, to infinite

are using compact

If

X, cap product

÷ Cn_q(X) @ A ~ Cn_q(X)

cq(x) ~ HomA(Cq(X),A).

chains in

and cohomology.

is a A-module,

~N:Hq(X) + Hn_q(X) define

~:Hq(X;M)

for

~ E Hn(X).

+ Hn_q(X;M)

by

the composition HomA(Cq(X),M) If

f:X ÷ Y

is a map,

~ cq(x) @ ^ M ÷ C n _ q ( X ) @ ^ M . f#:Wl(X)

~ Wl(Y) , then define

Ki(X;M ) = ker(f,:Hi(X;M ) ÷ Hi(Y;M)) Ki(X;M) The condition suffice

~ coker

wI(X) ~ Wl(Y)

f:X + Y

homolo6~equivalence is an isomorphism.

i.

isn't necessary,

with

over X

and

tFpe if there is a sequence

each

÷ Hi(X;M)). but will

for our purposes. A map

homology

(f*:Hi(y;M)

equivalences

over

R

f#:Wl(X) if

Y

~ Wl(Y)

f,:H,(X;Rw)

is a

+ H,(Y;Rw)

have the same R-homology X = Z0,ZI,...,Z m = Y

R~ Z i ÷ Zi+ I

or

and

Zi+ I ÷ Zi~ for

1.3. A-Sets. Let standard

A

n-simplex~

the face maps A

be the category

~i n.

to the category

similarly. between

n=O,l,..., A A-set

X

of k-simplices associated

We define

between

A-sets

is a A-set, of

X.

then

An ordered

to it a A-set

set of k-simplices

of

D(K)

A-groups,

is a natural

functor

by from

etc., transformation

from A-sets

homeomorphic let

X

is called the set

simpllcial

complex

has

by

D(K)(A k) = the

an inverse

to

De i.e. a functor

K, for

K

spaces

so that

a simpllcial

S(D(K))

complex.

be a A-set and form the disjoint

= ~ ] X ( A n) x A n ' where n=0 and we regard

K

defined

to topological

to

X(A k)

K.

We can also define

this,

generated

the functors. If

S

and morphisms

A n , the

is a contravariant

of sets.

A A-map

with objects

X(A n)

is

To do

union

has the discrete

topology

A n = {(t0,...,tn+l)~ R n + 2 1 0 = t 0 ~ t I ~ . . . ~ tn, 1 = i}.

The maps then defined by

~in: An-i ÷ A n

3i n(t0' .

6in+l:A n+l ÷ A n

and

.'tn) . .= .(to' . .

.'ti'ti' .

,t n)

are and

~in+l(t0 ' .... 'tn+2) = (t0'''''ti .... 'in+2)" Then we let relation

defined

(3in+lxn'an+l) an+l~

An + l

'

by

S(X) = X/~, where

~

is the equivalence

(6i n Xn'an 1 ) ~ (Xn'~ n i an-l)'

~ (Xn'Si i=O,...,n.

n+l a

n+l

)

for

Xn6

X(An),

an_l E

A n-1 ,

S(X) See M i l n o r

is called the geometric

[AI~, Gabriel,

Zisman

realization

of

X.

[A6] for a complete

discussion. A A-set

X

admits an e x t e n s i o n to important

An .

The p r o p e r t y

and we describe now a process

A-set into a h o m o t o p i c a l l y

equivalent

If

X

let

by

A(X,Y)(A k) = the set of A-maps

and

ExI(x) Then

Y

are A-sets,

= A(*,X), Ex~(X)

Exk(x)

and Sanderson

[Bl~.

Let

structed of

KH(

to

BH.

H

If

of p r i n c i p a l

a

(Kan [A9])-

be the A-set defined

X x A k ÷ y. Ex~(X)

This is expounded

Define = lim Exk(x).

KH(X)

denotes

H-bundles

over

and

E

H-bundle

has a free classes

X, then in [B10] there is con-

space

if

A-fibration

so that

complexes

here.

the set of i s o m o r p h s i m

BH

and a natural

[ ,BH], the set of homotopy

define a p r i n c i p a l ~:E ÷ X

A principal

~:E + X, where

X.

fully in Rourke

some d e f i n i t i o n s

be a Kan A-group.

More generally,

A-sets

A(X,Y)

We repeat

a classifying )

Kan A-set

any n o t i o n used for simplicial

is an orbit map

H-actlon.

Kan is

for converting

= Exl(Exk-l(x)),

can be used for A-sets.

X

of being

is Kan and has the same homotopy type as

In general,

over

.~n nAn-1. [ -Bi ) ÷ X

is Kan if any A-map

A

equivalence

classes

of maps

is a Kan A-monold,

then we

to be a A-map of pointed

(a)

w

is a Kan fibration

lifting property

(i.e.

satisfies

with respect

the

to the pair

(An,An-~inAn-l)) '

(b)

-l(,)

(c)

there

= A, is an action of

A

on

E, E × A ~ E,

so that E × A

"~ E

proJ. E

,

X

1[

commutes. Again there is a c l a s s i f y i n g equivalence principal

of

H

BH ÷ BA

A-fibrations

over

if

a submonoid is

BA

H

of

X, and

study bundles. Define

[X,BA].

is a A-group and

A

is a A-monoid

A, then the fiber of the map

Let Hq

H

the A-groups be one of

~:A k × ~ q ÷ A k x

~q.

and A-monoids

TOP,

PL,

or

to be the A-set such that

the set of zero and fiber p r e s e r v i n g

~

classes of

A/H.

We now define

and

and a natural

hA(X) , the set of homotopy

As usual, with

space

This means

commutes with p r o j e c t i o n

DIFF. Hq(A k) =

H-homeomorphisms

olA k × 0 to

needed to

~q.

is the identity

Define

Hq(A k) = the set of zero and block p r e s e r v i n g

Hq

by

H-homeomorphlsms

s:A k × I q ÷ A k × I q subcomplex

(i.e.

q(K x i q) = K × I q

K C Ak). Let

R

be a ring and define

the set of zero and fiber preserving over

R

(i.e.

of pairs

with block preserving Define Hq, Hq, H, H

for all

instead

are A-groups,

q.

~ PL = "PL,

According

1.4. Micr0bund!es~

with structure

let

and relate H = TOP,

Definition. complex. space

and

Gq(R),

Hq

and

Gq(R)

or

~ Gq

= GL(q~R)

and Spherical

(R)

= 0q.

F ibrations.

we defined principal or

geometrically

Hq, or bundles

Gq(R).

In

associated

to

As before,

DIFF. [Bg]).

An H-block bundle over

(I)

are A-monoids.

them to the A-set definitions.

PL

but

We have

[B8], DIFFq

section,

groups

similarly

Gq(R)

DIFF = ~D I F F ,

(R0urke ~ Sanderson

E(~)~K

equivalences

G (R) q

H = ---+ lim H q .

Block Bundles

this section we define these

Gq(R)(A k) =

of fiber preserving.

to Milnor

In the previous bundles

by

homology

Define

H = lim ---~ H q ,

TOP = TOP,

Gq(R)

~:(A k × ~ q , A k x 0) + (A k x ~ q , A k x 0)

~-l(Ak × 0) = A k x 0).

Also,

for each

Let

K, written

K

be a simplicial ~q/K,

is a

so that if

~ eK

is an n-cell,

then there

exists

an

H

(n+q)-ball (2)

E(~)

BcE({)

= k.JB . g~K ~

so that

(B~,~)

~ (In+q,In).

(3)

Int(Bol ) n I n t ( B o 2 ) = Z

(4)

Bol N B o 2

\~/

The trivial b l o c k bundle If

~q/K

If

~q/K

and

~qlL nP/L

The W h i t n e y

sum of

If

~q/K

~

over

A(K)cK

~q

and

isomorphism

so that

fl K = 1 K o

x

nq

× K

is the diagonal.

isomorphism

to be

of

= Be(n)

eq/~

and

T h e o r e m i.

([B9])

neighborhood

of

Thus if tangent of

A(M)

M

b l o c k bundle in

is a H - m a n i f o l d

M~ then

M × M.

N

M,

L.

A

f:E(~) ÷ E(n) for all

~q[o.

and

o a K. ~

if

A maximum

N

a regular M'

we can define the

TM, to be a regular n e i g h b o r h o o d

(see [BII] for the case

There are some d i m e n s i o n a l

K.

is a H-block bundle over

is a H-manifold, of

over

o ~ K, is a chart for

of charts is called an atlas. M

L × E(~q)IG(f)

is identified with

collection

If

f:L ÷ K, then

be block bundles

f(Bo(~))

× E(~).

is d e f i n e d by

is defined

I q ÷ E ( ~ I ~ ) C E(~),

it is a bundle

K

is an H - h o m e o m o r p h i s m and

E(~ql L) = L.JB . q~L °

E(~ × ~) = E(~)

= y}

then

then define the

is a block bundle and f*~q/L,

E(~ q) = K x I q.

is a subcomplex,

is defined by

G(f) = { ( x , y ) e L × Klf<x) Let

A map

L

and

where

the induced bundle,

bundle

LCK

~ × n/K x L, by

~

~ = ~ × nIA(K)

where

is d e f i n e d by

are bloc]< bundles,

product b l o c k bundle,

~ ~

eq/K

over

o I ~ o 2.

BO.

is a block bundle and

the r e s t r i c t i o n

if

considerations

H = TOP.

in this case).

We now show how to associate a principal to an H-block bundle. X = D(K).

We define

Let

~q/K

D(~)

Hq-bUndle

be a block bundle and let

to be the bundle

E + X

E(A k) = {hlh:A k x I q + E(~I~ )

where

is a chart,

¢ K, where hlA k × O:A k × 0 + ~ E(A k) ÷ X(A k) action

sends

h:A k x I q + E(~I~ )

E(A k) × Hq(A k) ÷ E(A k)

(in our notation,

Hq(A k)

self H-isomorphisms

of

Conversely,

then

S(~)

eqI~i~

and

by

~, and the

is the set of block preserving

eqlAk). if

~q

s:K + S(E).

~is(~) = s(~ic)'Fi, ~

Define

to

is defined by composition.

is a principal

w:E + D(K), we define an H-block bundle choose a section

is the identity),

If

~

H q -bundle

S(~)/K.

To do this,

is an n-simplex of

for some unique

K,

F £ Hq(An-l).

E(S(<)) = k-#zql~, where we identify cEK

(eql~)l~i ~

by the isomorphism given by

Fi, ~

and the ordering of vertices. These constructions

give the following:

Theorem 2.

There is a I-I cprrespondence

classes of

Hq-block bundles .0.ver

H q-block bundles over Corollary 1.

between isomorp~is ~

K......and ... principal

D(K).

The functor which associates to

K

of isomorphism classes of H q-block bundles over naturally equivalent to

[K,B~q].

the set K

is

Defi.nition.

(Milnor

[B8]).

diagram

K i E j K, w h e r e

and

x * K

for

i(x)

in

E

An H q - m i c r o b u n d l e i,J

are

H-maps,

t h e r e exist neighborhoods and

an

U

H homeomorphism

over

K

is a

so that

of

x

in

joi = i K K

and

h:V + U × ~q

V

so t h a t

V

U

h

U xNq

commutes. The t a n sent is d e f i n e d

by

the

Theorem

same m a n n e r

following

3.

The

isomorp,hism

(see

to

Spivak

as above.

An

fibration

w:E + K

F

R-homology

has

the

of p r o v i n g

which

0,,f ..... H

equivalent

Definition.

of an H - m a n i f o l d

M, TM,

with

associates

to

q-microbundles

2, we can

K

the

over

set of

....K,, i s

[K,BHq].

[El9] for the

R q-spherical

We d e f i n e

Theorem

theorem.

functor

classes

naturally

microbundle

M + M × M + M,

In the prove

U

fibration

fiber

type

Whitney

case

of

F

R=Z).

over

so that

K Wl(F)

Let

K

be

is a = 0

and

S q.

sum,

induced

usual.

10

bundles,

etc.,

as

of

4.

Theorem

The functor

fiber

hom0topy

over

K

1.5.

The

equivalence

associates

classes

is natura!..!y equiyalent

Immersion

In this DIFF.

which

and assume

the r e a d e r

of immersions. locally

flat

Submanifolds in the TOP and

Let a manifold R(TM,TN)

with

M

set of fibrations

Theorem.

may

definition

is f a m i l a r

the

[K,BGq(R)].

manifolds

We use the m i c r o b u n d l e

K

R q-spherical

to

Classification

section,

of

to

be TOP,

PL, or

of tangent

with elementary

will

always

bundle

properties

be a s s u m e d

to be

PL cases.

be a compact dim N = n.

submanifold

Define

of

A-sets

~n

Imm

and

N

(M,N)

and

by I m m ( M , N ) ( A k) =

the

set of germs

f:A k x U + A k x projection,

R ( T M , T N ) ( A k) =

of immersions N, c o m m u t i n g

where

neighborhood

of

identify

and

f

U M

on

Ak x (U~U'),

the

set of germs

projection, of

M

in

U ~u,

f'

~n

and we

if they

of bundle

agree

monomorphisms

commuting

with

an open n e i g h b o r h o o d so that

A k × TU + A k x U x N

11

is an open in

F:A k x TU ÷ A k x TN,

with

the map given

by

(t,u,u') tEA k,

+

(t,u,nF(t,u,u')),

( u , u ' ) c TU,

where

~:TN ÷ N

is an immersion, is the bundle

projection. Choose

an i m m e r s i o n

f:A k × U ÷ A k × N, M C U C ~

df:A k x TU ÷ A k × TN f(x,u)

= (X,fxU).

d:Imm(M,N)

Lees

case, [F2]

surgery

This defines

= (X,fxU,fxU')

a differential

where

map

following

theorem,

Haefliger

and

due to H i r s c h

Poenaru

in the t o p o l o g i c a l

[B3]

case,

[B4]

in the

in the

PL case and

is an i m p o r t a n t

tool

in

theory.

Immersion

Classification

decomposition homotopy

df(x,u,u')

÷ R(TM,TN). The

smooth

by

n, and define

with

Theorem.

all handles

If

M

of index

has a handle
d

body is a

equivalence. We will

messy

details.

Lemma

i.

Let

give a sketch

k < n.

of the proof,

omitting

Then the maps

¢:Imm(D k × Dm-k,N)

÷ Imm(S k-I

× Dm-k

N)

~:R(T(O k × om-k),TN) ~ R(T(S k-I × om-k),TN) defined

by r e s t r i c t i o n

Proof:

This

lifting

property.

is Just

are f i b r a t i o n s .

a matter

See Lees

of v e r i f y i n g

[F2].

12

the h o m o t o p y

the

Lemma 2. Proof:

The theorem is true for

M=D n.

See [B2] or [B3]. Let

¢ g Imm(S k-I × D m-k)

be the fiber of over

¢

over

@

and

and let

imm@(D k x Dm-k,N)

R¢(T(D k × Dm-k),TN)

fiber of

~

Lemma 3.

d:Imm¢(D k × Dm-k,N) ÷ R¢(T(D k x Dm-k),TN)

the

de.

homotopy equivalence . for

k < n.

Proof:

k, using Lemma 2.

By induction on

is a

Proof of Theorem: The proof is by induction on handles in Suppose

M = M 0 U D k × D m-k, M 0 ~ (D k x D m-k) = Sk-I × D m-k,

and that the theorem is true for flat in

A n, D k x D m-k

neighborhood of and

R n.

f cImm(M0,N)

Since

are flbratlons, is

where

@

Imm~(D k x Dm-k,N) is the image of

We have a commutative diagram Imm@iDk × Dm-k,N)--, R~(i(Dk × om-k),TN)

1

Imm(M0,N )

is locally

and the fiber over

imm(M0,N ) + imm(S k-I x Dm-k N).

Imm(M,N)

M

Thus by Lemma I, Imm(M,N) + Imm(M0,N)

(dfe R(TM0,TN))

(R@(T(D k x Dm-k),TN)),

M 0.

is contained in a coordinate

R(TM,TN) ÷ R(TM0,TN)

the map

M.

* R(TM,TN)

;

, R(TM0,TN)

13

f

under

where the v e r t i c a l maps are fibrations. Since by Lemma 3 and induction, maps are homotopy

equivalences,

1.6.

Numbers.

Intersection Suppose

connected

Nn

finite set of points,

gx

*

x

to

from

,oN

Let at

*.

* v ~ V, *w a W. pv~BPw

*v

to

-i

x

in B

a c c o r d i n g to w h e t h e r or not the loop at

k(V,W) We can define manifolds

=

k(V,W)

if

V

define

x~ VNW, where

~

is

is a path in

W

from

gx

Let

Sx = ±I

preserves

the local

number

and

W

of

V

and

W

by

~ = ~I(N,*). are immersed

sub-

in the same manner.

need to define the h o m o l o g y Is closed

where

For

~ gxgx E ~ , xEV~W

To give an algebraic

define

pv,Pw

*.

Define the i n t e r s e c t i o n

N

Choose paths

Wl(N,*)

and

in a

be a basepoint

*w' m i s s i n g all other intersections.

orientation

are simply

v+w=n.

to be the element V

vV,w w

transversally

to b a s e p o i n t s

a p a t h in

and

intersecting

and choose a local o r i e n t a t i o n from

the middle map is also.

is a m a n i f o l d

submanifolds

the top and b o t t o m

for simplicity.

ui-u2 ¢ ~w

to be

U l ' ~ Hn-V(N),

interpretation

intersection For

p a i r i n ~. We assume

~ Hn(N)

correspond

under Poincare duality.

14

k, we

U l g Hv(N) , u 2 ~ Hw(N) ,



u 2 ' ~ Hn-W(N)

of

to

= ~w, Ul,U 2

T h e o r e m i. Let

f:V v ÷ N~

~:W w ÷ N

be immersions.

Then

x(v~w) = f~[V]-g~[W]. Proof: and

Let

R~S

be r e g u l a r n e i g h b o r h o o d s

T ( R ) E H v (R,R-f(V)),

classes.

Let

T'(R)

of

V,W

T(S)(HW(s~S-g(V))

be the image of

T(R)

in

N,

the Thom under the map

HV(R,R-f(V)) ~ HV(N,N-fl(V)) J-~ HV(N) and

T'(S)

similar.

Then

f,[v]

= f,([R]nT(R)) = [N]NJ*T(R) =

Thus

f,[V]'g,[W] Let

= .

i:N ÷ ( N , N - f ( V ) N g ( W ) ) .

[ N ] x ~ Hn(N~N-x) ix[N] =

[N]nT'(R).

Then

= .

Also

is a local orientation,

~(¢x),[N]x,

where

~f

then

Cx:(N,N-f(V)~g(W))

+ (N,N-x).

x

It is clear that

~xg x = < r * T ( R ) U s * T ( S ) , [ N ] x > where

r:(N,N-x)

+ (N,N-f(V)),

s:(N,N-x)÷(N,N-g(W))

= . Thus

~xgx

= [ = f,EV]-g,Ew].

If define

V vC N

is an immersed

self-lntersection

simply connected

numbers

similarly.

15

submanifold

we can

Note we must have 2v=n.

By general position, of

V

in

N

are transverse

a self-lntersection branches

of

is the transverse

x~ V

number

first.

((-l)Vw(gx)Cx)gx -I, where

exg x

first,

W:~l(N)

in

generated

Z~/! v

of

Z~

a-

(-l)Vw(a)a -I = a -

2 (Whitney Assume

t o ..... W;, ,s,,,O,,,,tha, t , .

Since

Let

P,Q

taking one

the answer will be is the

,,of

N

number

Iv

is the ideal

N~V~W,,

satisfY

takin~

V ' ~W, = v

p(V) = [Sxgx

a~.

Let

x,~¢V~W

-

be as,,, above,,,with

~xgx = -Eygy.

V

to

Then

V',,~,,,transverse

{x~,y}.

be paths t h r o u g h

~xg x = -~ygy,

Since

We can

of the form

(-l)Va *,

Lemma).

Thus there is a map pQ-l.

where

by elements

t.,h,ere is an isotopy

Proof:

point.

homomorphism.

is w e l l - d e f i n e d

v,w ~ 3.

of two

the i n t e r s e c t i o n

+ {±I}

Thus the s e l f - i n t e r s e c t i o n

Theorem

locally

as above,

If we compute

taking the other branch

orientation

i.e.

intersection

be a s e l f - i n t e r s e c t i o n

the i n t e r s e c t i o n

branch a r b i t r a r i l y number

double points,

V.

Let compute

we can assume the s e l f - i n t e r s e c t l o n s

V,W,

the loop

pQ-i

f:D 2 + N

so that

is trivial

dim N > 6, we may assume

16

Joining

f(S I) f

x in

and

y.

N.

is the loop

is an embedding.

W

V

Let

R

be a r e g u l a r

this

to c o n s t r u c t

side

of

W.

For

an i s o t o p y details

or R o u r k e

& Sanderson

Corollary

i.

immersions points , homotopic

(2)

~(V!. = 0~ t h e n

the

with

following

to T h e o r e m

intersect

f:v...v.....+.... N n

I(V~W)

= ft.

so t h a t

is a r e l a t i v e boundary,

will

N.

Use

to the

other

6.6

~,:Ww ÷ N

are

in ,,a,f i n i t e

Then

=..¢.

> 6, is an i m m e r s i o n

we

be useful.

2.

17

with

to. an e m b e d d i n g .

of W h i t n e y ' s

leave

s,et,,o,f

f ......is ... r e g u l a r l y

f'(V)Ng(W)

version

which

in

Theorem

and

tra,nsversally

f:V v ~ N n, 2 v = n

corollary

V

5.12.

fi........is .. r e g u l a r l y . h o m o t o p i c

There manifolds

pulls

[B12], T h e o r e m

f':V~ ÷ N If

which

f(D 2)

[DI~,

~ 3, and

to

of

see M i l n o r

(i) .....S...u p p o s e

which

v,w

neighborhood

to the

Lemma

reader.

The p r o o f

for However,

is s i m i l a r

Corollary

2.

Suppos e

Nn

the i.nclusion, i nducin $ g:(wW,~w)

+ (N,8N)

transversally, Then . f that

and

Let

Proof:

of

M

Then there from

which

to

f:(vV,~v)÷(N,SN),

intersect

simply

c0nnected.

f':(V,~V)

+ (N,~N)

s_~o

be disjoint ................... embeddinss

are regular

f,g

intersections

~ E Wl(M). by paths

by the path

Let

homotopies

to embeddinss

f',g'

so

= x.

Since

x = ±~,

n ~ 6, with

= ¢.

F,G:S k x I ÷ M x I ~(F,G)

hom0topic

f,g:S k + M 2k+l,

x ~ ~Wl(M).

that

~ Wl(N).

v,w ~ 3, V , W , ~ V , ~ W

is r e g u l a r l y

3.

~I(SN)

be immersions

f'(V)~g(W)

Theorem

is a manifold,

Join

PI

f(S k)

and

P2 -I ~ PI"

are additive,

P2" Let

and

g(S k)

We Join F

we can assume

f(S k)

be a regular

to the basepoint to

g(S k)

homotopy

which

deform a small d i s c C

f(S k)

along this path and across

a disc transverse

g(sk).

Let

of

x

is

-I, then reverse

It also follows arbitrary

to

G = g x i.

orientation

g f that we can find regular

self-intersection

number.

18

If the sign

in the disc.

-i homotopies

g with

1.7. Algebraic

K-Theory.

Let

A

be a ring.

which are both useful for

L-theory

Definition.

in geometry and will serve as a model

in surgery. Let

We describe here some functors

K0(A)

A good reference

M

is a projective

isomorphism

class, with relations

define a ring structure (I)

If

A

on

X + ~,

If

X compact,

K-functor.

then

[M]

is its

by

We

[M].[N] = [ M ~ N ] . ideal domain,

then

[A]. the ring of continuous

functions

K0(A) ~ K(X), the topological

The correspondence ~

is given by

to the

g~+ r(~),

EX-module

of continuous

F(~). There is a natural

defined by rank, and

~0(A)

classes of projective

modules

relation

and

[M] + [N] = [ M ~ N ] .

Ko(A)

A = ~X,

sending a vector bundle sections

A-module,

is a principal

K0(A) ~ ~, generated by (2)

[C8].

be the abelian group with generators

[M], where

Examples:

is Milnor

[M] ~ [N] The reduced

if

splitting

to finiteness

by a finite

CW-complex,

~

is the set of isomorphism [M], with the equivalence

M ~ Ar ~ N ~ A

K-functor

obstruction

K0(A) = K0(A) ~

s

K0(Z~) measures

for a CW-complex ~ = ~l(X),

obstruction to finding a boundary

for some r,s.

Wall

X

the

dominated

[C10], and the

for an open manifold

Siebenmann[C9].

19

M,

Let matrices

Lo

over

GL(n,A) A.

be the group of invertible

We regard

lj

GL(n,A) ~* GL(n+I,A)

n×n by

n

linear group. Let (i,j)-th

Eij , i ~ J, denote

the matrix

spot and zeroes elsewhere.

~ A, is called elementary. generated

by elementary

The matrix

The subgroup

matrices

with i in the

is denoted

eij = I + ~Eij,

of

GL(n,A)

E(n,A).

Let

E(A) = ~ J E ( n , h ) . n

Lemma. (Whitehead). Proof: matrix in

We have

E(A) = [GL(A)~GL(A)].

1 1 I -~ -i eij = eikekjeikekj , so every elementary

is a commutator.

Conversely,

if

A,BEGL(n,A),

then

GL(2n,A), 'ABA-IB-Io]

=

0

B-1 Given any matrix

0

BA

X, we can write

[: :1 Ii:x :} and [~

~]

i s in

E(2n,h).

Thus

20

E(A) -- [GL(A),GL(A)].

Define the Whitehead

group of

More categorically, the abellan f:M ÷ M

A

by

Kl(A)

= GL(A)/E(A).

we can define

group with generators

Kl(A) to be

(M,f), M

a free A-module,

a A-Isomorphism, with relations (M,f) = (N,g)

if there

is a A - i s o m o r p h i s m

h:M + N

with

(M,f) + (N,g) = ( M ® N ,

hf = gh

f Og)

(M,f) + (M,g) = (M,fg). See Bass

[B2] or Swan

Definition. M

A free and based

is a free A-module

basis

for

M.

Then

A

Define Examples:

p-group

of rank

elementary If

Z

~

GL(n,A)

is an ordered M

bi~-* bi' , is elementary. let

1

for

SL(A)

be the matrices

and define

SKl(A)

= SL(A)/E(A).

SKl(A). Kl(A)

= KI(A)/{±I} , where

(1)

If

A

(2)

If

A = Zw, ~

k, p

then (3)

and

M + M,

÷ Kl(A)

is

A

{ ± l } C A" = GL(1,A).

or a field,

of rank

SKl(RW) If

Z

= 0

then

SKl(A)

pk-1/p-1 for

w

is a skew field,

is the determinant.

21

then

an elementary

an odd prime,

abelian p-group Rc~,

(M,B), where

B' = {bl',...,bn'}

is commutative,

= A" @

is a pair

B = {bl,...,b n}

with determinant

Kl(A)

A-module

basis

if the map

If GL(A)

and

Another

is preferred

in

[BIi].

SKl(A)

= 0.

abelian is an

_[p+~-l]' ~ ~

as above. then

See

Kl(A)

=

[C1].

^ I[A" ,A" ]

The m a i n a p p l i c a t i o n of K I is to torsion. Let 3 3 C:C n + Cn_ I + . . . + C O be a chain c o m p l e x w i t h C i a free and based

A-module.

T h e n by S p a n i e r so that

Suppose

[AI5], T h e o r e m

36 + 53 = 1 Let

and

is acyclic,

4.2.5,

and

is an i s o m o r p h i s m

determines

torsion

of

C.

in fact

define

of free

is i n d e p e n d e n t

Hi(C)

+ C + C

complexes

sequence

of the above

Then

H

is acyclic.

Lemma

i.

Proof:

T(C)

as above,

above

s-free

Definition. If

simple

H

of

6.

the We can

as in [D20]. sequence

be the long exact

regarded

as a chain

complex.

÷ T(.H).

Theorem

3.2.

modifications,

A chain

cone

complex

is a chain ~

we can do all of the

C

is simpl e

homotopy

A

involution

if

T(U)

KI(A)

We say

T(C) then F

= 0. its

is a

= 0.

has an i n v o l u t i o n on

if

equivalence,

is free and acyclic.

equivalence When

induced

C+,C -,

A-modules.

F:C ÷ C'

mapping

of

called

be an exact

and let

+ T(C")

[D20],

With minor

is free, + 0

sequence,

= T(C')

See M i l n o r

Kl(A),

since

TT

0 ÷ C

of chain

in

Then

the bases

The t o r s i o n

!

Let

Using

~(C)

if

= 0.

6:C i + Ci+ 1

A-modules,

an element

~(C)

exists

C- = ~ C 2 i + i .

(3+6) 2 = 32 + 36 + 63 + 62 = 1. 3+6

there

H,(C)

62 -- 0.

C + --~C2i

3 + 8:C + ÷ C-

C

,, then there

also d e n o t e d

22

, ,

is an

defined

by

(aij)* -- (aji*). Define

the Steinberg

with generators

grou p

St(n,A)

for

n ~ 3

to be the group

x ~ij' I ! i ~ J ! n, ~ C A , and with relations x I x ~ = x X+~ ij iJ iJ ~ ~ [xij, x j£] = xi£,

i ~

[x j, x k£] = I, J ~ k, i @ £. There are natural St(A)

= ~JSt(n,A). n

and let Lemma Proof:

inclusions Define

¢:St(A)

K2(A) = ker(¢:St(A)

2.

K2(A) Milnor

S t ( n , A ) C St(n+l,A); + E(A)

let

X ¢(x j) = eij,

by

÷ E(A)).

~ H2(E(A);Z). [C8], Theorem

5.10.

1.8. Localization. Let

A

be a ring and

C(A) = {X~ AIXW = wX be m u l t i p l i c a t i v e l y Define S-1A, if

to be

A × S

There

closed and contain

modulo

so that

the equivalence

class of

on

S-1A

is a natural

by

of

Let

A

(Xt-ws)u (k,s).

.away ... from

23

S,

(X,s) ~ (~,t) = 0.

Let

Define

X/s + ~/t = Xt+~S/st,

ring homomorphism

S ~ C(A) - 0

1.

the relation

u 6S

structure

its center,

w ~ A}.

the localization

there exists

denote

for every

C(A)

A + S-1A

k/s

a ring

~/s.~/t

= ~/st.

given by k ~ k / 1 .

If where

M

S-IA

is a A-module,

we define

has a left A-module

S-IM = M ~ S - ~ ,

structure

defined

by the map

A÷S-IA. Lena

i.

If

M' + M ÷ M"

is exact~ then

S-IM ' ÷ S-IM + S-IM"

is also exact. Proof:

The proof given in A t i y a h - M a c D o n a l d

3.3, works Examples:

since (I)

ICA

be a prime

multlpllcatively

closed

and we let

the localization

of

at

ai~ A,

ki

If

A ACA

For e x ~ p l e ,

away from ~

ideal.

A

Then

A I = (A-I)-IA,

A-I

is

called

I.

is a subset,

a non-negatlve

localization

Proposition

S c C(A) - 0.

Let

(2)

[AI],

integer}. by

let

S = H(A)

Define

kI kn ={a I ...a n I

the

A A = S-!A.

= Z[ ], Z(2 ) = Z[~,~

...3, Q = ~

where

is the set of all primes. Definition. primes. f:X ÷ Xp

Let

X

be a

A localization so that

f

CW-complex of

X

localizes

and

is a space homology,

~,(x) f*,,H,(Xp)

\ f commutes.

24

2

a set of ~

i.e.

and a map

2

Equivalently, A space a

Y

f

localizes homotopy.

(Sullivan [Gi5], Theorem 2.1).

is said to have local homol0~Y if

H,(Y)

is

H(~)

and

B-module. Let

choose a map n-sphere

kl,k2,.., di:Sn ~ S n

S2

be an enumeration of of degree

k i.

Define the local

to be the infinite telescope

of mapping cylinders.

The inclusion

Mdlk~Md2~J...

S n + Mdl ÷ $2 n

localizes homology. Theorem 1. (Sullivan ~15]). Let 0-cell and no 1-cells (e.g.

X

be a CW-complex with one

~l(X) = 0).

Then

X

has a

localization. Proof:

If

X

is a 2-complex, then

2-spheres, and

X ÷ X2 = VS22

X = VS 2, a wedge of

is a localization.

Inductively assume the theorem is true for complexes of dimension

< n-1.

Let

X (i)

be the i-skeleton; write

X (n) = x(n-1)kJf

c(vsn-1), where

f:VS n-1 ÷ X (n-l)

is the attaching map.

to

f2:VS2 n-! ÷ x2(n-l)

c

and let

denotes cone and We can extend

x2(n) = x2(n-l) ~J

Then we have by the Puppe sequence

25

f c(vs2n-I).

([A25]), an exact ladder

VS n-I ~

x(n-l)_+ x(n)_., VS n ~

i

L

VS~-I -'-~f24n-l)-*

I

L

x(n)--* VS2 ~

Since all spaces on the bottom row except homology, that

x2(n)

must have local homology,

X (n) ÷ X2 (n)

SX (n-l)

is a localization.

s4n-l) X~ (n)

have local

and it follows

Now define

Xp = kJX~ (n), which gives the result. n The following theorem is useful

in studying torsion;

it

follows in the same manner as Theorem i, but by considering only those cells in Theorem 2.

Let

~l(X) ~ ~I(Y).

(Y~X)

not in

X.

be a CW-pair with inclusion

Then there is a pai.r

f:(Y,X) ÷ ((Y,X)R,X) homology,

Y

so that

f

localizes

i.e. H, (Y,X)

\

((Y,X)2,X)

f*~ H, ((Y,X)~,X)

/

H,(Y,X) ® Z2. commutes.

26

inducin 5

and a map

relative

Let

X

be simply-connected

Define the colocalization

away from

P

X + X(p).

X ÷ Xp.

We list some elementary properties:

XP

=

Xp

If

wi(X)

(since then P

a set of primes.

P, X P, to be the fiber of

the map

(a)

Similarly let

and

X (P)

be the fiber of

is finite for all

i, then

X = H Xp). P

(b)

If

is the set of all primes, then

(c)

In general,

wi(X P) = rkwi+l(X)

(P-torsion in

Hi(X)) @ (~p,/~)

of primes not in

P

and

X P = *.

rk

, where

denotes rank.

P'

is the set

Thus

~i(x P) ® ~)= 0

~i(xP) ~ ~ = [(~p,_q/z)~kq+l(X)

and

X(p)

(d)

(xP)(p)

(e)

If

= ,;

w2(X)

(XP)p = ( X ( p ) ) ( P ) ;

is finite, then

is the cofiber of

X P ÷ X.

27

q ~ P.

(Xp) (P) = *.

H,(X,X P) ~ H,(X(p))

(see [Kll]).

Chapter Let

w

a homomorphism; natural

(Rw)"

Definition.

let

R

j:~ + R.

generated

by

Wh(w;R)

H C(Rw)" More generally, homomorphism,

if

w

i.

Let

and

~roup of

w

is a

be the sub<w,lm(J)~R'>.

with coefficients

in

where

is a ring and

and S h a n e s o n [ K 5 ] ,

w:w ÷ {±I}

There H

= GL(I;Rw) 6-+ GL(R~)

A

and

Im(J)NR',

= KI(RW)/H

~ KI(R~).

~:~w

+ A

is a ring

Wh(~).

This has been

though

their definition

from the one given here. The homomorphism

on

by

then we can define

done by C~ppell

group

be a ring with

Th9 Whitehead

is defined

Torsion

be a multiplicative

ring homomorphism

group of

differs

2. Whitehead

w

induces

an involution

Wh(w;R).

Examples:

(i)

Wh(l;Zp)

KI(~ P) ~ ~ p ~ ) S K I ( ~

I m ( J ) N : g ; = -+]I(P),

and (2)

Wh(~;Z)

(3)

If Thus

-- 0 : For

,~ if

P h 3,

-- Wh(~) is

as defined

abelian, w

is

of rank

so

then

28

H -- N •p . in Milnor

I'D20].

Wh(~r;R) ~ ( ( R ~ ) '

an elementary n h 3,

P)

then

abelian

/ H) @ SKl(mr).

p-group,

Wh(W;~p)

= (2~p~) / < ~ , w > , this

Definition. R,

Let

~ = Wl(Y).

p ~ ¢.

is true for all

f:X ÷ Y Define

If P.

be a homology

the torsion

of

to be the torsion of the chain complex where

Mf

is the mapping f

if

~(f;R)

• (f;R)

cylinder

is a simple homo!pgy = 0.

In case

is denoted

Lemmas

~(Mf,Y;R)

(2)

If

f Z g

(3)

If

g:Y + Z

z~R~,

f. over

is an inclusion

then

R

X C Y,

are easy to prove

~(f;R)

(see

R=~):

= T(g;R).

is a homology

T(gf)

equivalence

= T(g) + g,T(f), ÷ Wh(Wl(Z);R)

can be defined

for

f

over

where

is the natural if

map.

H,(Mf,X;R~)

(or s-free). The following

Theorem

C,(Mf,X) ~

= 0.

g,:Wh(nl(Y);R)

is free

over

T(f;R)6 Wh(w;R),

equivalence

properties

(I)

torsion

f,

7.5 - 7.8 for the case

R, then

As before,

of

equivalence

T(Y,X;R).

The following Milnor [D20]

f

n < 2, then

i.

= Wl(X),

Suppos e

X

theorem will be useful

in Chapter

is a finite

CW-complex,

connected

and (i)

Hi(X;Rw)

= 0

for

i ~ r

29

4.

(2)

Then

Hr+I(x;M)

Hr(X;R~)

Furthermore,

=

0

is a f i n i t e l y

we m a y

T(C,(X) ~R~;R)

=

every

R~-module

senerated,

our b a s e s

s-free

M.

Rv-module.

so that

O.

Proof: B i = Im(Bi+l),

choose

for

Let

C i = Ci(X) ~ R ~

H i = Hi(C,).

We h a v e

, Z i = ker(~i) , exact

sequences

0 ÷ Z i ÷ C i + Bi_ I ÷ 0 0 ~ B i ÷ Zi ~ Hi which

gives

the

exact

~ 0

sequence

0 ÷ Z i ÷ C i ~ Zi_ I + 0 since

Hi = 0

for

Since induction.

i < r.

Z 0 = CO,

Thus

C,

standard Let

so that

and

~(B'r+l)

is p r o j e c t i v e homotopic

÷ Z

r

for

if

be the

r

This

denotes

= ~'r+l

inclusion

defines the

÷

o ~r+2

-- 0.

30

by

0

4.2.5.).

and

an e l e m e n t dual

i < r

to

( S p a n i e r [AI5], T h e o r e m

3r+ I = i ~ ' r + I.

H o m (,C r _+ l ,_B r ) ~*, t h e n

is c h a i n

proof i:B

Zi

: . • .+ Cr+ 2 + Cr+ I + Z r

C,' by the

i < r

~'r+l:Cr+l in

homomorphism

÷ Br

Thus so

~'r+l ~'r+l

is a cocycle.

But by

is a coboundary.

Thus

(I),

Hr+l(HOm(C,,Br))

~'r+l = f o ~r+l =

Since

2'

is onto,

r+l So

Br

Z r ~ Br ~

Hr'

Cr ~ Zr~

Zr-i

finitely

projective. H(C,"')

is projective.

and

Cr

is finitely

= 0

Since

generated,

Hr

is

: ...÷ Cr+ 2 + Cr+ I ÷ B r + 0

0 ÷ H

+ 0.

r

since

0 ÷ Z r ÷ Cr_ I ~...÷ C,"'

Br

is contractible.

the sum of the even

to the sum of the odd terms.

Zr-i @ i ~ 0

Thus

Cr+2i ~ "~Cr+2i+l'l

to both sides,

~Cr+2i

-~ H r •

@Cr+2i+l.

is s-free. We may choose = 0

an s-basis

by Lemma 1.7.1.

31

for

H

r

is

Let ~

®c,"'@~:~.

is contractible,

has

C O ÷ 0.

Then

c~ ~ c,"

:

is isomorphic

Br @

be

and so

terms

Hr

C,"

C,"'

C," ~ C,"'

T(C,;R)

Zr, and

and so is chain contractible,

Since

r

of

Hr

c,'

H

summand

Thus

Let

the complex

Thus

r+l"

generated.

zero homology

Adding

for some f:Cr÷B r

fi = i.

is a direct

The complex

Then

fi~'

= 0,

so that

be

The result remains true for a finite CW-pair

(Y,X).

We now give a geometric characterization of torsion over

~.

degree

Let

r ~ ~(P)

r; let

cn(r)

Definition:

and fix a map

~r n :sn-I ÷ S n-I

denote the mapping cone of Let

(Y,X)

of

Cr n, CCr n.

be a finite CW-pair and

suppose Y = X ~.~ so that there exists a map

cn(r) ~-~cn+l(r)

f:C~r n

^ n Cr :(Dn,Sn-l) + (Dn'Sn-l)

+ Y, where

is of degree

(i)

flDnvCsrnISn_ I

(2)

fIS n-I

r, satisfying:

is the attaching map for

is the attaching map for

cn(r), and

(3) f(Dn)cx. We say Y ~

X

is obtained from

X, or

P-expansion,

Y

is obtained from X

~

We write X = Y0,Yi,...,Ym = Y Yi ~ ~p

Y

by elementary P-collapse , X

by elementary

Y. X ~

Y

if there exists a sequence

so that either

Yi ~

Yi+l

or

Yi+l' and we say there is a formal deformation over

from

Lemma i.

X

If

~n(Y,X) ® ~

to

Y.

Y ~

X, then

~i(Y,X) = 0

= O.

32

for

cn+l(r),

i ~ n

and

A deformation over

~

is a sequence of maps

{f0'''''fm )'

fi:Yi ÷ Yi+l

the inclusion if

Yi ~

fi:Yi+l ÷ Yi

the inclusion if

Yi+l'

Y0 = X,

Ym = Y"

If

Z

fi[Z = i, then we say relative to Lemma 2.

Y /~

Proof:

Z

f

(Y,X)

X Q Y

Yi

is a deformation over

and

~p

rel X

be a connected

Inductively assume

Wm(Y,X) @ ~ p

Y ~

of type

= 0,

Wm(Zm,X) @

there exists a map cm(r) + Zm

qW-pair so that the

is a homology equivalence over ~ . Then k k where Z = X U ~ ] C n (r i) u k # C n+l (sl). i=l i=l

Zm = x u c m ( r ) U c e l l s

Zm rel X

so that

~p = 0.

f(~D m+l)CX.

cm+l(r)ccm+2(r)

Zm

W.

Here we let

where

Ct(s), t ~ m.

f:cm+l(r) ÷ Zm

where ~

is a subcomplex of each

Yi+l'

Z.

Let

inclusion

Yi ~

Since

It follows that

extending the map Let

W = Zm u f C m + 2 ( r )

is the natural inclusion. D_k (resp. D+ k)

Clearly

denote the lower

(resp. upper) part of the standard k-disc, and

~ D k = ~D k

~+D k = ~D+ k. Let Then since

W' ~ W'

Em+l = ~+D m + 2 c C m + 2 ( r ) C W , X; so

~

X.

Y /~

Zm ~p

W ~

Thus by induction,

33

and

W' = x u c m ( r ) ~ E m+l.

x~cm+2(r)Uhigher

cells,

Y ~

Since

H,(Y,X;~)

Lemma 3.

Suppose

Wn(Y,X) ® ~ where

k cn+l (si). Z = X U k_7 C n (r i) U ~ i=l i=l = O, we must have

Y = xucn(r),

is a free

k=~.

n h 2,

~w-module

with basis

+ (cn(r) ,sn-I ) $

^ n :(Dn' sn-l) Cr

w = Wl(Y).

extends

Then

[$r n] @ I, Cr n

(Y,X) Proof:

It follows easily that

of fundamental is a free

groups.

~w-module

In E Hn(Dn,Sn-l;Zp) is a lift of

We have that with basis

Hn(Y,X;Zp~)

($rn),(In)

= Hn(Y,X;~ P)

where

Srn:(Dn,Sn-l) ÷ (Y,X)

Sr n.

is an isomorphism

of

Wn(Y,X) ÷ Hn(Y,X) , [@]~+ @,(in),

Zw-modules,

~ Hn(Y,X) (9 ~

The lemma is true for S n-I + X

induces an isomorphism

is a generator and

The Hurewicz map

Hn(Y,X;~)

XCY

n=2

and so

~ ~n(Y,X) ~ ~

m ~n(Y,X) ~

provided the attaching map

is a point. Suppose

(Y,X)

is a finite CW-pair with

a homology

XcY

equivalence over ~ . Then by Lemma 2, we can k k assume Y = X U k_7 cn(ri )U ~J cn+l(si ). Let i=l i=l k YI = X U k.7 cn(ri ). By Lemma 3, Wn(Yi,X) ~ ~ and i=l

34

~n+I(Y,YI) ~ ~

are free

[~l],...,[$k ], and ~i,~ i

[~l],...,[$k ]

The matrix of

~i,~i )

~n+I(Y,YI) ~

~

(Y,X)

÷ ~n(~,X) ~ ~p

w,(Y,X) @ ~

Lemma 4.

@i,¢i.

(Y,X)

Then

A

i-th row, column of

X

rel X.

"elementary" ~w,

where

operations Ri,C i

denote

g ~ w, a ~H(P)

(2)

Ci~+ +-aCig

(3)

R i~* R i + xRj

(4)

Ci~* C i + Cjx

A = CBD

is the identity

A:

R i ~+ -+agRi

can be transformed

Y /~

over

(i)

(1) - (4), then

This matrix is invertible

= 0.

on an invertible matrix

A

(with respect to the

(Y,YI,X).

Consider the following

If

where

coming from the exact

Suppose the matrix of

for some choice of

the

respectively,

is defined to be the matrix of the map

sequence of the triple since

with generators

are the attaching maps.

Definition. maps

~-modules

to

x~ ~p~

B

where

by operations of type C,D

matrices.

35

are elementary

Lemma 5. A

Let

A

be the m a t r i x of

can be t r a n s f o r m e d

type

(i) - (5).

Proof:

(I),

B

Then

Y ~

is o b t a i n e d

from

and the m a t r i x of Suppose

operations

A

of type

homotopy

A

Z

is

so that

to

I

X

Z p~

and the m a t r i x X

rel X, and so

Definition. X

and

there

Y

Let

X

and

(Z,X)

Y p~ Y

is

Z I.

or

so that

y /~

Z

By Lemma 4,

be finite CW-complexes. type over

X = Z 0 , Z I , . . . , Z r = Y, each

finite c o m p l e x and simple h o m o l o g y ~p, Z i ÷ Zi+ I

by

X r~l X.

have the same simple h o m o l o g y

is a sequence

Z

o

there exists

of

Y /~

Then

e oetar

rel

of type

B.

can be t r a n s f o r m e d

(i) - (5).

constructions

by o p e r a t i o n s

(Z,X)

Therefore

0f

X.

(3) or (5), then there exists

rel X

and suppose

to the identity by operations

It follows by elementary

that if

(Y,X)

Zi+ I ÷ Z i.

36

equivalences

Zi over

Then ~p a

if

This gives the geometric characterization of torsion: Theorem 2.

Suppose

(Y,X)

is a finite CW pair with

a homology equivalence over iff

Y @

X

and

Y

Then

If

X

and

Y

are finite CW-c0mplexes, then

hav.e the same simple homology type over

and only if there is a deformation from

Definition. 3N = M + U M over

M+CN, N

~p

T(Y,X;~p) = 0

tel X.

Corollary i. X

~.

XCY

Let ,

M+,M_

between

M c N

(N;M+,M_)

and

to

Then M_

N

Y.

is an h-cobordism

provided the inclusions

are homology equivalences over

is called an

i__ff

be a manifold triad,

closed. M+

X

~p

s-cobordlsm over

37

~

if

~.

T(N,M+;~)

= 0.

Theorem

3.

Let

Mn

be a closed

x £Wh(~iM;~p).

Then there

(N;M,M')

T(N,M;Zp)

Proof:

with

Let

and let maps

x

NI

M x I ~

S I x D n-I ÷ M x I Each row of

~2(NI,M). Ri

Since

represents

exists

an h - c o b o r d i s m

k

by a

k x k

2-handles,

are trivial. A, Ri,

~2(M!).

and M i l n o r [D20], T h e o r e m

ii.I,

Ri

by a trivial

Use these

over

~.

to

embedding. NI

to get

See also

Lemma

~p

Let

N.

This

6.2.1.

38

A,

the a t t a c h i n g M ! = ~+N I (M = ~_NI).

an element

N I = M I x I U (n-2)-handles,

an element

over

matrix

where

represents

in

3-handles

n > 5, and

= x.

be r e p r e s e n t e d

be

manifold,

and

By general

of n-2

> 2,

position

is r e p r e s e n t e d embeddings

to attach

is the desired

h-cobordism

Chapter 3.1.

3. P o i n c a r e

P0.incare Dualit~

Let

X

be a f i n i t e

= ~I(X,*)

, and

c E Cn(X)@~. define

c ~

w:~

:cq(x)

tr:C

If

is i n f i n i t e ,

~

n

are u s i n g

Definition exists

X

and c n

(X)

÷ H

X

n-q

is

acts

the

X

; we

give

Definition image

has

[X]

(X;R~)

of

is the m a p p i n g

on.

e H

n

with

([A2],

chaini

formula

over (X;~

~ ,

R

tr(c) pg.

but

since

holds.

if t h e r e

so t h a t

is an i s o m o r p h i s m .

complex are

over

chain

a basis

complexes

of

cycle

given

X

T(D,;R)

e KI(R~)

cone

c ~

of

.

of

is a c h a i n

the d u a l

torsion

R , then

The

for

since free

C,(X)@ARz R~-modules,

equivalence,

[X]

The

by c h o o s i n g l i f t s

free

of c e l l s

basis.

over in X

R

is d e f i n e d

Wh(~;R)

, where

is a s i m p l e

39

to be D,

Poincare

,

243).

n

Cq(X)

The

same

complex

is a r e p r e s e n t a t i v e Cq(X)

transfer

* ,

Given

trivially

is an i n f i n i t e

:HomA(C,(X),Rz ) ÷ C,(X)SARz c

basepoint

to be c a p p r o d u c t is the

class

HomA(C,(X),R~ )

module

the

(X)

is a P o i n c a r e

is a P o i n c a r e

where

in

of

n

~

supports,

a fundamental

dimension

n-q

tr(c)

compact

[X] (% :Hq(X;Rz)

If

÷ C

with

a homomorphism.

, and

(X)@ ~ ÷ C

X

CW-complex

÷ {+l}

, A = ~

where

we

complexes

complex

over

Theorem

i.

simple

R

if its t o r s i o n

If

Mn

Poincare

Proof:

is a c l o s e d

complex

Our proof

be a f i n i t e the

sheaf

U ~

H,(U;~)

~,(Y;~)

over

a more

CW-complex

and

homology

The

stalks

= H,(Y,Y-y;~)

n

over

~,(Y;~)

"~,(Y;~)

According

M

is a

a n y ring.

result.

a sheaf

groups

then over

general

~

of

is zero.

manifold,

of d i m e n s i o n

gives

of l o c a l

R

Let Y

.

Y Define

by the p r e s h e a f

are g i v e n

to B r e d o n

by

[A2],

pg.

208,

Y there

exists

a spectral

sequence

Ep'q = HP(X;~q(Y;Z~))=~Hq_p(Y;~)

Y if

is c a l l e d ~p(Y;R)

constant M

a homology = 0

with

stalks

is a h o m o l o g y Let

If

Y

for

B = R~

manif©id

p ~ n

and

isomorphic

manifold

over

, ~ = ~I(Y)

is a h o m o l o g y

over

manifold

R

of d i m e n s i o n

~ = ~n(Y;R) to

R .

n

is l o c a l l y

In p a r t i c u l a r ,

R . , and over

~ R

as in S e c t i o n

1.2.

, then

H P ( y ; ~ t) & H P ( Y ; ~ )

E p,n ~ E p,n ~

This

isomorphism

simple

is in f a c t g i v e n

on the c h a i n

level.

Hn_p(Y;'.~) by c a p p r o d u c t

See B r e d o n

40

[A2],

and

is

Corollary

10.2,

W a l l [HI9], T h e o r e m Let i.e.,

A C Wh(~;R)

A* = A

C0rollary homology X

1.

.

be a c o n j u g a t e - c l o s e d

(A

If

is a l s o

Mn

Definition. R

called

complex

A finite

R

R

CW-pair

of d i m e n s i o n

n

and

so t h a t

over

f:M ÷ X T(f;R)

with

(Y,X)

if t h e r e

subgroup,

self-dual.)

is a m a n i f o l d

equivalence , over

is a P o i n c a r e

over

2.1.

~ A

torsion

is a , then

lying

is a P o i n c a r e

is

[Y,X]

in

A

.

pair

e Hn(Y,X;Z)

so t h a t

[Y,X] •

:Hq(Y;R~)

is an i s o m o r p h i s m , complex

over

between

Poincare

Theorem

2.

dimension

R

~ = ~I(Y)

so t h a t

X,Y

and

, and =

be P o i n c a r e

(a)

[X] ~ : K q ( X ; R ~ )

(b)

there

(Y,X;R~)

X

÷ K

split

is a P o i n c a r e

[X]

is of d e g r e e

f:X ÷ Y

exist

n-q

~[Y,X]

complexes

Let n

÷ H

A map 1 if

complexes

a 1-connected

n-q

(X;R~)

short

exact

f,[X]

over

÷ Y =

R

[Y]

of

d e g r e e ' 1 map.

is an i s o m o r p h i s m , sequences

f* 0 ÷ Kq(X;R~)

÷ Hq(X;R~)

÷ Hq(Y;R~)

+ 0

0 ÷ Hq(Y;R~)

f~ ÷ Hq(X;R~)

÷ Kq(X;R~)

÷ 0

41

f:X

Then

and

Proof: For

(a) (b),

this

identity.

also

C,(f)

X

H

of

Y

= K

r

is s u r j e c t i v e

holds

in the

torsion

assume

q = n - r + 1 . (X;R~)

and

dimension , X ÷ X'

X'

X

n > 5 . , and

torsion

Then

X

in

case.

Now

Wh(#;R)

Define

satisfies

the h y p o -

= X

there

~JM

Hq(f;R~)

(on the c h a i n

case,

with

complex

equivalence

where

~M

42

M

C,(f)

over

is a P o i n c a r e

homoloqy o

We h a v e

A

be a P o i n c a r e

a simple

, so that

is i n j e c t i v e ;

~ Kq(X;Rz)

÷ Kr_I(N;Rz)

in the r e l a t i v e

Let

f

[X~q(X;R~)

furthermore,

This

3.

A C

(Mf,X)

with

is true

and

relative

in

is an i s o m o r p h i s m

also

Hn_q(X;R~)

2.1.

[X] (~:Kq(N;R~)

Theorem

of cap product.

f* --~

Hn-q(Y;R~)

f,

, and

Theorem

(f;R~)

l

have

Let r

the n a t u r a l i t y

splittings.

result

and

Remark. and

the

= C,(Mf,X)

thesis

~

Thus

defines

A similar suppose

from

the c o m p o s i t i o n

f, -~ Hq(Y;R~)

Hq(X;R~)

is the

follows

~

complex

level).

= C,(Mf,X

of over

over

is a m a n i f o l d

U ~Y)

f r o m ....D n

obtained

Furthermore,

by addin@

~I(M)

1-handles

÷ ~I(X')

Proof:

Assume

X

has one

0-cell

io

say

T h e n we

k

1-cells

can a s s u m e i'

'

attached

at

4°

~i e ~I(X) there

We h a v e

~ , i = ai 4o - ~o

by

4~

with

-

2

.

X

both

ends

, where

By d u a l i t y ,

n-cell Zn and k k Di n = ~ w ( ~ i) £i n-I - ~i n-I i=l

so t h a t

n

i

is d e f i n e d

exists

d i m X ° _<

is s u r j e c t i v e .

is c o n n e c t e d . and

, and

one

we can

assume

(n - l ) - c e l l s

in-1 l coefficients).

(~

,

!

Define of

X

with

X ,

as

X

~ M , where OSM M = Dnu k 1-handles

X

is the (n - 2 ) - s k e l e t o n o n-i , so t h a t ~4n = E w ( ~ i ) ~ i £i

n-I - 4i

~-coefficients.

If

(X,~X)

theorem

remains

t y p e of

(X',~X)

is a P o i n c a r e true,

with

, where

pair

over

(X,~X) X'

~p

having

= X O ~J M ~M

,

, then

the

the

~p-homology

~X C X O

,

d i m ( X 0 - ~X) ~ n - 2 .

Theorem

4.

Let

dimension homology chain

n

so t h a t

equivalences

equivalent

T(Y,X+;R)

(Y;X+,X_)

=

(-i)

the over

be a P o i n c a r e

inclusions R .

Assume

to a c o m p l e x w i t h n-i

~(Y,X_;R)

X+ C

*

43

triad Y

,

over X_ C

C,(Y,X+)

2 non-zero

@ R

terms.

R Y

of are

is s i m p l y Then

Proof: with

Let

matrix

given

by

We may

M

be an

R~-module

A =

(aii)

Then

C.(Y,X+)

0 ÷ C k ÷ Ck_ 1 ÷ 0 . over

R ,

represents

~

(-i) k-I

(-i) n - k on

8 R~

T(Y,X_;R)

Wh(z;R)

(-I) k-I

sends

T(Y,X+;R)

A

Wh(~;R) to

A *t

= (-i) n - k

equivalence

the m a t r i x

is c h a i n of

Since ,

T(Y,X_;R)*

44

+ M*

(a~i)

of

Wh(~;R)

÷ 0

so the m a t r i x

in

is a h o m o l o g y

in

0 ÷ C* 2"÷ C* n-k+l n-k

A *t =

f*:M*

form

Furthermore,

T(Y,X+;R)

, and

the

X+ C Y

is i n v e r t i b l e .

Dually, C.(Y,X_)

has

a homomorphism

homomorphism

has m a t r i x

8 R

Since

f:M ÷ M

the d u a l

f*(x) (m) = x ( f ( m ) ) *

assume

and

2" the

equivalent

to

represents involution

*

3.2.

If

Spherical

~

is

a

R-spherical

is h o m o l o g y

equivalent

then

the

define

Fibrations

Normal

fibration

over

Thorn s p a c e

and

R

to

of

~

Maps

(i.e.

a sphere) by

T(~)

the

p:E

fiber

÷ X

= C(E)U

,

X

,

P where

C(E) We

is

now

prove

Spivak

[EIg].

Theorem

i.

over , so

.

X

, with

Then

the

spherical.

The

Proof:

Let

of

, so

~X

N

M =

~i(N

- X)

for

i = k - 1

~i(N,No)

~N ° = ÷

.

We is

an

Since

for

i ~ k

an

theorem

pair

over

~

%-spherical

equivalent

unique

up

to

of

fibration

to

Hn+k(T(~),T(~I~X); is

of

S k-I ~)

over is

stable

fiber

P

neighborhood

S n+k-I

Let

~i(N)

= 0

of

a regular

.

the

a Poincare

~

M = N N

~M

of

homology

over

S n+k-I

N o~

.

exists

fiber

be

that

in

there

fibration

equivalence

> 4

E

be

generator

homology

k

of

(X,~X)

n

that

cone

a generalization

Let

dimension

p

the

N

=

o

have

N

is

;

an

isomorphism

for

N

is

o

÷ N

- 1

- X

X

a regular

~N - M is

of

in

D n+k

,

neighborhood

then (n + k ) - m a n i f o l d i

< k - 1

a homotopy

and

and onto

equivalence,

.

!

Let NO

÷ N

t If

:E'

÷ N

be

F

is

the

a fibration fiber

of

equivalent t'

45

, then

to nl(F)

the ~

inclusion ~2(N,No)

=

0

•

Let

~ = Wl(X)

so t h a t

Then

[N,~N] ~ :HP(N,No)

Since

(N,M)

so t h a t

~

(X,~X)

[X,~X]~

isomorphism. [X,~X]

isomorphism, ([N, SN] N

exists

U

covers

~)

P

n

(N,M;~)

is an [N, ZN] ~ U =

÷ HP+k(N,No;~

[X, SX] N

Y

t':E'

÷ N

by a s p e c t r a l

e Hn+k(N,~N;Z)

e H

so t h a t

y £ HP(N;%w)

£ H k ( t ') so t h a t

It f o l l o w s

[X,~X]

U s H k (N,No;~)

for

[N,~N]

is an i s o m o r p h i s m .

÷ Hn_p(N;~

Uu:HP(N;~_z)p

U) K3 y =

of u n i v e r s a l

exists

:HP(N,M;~_z)p

since

exists

÷ Hn+k_p(N,M)

, there

Choose

Then

there

.

Now

,

consider

the

fiber

.

with

U u:HP(~;~p)

sequence

z) is an P [N,~N] ~ (y U

F

Y) =

fibration Then

there

~ HP+k(t'~).

argument

that

F = S k-I

mod

C , where C is the S e r r e c l a s s of a b e l i a n g r o u p s w i t h nI nr exponent Pl "'" Pr ' Pie P . (A p r o o f is g i v e n in Browder

[G5],

homology

Lemma

equivalent

1.4.3. over

for ~p

P = 0 ). to

S k-I

Thus Let

F

is

f:E

÷ X

he

!

the p u l l b a c k Spivak

of

t

fibration;

We h a v e

T(~)

by the denote

(Dn+k,s n+k-l)

c ~n+k

(T(~)'T(~I~X))

÷

and (N/No, The

Zn+k(T({),T(~lSX)) ® %

sends

~ ® 1

X c N

.

This

is the

it by

= N/N °

collapse

inclusion

T(~I~X)

= M/~M

M/~M)

defines

the H u r e w i c z

, and the

map

÷ Hn+k(T(~),T(~ISX);Z ) P

to a g e n e r a t o r .

46

To s h o w

uniqueness,

suppose

El:E 1 ÷ X

satisfy

the

(assume

~X = @

form

the b u n d l e

the m a p

F

The m a p Wall

A

a map

Theorem

- ~2 )

be the

"

by

suspension

3.3,F

trivial

T(~I

points

over

~p

homology

type

In S e c t i o n classifying

for

over

fibration

with

Corollary

4.3.,

localized

space

× X

Stably~

and c o n s i d e r

X + X

is a d u a l i t y

which

over

by

map ~p

induces

each

- ~2 )

is o b t a i n e d

E

so

over

we

E E1

over

~ m

by

~p

~2

constructed

fibrations

on coLet

E ÷ X

in

identifying homologically have

the

same

monoids

for p r i n c i p a l with

fiber

a fibration

Gn(~p%

and

Gn(~p)-bundles,

homology

equivalent

~

Such

these

S n-I , and a c c o r d i n g to S u l l i v a n P f i b r a t i o n s are c l a s s i f i e d by the

(BGn) P ,

is

~p .

B G n ( ~ P)

P fiber

in

and

there

fiber

is fiber and

.

- ~2 )

As

isomorphisms

suspending

and

× X

, i.e.,

in d e g r e e s

Therefore

1.3,

spaces

equivalently S n-I

in .

both

simplicity).

x

the d i a g o n a l

obtained

Then

+ X

x (El - ~2 )) = T(~2) /~ T ( ~ l

~p-Coefficients

fibration

for

over

is c o - r e d u c i b l e

with

E1 - ~2

to

A ÷ T(~2

T ( ~ 1 - ~2 ) + S m

homology

fiber

T(~I)

~2:E2

by

is d e f i n e d

[E21], T(~i

~2 × (El - ~2 )

defined

al ÷

S n+k

so

theorem

,

G n = Gn(~)

47

is e q u i v a l e n t

Thus

to a [GI5]

B G n ( ~ P) ~

(BGn) P

Let

SG

be the s u b m o n o i d

n

equivalences cover of

of degree BG

homotopy

of

1.

groups

of

BSG

defined

n

Then

and stably,

n

G

BSG

BG ~ RP

are

is

n

by h o m o t o p y

the

x BSG

finite

universal

.

Since

(~i(BSG)

~ l i÷m

i > 1 ) ,

BH ÷

H = TOP,

(BG)p.

Theorem Then

Let

2.

"]-[ (BSG) p{p (P)

PL or 0. Gp/H

Let

Gp/H ~

Proof:

denote

SGp/SH

(SGp/SH)

Then there

be the fiber of

x 4 +

@ ~p ~ ~i(G/H)

~i(SGp/SH)

® ~(p)

....

÷ ~rl(BH)

÷

Furthermore,

that

~i(SGp/SH)

~I(SGp/SH)

the s e q u e n c e

0 + ~I(BH)

reduces

(BSH) p.

® ZZp,

ladder

÷ ~i

and it f o l l o w s

BSH ÷

~ z i ( B S H ) Q ~(p).

÷ ~i (BSH)

~i(Gp/H)

map

and

~i(SGp/SH)

We h a v e an e x a c t

is a n a t u r a l

the fiber of this map.

"'" ÷ ~i(SGF/SH)

Thus

gi+k+l(Sk),

k

(BG) p ~- K(2~p,l)×

Let

the

(BSG)p)

÷''"

l

Trl((BG) P) ÷ . . . & ~i(Gp/H)

= ~I(Gp/H)

= 0, and

for

i > 1.

~0(SGp/SH)

to

= Z / 2 Z ÷ Zl((BG)p)

48

= ~

÷ ~0(Gp/H)

÷ 0,

= 0.

and so -i ~

~0 (Gp/H) = ~'+p (since the map

-i).

Therefore

Gp/H ~ ( S G J S H )

Break the fibration composition of

Since

(SGp/SH) zi(G/H)

up into the

(BSH)p ÷ (BSG)p.

The

(BSH) (P)~ the colocalization,

fiber of the second is a fibration

and

sends

× ~p+.

BSH ÷ (BSG) p

BSH ÷ (BSH) p

fiber of the first is

~/2Z ÷ ~

(SG/SH) p & (G/H)p.

and

contain one factor of

~i(BSH)

ZZ if

Thus there is

with fiber

÷ (G/H) p

and the

(BSH) (P)

are finite if

i~4k and

i=4k, the long exact sequence

of the above fibration reduces to 0 + zi(BSH) ~ ~(p) + ~ i ( S G J S H )

i = 4k+l, 0 ÷ ~4k+3((BSH) (P))/(~p/~)

÷ Zi(G/H) @ ~p + 0

4k+2 ÷ ~4k+3(SGp/SH)

~4k+3(G/H )

0 + ~4k((BSH)(P)) since any map necessarily

÷ ~4k(SGJSH)

~i(G/H) @ ~.p ÷ ~i_I(BSH) ® ~(p) 0

is

(i~4k).

~4k+3((BSH) (P)) ~ W4k+3(BSH)

also true for

® ~p ÷ 0

+ ~4k(G/H) ® ~p ÷ ~p/~ ÷ 0,

This clearly implies the result for Since

÷

i=4k+3.

® ~(p)

i=4k+l, ~/~,

4k+2. it is

Finally we have

0 -+ ~4k((BSH) (P)) ÷ W 4 k ( S G / S H ) / ~ . ÷ (~4k(G/H)/~)® which concludes the proof since * See Theorem 4.4.3. 49

~

÷ 0

W4k((BSH)

Note:

(P)) @ Z{(p) ~ Z4k(BSH)

In the c a s e

We n o w

show

how

space

Gp/H

P = ~

to S u l l i v a n . Let

n.

P = all

apply

(X,~X)

An H - n o r m a l

the

primes,

Spivak

to s u r g e r y

is d e f i n e d

Z(p).

Gp/H

fibration

theory,

be a P o i n c a r e map

@

pair

due

over

= BSH

and

to be a d e g r e e

50

classifying

in the

R

× ~'+.

case

of d i m e n s i o n 1 map

¢: (M,~M) with

an

for

some

bundle

÷

(X,~X)

, where

eguivalence

M

is a h o m o l o g y Two

oval

in some

large

equivalence

normal

maps

if t h e r e

M1 n

~M 1 =

M2 =

is an

X

over

R

9,

= #i

B:°~N ÷ ~

b:~ M +

~M

is the n o r m a l

We a s s u m e

~I~M:~M

are n 0 r m a l l y

so t h a t

, with

so t h a t

~N = M I U

The

to

BI~M.:OrM.

First

of

the

X .

3.

(2)

and

let

the

Let

case

~

is d e n o t e d

invariants

NIH(x;R)

over

R

of

and

X

~X =

denote

NIH(x;~)

~ ~

the

over

to an

and

(i) ~

~

in this

Let

iff

Spivak

fibration

homology

equivalence

over

c:sn+k--9

T(T)

to

X

is f i b e r

over

so t h a t Making , we get

P)

over

X

homology

over

NIH(x;~)

~ e Hn+k(T(~;~ H-bundle

is a g e n e r a t o r .

o/

H-bundle

case,

be an

regular

is

Then

(i)

Proof:

classes

set of n o r m a l

consider

Theorem

÷ ~i

1

b.

set of c o b o r d i s m

is c a l l e d

M2 ,

an e q u i v a l e n c e

1

equivalent

÷ ~X

.

~:N ÷ X

~IMi

together

of

, ($2,b2)

is a m a p

of c o v e r i n g s

, where

sphere.

(#l,bl)

~M 2 ,

H-manifold,

of c o v e r i n g s

~

cobordant

class

class

H-bundle

of

M

X

+~

a generator,

h:~÷

~p

The

collapse

C , ( i n + k)

= ~

, where

T ( h ) C : S n+k a normal

map

51

÷ T(~)

,

[X,Gp/H]

denote and

equivalent

~

a fiber

defines

a map

_n+k in+ k ~ H n + k ( b ;~p)

transverse

, where M

M =

(T(h)C)-Ix.

;

b Conversely,

let

~ M

CM:S n+k

÷ T ( ~ M)

fibration

be

over

be a n o r m a l

the c o l l a p s e .

~

generator

(2)

It is e a s y

pondence

with

and

let

Then

~

is a S p i v a k

since

T (b), (C M) , :Hn+ k

sends

map

%X

(sn+k;~)

+ H

(T(~);~)

n+k

to g e n e r a t o r .

the

to see

that

NIH(x;~)

set of h o m o t o p y

is in i-i

classes

of

corres-

lifts

BH I

X "~

,>

(BG)

f

where

f

lifts

g0,g 1

denotes

Then

the

denote

, G:X

E ÷ X fiber

the c l a s s i f y i n g

are h o m o t o p i c

X x I ÷ (BG)p f i = 0,1 . Let

P

if t h e r e

x I ÷ BH

denote of

E ÷ X

the a c t i o n

of the

map

fiber

~p/H on

52

~/ , a n d two

exists

, so t h a t

the p u l l b a c k is

for

GIX

E .

x i = gi'

BH ÷

of

;

a l i f t of

let

(BG)p

T: (Op/H)

Clearly

by

f .

x E ÷ E

NI H ( X ; ~ )

is the

set of h o m o t o p y

Since Define

NIH(X;~p)

F: (Gp/H)

is a h o m o t o p y

must over

~X ~ ~

have

an

equivalent and

to m a p s

, then

~X .

NIH(X;~p) over

~p

If

so t h a t

M

equivalence

is a n o r m a l

, there by

F(y,x)

and

so s e c t i o n s

in o r d e r M

# ~

tel ~ I ~ X

R

map.

~ to an

Then

F

corres-

~ ~

homology

~I~X

, we equivalence

is an

H-bundle classes

maps

f:M ÷ X

is a l w a y s

53

.

H-bundle.

is f i b e r h o m o l o g y

, then we will This

s:X ÷ E

E ÷ X

NIH(X;~p)

assume

set of h o m o t o p y

and

of

.

.

for

~X ÷ X ÷ G p / H

E ÷ X

= T(y,s(x)).

and a normal

iff

of

is a s e c t i o n

X ÷ Gp/H

is a m a n i f o l d over

of s e c t i o n s

Thus we may

NIH(X;~p) < >the

X ÷ Gp/H

f

× X ÷ E

H-manifold

~p~ M ÷

Now we have

~ ~

equivalence

pond bijectively

If

classes

over

X

,

of m a p s

to the b a s e p o i n t .

is a h o m o l o g y

henceforth the case

assume if

that

R = ~ .

Chapter 4.1.

4. S u r s e r y

Coefficients

Surgery.

Let an e m b e d d i n g .

where

f,

Mn

be a c l o s e d

Form

M'

We

embedded

say

M'

sphere

If

is o b t a i n e d

from

M x I

Lemma

1.

M',

the t r a c e

k+l

called

and

If the

trace

where

space.

iff

~f0

a

x Sn-k-l,

if

surgery

on

Rourke

Then N

and

say

M N

to

M',

is o b t a i n e d

[J15]

if

that

there

theory.

for

~:M ~ X ~

54

to

H = PL,

This

trace

is

~T

f, t h e n

be a map,

admits

is n u l l - h o m o t o p i c .

and

See M i l n o r

[B12] if

H = TOP.

by

M

is a s e q u e n c e

is an e m b e d d i n g

defined

Let

manifolds

fl,...,fr.

Sanderson

f:S k x D n - k ÷ M

It f o l l o w s

from

is H h o m e o m o r p h i c

by

of M o r s e

decomposition

x 0.

We

between

defined

essence

surgery

we r e g a r d

(k+l)-handle.

surgeries

Siebenmann

of the

U D k+l f,

angle

by

where

surgery.

so that

f0 = f l S k

any

of the

fl,...,fr

a handlebody

M

is a c o b o r d i s m

H = TOP.

H = DIFF,

Kirby

N

be a c o b o r d i s m

is the

the

from

x D n-k,

__if

of the

This

[DI9] if and

N

d i m N _> 6

of e m b e d d i n s s

Proof:

trace

by a d d i n g

Let

with

the

f:S k x D n - k ÷ M

f(S k x 0).

N = M x I •D f

is c a l l e d

and

straightening

f:S k x D n - k ÷ M x l, t h e n and

H-manifold

= (M-f(Int(S k x Dn-k)))

~ flS k x S n - k - l ,

H = DIFF. the

with

and

N

is

N = M ~-/ D k+l, f0 where

an e x t e n s i o n

X

is

~:N ÷ X

Define of c o m m u t a t i v e

Wk+l(¢)

group

of h o m o t o p y

classes

diagrams Sk

Dk+l

C

M

There

to be the

)X

is a long exact

.

sequence

¢# ... ÷ ~k+l(¢) Equivalently,

Wk+l(¢)

an inclusion.

Clearly

from an element If two cases,

in

M

admits

In the

first

into the

× I)UM',

rglative

definition:

cobordant

if there

Lemma

2.

attachin5 to

M × I

fixed

by

f0

comes

interior N

then we c o n s i d e r

or doing

surgery

on

assume

f:S k × D n-k ~ M

of

Do surgery

M.

is a m a n i f o l d

~M' = 3M × I.

This

as

with is called

and

handles

the boundary,

(M!,~M I)

and

is a m a n i f o l d

N

we need

(M2,~M 2)

the

are

with

~P = ~ M I U 3 M 2.

Any c o b o r d i s m

rei

iff

¢

to the boundary.

following

2

÷...

we replace

an e x t e n s i o n

case,

If we w i s h to change

~N = M I U P k 2 M

where

boundary,

the b o u n d a r y

T h e n the trace

~N = M U ( ~ M surger~

¢

has n o n - e m p t y

is an e m b e d d i n g before.

= ~k+I(X,M),

+ Wk(¢)

Wk+l(¢).

leaving

the boundary.

÷ Wk(M)----+~k(X)

to

of

3M × I

(M,3M)

c a n be r e a l i z e d

followed by

~M × I.

55

attaching

bY handles

Proof:

Let

N

~N = M U P U M ' ,

~Q = BM' (Q,~Q)

and

be a c o b o r d i s m

from

~P = ~ M U ~ M ' .

Define

Q × I

is a c o b o r d i s m

(M,~M)

to

Q = M

from

(M',BM'),

UP; ~M

(M,~M)

then

to

since

B(Q x I) = Q x O U ( B Q

x I)taQ

= M x 0 U(PU3M'

Also,

N

is a c o b o r d i s m

to the

boundary

from

(Q,~Q)

x 1

x I)%JQ

to

x I.

(M',~M')

since

~N = M ~ O P U M '

= QUM'

= qu(aq

By L e m m a

i,

Q x I U

N

x I)UM'.

is t h e d e s i r e d

56

cobordism.

relative

4.2.

The

Problem

of S u r g e r y

Let

be a finite

considered homology

here

type

over

is the

over

following:

R

When does

X

have

the

consider

the

of a m a n i f o l d ?

related

problem:

if

is a map,

i_~s ¢

Mn

is a m a n i f o l d

and

cobordant

to a homology

equivalence

R? simplicity,

find a map

¢':M'

To do this,

we would

~k(¢)

÷ X

Since

bundle

equivalent

!.

¢

S k x D n-k ~ M

is p a r a l l i z a b l e , f0*TM

is a bundle of

M

to saying

Let

is n o r m a l l y

to

~

over

in some

large

that

¢:M n ÷ X cobordant

¢

= 0.

in

and do surgery. with

f0

~f0

extends

be trivial. X

sphere.

be a normal

map,

¢':M'

÷ X

to

¢*~

This map,

trivial.

We can do

so that

is a normal

to a m a p

wish to

~i(¢') ® ~ p

elements

if

must

We thus

with

like to r e p r e s e n t

f:S k x D n-k ~ M, then

normal

R = Zp.

f0:S k ÷ M~ be an e m b e d d i n g

S k x D n-k

if there

assume

cobordant

by e m b e d d i n g s Let

Lemma

The m a i n p r o b l e m

on this problem,

For

this

CW-complex.

To get a t o e - h o l d following ¢:M ~ X

X

with Coefficients.

is the

is

as in Section

n ~ 5. with

Then ¢'

[n/2]-connected.

The proof

over

R

will

be given

below.

By C o r o l l a r y

3.1.1,

of a m a n i f o l d ,

then

if X

X must

57

has the h o m o l o g y be a Poincare

type

complex

3.2.

over

R.

degree

Furthermore,

1.

Our p r o b l e m

i)

When

is the

2)

When

is

¢:M ÷ X

Question

1 is best

lifting

X ~

of these

notes.

Wall [HI9]

let

subcomplexes

of

there

a handle ¢i:Ni

Assume

X

~ Xi

Cr+l:Nr+l

cobordant

have

to a h o m o l o g y


the o b s t r u c t i o n

2 will

the

Ni,

NO = M

Ni_ I

Assume

and a h o m o t o p y

We construct

i ~ r.

cells

N

the rest

of

of

X-M

by i n d u c t i o n

I, o b t a i n e d

x

to

Following

be a sequence

one at a time.

to

occupy

is an inclusion.

by a t t a c h i n g

manifolds

for

must

an H - b u n d l e ?

XI,X2,...,X m

formed

of index

X

Question

¢


exist

¢

to two questions:

by a n a l y z i n g

BH.

M = X0,

of d i m e n s i o n that

to

i:

of

that

R?

answered

(BG)p

of Lemma

fibration normally

over

shows

is now r e d u c e d

Spivak

equivalence

Proof

a calculation

by a d d i n g

equivalence

and

r+l

÷ Xr+ I. Suppose

8N r = M U M r ,

determines

Xr+ I - X r

the d i m e n s i o n

an element

of that

Since

N

handles

of index

_

handles

of index

> (n+l-k)

is i-connected, a ' g wi(¢r )


and

a m ~i(X,Xr),

cell

where

i

is

cell. is formed

r

The

Cr = CrlMr"

so

correspond

from

it is formed > k+l

M x I from

Mr

> i+l.

~i(¢r ) ~ wi(X,Xr). to

~, and let

58

by a t t a c h i n g x

I

Thus Let

by a t t a c h i n g (Nr,M r )

CJ

S i-I

Di

I g0 ,X

M

be a representative Since X

so that

f0'

Classification

i-1 < [n/2],

equivalence.

f

Continuing

the k-skeleton

of

Therefore

(Nm,M m)

~m:Mm ~ X

is k-connected.

are actually

Corollary (Y,X)

i.

is s-trivial.

is a finite

is normally (a)

if

if

f

an

is an embedding. + Xr+l, we get

(X,X m)

a homotopy X m = M U X (k),

is k-connected.

and it follows

that

It is easy to see that

~

and

cobordant. + (Y,X)

CW-pair,

X a finite

to

n = 2k,

~'I~M' (b)

flS i-I x 0 = f0'

#:(Mn,~M)

cobordant

This

T(S i-1 x D n-i+l) ~ TM.

inductively,

normally

Let

is contractible,

so that

is k-connected

over

back by

this defines

~r+l:Nr+l

X, so

Di

~

Theorem,

we may assume

to get

is a bundle Pulling

But

f0*TM

f:S i-1 x D n-i+l + M

Do surgery on

~m

trivial.

is s-trivial. Thus

there

a stable bundle m o n o m o r p h i s m

immersion

X (k)

is stably

is trivial.

By the Immersion

Since

Is a normal map,

J'g0 *~

J'g0* ~

defines

~

T M ~ ~*~

f0*TM~

so

of ~'

be a normal map, where CW-complex.

~':(M',3M') ~'IM'

+ (Y,X)

is k-connected

Then

where and

is (k-l)-connected.

n = 2k+l,

~'IM'

k-connected. 59

and

~'I~M'

are

4.3.

S ursery O b s t r u c t i o n We describe

L-theory Let

A

Groups.

here the functors

w h i c h provide

us with surgery o b s t r u c t i o n

be a ring w i t h i n v o l u t i o n

ideal generated

by

Definition.

(-l)k-Hermitian

(G,k,~) X:G

×

A

where

G

A

Ak

in

direct

~:G ~ A/i k ):G ÷ A

form over

be the

are maps

A

is a triple

A-module, so that

k(x,

(b)

k(y,x)

= (-!)kk(x,y) *

(c)

k(X,X)

: ~(x) + (-l)k~(x) *

(d)

k(x,y)

= ~(x+y)

(e)

~(xa)

(f)

Ak:G ÷ G*,

is a k - h o m o m o r p h i s m

-~(x)-~(y)

: a*u(x)a.

of the form

Ak(x)(y)

(G,k,~)

= k(x,y)

is an isomorphism.

is defined

to be the t o r s i o n

e,f

~(e)

KI(A).

~ I, then

G

has a basis

(G,k,~)

sum of standard Let

be a self-dual

subgroup.

Definition.

(G,k,~)

a free and based

A

is called a kernel.

form is one with torsion

Let

~ ~(f) = 0,

is called a standard plane.

planes

ACKI(A)

with

(-l)k-A-Hermitian

HCG

Ik

groups.

xeA

(a)

In case k(e,f)

and let

is a free and based

and

The t o r s i o n of

*

x + (-l)k+Ix *,

G

of algebraic

in

A.

be a (-I) k A - H e r m i t i a n

submodule.

Then

60

H

An

form and

is called a

subkernel

of

(G,I,~)

preferred basis for I(H x H) = 0, by

I

if the basis of

G, making

G/H

H

based, and

~(H) = 0, and the map

G/H ÷ H*

is an isomorphism with torsion in

Lemma i.

(G,k,~)

extends to a

defined

A, an A-isomorphism.

is a kernel iff there is a subkernel of

(G,I,~). Proof:

If

(G,I,~)

is a kernel, then

el,...,en,fl,...,f n 1(ei,f j) = 61j ,

so that

Conversely, Then, as

G/}I ÷ H*

basis for

el,...,e n. let

el,...,e n

= 0.

Then

Then is a basis for

be the

is a subkernel. H.

we get a H*.

Let

of this basis in

1(ei,f j ' ) = (-i )k~ij , and G.

H

be the basis for

is a A-isomorphism,

be representatives

Let H

G/H, using the dual basis for

fl',...,fn'

has a basis

u(ei) = ~(fi ) = O,

1(ei,e j) = X(fi,fj)

submodule generated by

G

G.

el'''''en'fl

'''''fn'

Define

fi = (-1)kfi ' + (-l)k-l(ei~i

+

~ e.1(fj',fi')) i<J j

where

~i E ~( f'i ). Then shows

G

is a basis for

G

and

is a kernel.

Corol!ary i. a subkernel of H + H'

el,...,en,fl,...,f n

Let

H

be a subkernel of

(G',I',~').

(G,k,~)

and

H'

Then an A-isomorphism

extends to an A-isomorphism

61

(G,X,p) ~ (G',k' , p')

J

Definition.

Define

(-l)kA-Hermitian direct

LA2k(A)

forms over

A, with addition defined by

sum, modulo the equivalence relation

there exist kernels Lemma 2. Proof:

to be the semi-group of

LA2k(A)

K,K'

so that

if

G ~ G'

G ~ K ~ G' ~ K'

is an abe lian sroup.

All we need show is that

kernel.

Let

ei',ei"

the corresponding

el,...,e n

(G,~,U) ~

be the basis of elements in

(G,-k,-~)

G.

is a

Denote by

G ~ G,

e i' : e i + O,

el" = 0 + e i • Let

HCG

~ G

be generated by

~(e i' + ei",e j' + ej") = O, Furthermore,

the map

G ~ G/H ÷ H*

ei'~-~ ~(ei' , and so has matrix k(ei,ej). H

(aij)

Then

u(e i' + el") = O.

is given by

)

where

But by hypothesis

e i' + ei".

aij = ~(ei',e j' + ej") =

this matrix is in

A.

Thus

is a subkernel.

The group

LA2kCA)

is called the

To define odd dimensional

(2k)-th Wall sroup of

surgery groups, let

A.

Kn

denote the standard kernel with basis el,...,en,fl,...,fn,

~(e i) = u(fi ) = 0,

~(el,e j) : ~(fi,fj) = 0, Let

Uk(n,A)

~(ei'fJ) : 6iJ = (-1)k~(fj'ei)"

denote the group of isometries of

the group of A-automorphisms

Kn ÷ K n

62

preserving

Kn, i.e., k

and

~.

There is a map matrix; let

Uk(n,A) ÷ KI(A)

UkA(n,A)

Let

sending an isometry to its

denote the inverse image of

EUkA(u,A)

be the subgroup of

ACKl(A).

UkA(n,A)

generated by (a)

isometries which fix the subkernels generated by

{el,...,e n}

and

{fl,...,fn}, and induce

automorphisms with matrices in (b)

isometries which fix

(c)

the isometry

an

Then

EUkA(u,A)

el,...,e n.

defined by

en(fl ) = (-l)kel '

A.

an(e I ) = fl'

~n(ei ) = el'

~n(fi ) = fi' i > i.

can be regarded as the group of matrices

generated by matrices of the form

(a)

IP°l

(c)

the

Embed way; then

0

p,-1

'

PEA'CGL(n,A),

0

t

(-i) k

0

matrix

UkA(n,A)

the inverse image of

0

in

UkA(n+I,A)

EUkA(n,A)C EUkA(n+I,A).

63

in the obvious

A,

Define

UkA(A) = lim UkA(n,A),

The (2k+l)-th Wall 5roup 0f

A

EUkA(A) = lim EUkA(n,A). is defined by

UkA(A) LA2k+l(A) =

/EUkA(A).

We will show in Corollary 5.2.1 that abelian group for

A = ZpW.

LA2k+l(A)

is an

It is in fact abellan for all

A, as is shown algebraically in Wall [H19]. Consider the following surgery hypothesis: (M,~M)

be a manifold pair with

dim M = m > 5,

a connected Poincare pair over ¢:(M,SM) ÷ (X,~X) ACWh(w;~), (X,~X)

of dimension

a normal map.

Let

(X,~X) n,

Suppose that

w = ~l(X), is self-dual and the torsion of

is in

A;

assume

equivalence over

~

¢I~M:~M + ~X

with torsion in

is a homology A, under the map

Wh(~i(~X);Z P) ~ Wh(~I(X);Ze). Theorem I.

There is defined an element

~(~) E L n A ( ~ w ) ,

which depends only on the normal cobordism class of so that

0(¢) = 0

if and only if

¢

¢,

is normallY cobordant

relative to the boundary to a homolosy equivalence over with torsion in Proof: ¢ for

Case 1.

n=2k.

is k-connected. i < k

Ki(M;~w) i 9 k.

and = 0

A. By Lemma 4.2.1, we may assume

By the Hurewicz theorem,

Kk(M) a ~k+l(¢). for

i > k, and so

By Theorem 2.1,

By duality Ki(M;Zp~) = 0

G = Kk(M;~)

64

Ki(M) = 0

for

a Wk+l(¢) @ ~

is s-free.

Then

G* ~ K k ( M ; ~ )

an i s o m o r p h i s m following

G ÷ G*

Theorem

k:G x G ÷ Zp~, numbers,

as every

immersed

k-sphere.

then we replace G

is r e p l a c e d

class

of

by adding

¢,

G ~

¢':M'

K1,

handles

f

show that

¢:N ÷ X

function

class

we may

assume

sending

x

By C o r o l l a r y

to

= 0,

4.2.1,

and c h a n g i n g

exact

sequence

of

(N,~N)

0 ÷ Kk+l(N,~N;~) we can assume

e

and

Thus

is free.

Let = 1.

in

M,

is the

1 x S k. G

@(M')

Routine form.

LA2k(~).

cobordism

class

between

¢

and

~:N ÷ I

be an

T h e n the map

(X x I;X x 0,X × i)

is a degree

we can assume Kk(M')

of

cobordism

with

(¢(x),¢(x))

where

only on the

x I U M'.

÷

in

M # ( S k x sk),

(G,k,~)

~N = M U ~ M

fixed

where

of

be a normal

¢:(N;M,M')

by an

is a ( - 1 ) k A - H e r m i t i a n

depends

~(M)

by i n t e r s e c t i o n

(k-1)-sphere

sum

is the

(G,X,U)

c(¢)

defines

by s e l f - i n t e r s e c t i o n s .

K 1 = <e,f~,

to be the class

÷ X, with

Urysohn

connected

gives

by the remarks

is r e p r e s e n t e d

on a trivial

trivial

let

G

duality

is given

surgery

and

o(¢)

1.6.1

u:G ~ 2P~I k

by the

by

A

Define

M

To show of

of

in

isomorphism

by T h e o r e m

Sk x 1

calculations Define

This

element

If we do

Poincare

with m a t r i x

3.1.2.

which

and

Kk(¢)

by adding reduces

map.

= 0, k e e p i n g

a kernel.

The

~ Kk(N;~)

long

+ 0,

free and the t o r s i o n

G5

M

to

~ Kk(BN;~)

all m o d u l e s

i normal

of the

complex

above lies in

Kk(M;~w)

- Kk(M';~w)

show that that

spheres.

,

~aD

k

p'

intersections

of

ax

with t o r s i o n @

and

in

¢

with torsion

say

on this e m b e d d i n g and

N

X(x,x')

ax'.([B12]) Thus

~

Kk(M;~)

p

= k(ax,ax,)

Similarly

is a c o b o r d i s m equivalence

: 0, we have

~(~x)

= 0,

in

= 0, and

invariant. over ~(@) = 0

equivalence

if

~(@) = 0, then

Zw ÷ ~ w

Since

Kk(M; ~ ) N(e m) : 0,

by Corollary

1.6.1.

is InJective.)

over

as in Lemma 4.2.1.

Let

N = M ~ D k+l = M' u

66

D k.

is a em (This

Do surgery

M'

be the result

the trace of the surgery. We have

restricted

transversally

to a h o m o l o g y

by an embedding

uses the fact that

so that

intersect

e l , . . . , e m , f l , . . . , f m.

is r e p r e s e n t e d

is a

A.

Conversely, kernel,

ax', then p'

cobordant

in

D k) ÷ N

is a homology

A, then

is n o r m a l l y

and so there

and paths with ends r e p r e s e n t i n g

is a kernel. Since if

so for

ax.

and

set of circles

Using

by a sum of maps of

Wk(N),

represents p

shows

has a basis

of spheres,

o:(S k+l - U I n t

since we can assume

Kk(aN)

in

represents

If

if

0

Duality

Kk(aN)

is r e p r e s e n t e d

represents

, and so we need only

~ Kk(N;~w)*.

throughout,

This sum is

Kk(aN;~w)

is a subkernel.

~ Kk(N;~w)

ax

framed immersion

a finite

LA2k(~W)

by framed immersions

X~Kk+l(N,aN)

so

in

coefficients

represented

to

Now, the form

Kk+l(N, aN;ZpW)

Kk+l(N,aN;~w)

A = ~w

A.

The map

j,:Kk(M;ZpW) given by

÷ Kk(N,M';~)

j,(x) = l(em,X).

l(em,f m) = 1.

0

Kk+l (M')

Kk+I(N)

o

o ~

/\/\

Kk(N)

Kk_l (M')

Kk(N,~N)

0

\/\ 0

Kk(N,SN)= Kk_I(M') = 0.

0

Furthermore,

N = M U D k+l, and the map

sends the generator to

represents the attaching map.

Thus

~

em, since is inJective

Kk+l(N) = Kk+l(M') = 0. A basis for

el'''''em'fl .... 'fm-l' these elements.

Inductively, Zp.

since

em, and so

is given by

ker(J,)

is generated by

Kk÷l(N,M ) ÷ Kk+l(N,~N)

sends

Kk(M') ~ <el,...,em_l,fl,...,fm_l>.

we can do surgery to get a homology equivalence

Also, we chose bases so that

took care that Case 2.

Kk+l(N,~N)

Also, the map

the generator to

that

since

(~w-coefficients)

Kk(N,M' )

Kk(M') ~

8:Kk+l(N,M) ÷ Kk(M)

and so

is surJective,

/\

Kk+l(N,M) ~ Zp~, since

over

J,

Kk(M)

Kk+I(N,~N)

J\/ we see that

Thus

is easily seen to be

Looking at the exact braid

K•N•

°/'

em

~ Zp~

~(C,(~'))~ A.

n = 2k+l.

embeddings representing

Let

¢

is k-connected,

gi:S k × D k+l + M

a set of generators of

67

and

Thus we have the result.

Again assume

Kk(M) ~ ~k+l(~).

~(C,($))~ A

so

be disjoint Wk+l(~).

Let

U = Ugi(sk

x Dk+l),

M 0 = M - Int U.

By Theorem 3.1.3,

we can assume there is a Poincare pair over torsion in ¢

A, (X0,Sn-1) , so that

by a homotopy,

triads

with

X = X O u D n.

we can assume

Changing

is a degree 1 map of

(M;M0,U) ÷ (X;X0,Dn). The modules

bases given by

Kk+l(U,SU;~pW)

and

Kk(U;~pW)

e i = (gi(1 x Dk+l), gi(1 × sk))

fi = gi (Sk × 1).

have

and

Then in the sequence

0 ÷ Kk+l(U,~U;*pW ) ÷ Kk(~U;~p~ ) ÷ Kk(U;~p~) ÷ 0 these

combine to g i v e a b a s i s

In fact,

Kk(~U;~w)

a subkernei.

{ e i , f i i}

is a kernel

and

We h a v e t h e d i a g r a m

Kk+l (M)

Kk+I(U,~U;~w)

is

Kk(M0 )f'---''~ 0

Kk(~U)

Kk(M)

/'\ / \

0k....w~ Kk+I(M,U)

= Kk+I(M0,~U)~__.~

By theorem 2.1,

Kk+I(M0,~U)

can assume it is actually is a subkernel. Then the map

Let

~:Kk(~U)

~

in

As above,

be a basis for

÷ Kk+I(M0,~U) , ÷ Kk(~U)

the matrix of

be the class of

Kk(U)~.._._~

a

0.

is s-free and s-based.

free.

{el*}

Kk+I(U,SU)

to an isometry Furthermore

Kk(~U;ZpW).

(~pW c o e f f l i c i e n t s )

0 f-''~ Kk+I(M,M 0) = Kk+I(U,~U) ~ ' ~

\/ /\

for

is in

LA2k+l(~W).

68

Kk+I(M0,~U) Kk+I(M0,~U).

ei~-* el* , extends

by Corollary A.

We

Define

4.3.1. o(¢)

to

To show

c

is a c o b o r d i s m

that surgery on the e m b e d d i n g and

gl

g!(! x sk),and replaces

unaffected

by surgeries

a

with

then we can assume

and so the pairs

(N,~+N),

(N,3_N)

Theorem

2.1, we can assume

based, w i t h g e n e r a t o r s Let

Then

s-cobordism

~

homology o(¢) = 0.

where

a0

equivalence Then

~_N'

class of

over

if ~.p

form

or

If

corresponding

to

(-i)k

gi

replaces

= (-l)kc -I, we may write

~_N.

and so

¢

to

N"

~+N.

is an

Thus, c(¢)

over

[o 1 R*-I

to a

A.

Now suppose

a = a0-a I ...a m,

and

ai

is either of

'

(-1) k

'01

' the surgery on the

a

~p,

depends

is cobordant

Write

(stably)

am =

By

is free and

with t o r s i o n

0

the

>k.

¢.

ag EUkA(r,~w).

is of the form

Thus

which generate

N = N' t9 N", where

c(@) = 0

is a

are k-connected.

need be considered,

Clearly

is

is (k+l)-connected,

Kk+I(N,8+N;Ep~)

from

only on the c o b o r d i s m

~

~:N ÷ X

given by handles a t t a c h e d

Kk+I(N,~+N;~).

only k-handles

If

c(¢)

of handles of index

N' = ~+N x I k A k - h a n d l e s

over

gl(S k x i)

ae n. Thus

on k-spheres.

has a handle d e c o m p o s i t i o n

note first

interchanges

normal cobordism,

N

invariant,

with

a = a 0 ...

68

ao.

But as

am_ I.

fl Q1

m

If

a matrix over

lj thonorsome

Zw.

Replacing

h:S k × D k+l ÷ S k x Int D k+l

replaces

a

with

with

is

gi o h, where

is an embedding of degree

p,

01 lq 0I

-I p

Io As

gi

pQ

I

pI

m lq

?I

is a simple isometry, we can assume pI

~m = [i

~]

where

Q

is a matrix over

~

with

Q = QO + (-l)k+iQo*" Let

H

be a regular homotopy of

U g i : W S k x D k+! ÷ M

to disjoint embeddings, where the intersection matrix of the

immersion

H

matrix given by

1.6.3.

[~-~0).

The spheres

bound disks in

nd el Intersectlon

is given by

M

This can be done similar to Theorem

gi(l x S k)

are unchanged, as they

but the sphere

by the other end of the homotopy.

gi(S k x i)

is replaced

This replaces

70

a

by

~.

= s0

am_ I.

. . .

Thus by finitely many surgeries we can assume

m =

But

R

has image in

A, so by a change of

R,_ 1 •

basis, we can assume assume

m

is of the form

m(el) = (-l)kfi ,

fi*

for

By surgeries on the

gi' we can

±e ~ . . . ~ a, i.e.,

a(f i) = e i.

The bases of a basis

a=l.

Kk+I(M0,~U)

Kk(M0).

and

Kk(~U)

determine

Thus after all surgeries, the

maps in the diagram above become

(Zp~ coefficients):

Kk(~U) ÷ Kk(M0):ei ~-+ fi*,fi ~-+ 0, Kk(BU) ÷ Kk(U):fi~-* fi,ei w-* 0, Kk+I(M,U) ÷ Kk(~U):ei*~-~ (-1)kfl, Kk+I(M,U) ÷ Kk(U):ei*~-* (-l)kfi ,

Thus the map torsion in

Kk+I(M,U) ÷ Kk(U) A, and so

is an isomorphism with

Kk+i(M;Zp~)

• (C,(¢);Zp~) ~ A.

71

= Kk(M;ZpW) = 0, with

T h e o r e m 2.

(Realization

compact manifold, Then there

n ~ 6; let

is a m a n i f o l d

$:(X;~+X,~_X) ~I~_X

X

Zp

Proof:

Suppose form

homotopy

(G,l,~)

~Is+x

Then

fi'

el,...,e n.

Let

into disjoint Fi

by a

discs,

be a regular l(Fi,F j) = P i j l ( e i , e j )

P i j , P i ~ H(P). attached

by the maps

we can extend

to a normal map

¢:(X;8+X,~_X)

+

to

(G,I,U),

~

x, where

K

trivial

Thus

Thus

with t o r s i o n

in

we can again extend

Kk(X;~w)

G ÷ G*

= 0.

in

to

~:K + K

of d i m e n s i o n M.

Let

since the embedded 1M

is

is given by

is a h o m o l o g y

n+l = 2k+l, and let

(k-1)-spheres

(M x I;M x 1,M x 0 u ~M × I).

Kk(~+X;~,)

¢12+X

fi'"

Since

equivalence

A.

is the kernel

the trace;

form on

and the map

÷ Kk(X,8+X;~). ~l(~+X ) m ~.

X'

equivalence

is r e p r e s e n t e d

let

X = M × I~handles

Now let

and

so that

were trivial,

Kk(X;~)

r

xCLAn+l(~Z).

fi

isomorphic

over

x I)

so that

Then the i n t e r s e c t i o n

k > 2,

x

1.6.3,

for some

Since the embeddings

Let

~($) = x.

with basis

to an e m b e d d i n g

be a connected,

is a h omolosy

A, and

As in T h e o r e m

Let

~ = ~l(M).

be embeddings

~(F i) = pi~(ei),

l:M ÷ M

in

n+l = 2k.

fi:S k-! x D k ÷ Int M i=l,...,r.

Mn

÷ (M × I;M x 1,M x 0 ~ M

with t o r s i o n

Hermitian

Let

and a normal ma p

is the identity map,

over

and

Theorem).

¢':X'

72

2r.

M'

represent

Do surgery on

be the resultant

spheres were trivial,

+ M x I.

Then b x the

2r

Kk(X';~)

spheres

is a kernel,

S k x i,I x S k.

images of

1 x Sk

under the map

v i = ui~a

i

u i E Kk(X')

ui

for

are r e p r e s e n t e d

on these embeddings X = X'~X"

and

by disjoint and let

a.

X"

Let

with basis given Vl,...,v r

We can write a i £ • P.

The elements

framed embeddings; be the trace.

is the desired manifold.

73

be the

do surgery

Then

4.4.

The Simply .Connected Case. We assume

w = 1

not taken into account,

and compute

as

Wh(l;~p)

Ln(~).

Torsion is

= 0.

Theorem i. 0

n odd or n=4k+2

Ln(Z P) = ~/2E W

P

n=4k+2 and

and

2 eP

2 ~ P

n=4k

P

where

Wp P =

Z/2Z ~) 7J2Z [7./47.

Proof: Suppose K r.

Case i. A

n

odd.

represents

Assume

or

O

~Z/2~ p = 4k+3

that all endomorphlsms

mod E U ± I ( r - I , ~ )

A

is equivalent

with the element

column

of

of

given in [GI],

mod EU_+I(r, ~ )

P

Kr_ I

to a matrix of the form

to a matrix

in the first row and column of

to i, all other elements

[GI].

of the standard kernel

According to a matrix calculation Lemma 2.1,

p = 4k+l

This proof follows Bernstein

an automorphism

inductively

are equivalent

or

P

[~

equal

in the first row or first

0, and the first row and column of

74

R

equal to

0.

~]

Let

A0

be the matrix obtained by deleting the

first rows and columns of we can assume

P,Q,R,

I:

A0 =

and

Thus

S.

By induction,

U±I(r,Z P) = E U i I ( r , ~ )

C0

and so

L2k+l(~)

Case 2.

n=4k.

prime, and

Fp

= 0.

Let

~p

denote the p-adlc numbers

the field of

a field,

p

elements.

for

F

p~2

there is a residue homomorphism

p

a

Then, since

L4k(F) = W(F), the Witt ring of

(called the second residue in

for

F, for

L4k(Q p) ÷ L4k(F p)

[AI0], Theorem 1.6).

Composing this with the functorial map obtained from ÷ @p,

p & P, we get

L4k(Z P) ÷ ~5'2~ if where

~

L4k(~)

2G P

+ L4kdFp).

by sending

Define

(G,l,~)~-+ ~(det l) mod 2,

is the 2-adic valuation. Then there is an exact sequence

0 ÷ Z ÷ L4k(~)

-~ Z / 2 Z ~ ) 0

L4k(]Fp) -~ O.

This is proved in Lam [A!0], Theorem 4.1. L4k(Q) + ~

splits this sequence.

The signature map

The Chevalley-Warning

Theorem ([A24]) and some elementary number theory show that the signature and in

~p,

p 6 P, define an embedding of

~ • ~) L4k(~p). peP

75

W(~p)

~P P

We let

ape ~+

so that

= image of

~p:L4k(Z~p) + L4k~Fp) , and

a p ~ = image of the signature

W(~)

+ ~.

We have if

2 e p

if

2 ~ P, but there the form

is a prime of 4k-i

in

P

ap = if neither

hold, but there prime 4k+l

if

is a

of the form in

P

P = ¢

and

P

Minkowski [K12],

p=2

~/4~

P-3

mod

(4)

Z~/2~ ~ Z~/2~

p-i

mod

(4)

I

~P

The e l e m e n t s

~/2~

8p(G,l,~) & L4k(Fp)

invariants

of the form

for proofs of these

of H a s s e - M i n k o w s k i

are called the Hasse(G,l,U).

statements

invariants.

76

See Anderson,

and a d e t a i l e d

study

Case 3.

n=4k+2.

Suppose

(-l)-Hermitian

form,

anti-symmetric

billnear

so that

= a2~(x),

k(x,x)

~(ax) = 0.

Since

2 J P.

i.e.,

and

form and k(x,y)

2 ~ P,

= W(x+y)

argument

so that

~(e i) -- w(fi ) -- !

consecutive

pairs.

and similarly

to get a sympletic

with

k(ei,e j) = k(fi,fj)

Then G, so

G

G = <e,f>,

argument

is even.

Group these

el' = el + e2

fl' = fl

e2' =

f2' = fl + f2'

e2

e2',...,er',f2',...,fr'

~(e i) = 0

or

is a kernel.

i

in

c(G,k,~)

2 g P, then

shows every

such

, leaving

~(fi ) = 0. is a subkernel

The form

~(e) = ~(f) = I,

If

by the

Then the number of

< e l ' , f 2 ' , . . . , e r _ l ' , f r '>

= i, has

~ ~/2Z

Apply the transformation

obtain

fixed those with

k(e,f)

= 0.

- ~(y),

= Z/2Z.

Define c:L4k+2(Zp) r c(G,k,W) = ~ ~(ei)~(fi). i=l c(G,k,~)

a map

- ~(x)

k(ei,f j) = 6ij.

Suppose

be a

non-degenerate

~:G ÷ Zp/(2)

Zp/(2)

for G, e l , . . . , e r , f l , . . . , f r

Arf invariant

of

(G,k,W)

k:G x O ÷ Zp

Apply the standard basis

Let

~(e,e)

: i, so ~/(2) G

c

= ~(f,f)

= 0,

is an isomorphism.

: 0, and the same

is a kernel.

77

= 0

The groups Kervaire-Milnor construct

[FI].

the Milnor

Let and

np ~ Wp P

an element a degree

Ln(Z)

n

See also

Browder

p e P.

x ~L4k(2p) ,

We now

manifolds.

be an integer,

for

[G5].

in

with

apIn ,

Then

(n,(np)p e p)

k > i.

By Theorem

determines 3.3.2,

there

is

I map

so that

0($) = x.

equivalence

over

The cone of

boundary

~+M.

the boundary, similarly

Theorem

first considered

and Kervaire

~:(M;~+M,~_M)

~.

were

2.

÷ (S 4k-I x I,S 4k-I x I,S 4k-I x 0)

Now ~ ~+M,

~+M ~ S 4k-I x i

and so

~+M

C(~+M),

is a homology

is a Poincare

Define the Milnor

Poincare

C(3_M)~MUC(~+M).

using

Define

sphere

complex

over

over

complex

by coning

Kervaire

manifolds

x~L4k+2(~).

There are Poincare

k > l, and normal maps Hasse-Minkowski

is a homology

complexes

M + S 4k

invariants

with

np, and

k > 0, and normal map_ss K + S 4k+2

78

over

~,

signature PL-manifolds

M 4k, n

and K 4k+2,

with Arf invariant

I.

~

with

Theorem

3.

Let

H

be PL or TOP.

Then

~n(G/H)

a Ln(Z),

n > 5. Proof:

By T h e o r e m

is generated

3.2.3,

by the Milnor map

and the Kervaire map 0

if

n

~n(G/H) *-+ NIH(sn). M 4k ÷ S 4k

K 4k+2 ÷ S 4k+2

is odd.

79

if

if

But this n=4k

n=4k+2,

and is

4.5.

The Exact

Sequence ' of Sur~er~.

An important of Section closed,

4.3 is the exact

simply connected

Sullivan

~pH(X)

M

if there

is an h-cobordism

between

M0

fl" in A

Here

and X

case,

modulo

the relation

over

i.

provided Proof:

X

~pH(X)

as elements

maps.

The map

let

¢la+M

A,

F:W ÷ X

f0

and

surgery

over

Zp

with torsion The subgroup

of

sequence

÷ [X,Gp/H] ~pH(X)

obstructions. Then there

equivalence

÷ Ln(Zp~)

of dimension

n >. 5.

is defined

are represented in general)

To define

by Theorem by normal ~

is defined

~,

is a normal map

M) ÷ (X × I;X × I,X × O)

a homology

extending

to be normal.

(not a homomorphism

x g Ln+l(Zp~).

¢:(M;a+M,a

in

÷ [X,Gp/H]

is an H-manlfold

3.2.3,

by taking

f0 ~ fl

Zp, W, with torsion

complex

is an exact

+ ~pH(X)

The map

~ = HI(X),

from the notation.

There

Ln+l(~pW)

[HI9].

equivalences

AEWh(~;Zp),

A, and all maps are assumed

Theorem

In the

P = ¢, is due to Wall

in

M1, and a map

is a Poincare

is suppressed

of surgery.

be the set of homology

is an H-manifold,

theorems

P = ¢, it is due to

Zp, f:M ÷ X, with torsion

where

of the surgery

sequence

case,

[GI3]; the general Let

over

corollary

over.

80

with Zp

~(¢) = x

with torsion

and in

A.

~(x) = the class of

Define

¢la+M

in

~pH(X).

In fact, this procedure defines an action of Ln+l(~) over

on

~pH(X),

by taking a homology

Zp, f:N ÷ X, and doing as above to

homology

equivalence

a+M ÷ N.

equivalence

N, getting a

Composition

defines the

action. Exactness

in the theorem then means that

m

induces

a bijection of the orbits of the action to the kernel of o.

This is the content

of Theorems

81

4.3.1 and 4.3.2.

Chapter

5.1.

Handle S u b t r a c t i o n

5.

Relative

and Applications.

In this section we use handle dual to surgery,

Surgery.

subtraction,

to prove a general relative

w h i c h forms the basis

for the geometric

an operation

surgery

formulation

lemma of surgery

groups. Let ~r(@)

~:(N,M)

÷

(Y,X) be a map of pairs

to be the set of homotopy

classes

~

(Dr,D+r-I )

(N,M)

-~

(Y,X).

then each ~ ¢ ~r+l(@) of immersions

contains

determines

M and @ is a normal map, a regular homotopy

f:(DrxDn-r,sr-lxDn-r)

by the relative class

of diagrams

(Dr-I S r-2)

If N n is a m a n i f o l d w i t h boundary

immersion

÷ (N,M)

classification

an embedding,

and define

class

for r < n-2

theorem

([B3]).

If this

let

N O = N - Int f(DrxD n-r)_ M0

:

Since ~ s ~r+l(~), more

~ induces

~0:(N0,M0)

~r+l(~0 ) = Wr+l(@)/<~>.

attaching

Theorem where

~ N O.

i.

an (n-r)-handle

triad over

(Y,X).

Further-

N and N O are cobordant

by

to N0xl.

Let ¢:(Nn;M,M+)

(N;M,M+)

÷

~s a m a n i f o l d

~ p with torsion

÷

(Y;X,X+) triad,

be a normal map,

(Y;X,X+)

in A C Wh(w; ~ p ) ,

82

is a Poincare w = Wl(X)

~I(Y), over

induced by inclusion, ~ p with torsion

r el M+ to a homology

Proof:

in A~ and n ~ 6. e~uivalence

Even-dimensional By Corollary

4.2.1.,

By Theorem

~p~

theorem,

~:(N,M)

immersions

÷

otherwise

el,...,e r. ~ ~k+l(~) ~ ~ p ,

Thus the elements

f!:(DkxDk,sk-lxD k)

qe i s ~k+l(~)

Wl(N) , the maps embeddings

+

(N,M);

where

U =

fi' by Corollary

subtraction:

~fi(DkxDk).

and H,(~)

•

..

Since

homotopic

Let

Let C,(~)

~I(M) m

to disjoint

N O = N - Int U, M 0 = ~N 0, be the chain complex of 4,

of ( Y , N U X )

= H(C,(~))

if using coefficients). exact

f! rep-

1.6.2.

given by the chain complex inclusion,

ei

i

for some q s H(P).

fl! are regularly

Do handle

noted.

By adding trivial handles,

Kk(N,M)

(Y,X).

= 0 for

i

resents

in A.

~IN is k-connected

Ki(N,M)

unless

is s-free.

By the Hurewicz

determine

~ p with torsion

By duality,

it is free with basis

where we regard

Then ~ is cobordant

we may assume

coefficients

2.1, Kk(N,M)

we can assume

over

equivalence

case~ n = 2k.

and ~IM is (k-l)-connected. i # k and we use

~IM+ a homology

i~ ~ is replaced by an

(tensoring

as in Section

For any coefficients

1.2

there is an

sequence

÷Hi(N,M)

By Theorem

>Hi(Y,X)

~ Hi(C)

3.1.2, Hi(~)

÷ Hi_I(N,M)÷.--.

= Ki_I(N,M).

chain complex

defined by

D i = Ci+l(~) ®

Hk(UUM,M)--~

Kk(N,M) , and it follows

83

~p~.

that

Let D, be the We have

C,(UUM,M)®~pW in A.

As

÷

(N0,M 0) ÷

equivalence in A.

By P o i n c a r e

C,(X) ®

over

Odd-dimensional We

over

~p

with

case~

so

~(~IM)

obstruction

we have

÷

with

C,(Y,X)

® ~pW

torsion

÷

that

@0:(No,Mo)

÷

a chain

torsion

C,(Y) ® ~ p W C , ( M O) ® ~ p W

(Y,X)

÷

is a h o m o l o g y

in A.

~ as a n o r m a l = 0, since

torsion

in A.

Let

has

from

~IM to

is a h o m o l o g y

~ denotes ¢:Q ÷

equivalence

~ Y~Xxl

cobordism

~IM+

Here

in L An_l ( ~ p ~ ) .

@ U@:NUQ

torsion

n = 2k÷l.

to a h o m o l o g y

Then

with

is an e x c i s i o n ,

it follows

Thus

can r e g a r d

with

cobordism

~p

equivalence

C,(N O) ® ~ p W

and

is also.

equivalence

and

duality,

equivalence

~pW

~IM+,

(N~UuM)

C , ( N o , M O) @ ~ p W

is a c h a i n

A.

D, is a chain

the

equivalence

surgery

Xxl be a n o r m a l

over

~p

with

a well-defined

torsion

in

obstruction

x ~ L~(~p~). Let Then is

@:R ÷

~(@v¢~@)

relative

over

~p w i t h

as a c o b o r d i s m over

~p

with

For [GI8] The

for

proof

Shaneson

a normal

= x - x = 0, and

cobordant,

alence

Xxl be

of

the

the

case

case

of the

torsion

with

~(~)

= -x.

so C u C u @ : N U Q U R

to the b o u n d a r y ,

(N,M)

torsion

map

in A.

~ Y~Xxl

to a h o m o l o g y

This

equiv-

can be r e g a r d e d

rel M+ to a h o m o l o g y

equivalence

in A.

P = ~, this

~ = I, and

theorem

[HI9]

odd-dimensional

for

case

[K5].

84

is due the

is due

to Wall,

general to

case.

Cappell

and

5.2.

Geometric Definitions

of Surgery Groups.

In this section we define surgery groups in a more general context and relate them to the algebraic definitions given in Section

4.3.

To do this we need the notion of an

n-ad. Let

C

an integer.

be a category of spaces and maps and n ~ 2

Define

~(n) to be the category with objects

X = (IXI;X1 . . . . ,Xn_ 1) Xi C

IX], and

X(a) = O

c { 1 ..... n-l} and morphisms

between

, is an object of

X and Y given by a map

so that f(X(a)) C Y ( ~ )

for each

a.

f:IXi

We let

~. : ~(n+l) m

÷

~i' 6i, and Sn, k by:

e(n), ~ X i ~ Xj

l~iXl = Xi" (~iX)j = 6. : ~(n+l)

[XiNXj+ I

j > i,

.~ ~(n)

Z

J l$iXl = IX1, (6iX) j =

(~(n) Sn,k:

j < i

Iv j j+l

<

i

j > i,

c(n+k) ÷ ISn,kXl = IXI'

(Sn,kX)j

=

{~j

jj <> nn.

n-t Define

~X(a) =

~

X(6).

In particular,

85

~X = ~ X . .

C

* IYI

X{1,...,n-i}

This is called the category of n-ads associated to Define functors

Xi,

G.

= IXI.)

If f:X ÷ Y is a map the

induced

n-ad

and

map;

similarly

Y is a space,

of n-ads, for

define

6i'

let

~ and

an n - a d

$if:~i X ÷ ~i Y be Sn, k.

XxY by

If X is an

IXxYI =

IXlxY,

(XxY) i = XixY. Let ad X' by [C;A,B]

X be

IX'I

Wm(X,Xo)

manifold

with

a ring)

a Poincare

~ A,

Define

an

(n-l)-

X!l = [Xi+ l ; X l ~ X i + l , X o ] ~(i)

M is a m a n i f o l d

boundary

~ B]

for A , B C C .

where

(-~

A map

, where Define

X'o is the

constant

= j,

class

X is a P o i n c a r e

[X(~),ZX(~)]

k ~ (-l)t[x(~-it)], t=l

Poincare

= [Y(~),~Y(~)]

~/2~).

category

of n o r m a l

manifold

n-ad

M and

functors

further

is a n-ad

~ ={i I ..... ik} , ( X ( ~ ) , ~ X ( a ) )

fundamental

Let ~ be a p a i r

omit

M(~)

is

so that

where

~X(~)°

if ¢ , [ X ( ~ ) , ~ X ( ~ ) ]

the

if each

An n - a d

}:X ÷ Y b e t w e e n

and w ~ H I ( I K I ;

n-ad

~M(~).

if for each

pair with

k j: ~ J X ( ~ - i t) t=l

we

s X(~).

O

= Wm_l(X',x~)

~[X(~),ZX(~)]

under

x

at x o and m _> n - i. An n - a d

(over

with

= [IXI;Xl,Xo],

= {~ s CI:~(0)

inductively path

an n - a d

maps

for e a c h

is of d e g r e e

where

Let

be a s u b c a t e g o r y

C

of d e g r e e

2, ~i"

Here

K is a CW

(n-l)-ad of the

~ ~:M ÷ X, b e t w e e n

n-ad

X over

H = TOP,

of it.

86

i

~.

(K,w),

a Poincare

mention

n-ads

a ring

PL,

an H-

R,

or DIFF,

closed and

Define ~ ( ~ )

to be the cobordism group of C

~, where we regard a map w : X ÷ Sn_l,iK

so that

(~:M ÷ X) ~ ~ as a map of n-ads

Wixi = ~*w, where Wlx I is the o r i e n t a t i o n

class of IXI, and we use the boundary

operator

Thus M I ~ X I + ~ and M 2 ÷ X 2 ÷ ~ are bordant maps

of (n+l)-ads

N ~ Y ÷ Sn_l,2K as above

~n N ÷ ~n Y + ~nSn_l,2 K = Sn_l,iK and similarly

applying

~iso require

compatible

~m(~)

denotes

category.

~n-i yields

in A.

Define ~m(H) and

~m(H)

so that

relation ~

the full of

IMI.

R is a ring and A is a selfDefine

of homology

Qm(H)

= ~(~)

equivalences

= ~IhHQm_I(H),

where

Define

where

over R with

is a natural map hH:Qm(H)

= h~l~n~lhHQm_l(H).

Note we

for Y as above.

the d i m e n s i o n

of W h ( w I ( I K I ) ; R ) .

There

if there are

to M I ÷ X I ~ Sn_l,l K

the group defined above with

is the s u b c a t e g o r y

~n-i + ~n"

M 2 ÷ X 2 ÷ K.

orientation

Let H = (~,R,A) where

torsion

is equal

The integer m denotes

dual subgroup

over

÷ ~m(~).

~n_l:~m(~)

~(H)

and

÷ ~m_l(~),

~(~)

similarly by r e q u i r i n g that ~ : 6 m _ i X + K induce isomorphisms on f u n d a m e n t a l responding

groupoids

intersections

on each component. also assume

(that is, for each ~ the corhave

isomorphic

This will be made

X is connected,

fundamental

clear later.);

and w~ require

groups we

the same for

cobordisms. There is a natural map Lm(H)

to be ~m(H)/image

There

is a natural map L~(H) ÷

no natural

group

of

~m(H) ÷

~m(H);

87

and we define

define L~(H)

Lm(~);

structure.

~m(~)

similarly.

h o w e v e r L~(~)

has

The the next

n-ad

Suppose

Let K be an

(n-l)-ad

the

corresponding

in 6 n _ i X

(and c o n s i s t e n t classified

orientation

class

Theorem

i.

With

m - n ~

3~ then

equivalence if the

Proof:

class

Suppose

a homology

first

that

~N = M U M + ,

f:N -

f(M+)

with

n = 2. ~p,

~M + I be a U r y s o h n

= i.

Define

Let

if

the

class

of

Now there

~

is

are

of the

spaces

Then an

di m M

= m~

and

to a h o m o l o g y in A if and

only

vanishes.

Assume

~ is c o b o r d a n t

~ X, by

to

a cobordism

= ~M = ~M+.

(x,0) ~ (x~t) function

for x E ~X, t ~ I.

with

f(M)

= 0,

~:N ÷ Y by

~(x)

@:(N;M,M+)

that

maps).

I(~'(x),f(x))

Then

torsion

H = (K,w, ~ p , A ) .

torsion

~+:M+

MAM+

where

torsion

subspace

groups

in

between

S n _ l ~ i K , and d e f i n e s

i_nn L~(H)

over

with

inclusion

qRbordant

~p

M ~ X + K

Let Y = X x l / ~ Let

X ÷

map

with

and e a c h

as above~

over

equivalence

~ ' : N ÷ X, w i t h

the

¢ is n o r m a l l y

of

is s h o w n

(Wl(~n_iX),l),

w in H I ( I K I ; ~ / 2 Z ) .

of n - a d s

~p

fundamental

with

by a map

~p

over

space

to the

notation

L~(H)

over

of type

total

K(~,l)'s

W i x I is

n-ad

equivalence

(n-l)-ad

so that

set

@ : M ÷ X is a n o r m a l

and a P o i n c a r e

~n_l @ a h o m o l o g y

in A. an

of the r e s t r i c t e d

theorem.

a manifold in A,

value

=

(Y;Xx0,Xxl)

M ~ X vanishes suppose

is a c o b o r d i s m

the

class

x I

~'(x),0)

is a n o r m a l

~H

x ~ ~H.

map

and shows

that

is L~(H). of M ÷ X ÷ K v a n i s h e s .

@:(N;M,M+)

÷ 88

(Y;X,X+)

Then

to a h o m o l o g y

equivalence

@+:M+

Furthermore, that Wl(X)

+ X+ o v e r

Wl(X)

m Wl(K)

surgery

suppose

and

in A).

it follows

easily

by i n c l u s i o n .

we

over

remainder

surgery

the

can do s u r g e r y

~p.

on n-ads. torial

has

of the

been

procedure

Thus

Lemma

torsion

on ~ to get

In p a r t i c u l a r ,

we

a

can do

on ~ : M ÷ X. The

apply

5.1.1,

equivalence

(with

~ Wl(Y),

m Wl(Y) , i n d u c e d

By T h e o r e m homology

~p

We now

properties

i.

Let

Poincare

pair

show

¢:(W~V) over

extension

÷

on M(B)

that

L'(H) m

has

L~(H)

÷ (Y~X)

Wy

= wo~#:~l(Y) Then

(Z;Y,Y+),

theory

for

nice

group

be a n o r m a l

with

torsion

there

in A~

map~

Then

surgery and

func-

(Y~X)

a

dim W = m ~ 5,

+ Z/2Z.

= V, Y ~ Y +

of ~, ~:Z ÷ K so that

~.

is a b i j e c t i o n .

Assume

is a n o r m a l

W~W+

B c

(M(~),~M(~)).

~ Lm(H)

~p

by i n d u c t i o n :

for each

to the p a i r

by p r o v i n g

2-skeleton.

¢, @:(U;W,W+)

follows

is an o b s t r u c t i o n

and ~:Y ÷ K so that a finite

done

above

L~(H)

theorem

K has

cobordism

= X, and

( ~ I Y + ) # : w I ( Y +)

of

an

÷ Wl(K)

is an i s o m o r p h i s m .

Proof:

By T h e o r e m

dim(Y0~

X) ~ m - 2, and H is o b t a i n e d

1-handles. If

The

inclusion

~ is the

TH + ( ~ I H ) * ~

3.1.3,

line

is t r i v i a l ,

we

can

assume

induces bundle so we

Y = Y0 V H H'

from

D m by

a surjection

over can

8g

K defined

do s u r g e r y

XCY0,

adding

~I(H)

+ ~I(Y).

by w,

then

on

~IH:H

~ K

to get H' ÷ K w h i c h

induces

an i s o m o r p h i s m

on f u n d a m e n t a l

groups. Let

J be the

Z0 = Y0xlUJ, for

Y+

Since

thus

is also

and

U, we

onto

consider

1-handle sphere

Z = Z0/~

extends

S O and

of each

(~IY+)# so

and

define

, where

over

(x,t)

J, we

~ (x,0)

get

a map

such

pair

2-handle

but and

case.

r ~ K(P)

wI(K)

is an

SO =

{a,b} with

S ~ @-I(s0),

÷ Wl(Y+)

one

To

regular

and so the

construct

at a time. to the

T = ¢-I(H).

and

choose

total

embedded

Then

multiplicity

Xl,...,Xr8

multiplicity

i.

@.

of S in c o m p l e m e n t a r y

opposite extend Let

components

by paths,

S = @-I(sI)

can assume

S I x D m-I

degrees.

Attach

We

handles

can now

pairs

to

arrange

having

a handle

we

the

(assuming

W is c o n n e c t e d , can

assume

r ~ H(P)

+ S l x l n t ( D m-l)

gives

Add

@ -l(a),

the

along

each

@.

÷ S I is of degree

which

taking

xi,Y i and e x t e n d

We

to Wxl,

~I(H')

of S O is r.

regular).

embedding

~#~

But

@ is t r a n s v e r s e

degree

points

image

of J,

Let

containing

same

~I(Y+)

is an i s o m o r p h i s m .

to H.

y l , . . . , y r s @-l(b)

other

+

is onto.

(~IY+)#

Assume

component Write

~I(H')

the h a n d l e s

case.

@IT:T ÷ H has

@IS:S

~IH

constructions

isomorphism;

the

and

surgery,

K. By

discs

of the

= YO U H ' ,

x ~ X, t a I.

2:Z ÷

trace

so by j o i n i n g

so is h o m o t o p i c

of d e g r e e

90

and

S is c o n n e c t e d .

and

result.

@ is t r a n s v e r s e

r.

Add

Then to an a handle

Theorem

2.

If m - n ~

then Lm(H) + Lm(~)

Proof:

Follows

Corollary if m >

i.

3 and

IKI has a finite

2-skeleton,

is a bijection.

immediately

L~(K,w; ~p)

from the lemma.

is isomorphic

to L ~ ( ~ p W l ( K ) )

5.

It follows isomorphic abelian. H = TOP, Theorem

defined

Note also that though

~n(H)

4.3.2,

2.

Lm([)

we need only

space

modules;

group,

consider

since which

it is is

and f~n(Z) depend

is independent

of H.

on

Also, by

normal maps with

a manifold.

Let R be a principal

and R ~ Z / p Z

is abelian,

to the geometrically

PL, or DIFF,

the target

Lemma

A ~p~) that L2k+l(

= 0}.

ring,

and P = {p:p a prime

If C is a chain complex o f

then C(9 Z~p~ is acyclic

free

ZZ~

if and only if C ~ R ~

i_s_s

acyclic. Proof: C®R

free,

Since

C~R~

is acyclic,

--- ( C ~ g R ) ~

and similarly

Now suppose

C~

0

Z~p)

= Hi(C~

Z~,

we need only show that

for

Zp.

2Zp is acyclic. --- Hi(C) (~ Z~p.

Since

Thus H i ( C ) ~ R

So we have Hi(C~R)

"- Hi(C) ® R

(~

= Hi_l(C)*R.

91

Z p is torsion

Hi_I(C)*R

= 0.

Thus

the p r o o f

is r e d u c e d

A,R

= 0 for A a f i n i t e l y

and

* commute

exact

with

to s h o w i n g

generated

direct

sum,

abelian

we

A@R

= 0 implies

group.

can a s s u m e

Since

A = ~/n~.

The

sequence n

0 ÷ is a free

~

÷ ~

+

A

presentation

~

0

for A so we h a v e n

0 Thus

+

A*R

~

R

÷ R

÷

A®R

~

0.

A*R Z R/nR A®R.

Conversely, so H i ( C ) ~

Zp

= 0.

if C ® R So

is a c y c l i c ,

C @ Zp

= @,IA,

unique

where

¢,:Wh(~;

Define

LmA(w,w;R)

denote

L A

for

m

We n o w promised

where

clarify

is for e v e r y

commute.

~: ~ p

the

Let

e

The m a i n

~ c

+ w(~)

example

= Wl(K(~))

K(w,l)

the

so that

is i n d u c e d

We

•

let

nonsense

category

2 n is

by

the

Lm h, L m s

for

of f i n i t e l y

B C~

so that an

Wl(K(w,l))

92

2n = w.

in

is a g r o u p o i d

generated ~(n+l), w(~)

and w = Wl(K)

÷ Wl(K(~)).

then there This

as

all d i a g r a m s

(n+l)-ad

f ~ = i#:Wl(K(B))

of type

of n - a d s

an o b j e c t w

there

is if K is and

let

0.

(l,...,n)

If ~ is a g r o u p o i d (n+l)-ad

Wh(w;R)

algebraic be

of W h ( w ; R ) ,

+ R.

of type

f B:w(B)

~I(K)(~)

+

= Lm A'(K(w,I),w;~)

A groupoid

and m o r p h i s m s

subgroup

A = Wh(w;R),

earlier.

groupoids. that

~p)

ring homomorphism

= 0, and

is a c y c l i c .

So if A is a s e l f - d u a l A'

Hi(C) ~ R

is an

is d e f i n e d

as

follows:

the

K(~,I)(~)

=

components ~_~ i

of w(~)

K(Gi,I)

are g r o u p s

and m a k e

the

our main

theorem:

Gi,

so let

corresponding

maps

inclusions. We

T h..e.... o..r e m rin~

can n o w

3.

Let w be a ~ r o u p o i d

and ACWh(w{I,...,n-2};R)

there

are

@:M ÷ X ension

surgery

s L m(W;R) A

Zn_l M to

torsion

Assume

the K.

are

~(~)

the

2 n-2 ~ R a p r i n c i p a l subgroup.

L~(w;R)

a manifold

X over

equivalence

so that

rel

R with

Proof:

n-ad

over

= 7, m - n _> 3, t h e n

cobordant over

srqups

map b e t w e e n

m and a Poincare

Wl(6n_iX)

of type

a self-dual

obstruction

is a n o r m a l

~n_l @ a h o m o l o g y

~(¢)

state

R with

R with

there

so that n-ad

M of d i m -

torsion

torsion

if

in A,

in A,

is an o b s t r u c t i o n

= 0 if and

a homolo@y

Then

onl~

if

equivalence

% is n o r m a l l y of, n - a d s

in A.

n = 2.

Define

components

L~(~;R)~

=

@

A L~(Ki;R)

where

i of t y p e K ( w , l ) .

of a s p a c e

i

If R =

~p,

the

R arbitrary,

result

the

X is a P o i n c a r e Let

C, the m a p p i n g Poincare and over

~p

surgery

and

will

over

s Hm(X;~) cone

complex

so C, @ ~ p

result n-ad

[X]

follows

chain

over

be

over

the

complex

is a c y c l i c .

problem

follow

i and

from Lemma

2.

For

2 provided

~p.

R with

is e a s i l y

from Theorems

seen

(w,R,A)

fundamental of

[X]~.

torsion

in A,

Thus

X is

to h a v e

Since

torsion

and X is a

C , @ R is a c y e l i c

a Poincare

is e q u i v a l e n t

93

class,

complex

in A'.

Thus

to a s u r g e r y

a

p r o b l e m over (w,~P,A').

This gives the result.

The n-ad case is similar. h o m o m o r p h i s m w was s u p p r e s s e d

The t h e o r e m works

Note the o r i e n t a t i o n

from the notation.

for any ring R that satisfies

Lemma 2. Torsion example, ~/2~ •

5.3.

for arbitrary

let R = ~'/ ~',

rings

~[x,y]/(x2+y2-1). but Wh(l; ~p)

Classifying

Spaces

spaces

groups,

groups,

notably Let

~

Then Wh(I;R)

for Surgery.

classifying

as was first done by Quinn

to painlessly

For

= 0 for any P ([D20]).

In this section we define surgery

can be bizarre.

derive

sequence

be a small c o b o r d i s m

for

[H8], and use these

some properties

the long exact

spaces

of surgery

of surgery.

category

(Stong [AI6])

and define a A-set by ~,~(A k) = the set of (n+2)-ads maps If

C

is graded

in

~, with face

induced by face maps

(e.g. manifolds),

of objects.

then define A ~

~n (Ak)

= those elements

in ~,G(Ak)

of d i m e n s i o n

k+n. De fine ~n e

= Sx~(~ne)

According

to Prop.

1.4.4 of [H8], ~r(~ 94

7) ~

an+r(C),

the (n+r)-th cobordism group of the category

~.

~ n~ n -~ -i" It follows that given H = (~,R,A) as

Also

C n

is an infinite loop space, ~

5.2, there exist classifying spaces ~m(~)

~- ~m+j(H)

in Section

~Hj and ~'3 so that

and ~m(#~Hj) = Image(~ m+j(~) ÷ ~m+j(g))"

There is a natural map H+~H ~J j" Let ILj(H) deonte the fiber of the map ~ Hj-i ÷ ~ j -i"

Theorem i.

~j(H)

is an infinite loop space with

Wm ( ~ j ( H ) )

~ Lm+ j(H) .

Define 3i H = (3iK,R,A) where

~i ~ = (3iK,WlWl(l~iKl))

and similarly for 6.K. Then there are natural maps i ~j(~i H) ~ Lj(6iH) ÷ ~j(H), which is, up to homotopy, a fibration.

Thus by the long exact homotopy sequence of

a fibration, we have

Theorem 2.

There is a long exact sequer~qe

• ..÷ LA(~iK;R)

÷ LA(~i X;R) ÷

LA(K;R)

÷

LA_I(~i ~;R)~''"

These ideas are more fully expounded in Quinn's thesis,

[HS]

and in an article in the Georgia Conference on the Topology of Manifolds.

See also section 17 in Wall [HI9].

95

5.4.

The Periodicity

T h e o r e m ~ Part I.

Let N n be a closed orientable xN:Lm(H)

+ Lm+n(H)

manifold

by sending M + X to MxN ÷ XxN.

easy to check that this is a w e l l - d e f i n e d

Williamson

[H20] and S h a n e s o n

show that

xCP 2 is an i s o m o r p h i s m The general

Recall that L mA ( Z p W )

T h e o r e m i.

[HI0].

[HIg],

In this section we

for the non-simple

case,

case will follow in Section

x@p2:L~(~,w;R)

÷

Lh

--

R =

in W a l l

6.3.

A ~p~). = Lm+4(

For m > 5,

an isomorphism;

It is

homomorphism.

For R = Z, this map is d e t e r m i n e d partially

A = Wh(w;R).

and define

(w,w;R)

is

m+4

--

coinciding with the i s o m o r p h i s m

above if

Zp.

Proof:

Even d i m e n s i o n a l

case~ m = 2k.

Let ~:M ÷ X represent Assume

as in Section

throushout, free.

5.2 that ]~ =

we can assume

Then,

representing

Zp.

x e ~L~(~,w;R). Using R~-coeffients

~ is k - c o n n e c t e d

algebraically,

form on Kk(M).

an element

and Kk(M)

x is r e p r e s e n t e d

by a H e r m i t i a n

Let fi:SkxD k + M, i = l,...,r,

a basis

be immersions

for Kk(M).

M u l t i p l y i n g by {p2, the only n o n - v a n i s h i n g groups

are Kk(MX@p2),

isomorphic

Kk+2(Mx@p2)

and Kk+4(MxCp2),

kernel all

to Kk(M).

Let j:S 2 ÷ {p2 be an embedding generator

is

of ~2(~P 2) ~

~.

Define

96

representing

gi:skxs 2 ÷

a

MxCP 2 by

(fi,J) > MxCP 2, and assume they

skxs 2 SkxDkxS 2 (x,y) ~-~ (x,l,y) are in general position.

It follows easily from Theorem 1.1.6 and Spanier Chapter 5, that X(fi,fi,)

= ~(gi,gi,)~ ~(fi ) = ~(gi ) since

j is an embedding representing a generator. gi(Skxl)

[AI5],

The spheres

are disjointly embedded and framed, so we can do

surgery on them, obtaining a manifold N. of the surgery

and 9:N ÷ X.

Let W be the trace

Then k(MX@p2)

I~

Ki(W,Mx@p2 ) = and so Ki(W)

i = k+l otherwise

= 0 for i # k+2~ k+4.

We have Kk+4(Mx@p2)

--" Kk+4(W ) +

Kk+I(W,MxCP so Kk+4(N)

Kk+2(N)

Kk+4(W,N)

2)

Kk+l(W,Mx~p2)

= 0, and the only non-vanishing kernel is

~ Xk(M). Surgery on the spheres Skxl yielded immersions

hi

:sk+2

~ MxCP

2

; furthermore

~(hi,h j) = ~(fi,fj),

~(fi ) since the spheres gi(Skxl) Clearly the maps h i represent

~(h i) =

are disjointly embedded.

a basis for Kk+2(N)

and

correspond to the fi under the isomorphism Kk+2(N ) ~ Kk(M). Also, Kk+2(N ) ~ ~k+3(~ ) and so the h i are framed.

Thus

the surgery obstruction for Mx~P 2 ÷ Xx@P 2 is represented by the Hermitian form on Kk(M). 97

Odd-dimensional

case~

x s Lh(w,w;R),

R = Zp,

algebraically Kk~I(~U),

2k-l:

and as in Section

by the subkernels

Kk(U,~U)

generators

fi disjoint

and Kk(M0,~U)

in

in Mx@P 2 to get a manifold N. the surgery yields

from sk-lxs 2

÷ MxCP 2.

But Kk(M,U)

are disjoint

spheres

fi(sk-lxl)xpt.

Then Kk_I(N)

= 0, and as

framed embeddings

gi:S k+l +

N

Let W be the trace of the surgery.

~ Kk(MX@p2,u)

~ Kk(M0,~U),

framed spheres

re-

The maps gi generate Kk+I(N) ~

Kk+I(MXCP 2) ~ Kk_I(M). Then Kk(W)

embeddings

of Kk_I(M) , M 0 = M - Int(U).

Do surgery on the embedded

above,

4.3, x is represented

where

r U = ~ fi(sk-lxDk), i=l resenting

Let ¢:M ÷ X represent

m Kk(M,U);

so Kk(N)

representing

is free.

a basis

from the gi(sk+l);

also Kk(N) ~ Kk(W). Do surgery

on

and assume these spheres

let Q be the resultant.

Clearly

~

i = k+l, k+3

LO

otherwise.

Ki_2(M)

Ki(q)

=

The embeddings

gi determine

embeddings

gi ) in Q; these maps generate Kk+I(Q) r V = ~gi(sk+ixD i=! fi(sk-lxl) isomorphism

k+2) C Q.

~-~ gi(sk+Ixl), of kernels.

(also denoted

~ Kk+I(N).

Then the map Kk_I(~U) fi(ixsk-l)~

is an

sends Kk(U,~U)

Kk+2(V,~V) ; we must show it sends Kk(M0,~U)

98

÷ Kk41(~V),

gi(ixsk-l),

This isomorphism

Q0 = Q - Int(V).

Let

to

to Kk+2(Q0,~V),

To this end, note we have

Kk(M !,~U) ÷ Kk+2TM0xCp2, 2 )a2$~UxCP Kk+2(MxCP~UxCP 2) Kk+2(MxCp2,UxD) where

~ Kk+2(MxSP 2- UxD,~(UxD))

D is a regular n e i g h b o r h o o d

on the fi inside each component (the S 2 comes

of w h i c h has the homotopy

of the triple

and

We can assume V C Int(V'). induces

isomorphisms sequence

(V';V,V'-Int(V);~V).

+

~ Kk+I(~V')

Kk+I(~(UxD))

by the fact Kk+I(UX~D)

and the map

is an isomorphism.

= 0 since

(It is onto

~D + S 2 is the n o n - t r i v i a l

~D = S 3 and the homology

free of the same rank, identify

as before

seen by the M a y e r - V i e t o r i s

Now Kk+I(~(UxD))

sl-bundle,

sphere).

of ~V and ~V' in V' - Int(V)

) as is easily

Kk+i(~Ux@p2)

type of S2vSk+Ivs k+3

from D, the S k+l is c o n s t r u c t e d

The inclusion

Doing surgery

of UxD instead of UxCP 2 we get a m a n i f o l d V',

the S k+3 is the transverse

on Kk+l(

of S 2 in ~p2.

sequence.

it is an isomorphism.)

Kk+2(Mx@p2-1nt(UxD),~(UxD))

Since both are Now we

~ Kk+2(N - Int(V'),3V') K k + 2 ( N - Int(V),~V) Kk+2(Q0~3V).

This

concludes

the proof.

99

Theorem

2.

If K is an n - a d

an i s o m o r p h i s m

Proof:

L~(K,~;R)

Immediate

and T h e o r e m

by

~

and m - n > 3~ then

x~P 2 is

Lh (K,w;R) m+~

induction,

the

I.

100

five

lemma,

Theorem

5.3.2

Chapter

6.1.

The

Long

6.

Relations

Exact

Between

Sequence

of S u r $ e r y

Let w be a m u l t i p l i c a t i v e homomorphism.

For

is a s e l f - d u a l

subgroup

Def.

Let

structions and

0

I.

the

subgroup maps

and w:w

+ {+i}

a

Suppose

A

= L n ( l ; Z)

is an exact

÷ CH A +l(~;R) ~

of ob-

of L~(~;R)

M ÷ X with

(H = TOP,PL,

CPLn(I ,~

There

group,

Coefficients.

let ~ = (w,w).

by n o r m a l

X H-manifolds

For e x a m p l e ,

with

of W h ( w ; R ) .

denote

realizable

M and

Theorem

convenience,

CH~(~;R)

Surger Z Theories.

aM = ~ = aX,

or DIFF).

by T h e o r e m

4.4.2.

sequence

~ n H (~,A,R)

L~7(E;R).~ ~

÷ ~Hn(~',A,R ) -~ LI(E';R ) -~

Proof:

The

terms

follows

~(~,A,R)

exactness from

of this

Corollary

~ ~(Y,A,R).

We

sequence 5.2.1. show

at the

Let

that

last

B denote

ker(B)

0.

four the map

~

Li (Y;R)/CH~+I(~;R) n+l Let equivalence This ~

gives

is the

x s ker(8). over

x is r e p r e s e n t e d

R, M ÷ X, b o u n d e d

a well-defined

equivalence

8. is r e p r e s e n t e d 1

Then

map

relation

by N i + Yi'

by

ker(B)

~ normal

map

+ LA (~;R)/~ n+l

defined i = 1,2,

101

by a h o m o l o g y

by

N ~ Y. where

e I ~ 0 2 if

so that

aN I ÷ aY I

and

~N 2 ÷

3Y2 r e p r e s e n t

Suppose equivalence define

01 ~

over

maps

equivalence

over

Thus

R, the

81 -

02 ¢ c H A + I ( ~ ; R )

eI -

So we h a v e

derived

g:R ÷ R' be

g[f](rx)

rings; w,w'

÷

= g(r)f(x).

of the map Let

so that

9Y I and

g[l],A

C A'.

3Y2' the

W ÷ V is a h o m o l o g y of N ~ Y is

Conversely,

map

ker(B)

÷ LA+I([;R)

calculations

show

that

an i s o m o r p h i s m .

of a u n i v e r s a l A similar

Theorem

5.3.2.

version

sequence

a ring homomorphism,

where

of

can be

R and

let ~ and ~' ~e m u l t i p l i c a t i v e as u s u a l , = w.

Wh(~;R') We w a n t

g[l]:Rw

A, A' be

~N 2 ÷

) and e x t e n d

4.5.1.

so that w ' f

g[f],:Wh(w;R)

groups

if W + V is a h o m o l o g y

02.

and in fact

using

homomorphisms

homomorphism

eI

is sort

~nH(~',A,R).

obstruction

Straightforward

in T h e o r e m

are p r i n c i p a l

Since

a well-defined

sequence

for n - a d s

Let

map

~N I ÷

surgery

implies

is a h o m o m o r p h i s m

sequence

with

in

Then

62 ¢ cHA+I(-~;R).

cHA+I(~;R).

This

R'

bounds

to get N ÷ Y.

62.

the

class

02 as above.

R which

eI -

this

same

N = NILJWLJ(-N2) , Y = YIUVU(-Y2

respective

modulo

the

Then

let

f:~ + ~' be a

there

induced

by

to study

is a ~ / 2 E - e q u i v a r i a n t g[f]:Rw

the

effect

÷ R'w', on s u r g e r y

÷ R'~.

self-dual Let

and

groups

subgroups

H = (w,w,R,A),

102

of W h ( w ; R ) ,

Wh(w;R')

H' = ( ~ , w , R ' , A ' ) .

Using the n o t a t i o n hH~-IhHQm_I(H Define

of Section

=

~ m ( H ' ~H)/g, ~ m ( H ) ,

is the induced map. we can assume

By the remarks

all spaces

=

There is a long e x a c t

following

Corollary

÷ Qm(H') 5.2.1~

sequence L(g)

,LAm( ,w;R)

> ~ m ( g ; H , H ')

where g,:Qm(H)

involved to be manifolds.

j, ...

~m(H',H)

).

~m(g;H,~')

T h e o < e m 2.

5.2, let

, > ~ m _ l ( g ; H , H ') ÷.-..

Proof:

This can be proved using classifying

s h o w i n g that the fiber of the required properties;

~m(W,w,R,A)

÷

spaces and

LLm(W,w,R',A')

we give here an elementary

has

geometric

proof. Define L(g) by sending

to be the functorial map;

f:M ÷ X in L A ' ( w , w ; R ') to the class of ~f:~M ÷ ~X, m

a homology e q u i v a l e n c e

over R'.

be the class of f in L~(~,w;R).

We define j , [ f : M + X] to Elementary

show that these are well-defined. equivalence

L(g)j,

~,L(g)

We

Clearly

if f:M ÷ X is a homology 0 in Lm(W;R').

= 0: ~,L(g)[f:M ~ X] is r e p r e s e n t e d

~X, a homology

(iii).

homology

= 0:

over R', then f represents

(ii).

~f:~M ÷

assumed to be normal.

of A, A', and w throughout.

(i).

by ~f:~M ÷

considerations

Recall that a homology

over a ring is always

omit m e n t i o n

equivalence

2, is defined

j,~, = 0:

equivalence

over R, and thus is 0.

j , ~ , [ f : M ÷ X] is r e p r e s e n t e d

~X in Lm(W;R) , and f gives a b o r d i s m of ~f to a equivalence

over R. ]O3

by

(iv). normal

map,

cobordant f':M'

ker(L(g))

C Im(j,):

f a homology

to a h o m o l o g y

÷ X'.

Then

(v).

~f is cobordant

equivalence

equivalence

j,[f']

ker(~,)

cobordism

property

(vi).

(pg.

[f:H

=

45 in

ker(j,)

C Im(~,):

Then

~,[F]

This

Remarks:

(i) [K9]

L~(~)

to be the

to relate

to change

locally

and

the

now shows

that

in this

This [KS]. epic,

cobordism

by r a t i o n a l case

type was

and L ~ ( ~ ) , group,

to be

Then

the proof.

first

where

used by

he defines

except

h-cobordisms.

is shown

÷ X] = 0.

over R by a c o b o r d i s m

completes

of this

L~(~)

usual

= (~,w,g, Wh(~)),

are

= If].

A sequence

Pardon

and S h a n e s o n

[J22])

Let j , [ f : M

F:N ÷ Y.

(2)

R, the c o b o r d i s m

~ X].

equivalence

~m

If] E ker(L(g)

8X

to a homology

factor

over

over R', F:N ÷ Y,

f is c o b o r d a n t

allowed

i.e.

If ~,[f:M ÷ X] = 0, then

equivalence

equivalence

÷ XOY]

over R, and f is

over R',

C Im(L(g)):

given by a homology

L(s) EfUF:MuN 9N

f:M ÷ X is a

= [f].

to a homology

extension

Suppose

boundaries

The

~%m(H',~),

correction where

Z' = (~,w,Q,Wh(~,~)). is also r e l a t e d In fact, then

to the surgery

of Cappell

if g ~ ÷ R~ and g[l]:R~

~ m ( g ; H , R ') ~ F~+I(~) , where

÷ R'~ ¢ is

the d i a g r a m

and H = (~,w,R,Wh(~,R)) (3)

are

A similar

etc. formulation 104

can be done

for n-ads.

6.2..

The

Rothenber$

In this Rothenberg

with

the p r o o f

Lemma with

section

using

a map

2.3

2.3)

~p w i t h

q:W ~ M so that

Then

A be a kxk m a t r i x

Z~-module

to get

on the

~

identity

on g e n e r a t o r s

~2(~+I(W+,M+)),

so every

Let

A = (aij)

and

be the

over Zp,

factor).

i = l,...,k,

trace and

gives

4.

and M n a m a n i f o l d

Furthermore

representing

there

is

q is a h o m o l o 6 y

Then

assume

W ~ Mxl ÷ M is the

aij

let

Then

map

105

amounts

is r e p r e s e n t e d 5.

(if not,

~

a

=

multiply

representing

(~ixl;Mx0,Mxl)

~ is a h o m o l o g y

over Zp,

q.

this

~2(W+,M+)

on spheres

~:(W;M,M')

an h - c o b o r d i s m desired

E Z~

and

is a free

dim M+ = n ~

Do s u r g e r y

and

since

~2(W+,M+)

Also,

M+,since

Add k t r i v i a l

(W+;M,M+)

in ~ 2 ( W + , M + )

of the s u r g e r i e s . so is

triad

x.

(Mx±;Mx0,Mxl),

element

S2 C

k j~laijej,

of C h a p t e r

i:M C W;

e l , . . . , e k.

embedding

a suitable

= x.

map.

by a f r a m e d

A by

of these

We

is an h - c o b o r d i s m

a manifold

~+:(W+;M,M+)

to s u r g e r y

6.1.

~p.

to Mxl

map

there

qi = I, w h e r e

Proof:

normal

light

T(W,M;~p)

over

2-handles

first

of

to a r b i t r a r y

of S e c t i o n

x a Wh(W;~p)

equivalence

Let

The

sequence

[Hg]

sequence

in the

Let

= w, n _> 5.

over

the

and S h a n e s o n

of lemmas.

of T h e o r e m

~I(M)

generalize

the exact

a series

i (Theorem

(W;M,M')

we

(unpublished)

coefficients begin

Sequence.

T(W,M;~p)

equivalence = x, and

Lemma

2.

Let f:M n +

Zp between

manifolds,

f is c o b o r d a n t ~ homology

X n be a h o m o l o @ y

over

n _> 5; let a E W h ( ~ I ( X ) ; Z p ).

by a h o m q l o s y

equivalence

equivalence

equivalence

over Z p w i t h

Then

over Z p ,

torsion

to a

a if and only

T(f;Zp)

= b + (-l)n+ib * + a for some b s W h ( w l ( X ) ; Z p ) .

Proof:

Let w = wI(X).

groups

We i d e n t i f y

by t h e i r r e s p e c t i v e

maps

W h ( W l ( X ) ; ~ P) are i d e n t i f i e d Suppose

T(f;~p)

be an h - c o b o r d i s m the map F(w)

all r e l e v a n t

(e.g.

by f,).

= b + (-l)n+ib * + a.

in L e m m a

= (fq(w),@(w)),

Whitehead

W h ( W l ( M ) ; ~ P) and

over ~p with ~ ( W , M ; Z p )

constructed

if

i.

Define

Let

(W;M,M')

= b and q:W ÷ M F:W ÷ Xxl by

w h e r e ~ is a U r y s o h n

function.

Then

the d i a g r a m M

)

Xx0

~

Xxl

~

<2 Xxl

F W tJ M' commutes,

f,

(*)

so ~(f';~p)

= T(F;Zp)

+

T(W,M' ;~p)

= T(f;~.p) =

b

+

T(W,M;~p)

(-l)n+ib

*

+

a

+

-

b

T(W,M';Zp) +

(-l)nb

*

by T h e o r e m =

a.

Now s u p p o s e equivalences

over

T(f;Zp)

3.1.4

we have

~p,

a diagram

T(f';~)

= ~(F;~) = ~(f';~p)

= a.

(*) w i t h

f', F h o m o l o g y

Then

+ T(W,M;~p) - T(W,M';~p) 106

+ ~(W,M;Zp)

a + T(W,M;Zp)

=

Lemma

3.

Let

a homology with

a s Wh(W;~p),

equivalence

T(f;~p)

equivalence

+ (-1)n+I~(W,M;Z~) *

= a. over

over

~,

M and

= ( - l ) n + I T ( f ; ~ p ) *.

Proof:

Let + X',

n odd:

the

the

surgery

cobordant

above

matrix over

F:W ~ X'xl

as

is

X manifolds,

if f:M n ÷ X n is a h o m o l o g y

by

n _> 5, then

a matrix

A

aad

define

normal

maps

follows:

matrix

~p c o b o r d a n t form

obstruction

to a h o m o l o g y represents

representing

Hermitian

f0"

(G,~,~)

Ax =

Then

where

n even:

a normal Apply

map

over

f0:M0

we

can

o

I

~ X';

~p since Let B be a

f' is a h o m o l o g y construct

equivalence

a (-i) ( n + l ) / 2 -

with

/

<__~SfZ

.

.

with

su2gery

A to the basis

] .

.

.

.

.

.

.

.

.

i<~') -1

[o F be

normal

0 in L ~ ( w , W ; Z p ) .

T(f';~p), to

of some

equivalence

A

Let

there

f:M n ÷ X n, M and

X manifolds,

a be r e p r e s e n t e d

Then

The m a t r i x

represents f0 is

~,

C onversely~

T(f;~p)

f':M'

a = ( - l ) n + l a *.

obstruction

of the

107

standard

o/

(G,X,~). kernel

to get

a (-l)n/2-Hermitian this f':M'

form.

Then

÷ X' w i t h

Then

let

cobordant

torsion

f' + ~F:M'

To show h-cobordisms

a simple

f0 is

(G,I,~);

f0:M0

÷ X' r e a l i z e

to a h o m o l o g y

represented

by B.

equivalence

Let

F represent

in L ~ + I ( ~ , W ; ~ ) .

(B~)-

T(W,M;Zp)

form

the

over

+ SW ÷ X' + $(X'xI)

converse,

~p,

(W;M,M')

= a, T ( Y , X ; ~ p ) Poincare

construction

and

= -a.

complex

W

of

suppose

and

Y.

T(f;~p)

(Y;X,X'),

Then

the

~p.

This

over

has

Using

this

= a.

a.

Construct

with

space

we

torsion

N = W

follows

<-7 Y is f

from

the

get

(-l)na * = T(W,M';~p) = T(N,M';~p) =

(-I)nT(N,X,;z2>*

=

(-I)nT(y,x';Zp)*

=

(_l)n((-l)n+la*)*

-

a

by

repeated

use

of T h e o r e m

3.1.4.

We

can n o w

prove

our m a i n

theorem.

Theorem where

i.

Let

A C B C Wh(~;~p)

* is d e f i n e d

by w : ~ ÷

be

{+i}.

self-dual Then

there

subgroups is a long

exact

sequence • .. ÷ H n + I ( ~ / 2 Z ; B / A ) ÷ termin&tin~ Here

L~(~,W;~p)

÷

L~(~,w;~)

÷

Hn(~/2~;B/A)÷

,at H S ( ~ / 2 Z ; B / A ) .

H*(~/2Z;B/A)

denotes

cohomology

108

of g r o u p s ,

[A26].

...

Proof:

We show that

= Hn+I(~/2~;B/A)-" where

~n(I;K,H')

K = (w,W,~p,A), H' = ( w , w , ~ , B ) .

The result will then follow

from Theorem 6.1.2. {b s B: b = (-l)n+ib *} {b+(-l)n+Ib*:beB} + A

Recall that Hn+I(z/2Z;B/A) and define ÷ Hn+I(~/2Z;B/A)

¢: ~ n

by

[f:M ÷ X] = T(f;Zp). ¢ is well-defined and injective by Lemma 2.

(Note we need only consider cobordisms

F:W ÷ Xxl by the remarks

following Corollary 5.2.1, Theorem 6.1.2 and the five lemma.) ¢ is surjective by Lemma 3. We interpret L2kB ~ H 2k sends

the maps in the sequence as follows:

[G,l,p] to [Al];

H2k ÷ L2k_l A sends

[Q] to the class of [~

(9,)-I ;

B ÷ H2k-I L2k_l sends

[~] to [T(~;Zp)];

H2k-i ÷ L2k_2 A sends

[Q] to the (-l)k-l-Hermitian

form defined

by applying Q to the basis of the standard kernel.

6.3.

The Periodicitz Theorem~

Part II.

In this section we use the Rothenberg sequence to complete the periodicity

theorem.

109

T h e o r e m I.

For m > 5,

x@p2:L~(~,w;R)^

+

LA (~,w;R) m+4

is

an isomorphism.

Proof:

By T h e o r e m 6.2.1 and the five lemma we need only

show that

x{P 2 induces

Hm+4(Z/2Z;B/A), as in Section

an i s o m o r p h i s m Hm(~/2Z;B/A)

where we interpret

these groups g e o m e t r i c a l l y

6.2.

Let f:M ~ X be a h o m o l o g y

equivalence

C, the chain complex of f as in Chapter 2. cell d e c o m p o s i t i o n

e0~e2~e

But Hm(~/2~;B/A)

i.

is 2-torsion,

For m > 5,

over R and

Since

{p2 has a

4, the chain complex of

fxl~p 2 is given by C, + C,+ 2 + C,+ 4.

Corollary

÷

Thus T(fxl;~p)

= 3T(f;~).

so this is an isomorphism.

x~p2:~m(g;H,H')

~

~ m + 4 ( g ; H , H ')

is an isomorphism.

6.4.

Simple L i n k i n g Numbers.

Let ~ be a finite group and w:~ ~ {+I) a homomorphism. Let K(~)

denote the k e r n e l of the map Wh(~) ÷ Wh(~,~).

this section we will define a K - t h e o r e t i c linking

forms which

LK(~)(~,w) n

determines

÷ LS(~,w;~) n

In

group of simple

the kernel of

for n odd.

110

This also estimates

the

the kernel of the c o r r e s p o n d i n g

map with K(w)

Let G be a free and b a s e d ~ - m o d u l e Hermitian

a simple

and ~ a (-I) k-

form on G which is n o n - d e g e n e r a t e ,

is injective.

replaced by s.

i.e. AI:G ÷

We further assume that A ~ ® I : G ® ~

isomorphism.

Let A denote ~ / X ~ ;

~

G*

G * ® ~ is

let

Ik :

L e m m a I.

We can associate (i)

X0:KxK ÷ A is ( - l ) k - H e r m i t i a n

(iii)

and non-singular,

~0:K ÷ ~w/l k is an a s s o c i a t e d

Let K -- coker(A~);

rG* C Im A~.

(K,k0,~ 0) where

K is a torsion group,

(ii)

Proof;

to (G,~) a trip!e

Define

~$(x,y)

quadratic

form.

then there exists r so that

A S:G*xG*

÷ ~w by

= 12~(A~irx,A~Iry).

For x,y s Im A~,

(A Ix,

=

=

A ly)

x(A~IF)

£ Z~,

s o t h e map ~o:KXK ÷ A, k o ( f ( x ) , f ( y ) ) f:G* ÷ K, is w e l l - d e f i n e d .

= k$(x,y)

It is easily

rood ~ ,

verified

that

~0

is (-i) k-Hermitian. A p p l y i n g Homz~( 0

+

G

÷

,Zw) to the sequence G*

~

K

÷

0

we get 0 = ExtI(G*,Z~) Therefore

÷

coker(A~)

Extl(K,~w)

+ G*

÷ G**

÷ Hom(K,~w)

= 0.

-~ Extl(K,~w).

Applying Hom~(K,

) to

0

111

+ ~w

÷ Qw

÷ A

÷ 0 gives

0

÷ K*

÷ Hom(K,Q~)

and so Hom(K,Qz) Since

÷ Hom(K,A)

÷ Extl(K,Zw)

÷

Extl(K,~w)

= 0

m ExtI(K,Z~).

A1 = A~¢

, ¢:G

~

G**,

¢(g)(x)

= x(g)*, we have

K = coker(Al)

= coke~(A~) = ooker(A~) Hom(K,A). I0

This map

K ~ Hom(K,A)

is equal to

Ak0 , so

is non-slngular. Define

D0:K ÷

£w]I k by D0(f(x))

='~$(x,x)

mod I k.

We leave it to the reader to show that p 0 is well-defined, D0(xa)

= a*D0(x)a , and ~0(x,y)

D0(x+y)

- P0(x)

+ (-l)kl0(x,y) * =

- p0(y).

(K,10,D 0) is called a standard

If (K,X0,P0) D0:K

is a triple,

÷ ~ / I k an associated

(K,10,D 0) is a simple (i)

and based

e

~

form,

form.

A (-l)k-Hermitian, then we say

form if

(K,-IO,-D0)

is an injective

so that f ~ l : F l ~

linkin$

10:KxK

quadratic

linking

(K,10,D0)

(ii) there

simple

~

is standard,

map f:F I ÷ F2, FI, F 2 free

F2®~

is a simple

isomorphism

and K = coker(f). Define

the group

to be the monoid standard

simple Let

homomorphism

of simple linking

linking

linkin@

of Theorem

rational

~ 2 Sk _ l (W~W) ,

the monoid

of

forms.

6.1.2,

Then coker(~,) homology

forms,

forms modulo

~,:L~k(~,w;Q ) + ~ 2 k _ I ( I ; H , H ' )

H' = (~,w,Q,0). simple

of s i m p l e

where

denote

H = (~,w,~jK(~)),

is the cobordism

equivalences 112

the boundary

modulo

group

cobordism

of tca

homology

over

cobordism).

normal

Lemma

equivalence

2,.

Let

M a manifold

Then

The

exact ~

ExtI(K;Z~),

Kk(M;A)

and we

0

a map

(k-l)-

numbers

associated

define

quadratic

~

~

~ {~

~

A ÷

÷ Kk_l(M;{~)

As

in L e m m a

0 yields

= 0.

Let

I, H o m ( K ; A )

=

get ---* K k - I ( M ; A )

K

÷ Hom(K,A)

÷ Kk(M)

= K*

÷ 0.

and so a n o n - s i n g u l a r

map

~ A.

~ K be

disjointly in M.

represented in M;

Then

we

can d e f i n e

by

(k-l)-spheres

rx = 0 for some

X0(x,y)

=

pairing.

Similarly

satisfies

the p r o p e r t i e s

The

of o u r m a i n

first

i.

a homolosy linkin~

0

= Kk(M).

÷ Hom(K;A)

defines

Theorem

f is

" s e l f - l i n k ins"

÷ Kk_I(M)

Geometrically, x,y

equivalence,

Assume

on K k _ I ( M ) , w i t h

sequence

K = K k _ I ( M ) ; t h e n K*

~0:KxK

and

~ coker(~,).

homology

> 5.

(by any

A.

0 = Kk(M;£~)

This

2k-i

"linking" form

in K(~)

show ~ 2 k _ l ( ~ , w )

f : M ~ X be a r a t i o n a l

a (-l)k-Hermitian

Proof:

torsion s

We w i l l

of d i m e n s i o n

connected.

f.orm~ o v e r

~ with

Let

for

technical

f : M + X be

equivalence

form

on K k _ I ( M )

over

Sx~

where

follows: Sy

~ is

It is e a s i l y

alluded

as

let

embedded

r so rS x b o u n d s

~(Sy,~), U 0.

this

the

a k-chain intersection

checked

that

this

to above.

results

as above. ~ with

Then

torsion

is s t a n d a r d .

113

is:

f is c o b o r d a n t in K(~)

i ff the

to

Proof:

Suppose

homology

F:N + Xxl is a c o b o r d i s m from f to the K(w)

equivalence

k-connected

over Z~ f':M'

and the exact

÷ X.

sequence

We may assume F is

of the pair ~,M)

reduces

to 0

÷ Kk(M)

÷ Kk(N)

÷ Kk(N,M)

By adding trivial handles

~ Kk_I(M)

÷

we can assume Kk(N)

are free and of the same rank,

since Ki(M)

0.

and Kk(N~M)

is a torsion

group,

i = k-l, k. Furthermore,

Kk(N,M)

and so the middle map above

~ Kk(N,3N) defines

This is defined g e o m e t r i c a l l y simple i s o m o r p h i s m the linking

over ~w.

form on Kk_I(M)

Conversely,

0 where

basis

a pairing

in Section

X on Kk(N).

4.3 and A X is a

and so Kk_I(M)

induces

is standard.

assume we have an exact sequence

÷ G

~ G*

the form on Kk_I(M).

~ Kk_I(M) form,

~ 0

simple

over ~ ,

which

Let el,...,e m be a p r e f e r e d

for G and e~,...,e*m the dual basis. Write Ax(e i) = e~aji , aji ~ ~ .

so if Sj is an e m b e d d e d Sjaji

~ Kk(N)* ,

The form on Kk(N)/Kk(M)

I is an ( - l ) k - H e r m i t i a n

induces

~ Kk(N)

is a b o u n d a r y

(k-l)-sphere

in M, say

Then ~(e~aji)

which represents

~i

= Sjaji"

at the S i in Mxl to get (N;M,M').

We have

n i is the core of the handle lifts to the class fi a Kk(N)

at S i.

114

~(e~),

Attach handles 3n i = -Si, where

It follows

represented

= 0,

that Ax(e i)

by qjaji + ~i'

Here we are using Kk(N)

÷ Kk(N,M )

÷ Kk_l(M)

÷ 0

G ~ Kk(N,M') , G* ~ Kk(N,M) ~ K k ( N , M ' ) . Let ~p denote the result of moving qp a small distance so that qp and ~p are disjoint. epbpi, bpi = ~(~p~qjaji

=

Then the image of fi in G is

+ ~i )

-A(Sp,6i).

Let r be an integer with r~ = 0.

Then rS i = ~E i

for some k-chain E i and r~ i - Ejaji is a cycle. rbpi = r~(Sp

Therefore

,~i )

= ~(Sp,Ej aji) =G.a..

P3 Jl where Gpj = ~(Sp,Zj). Choose Kpj SO that re~ = A~(epKpj); construction,

~0(¢(e~),¢(e~))

i also equals ~Kpj.

by the geometric

= rl-~pj and by hypothesis,

Let Pij = ~(Kij i -- ~ij )" and subject the

spheres S.Z to simultaneous mutual intersections

disjoint

regular homotopies

~jaji + ~i; then the image of ~

i

Let ~i ~ Kk(N)

in G = Kk(N,M)

epSpi , where ~pi = bpi + Ppjaji I )aj = bpi + ~(Kpj - ~pj i .aj r pJ i"

= ~K ^

Thus AA(epbpi)

with

Pij to get embedded spheres Si"

Do surgery on the Si to get (N;M,~). represent

it

I

= Ax(r--epKpjaji)

= e~, and so the diagram

115

is

Q*

K

¢

K kc''N,M

Kk( M)~

"

17/ Kk(1~'~ ) ~ G

~

Kk-1(lgI)

commutes. Since i, @ ~, is onto i,A~ = m,, i, is onto. duality.

Since

(aji)

f:M + X has torsion

In the non-simple and our proof

(by Mayer-Vietoris)

Thus Kk_I(M) is e l e m e n t a r y

in K(~).

This

follows

= 0 and Kk(~)

= 0 by

over ~w, as is (<ji), concludes

case s this result

and

the proof.

is due to Wall

[HI8]

his.

We now prove our form of Connolly's r e a l i z a l i z a t i o n theorem

[K6].

Theorem

2.

Let

(K,A0,~ 0) be a simple

a m a n i f o l d with Wl(X) map

F:(W;~_W,~+W) i)

form,

X 2k-2

Then there is a normal

÷ (Xxl;Xx0U~XxI,Xxl)

so that

FI~ W is the identity,

ii) ~orsion

=w, k _> 3.

linking

FI~+W is a homolo6y

in K(~)~ (iii)

equivalence

over ~ with

and

the !inkin~ form on Kk_I(W)

the form on K. 116

coincides

with

Proof:

Let

GI,

exact

sequence

G 2 be free Z w - m o d u l e s

f 0 where and

÷ GI

~ K

a normal

map

This and

k to Xxl

~.

Let

el,...,em,

G 2 respectively.

Add m

to get

N and

a manifold

÷ Xxl.

Then Kk_I(~+N) x. = i - t h 3. Yi = i - t h

over

for G I and

of i n d e x

N

an

÷ 0,

isomorphism

el,...,e*m be b a s e s handles

we have

¢ ~ G2

f is a s i m p l e

trivial

so that

is a free Z w - m o d u l e

sphere

IxS k-I

sphere

sk-lxl

with

generators

i = l,...,m.

gives

Kk_I(~+N)

the

structure

of a s t a n d a r d

kernel,

so m

m

~( i=l [ xiai+Yib i) Here

p denotes Write

with

the

=

usual

f(e i)._

self-intersection

= e~a.. as b e f o r e j jz

a.. in the j - i - t h jm

0 otherwise.

[ b.*a.. z

i=l i

spot

if the

By h y p o t h e s i s ,

cij

e X0(¢(e~),¢(e~))

the

entries

of B are

bij

= cihahj

and

and

above

let A be the m a t r i x equation

A is e l e m e n t a r y

let

in ~ ,

number.

C = (cij);

since

= k0(~(e~),¢(e~)ahj)

over

define

for some

holds Q~.

B =CA.

and Choose Then

h,

m o d Z~

= k0(¢(e~),¢f(eh))

Also, and

=

XO(¢(e~),O)

=

O°

(A'B)*

= B*A

the d i a g o n a l

= A*C*A

terms

of

= (-1)kA*B, A*B

are

117

in

since Ik

C* = (-1)kc,

since

0 = po(@f(ei)) = PO(¢(e~aji)) = It follows

a~.jmcjj..aji mod I k-

that there is a matrix Q over Zw so that

A*B = Q + (-l)kQ *. m Let u i -- j=lT"(xjbji+yjaji) l(ui,u j) = 0, i#j,

in K k_l(~+N).

and so we can do surgery on embedded

spheres representing the classes u.. ! the surgeries

f >~l

0

~

>Ck(W,X)

> ~G2

G2

+ C k_I(W,X),

8g~(Dkx0) K

P

>Ck_l(W,X)

+ Ck(W,X) , e i ~-. g~(Dkx0)

extends gi:sk-lxD k

above).

Then

¢

where the vertical isomorphisms GI

Let W be the trace of

(including the initial surgeries

0

Then ~(u i) = 0

~k!=

> K _z(W)

= yjaji = f(ei).

~ 0

are given by

where g~:(Dk,sk-l)xD k-I

÷ ~+N which represents

e.*l~'~ Yi"

)0

+ (N,)+N)

ui, and

The diagram commutes

since

Thus we have an isomorphism

÷ Kk_I(W). Le5 I$, ~6 denote the linking and self-linking

on Kk_I(W).

We must show l$(p(yi),p(yj))

so that r~ = 0.

= cij.

Choose r

Then rG 2 C f(Gl), so

m re~ = f( 7 e.a!.) iL 1 m lj for some a!. mj £ K~.

ryj

Let A' = (a!.); then AA' = rl and lj

m [ g~(Dkx0) a! .

=

i=l

i~

m

= i[lgi(sk-lx0)aLj. 118

forms

So

l = yl(yi,

l~(p(yi),P(yj))

m [ g~(sk-lx0)a' .) h= I ,, hJ

i m ,gh(Sk_ix0 r h=~ I ~(Yi ' ) ) ahj m m i = r hX 1 qX1 ( l(Yi'Yq)aqh+l(Yi'Xq)bqh)ahi m _ i ~ ~i b a' - ~ h ih hj

= cij m6d ~

since ~BA'r = BA'(AA')-I = BA

= CA 2 and A 2 = al for some a ~ ~. self-linking

numbers.

fact that ~:Ck(W,X)

A similar

Part(ii)

~

follows

Ck_I(W,X)

result holds

for

immediately

has matrix

A.

from the

This

completes

the proof.

Our main theorem

is:

Theorem

3•

coker(~,)

Proof:

The constructions

Lk:coker(3,) jective

÷

~s

by Theorem

~ ~ s2 k _ l ( ~ w ) •

2k-I

in Lemma

(~,w) which

2 define

a map

is w e l l - d e f i n e d

i and surjective

by Theorem

and in-

2.

The only

thing we need to check is that the form on Kk_I(M) linking

form.

Assume

(M,~ M) has a handle k and k-i only.

M is a triple decomposition

The sequence 119

(M;3 M,3+M); with handles

is a

then in dimensions

0 satisfies

+ Ck(M,~_M)

÷

Kk_I(M)

÷

0

(ii) of the definiton.

To see that is standard,

notice

M + (-M) and apply

Corollary

~ Ck_I(M,~_M)

i.

(Kk_I(M),I0,~ 0) @

(Kk_I(M),-10,-~ 0)

that this is the linking Theorem

form of

i.

For w a finite group,

k > 3, there

is an

exact sequence 0

Proof:

÷ ~ S

2k_l(~,w)

Immediate

Corollary

2.


÷

-2k-l" ~,w)

L2k-l( ~,w;~).

from T h e o r e m 6.1.2 and T h e o r e m

As in Corollary

H2k(~/2~;K(~))

÷

ker(

We have the following

3.

i~ there is an exact sequence s ~,w) L2k_l(

s

Proof:

s

+

s ~ L2k_l(W,w;~))

H2k-l(

diagram

120

÷

H2k(l/2~;K(w)

jjjjl

) i[l],

0

)ker(i[l],) I

s - l(Z,w) ) L2k

2k_l(~,w)

(~,w;{)

¢

l

O___~s

s

> L2k_l

K(~), ~ L2k_l(~,w)

s

> L2k_l(~

,w;£)

"~ H2k-l(~/2Z;K(~) )

with exact rows and columns.

It is easy to construct

maps i n d i c a t e d by the dotted resulting

Remark:

sequence

According

the

lines and to show that the

is exact.

to Bass

[C2], T h e o r e m 7.5, if ~ is finite,

then there is an exact sequence

0 ~ SKl(~) It follows

÷ Kl(~W)

~ KI(Q~).

that

K(~) ~ SKI(Z~)/~ for ~ abelian, general

* Chapter

case

and this allows

(theoretically

us to compute K(w)

anyway).

XI.

121

in the

Appendix

A.

Torsion

for n-ads.

In Chapter

5, the torsion of a h o m o l o g y

over a ring of n-ads was

(implicitly)

of the map of u n d e r l y i n g

spaces.

without

A

a c{l,...,n},

that

is a

A A-homomorphism of rank

for

8 c a.

is d e f i n e d

are defined

Define

Kl(A)

of a (finite)

If

of

of the form

is an object

M

so

An(m)

l i n e a ~ group

of

The free

= A(m) n.

GL(n,A)

to be

An; the e l e m e n t a r F t r a n s f o r m a t i o n s Stably,

E(A) = [GL(A),GL(A)].

= GL(A)/E(A).

w by

is e q u i v a l e n t

simple h o m o t o p y

is a group of type (R~)(a) = Rw(a).

GL(1,~p~) f(m)(k)

Define the W h i t e h e a d

to the d e f i n i t i o n

tree, used by Farrell

the theory of infinite

subgroup

for each

in the obvious way.

as before.

This d e f i n i t i o n

R~

i e

and all maps are compatible.

n, An, is d e f i n e d by

E(n,A)

define

2n

A A-module

A(~)-module

the group of a u t o m o r p h l s m s

K1

this

is a ring and h o m o m o r p h i s m s

Define the 5eneral

of

We can g e n e r a l i z e

be a ring of type

A(~)

÷ A(a)

M(m)

A-module

d e f i n e d to be the t o r s i o n

too m u c h trouble. Let

ras:A(8)

equivalence

generated

= rgk,

types.

2n

If

group by

122

and

See R

R = ~p, let

by maps

ke~pW(a),

and W a g o n e r

[Dll].

is a

ring,

U

be the

f:~pW ÷ ~p~ geT(m),

in

r~.

=

Let be m a n i f o l d

M = (IMI;M1,

n-ads.

We say that

if there is an n-ad is a m a n i f o l d M CINI each

and ~,

Theorem then

i.

M' c INI

If

N

N ~ M x !

T(N,M)

induction.

are homotopy

M'

between

1' , . . . , M ' n _ l

are h - c o b o r d a n t so that

INI

Nn_l U M',

equivalences,

is an h - c o b o r d i s m

M(~)

between

M

and for and

M'(~).

and

M'

if and only if a t o r s i o n invariant vanishes.

is d e f i n e d to be the t o r s i o n of the

"chain complex n-ad" KI(~I(M))

and

MuN I m...o

is an h - c o b o r d i s m

• (N,M) e Wh(Wl(M)) Proof:

M

M' = ( I M ' I ; M

and

N : (INI;NI,...,Nn_I)

with b o u n d a r y

N(~)

. . . ,M n _ l )

C,(N,M)(~)

= C,(N(~),M(~))

, where the c o n t r a c t i o n The result

follows

$

is c o n s t r u c t e d

by the s - c o b o r d l s m

and induction.

123

in by

theorem

)

Appendix

B.

Higher

L-Theories.

In Chapter by geometric

5, we defined

construction.

is there an algebraic Chapter answer

is yes for

K

Ignoring denote

Ln(K(~,I)

Case i.

The following

definition

4 that the answer

of

is yes if

torsion

K

We saw in

is a space.

The

and orientation,

let

Lm(W÷w'; % )

÷ K(~',l);%).

4-tuples

(i) (ii)

K

%w

with subkernel (ii)

sl~

is a subkernel

of

~ (G' I' ~' K')

(G,k,~)

in Chapter

7.

where:

S

G ® ~ w P if there

-ff

form over %w'. is a kernel

so that

@ H @ (G',-I',-~')

= HI

is a kernel

SI, and any automorphism

= =.~'~

to

Ke

of

(s~

P

modulo

HI ® % w

~

<~')

Z~p~'

® ~'

taking

is in

EU(~pW').

p

L2k+l(W+w';%)

(G,X,~,K)

[Mig]

is a (-l)k-Hermitian

with subkernel

(i)

by Wall

(G,I,~,K)

(G,k,~)

(G,k,~,K)

over

Then

Ln(K;R)?

arises:

n = 2k+l.

We consider

H

question

groups

a 2-ad and we sketch the construction.

This is done in detail

Define

surgery o b s t r u c t i o n

is isomorphic

the equivalence

to the group of objects

relation

124

~.

Case 2.

n

2k.

=

This was determined Wall

[H19],

where

¢

W

Chapters

= <~,w'>. Define

xlj,

(see the end of Section

1.7).

module

of dimension X

r)

~pW'

the relation

I + (-l)kQx = A £ SL(r,ZZpw')

(ll)

P' - P = X + (-l)kx * + X*QX

+

~ (P',Q')

by

map

h:St(~p~')

= St U(~p~')/h(ker

the associated

(P,Q)

Q = AQ'A*.

st

St U(~pW')'

(on a free

so that

and we have a natural

split group,

Ur(~p~)

i1

forms over

modulo

and

to be the group of pairs

(1)

(iii) Define

by

St Ur(~pW')

(-l)k-symmetrlc

see also

W h 2 ( w ' ; ~ p) = K2(~p~')/ker

xijxji

of

is an

Let

[HI4];

St(Z~p~') ÷ E(~p~'),

generated

(P,Q)

if there

8 and 17G.

is the natural map

is the subgroup ET'

by Sharpe

¢IW),

+ St U(~p~'). where

~

Let

denotes

e.~.,

is the set of things

a + (-l)k+la * = ~*y,

125

,

,

b + (-l)k+ib * = ~*B.

¢IW,

T'

acts on

St

Let

St U ( % ~ ' ) '

U(~';~p)

P(~+wT;~p)

by conjugation and define

= T'XT,St U ( ~ p W ' ) ' / [ w ' , w ' ] .

be the p u l l - b a c k of

u(zp~) $ st Finally,

+

we have

L2k(W+w';%) ~ P(~÷~';%)/. Considerations like these can be used to give algebraic definitions of surgery groups to solve problems of the form "how to get a homology equivalence over Int(M)

and a homology equivalence over

R'

on

R

~M".

on This

is similar to the problems solved by Cappell and Shaneson's relative (with

F-groups,

2 ~P)

The condition

[KS].

Miscenko

([K8])

defines

L

groups

algebraically by an algebraic bordism procedure. 2 6P

(which eliminates self-intersection

problems) has been removed

(at least in some cases) by

Connolly.

126

Append.ix C.

L

Groups of Free Abelian

Groups.

n

In this

section

is i d e n t i c a l

to that

reader

for details.

there

T h e o r e m 1.

Proof:

every

map.

element

normal

map

Let

in ¢

(smooth)

of

exact

K

x ~,w;~)

then we get a c o m m u t a t i v e

is

Wl(K)

fI3X:~X

(L,3L)

of

: f-l(,)

0.

= G.

Then

by a

4.3.2.

* S 1. SI

÷

the

X ÷ K x I x SI ÷ S1

the b a s e p o i n t If

with

the

n ~ 7)

+ Lhn_l(G,W;~p)

by T h e o r e m

and also

M ÷ X ÷ SI

(for

is r e p r e s e n t e d

the c o m p o s i t i o n

so that

The p r o o f

and we refer

÷ Ln S(G x ~,W;~p)

be a m a n i f o l d

f

[H9]

sequence

+ Ln s(G x Z,W;~p)

L s(G n

fibration

by a h o m o t o p y value

a split

M * X ÷ K x I x S1 Assume

Lk(~;~p).

in S h a n e s o n

Ln S(o,w;~)

The map

functorial

given

There is

0 ÷ LnS(G,W;~p)

we compute

is a

Change

¢

is a r e g u l a r (N,3N)

= (f~)-l(,)

diagram

(N,~N)

÷

(L,~L)

(M,~M)

÷

(X,~X)

Sl . Clearly

~IN

is a normal

sequence

of a f i b r a t i o n

map and by a p p l y i n g

to the

fibratlons

127

the h o m o t o p y

above,

¢]~N:~N

~ ~L

is a homology equivalence over LnS(G x ~,W;~p) ~ Lhn_l(G,W;~p) of

¢18N.

Ln s

¢

This is a well-deflned homomorphism,

splitting is defined by to

~ . Define P by sending

to the class and the

xS l, as in Section 5.3.

This goes

by Kwun and Sczcarba [D16]. The next theorem allows us to apply this.

Theorem 2.

Let

group.

Then

Proof:

Assume

G

be a finitely generated free abelian

Wh(G;~p) = 0. G = ~.

Then

~p[G] ~ 2~p[t,t -1] , the ring of

Laurant polynomials over

E . According to Bass [C2], Chapter P XII, Theorem 7.4, there is an exact sequence Kl(2~p[t]) @ Kl(2~p[t-l]) ÷ Kl(Z~p[t,t-1])

*

K0(2~p)

+

But Kl(%[t,t-l]) Kl(~p[t]) and similarly for Thus

~

t -1.

Kl(%[t,t-l])/~

result follows.

Kl(~p) = ~Zp

-= K0(~ ~) ~ ~, and the

The general case follows by induction and

the fact

~p[G × ~] ~ ~p[G][t,t-l].

Corollary

i.

n

Ln(~k;~zp) -= @ L n _ i ( l ; ~ p ) . i=O

128

0

Appendix

Ambient

D.

Sursery

and Surgery

Leaving

a Sub m a n i f o l d

~ixed.

In this For

simplicity,

involke

the

problems

section

we will

considered.

Poincare

the general

g

is a simple ambient

because

so we can

of t r a n s v e r s a l i t y

only m a n i f o l d s

will

be

of t r a n s v e r s a l i t y

can be made

to get

be a simple

homotopic homotopy

to

n.

go

homotopy

If

X

so that

equivalence?

equivalence

is a s u b m a n i f o l d g01g0-1X:g0-1X

This

is the p r o b l e m

+ X of

surgery.

regular

F

Theorem

a simple

X

I.

g

There

obstruction

of

be a c o b o r d l s m

of

k > 5

X

in

Y.

is t r a n s v e r s e

to

X

neighborhood

of

so that

equivalence

~ = 0. (F,~F) ÷

Then

if

to a simple

in

~ = 0

129

be a

M = g-lx;

c ( g l M ) E L k s (Wl X;~)' g01g0-1X

and

0 ~ e.

that

the

vanishes.

homotopy

E

N.

g ~ go' with

it follows

(E,~E)

and let

and put

M

are two o b s t r u c t i o n s

+ WlY;~)

homotopy

Suppose

has d i m e n s i o n

neighborhood

be a r e g u l a r

ecLnS(Wl(Y-X)

Proof:

A = 0

for a d i s c u s s i o n

of d i m e n s i o n

Assume let

Also,

modifications

g:N ÷ Y

Suppose closed

[E12]

and

on submanifolds.

case.

manifolds

Y, is

P = ~

complexes,

Suitable

Suppose

of

some r e s u l t s

theorem.

(see Jones

obstructions.)

between

assume

s-cobordism

with

we give

Let

equivalence.

÷ X

surgery

f:(P,Q) Form

~

(E,~E)

the n o r m a l

map

@:N

defined

by

g

obstruction R

is the

Let

8

I:N

x

right

be the

I + Y

x

to d o i n g

x I <J P + Y x I

surgery

boundary

and

f:P ÷ E.

relative

of

set of all

I

x

P,

such

(P,Q).

lies

to in

0 6 e, t h e n

homotopy

equivalence

leaving

N × 0

W ~ N x I.

between

0 ~JR,

where

LnS(Zl(Y-X) for all

+ ~lY). cobordisms

-1

g'.

to

gives

X. Assume

F'(x,t) g-l(x

the

the

see t h i s ,

Let

F:N

F':N

and

÷

By the

to a s i m p l e

(Y x !;Y

x O,Y

s-cobordism

x i),

theorem,

result.

obstruction suppose

x I + Y

Then

g,-l(x

g = g'

ambient and

g'

and

x I) so the

is

between

is t r a n s v e r s e

F'-I(x

x i),

130

to the

be a h o m o t o p y

x I + Y x I

= (F(x,t),t). x 0)

is c o b o r d a n t

fixed.

gives

To

¢

~':(W;~_W,~+W) R

also

problem.

transverse

where

and

Thls

This

and

the

R

If

g

x

obstructions

N xM i

surgery

N

Then

to

X x I,

is a c o b o r d i s m obstructions

(which

are

cobordism

obstructions transverse

are i n d e p e n d e n t

to

wI(Y-X)

by T h e o r e m

~ ~i Y, e.g.,

A more and Shaneson knot

theory.

[G2].

are the

of which

same.

Thus

way we make

the

g

X.

Note if

invariants)

careful

5.1.1, if

Ln

(wI(Y-X)

~ WlY)

= 0

n-k ~ 3.

analysis

has been made

[K5]

in the

codimenslon

The

simply

connected

Se also Lopez

S

de M e d r a n o

2 case

case

[J22],

by Cappell

and applied

to

is due to B r o w d e r sections

111.3.2

and V!. 3.

A similar [A21]

can be used

As before, i.e.,

this

procedure,

to do

is e s s e n t i a l l y

we look only First

an example.

2 or

is a finite ~.

Let

By T h e o r e m

el,...,e r

4.3.2,

¢i:(Wi;~_Wi,3+Wi) the n-torus.

Wi

Let

Define

0 problem,

a homomorphism by A p p e n d i x groups

of

Ln+l(~n).

by

(T n x I;T n x 0,T n x i) p(e i) = the

C,

of order

be a set of g e n e r a t o r s ei

fixed.

neighborhoods.

sum of cyclic

represent ~

a submanifold

a codimension

as follows:

direct

by an idea of Wall

leaving

at the r e g u l a r

P : L n + l ( ~ n) ÷ L n + l ( ~ n-l) Lm(~)

surgery

modified

surgery

obstruction

D 2 x T n-I ~ T n x I k J D 2 x T n-I

wi

131

where

Tn

is

of

x 0 ~ D 2 x T n-I

Since

L n + l ( ~ n)

non-zero map has

element

in

W ÷ Tn x I surgery

leaving

to a s i m p l e N

so that

D 2 x T n-I

homotopy

is r e p r e s e n t e d

Thus

we

cannot

be a n o r m a l

map,

do

a

by a

surgery

n > 5.

where

Suppose

equivalence

and that

k

¢IM:M

so that

N

¢ M

and

Y

is c o b o r d a n t is s u b m a n i f o l d

+ X

is a s i m p l e

equivalence.

2.

There

so that

if

0 ~ e, t h e n

(i)

is c o b o r d a n t

to

is a s i m p l e h o m o t o p y

¢+:N+

÷ Y

with

equivalence.

¢+IM = } I M .

Let

~:Q ÷ Y

homotopy to

be a c o b o r d i s m

equivalence X

and

cobordism

between

Let

be r e g u l a r

D,E

¢

÷ Wl(Y);~)

M = @+-Ix

(iii)

transverse

8 C LnS(Wl(Y-X)

is an o b s t r u c t i o n

~+

(ii)

slmple

exists

W ~ D 2 x T n-I + T n x I k 3 D 2 x T n-I

of d i m e n s i o n

Theorem

Proof:

there

fixed.

of d i m e n s i o n

homotopy

This

0.

¢:N + Y

are m a n i f o l d s

@ Ln(~n-l),

ker(p).

obstruction

Let

of

~ L n + l ( ~ n-l)

let

¢IM:M

¢+:N+

between

+ Y.

Assume

W = ~-l(x). ÷ X

and

neighborhoods

¢

Then

W

¢+IM+:M+

+ X,

of

in

W,X

and ~

a is

is a M+ = ¢+-lx. Q,Y,

respectively. Then and

~

@

is a s i m p l e

It f o l l o w s

that

the

extends

to

homotopy

~:D ~ N+ x I + E x I ~ Y equivalence

obstruction

to d o i n g

132

on ~ _ D ~ ( N + surgery

on

x I,

x i).

rel lies

~_D ~ ( N +

in

LnS(~l(Y-X)

different

homotopy

Since is e o b o r d a n t

by

9',

Then

Now

9'

~l((Y

D

and

and

x !,(Y

homotopy

(E x I ~ Y x I;E

So

A'

x i).

on

Y × I) × I)

9 T'

to get

and

so

P m V x I.

equivalence x I,Y

x I;E)

is an s - c o b o r d i s m

is an s - c o b o r d i s m

Z

x I) x I;

equivalence

is an s - c o b o r d i s m

÷

x I)

× I ~

(E x I,E)

induced

((E x I <]Y

on

÷

U = V x I ~

of t r i a d s

homotopy

surgery

(E x I,E)

between

N+'

between = ~_B'

N+ x i = ~+B' Let

union

B'

Z0) ÷

to a s i m p l e

= V - (A'-C'). and

x 1 U

x I),

(A',C')

Let

map

x 0 uP (E x I)

P

÷

equivalence

× I) x l) ÷ ~ I ( ( E

As b e f o r e ,

C',

9':(A,C)

a normal

assume

A = (¢')-l(E

(E x I,E).

so we can do

is h o m o t o p i c

B'

homotopy

we have

(V;A',B';C')

where

x I),

is a s i m p l e

is an i s o m o r p h i s m ,

Thus

x i).

x I)

~T'

and

(P;P1,P2).

C = (9')-I(E ~ nl(E

to a s i m p l e

× I, and

Let

x 0 ~ A',B

(E x I ~ Y

equivalence

by c o n s i d e r i n g

is c o b o r d a n t

9 0 : ( Z , Z 0) ÷

9T':(U;V

V × 0 ~A',

~

to a s i m p l e

A x i.

8

E x 1 = E.

~I(E)

by a c o b o r d i s m

Define

9':V ÷ E × I uY

to

x I),

homotopy

Q.

equivalence

B = (9,)-l(y

a simple

~ ~lY).

0 6 e, t h e n

is t r a n s v e r s e

along

to get

cobordisms If

9'

i)

x

of

R = N x I ~ A' ~

N x 0, N

+

x I

and

B'.

Then

N" = N x i ~

133

8R ~A'

is the

disjoint

~ N+ x 0.

We can that

extend $ ~

$":P"

~ Y

is c o b o r d a n t

extend

our m a p s

~+

leaving

map

between to

is a c o b o r d i s m

Since to

to a n o r m a l

is a c o b o r d i s m

Since

and

our m a p s

to

~+,

$:R ~ Y.

~ + (-~+) ~"

and

It f o l l o w s ~":N"

is n u l l - c o b o r d a n t ;

bounding

~".

Let

÷ Y. say

T : RUP"

T ÷ Y.

A'

~nd

M

fixed.

B'

are This

products, concludes

134

~ the

is c o b o r d a n t proof.

Appendix E.

Homotopy and Homology Spheres}

In this section we describe the calculation of groups of homotopy or homology spheres over theory of resolutions of Let

(W;M,M')

is an

over

Zp

A closed

We let

classes of

H-manifold

H,(Z;~p) & H,(sn;~p).

H-cobordism over

H,(W,M';Zp) = 0°

This is applied to the

~p-homology manifolds.

H = PL or DIFF.

Zp-homology sphere if

~p.

~(H)

~p

if

zn

A cobordism

H,(W,M;~p) =

denote the group of

H ~p-homology

is a

n-spheres

H-cobordism

(addition given

by connected sum). A

~p-homology sphere

~I(Z) = 0. gp

We let

classes of

method of

8P(H) n

Z

gp-homotopy sphere if

denote the group of h-cobordism over

H gp-homotopy ~-spheres.

Generalizing the

[G20], we have

Theorem t .

Let

n _> 4

and

2 ,~ P .

Define a homomorphism sending

is a

x

to the

r' :L4k(l;~p)/L4k(1)

r:L4k(l;~ P)

8P(H) & ~P(H).

> 8~k_I(PL)

by

~p-homotopy sphere bounding the manifold

obtained by plumbing with guments show that

Then

r

x

(see Section 4.4).

defines injection > @~k_I(PL) .

iAdded May 1976

135

Surgery ar-

T h e o r e m 2.

For

n > 4, n~3mod (4 )

H

n=4k-1

@ L4k (i; ~p)/L4k w(k)-i

where

Hn

is a finite

of p a r t i t i o n s

of

P-torsion

group and

z(k)

is the number

k.

For the proofs of T h e o r e m s

1 and 2, see

[KI3].

In the smooth case, we have the results of Barge, Latour and Vogel can be d e f i n e d -P ~4k-I

[KI4]:

A left inverse

in the smooth h o m o l o g y

r'

above

sphere case; we let

~p P ~n = ~n (DIFF)

be the kernel of this map,

Then for

to the map

Lannes,

if

n~3mod(4).

n > 4,

~P ~{ ® ~p • ~Q ~n = ~n ~n @ ~(P)

Thus,

the c a l c u l a t i o n s

e n o u g h to c h a r a c t e r i z e In the case bP4k/bP4k

of ~n ~p ~n"

2 { P,

~ ap~/8~ ~

[G20],

it follows

@ WP where peP P '

the group of spheres w h i c h b o u n d One a p p l i c a t i o n manifolds.

Let

~@ ~n

[FI] and

from bP4k <

P 04k_I(DIFF)

%-parailelizable

be a s u r j e c t i v e

136

are

[K2] that

of this is to the study of

f:M .... > N

[KI4]

is the

manifolds.

~p-homology

PL-map between

polyhedra. for e a c h

Theorem Then M

f

is a

(Sullivan)

Let

We

say

~-resolution

if

~ " --'H,[f-l(x);~p) = 0

x e N.

3.

there

is a

if and o n l y

Nn

~-resolution if o b s t r u c t i o n s

be a f:M n • in

137

~p-homology > N

to a

H k ( N ; ~ k-i P (PL))

manifold. PL-manifold vanish.

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geometric

Induction in equivariant applications, in II.

K-theory

and

27.

Quinn, F., B(TOPn)~

28.

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29. Bull.

31.

I.

, Manifolds with free abelian fundamental groups and their applications (Pontrjagin classes, smoothness, multidimensional knots). Translations A.M.S. 71 (1968) 1-42.

26.

3O.

Springer-

and the surgery obstruction. Bull. A.M.S. 77 (1971) 596-600.

On homotopy tori and the annulus London Math. Soe. I (1969) 95-97.

, The topological in Topology of Manifolds.

theorem.

space-form problem.

Farrell, F. and Hsiang, W., Manifolds Amer. J. Math. XCV (1973) 813-48.

152

on

wit~ ~I

=

Gx Z.

K.

Sursery with Coefficients.

I.

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2.

Alexander, J., Hamrick, G., and Vick, J., Involutions on homotopy spheres. Inv. Math. 24 (1974) 35-50.

.

o

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and Shaneson, J., The codimension two placement problem and homology equivalent manifolds. Ann. of Math. 99 (1974) 277-348.

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Connelly, F., Linking numbers (1973) 389-410.

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153

P.,

INDEX

adJoint 60 Arf invariant 77 Atiyah, M. and MacDonald, I. 24 Ambient surgery 129

Haefliger, A. and Poenaru, V. 12 handle 54 handle subtraction 82 Hasse-Minkowski invariant 76 h-cobordism 38 Hermitian form 60 Hirsch, M. 12 homology equivalence 3 homology intersection pairing 14 homology manifold 40 homology type 3 Hudson, J. 38

Bass, H. 21, 121, 128 Bernstein, I. 74 block bundle 7 Bredon, G. 40 Browder, W. 46, 78, 131 Cappell, S. and Shaneson, J. 28, 84, 104, 126, 131 classifying space for surgery 95 cobordism extension property 104 Cohen, M. 37 colocalization 27 conjugate closed subgroup 41 Connolly, F. 116, 126 deformation 32 degree i 41, 86 dimension (of a Poincare complex) 39 dual (of a module) 1 duality theorem 43 A-map 4 A-set 4 elementary matrix 20 elementary P-collapse, expansion 32 Farrell, F. and Wagoner, J. 122 free and based module 21 free module 1 formal deformation 32 fundamental class 39 Gabriel, P. and Zisman, M. 5 Gauss, C. 75 general linear group 20, 122 geometric realization 5 group ring 2 groupoid of type 2n 92

immersion classification theorem 12 infinite simple homotopy type 122 intersection numbers 14 Jones, L.

129

Kan, D. 5 kernel 60 Kervaire, M. 38 Kerv&ire, M. and Milnor, J. 78 Kervaire manifold 78 Kirby, R. and Siebermann, L. 54 Kwan, K. and Szczarba, R. 128 Lam, T. 75 Lees, J. 12 linking forms, group of ll2 , simple ll2 , standard simple local homotopy (homology) 25 local n-sphere 25 localization, algebraic 23 , geometric 24 , relative 26 Lopez de Medrano, S. 131

112

manifold n-ad 86 microbundle l0 Milnor Poincare complex (manifold) 78 Milnor, J. 5, 7, 10, 17, 22, 28, 29, 54 and Husemoller, D. 76 Miscenko, A. 126 Morse theory 54

154

Symbol Index Ak

Ik

60

16, 60

Imm(M,N) (BG)p

47

BH, BA

5

CHnA(~,w;R)

i01

K0(A)

19

KI(A)

21, 122

KI(A)

21 23

C (n)

85

K2(A)

c,(x)

2

El(),

C,(f),

f

a map

C,(f>,

f

a map of pairs 42, 83

~i,~,6i

42

ii

K(~)

Ki( ) 3 llO

K(~,l), ~ a groupoid of type 2n

LA2k(A)

85

D_k,D+k,~_Dk,3+D k

33

62

LA2k+I(A)

64

Lm(H) , Lm'(~) E(u,A), E(A) EUkA(u,A) Ex

LmA(K;R)

20

87

92

~m(g;H,E ') 103

63

ZS2k_i(~,w)

5

i12

Lmh(K;R), LmS(K;R) GL(u,A), Gp/H

48

Gq(R)

7

GL(A)

A"

1

°~pH(X)

Hn ( Z~/22Z; )

Hq, Hq

20,122

109

for H=TOP, PL, DIFF 6

92

80

Wr(¢), ¢ a map

55

Wr(~), @ a map of pairs Wr(X), X an n-ad H(P)

Qm(~) 155

24

87

86

82

92

Rw

2

xN

R(TM,TN)

~m(n)

103

SKl(A)

Xp

24

Xp

27

41

21 % , Z~(p) 24

85 23

St(n,A),

St(A)

22

63 *

(on R~)

*

(on

TM

I0

TM

8

2

KI(A))

UkA(n,A)

22

63

28

Wh(~;R) Wh2(~;~p)

125

~¢ 94 n cm (n) 87

a~(g)

39

[x,~x]

87

~m(~',n)

Sn, k S-1A

[x]

ii

96

87

156

n-ad 85 normal cobordism 51 invariant 51 map 50, 57

subkernel 61 Sullivan, D. 25, 47, 50, 80 surgery 54 hypothesis 64 leaving a sub-manifold fixed 131 obstruction theorem 64, 93 rel the boundary 55 with coefficients 57 Swan, R. 21

Pardon, W. 104 periodicity isomorphism 96, ii0 plumbing theorem 72 Poincare complex 39 n-ad 86 pair 41 preferred base 21 principal H-bundle 5 A-fibration 5 projective module 1 ~-~ theorem 83

tangent block bundle 8 microbundle I0 Thom space 45 torsion for n-ads 122 of a chain complex 22 of a Hermitian form 60 of a map 29 of a Poincare complex 39 trace 54 transfer 39

Quinn, F. 94, 95 realization theorem 72 ring with involution i Rothenberg, M. 105 Rourke, C. and Sanderson, B. 5, 7, 17, 54

unitary Steinberg group Wall, C.T.C.

s-basis i s-cobordism 38 s-cobordism theorem 38, 123 self-dual 41 self-intersection number 16 Serre, J. 76 s-free 1 Shaneson, J. 96, 105, 127 Sharpe, R. 125 Siebenmann, L. 19 signature 75 simple chain complex 22 equivalence 22 homology equivalence homology type 36 Poincare complex 39 spherical fibration (over a ring) l0 Spanier, E. 22, 30, 97 Spivak, M. 10, 45 Spivak normal fibration 45 split group 125 stable basis i stably free 1 standard plane 60 Steinberg group 23 Stong, R. 94

125

19, 41, 47, 58, 64, 80, 84, 95, 96, 116, 124, 125, 131 Wall group 62, 64, 87, 92, 124, 126 Whitehead, J.H.C. 37 Whitehead group of a group 28 of a ring 21 , secondary 125 Whitehead lemma 20 Whitney lemma 16 Williamson, R. 96 29

157