Surgery with Coefficients (Lecture Notes in Mathematics, 591) 3540082506, 9783540082507

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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

X

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

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

X.

[A6] for a complete

discussion. A A-set

of

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

,

1[

X

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

or

Gq(R)

~ Gq

= GL(q~R)

(R)

= 0q.

F ibrations.

we defined principal or

Hq, or bundles

Gq(R).

In

associated

to

As before,

DIFF. [Bg]).

Let

K, written

K

be a simplicial ~q/K,

is a

so that if

~ eK

(n+q)-ball (2)

and

and Spherical

geometrically

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

E(~)

is an n-cell, BcE({)

= k.JB . g~K ~

then there

so that

exists

(B~,~)

H

an

~ (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 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

is

n-q

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;~) there

Y

gives

of l o c a l

a more and

homology

The

stalks

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

n

over

~,(Y;~)

"~,(Y;~)

M

is a

a n y ring.

result.

a sheaf

groups

then over

general

~

of

is zero.

manifold,

According

a spectral

R

of d i m e n s i o n

CW-complex

exists

over

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,

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*

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

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)

÷ ~I(X')

Proof:

Assume

X

has one

0-cell

io

attached

at

there

X

with

say

We h a v e

k

1-cells

by

X ,

!

as

d i m X ° _


f

are h o m o t o p i c

X x I ÷ (BG)p f i = 0,1 . Let

I

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(¢)

)X

is a long exact

Equivalently,

Wk+l(¢)

an inclusion.

Clearly

from an element If

in

M

the boundary.

¢#

÷ Wk(M)----+~k(X)

= ~k+I(X,M), ¢

admits

has n o n - e m p t y

first

into the

T h e n the trace × I)UM',

rglative

definition:

cobordant

if there

Lemma

2.

attachin5 to

M × I

fixed

we replace iff

¢

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

÷...

to the boundary.

following

2

+ Wk(¢)

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

In the

is an e m b e d d i n g

surger~

.

Wk+l(¢).

leaving

~N = M U ( ~ M

classes

sequence

... ÷ ~k+l(¢)

before.

of h o m o t o p y

Dk+l

C

M

two cases,

group

diagrams Sk

There

to be the

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

i.

can be regarded as the group of matrices

IP°l

generated by matrices of the form (a)

(c)

Embed way; then

0

p,-1

the

matrix

UkA(n,A)

'

PEA'CGL(n,A),

0

t

(-i) k

0

in

UkA(n+I,A)

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

63

the inverse image of

0

in the obvious

A,

Define

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

The (2k+l)-th Wall 5roup 0f LA2k+l(A) =

UkA(A)

A

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

/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 = k.

¢.

ag EUkA(r,~w).

is of the form

If

~_N'

class of

over

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

p

Io As

~]

with

is

gi o h, where

is an embedding of degree

p,

01 lq 0I

-I

I

pI

m lq

?I

is a simple isometry, we can assume

pI

~m = [i

gi

pQ

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,_ 1 •

R

basis, we can assume assume

m

a=l.

is of the form

m(el) = (-l)kfi , fi*

for

A, so by a change of

By surgeries on the

gi' we can

±e ~ . . . ~ a, i.e.,

a(f i) = e i.

The bases of a basis

has image in

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 where Wp P =

Suppose K r.

Case i. A

n

odd.

represents

Assume

are equivalent

or

or

O

~Z/2~ p = 4k+3

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

that all endomorphlsms

mod E U ± I ( r - I , ~ )

2 eP

2 ~ P

This proof follows Bernstein

an automorphism

inductively

and

and

n=4k

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

Proof:

P

P

n=4k+2

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 and so

L2k+l(~)

Case 2.

n=4k.

prime, and

I:

A0 =

Fp

P,Q,R, Thus

C0

S.

By induction,

U±I(r,Z P) = E U i I ( r , ~ )

= 0.

Let

~p

denote the p-adlc numbers

the field of

a field,

and

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

ap =

if neither

4k+l

and

P

The e l e m e n t s Minkowski [K12],

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

in

is a

P

p=2

~/4~

P-3

mod

(4)

Z~/2~ ~ Z~/2~

p-i

mod

(4)

are called the Hasse-

of the form

for proofs of these

P

of the form

~/2~

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

invariants

in

P = ¢

I

~P

4k-i

hold, but there prime

if

is a prime of

(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 = ,

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~

resents

embeddings U =

•

..

+

~fi(DkxDk).

and H,(~)

(N,M); Since

homotopic

ei

f! repi

~I(M) m

to disjoint

1.6.2.

subtraction:

Let

Let C,(~)

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

Thus the elements

for some q s H(P).

fi' by Corollary

noted.

~ ~k+l(~) ~ ~ p ,

i

given by the chain complex inclusion,

otherwise

el,...,e r.

Kk(N,M)

(Y,X).

fl! are regularly

Do handle where

÷

= 0 for

By adding trivial handles,

f!:(DkxDk,sk-lxD k)

qe i s ~k+l(~)

Wl(N) , the maps

theorem,

~:(N,M)

immersions

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) Z

[XiNXj+ I

.~ ~(n) l$iXl = IX1, (6iX) j =

Sn,k:

(~(n)

÷

j < i

Iv j j+l

j > 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

sequence 0 ÷

is a free

n

~

+

÷ ~

A*R

~

generated

direct

+

A

presentation

0 Thus

with

to s h o w i n g

sum,

~

abelian

we

A@R

= 0 implies

group.

can a s s u m e

Since

A = ~/n~.

The

0

for A so we h a v e

n

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)(~)

=

~_~ i

inclusions. We

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

components

are

@:M ÷ X ension

surgery

theorem:

so that

rel

R with

are

i

If R =

torsion

the

~p,

Let

C, the m a p p i n g

complex

so C, @ ~ p ~p

surgery

and

over

chain

over

be

n-ad

torsion

if

M of d i m -

torsion

in A,

in A,

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

if

equivalence

=

@

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

follow

A L~(Ki;R)

where

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

from Theorems

the

complex

R with

is e a s i l y over

so that

i and

from Lemma

2.

For

2 provided

~p.

is a c y c l i c .

problem

there

of a s p a c e

will

s Hm(X;~) cone

R with

L~(~;R)~

follows

result n-ad

[X]

R with

= 0 if and

Define

result

the

L~(w;R)

Then

in A.

components

X is a P o i n c a r e

Poincare

over

a homolo@y

n = 2.

the

R arbitrary,

~(~)

Zn_l M to

Assume

the K.

equivalence

maps

subgroup.

a manifold

X over

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

s L m(W;R) A

Proof:

n-ad

so let

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

a self-dual srqups

Gi,

corresponding

of type

map b e t w e e n

m and a Poincare

cobordant

over

our main

obstruction

is a n o r m a l

Wl(6n_iX)

and

the

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

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

over

state

are g r o u p s

and m a k e

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

there

~(¢)

K(Gi,I)

can n o w

3.

of w(~)

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 ~n (Ak)

in

~, with face

induced by face maps

(e.g. manifolds), = those elements

of objects.

then define A ~

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

~.

is an infinite loop space, ~ n -~ ~ n~ -i" It follows that given H = (~,R,A) as 5.2, there exist classifying spaces ~m(~)

~- ~m+j(H)

Also

n

C

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.

R =

in W a l l

[HIg],

In this section we

for the non-simple

case,

case will follow in Section

6.3.

A ~p~). = Lm+4(

For m > 5,

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

--

an isomorphism;

[HI0].

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

÷

Lh

(w,w;R)

m+4

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

is

--

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.

Then

I~

k(MX@p2)

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

Let W be the trace

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

i = k+l otherwise

We have Kk+4(Mx@p2)

--" Kk+4(W ) +

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

Kk+2(N) hi

:sk+2

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 ~ 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

Ki(q)

=

~

Ki_2(M)

LO

The embeddings

i = k+l, k+3 otherwise.

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).

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

equivalence This ~

gives

is the

x s ker(8).

over

Then

5.2.1. show

equivalence

map

relation

by N i + Yi'

at the

Let

that

by

ker(B)

ker(B)

~ normal

four the map

~

by a h o m o l o g y map

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

defined i = 1,2,

101

last

B denote

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

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

We

sequence

by

N ~ Y. where

e I ~ 0 2 if

so that

0.

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

...

~m(H',H)

=

).

~m(g;H,~')

T h e o < e m 2.

5.2, let

By the remarks

all spaces

> ~ m ( g ; H , H ')

following

Corollary

÷ Qm(H') 5.2.1~

involved to be manifolds.

There is a long e x a c t j,

where g,:Qm(H)

sequence L(g)

,LAm( ,w;R)

, > ~ 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).

ker(j,)

45 in

to a homology

equivalence

F:N ÷ Y.

Then

~,[F]

This

Remarks:

(i)

A sequence

[K9]

L~(~)

to be the

allowed factor

to relate usual

to change ~m

in this

= (~,w,g, Wh(~)), (2) and S h a n e s o n are

locally

This [KS]. epic,

case

type was

and L ~ ( ~ ) ,

cobordism

by r a t i o n a l

and

the

now shows

that

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~(~)

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

[J22])

Let j , [ f : M

f is c o b o r d a n t

Pardon

If] E ker(L(g)

~ X].

C Im(~,):

= If].

over

over R', F:N ÷ Y,

(pg.

[f:H

=

8X

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 F

W tJ M' commutes,

f,

)

Xx0

~

Xxl

~

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

exact

sequence 0

where and

GI,

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

f

÷ GI

¢

~ G2

f is a s i m p l e

handles

a normal

map

N

This and

IxS k-I

sphere

sk-lxl

so

Kk_I(~+N)

the

m

p denotes Write

with

el,...,em,

G 2 respectively.

Add m

to get

N and

the

structure

a manifold

usual

f(e i)._

e X0(¢(e~),¢(e~))

the

entries

of B are

bij

= cihahj

generators

of a s t a n d a r d

kernel,

[ b.*a.. z

i=l i

self-intersection

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

if the

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

cij

with

m

=

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

0 otherwise.

Let

i = l,...,m.

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

k to Xxl

~.

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

sphere

gives

an

÷ 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

we have

÷ 0,

isomorphism

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

trivial

~ K

so that

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.

. 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

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.

There is

0 ÷ LnS(G,W;~p) Proof:

functorial 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

n

we compute

given

Groups.

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 N

to a

Hk(N;~ P k-i (PL))

manifold. PL-manifold vanish.

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Topology

20.

and Orlik, P., On a codimension problem, preprint, IAS 1969.

preprint 12

i embedding

21.

Levitt, N., Fiberings and manifolds Bull. A.M.S. 79 (1973) 377-81.

22.

Lopez de Medrano, Verlag 1970.

23.

Novikov, S., Homotopy Equivalent Smooth Manifolds Translations A.M.S. 48 (1965) 271-396.

24.

25. 26.

S., Involutions

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geometric

Induction in equivariant applications, in II.

K-theory

and

Quinn, F., B(TOPn)~

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3O. 31.

I.

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Springer-

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=

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2.

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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

ii

47

BH, BA

5

K0(A)

CHnA(~,w;R)

C (n)

85

c,(x)

2

C,(f), C,(f>,

i01

19

KI(A)

21, 122

KI(A)

21

K2(A)

23

K(~)

llO

El(), f

a map

f

a map of pairs 42, 83

~i,~,6i

42

Ki( ) 3

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

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)

20,122

A"

109

for H=TOP, PL, DIFF 6

92

1

°~pH(X)

Hn ( Z~/22Z; )

Hq, Hq

87

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)

87

~m(~',n)

SKl(A) Sn, k S-1A

ii

[x]

103

Xp

24

Xp

27

St(A)

63

(on R~)

*

(on

TM

I0

TM

8

Wh(~;R)

22

63 28

Wh2(~;~p)

~¢ 94 n cm (n) 87

a~(g)

22

2

KI(A))

UkA(n,A)

41

% , Z~(p) 24

85 23

*

39

[x,~x]

21

St(n,A),

96

125

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 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

unitary Steinberg group Wall, C.T.C.

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