Seminar on periodic maps

Lecture Notes in Mathematics A collection of informal reports and seminars Edited by A. Dold, Heidelberg and B. Eckmann, ZLirich

46 P. E. Conner University of Virginia, Charlottesville

Seminar on Periodic Maps 1967

m

Springer-Verlag. Berlin. Heidelberg-New York

All rights, especially that of translation into foreign languages, reserved. It is also forbidden to reproduce this book, either whole or in part, by photomechanical means (photostat, microfilm and/or microcard)or by other procedure without written permission from Springer Verlag. 9 by Springer-Verlag Berlin " Heidelberg 1967. Library of Congress Catalog Card Number 67--31229 Printed in Germany. Title No.736

Contents

Introduction ......................................... 9

The

2. Real

index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . representations ..................................

3. E x t e n s i o n 4. The

and r e d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

functor

~.

(H;~,)

. 1 19 25 30

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

33

invariant . . . . . . . . . . . . . . . . . . . . . . . . . .

36

invariant ...................................

41

7. The local i n v a r i a n t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

44

5. Index as a b o r d i s m 6. The trace 8. P e r i o d i c

maps

on a R i e m a n n

9. The A t i y a h - B o t t 10. W e a k l y

complex

surfaces . . . . . . . . . . . . . . . . . . . 50

formula ...............................

60

involutions ............................

62

11.

The ring F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

68

12.

The r i n g ~ ( Z

80

13.

The

2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

r i n g ~ ( Z 2) of c o n s t r u c t i o n s . . . . . . . . . . . . . . . . . . . . . . . 92

14. A p p l i c a t i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.

Dimension

16. A local

of the fixed

invariant

for~

point

100

set . . . . . . . . . . . . . . . . . . . . . 103

U (Z 2) . . . . . . . . . . . . . . . . . . . . . . .

References ...........................................

107 116

Introduction These notes are taken directly from a seminar held at the University of Virginia during the academic year 1966-67.

They depart from the paper Maps of Odd Period~

[6]~ and represent an attempt to study the bordism ring ~.(H)~

the bordism ring of all orientation preserving

actions of the finite group H on closed oriented manifolds~ from a new viewpoint. In sections 1 through 9 we discuss the Atiyah-Bott fixed point theorem in a form to which it has been applied to maD~ of odd prime power period~ [i~ p. 21, Ex. 6]. Thus we are concerned only with one aspect of the AtiyahBott techniques and we have nothing more than this. In sections 1 through 4 we review the definition of the index of a triple (H,V~(.~.)) consisting of a complex representation of a finite group which preserves a nonsingular conjugate symmetric inner-product,

(v,w) = ( ~ ) .

Because we want our definition to resemble as closely as possible that of [i~ p. 21, Ex. 6] we adopt the following approach.

We find that the set of linear transformations

D : (M,v)

> (H,V), commuting with the action of H and

satisfying

(1)

(v~Dw)

(2)

(v~Dv) > 0 r v / 0

(s)

D 2 = Id

- (Dv,w)

forms a non-empty connected subset of GL(V).

Obviously D

The help of NSF~ through grant GP-6567~ is gratefully acknowledged.

-

2 -

is to play the role of the "star" operator in differential forms.

We split V = V+ O

V_ into the •

D and, so receive representations index (H~V,(.~.)) the Grothendieck

eigenspaces of

(H,V+) and (H,V_).

be the difference

We let

(H,V+) - (H,V_) in R(H)~

ring of complex representation

classes.

This does not depend on D~ and index is suitably multiplicative. A separate treatment of real representations which preserve a real, non-singular

product,

(v,w) = -(w,v)

is made.

(v,Jw)

(2)

(v,Jv) > O, v / 0

(3)

j2 = -Id

skew symmetric inner-

It is shown that the set

of real linear operators J : (H,V)

(1)

.

> (H,V) for which

i s a n o n - v o i d connected subset o f GL(V). complex s t r u c t u r e is

Thus we have a

on V i n which the g i v e n r e p r e s e n t a t i o n

complex l i n e a r .

complexification to ( H , V , ] )

(H,V,(',.))

Now index i s r e a l l y (H,V ~ R

- (H,V,-J)

C) and i t

e R(H).

a complex r e p r e s e n t a t i o n

is

defined for

shown t o be equal

This i s the d i f f e r e n c e

and i t s

the

conjugate.

between

Again a s u i t -

able p r o d u c t f o r m u l a h o l d s . In section orientation

5 we d e f i n e

preserving

i s done by c o n s i d e r i n g

Viewing Hn(M2n~c) jugation w - - >

index

action

a right,

o f H on a c l o s e d m a n i f o l d .

the l e f t

as Hn(M2n~R) ~

w in Hn(M2n~c).

product on Hn(M2n~c)

(M2n,H) e R(H) f o r

is given by

representation

This

(H,Hn(M2n|c)).

C we see there is a conR A conjugate symmetric inner-

-

(I)

3

if n is even, evaluating

-

v u w on the funda-

mental class (2)

if n is odd, evaluating fundamental

The representation

-v u iw on the

class.

of H preserves

index (M2n,H) e R(H) is defined.

this inner-product Incidentally~

so

if n is

odd we can put a complex structure J on (H, Hn(M2n;R))

so

that index

(M 2n H) = (H, Hn(M2n;R),J)

By the classical

-

(H, Hn(M2n;R),-J)

argument index is shown to be a

bordism invariant and with a little attention to tensor products of graded algebras we obtain a ring homomorphism index : ~ H )

> R(H).

Suppose K C H is a subgroup. homomorphism ~ H

:~(H)

into the reduced action homomorphism

> ~(K) (M2n,K).

r

: R(H) - - > KH ind RKH = rKH ind.

> R(H).

can

(M2n, H)

by sending

There is a similar

of the induced representation

That is, given a representation

let W be the linear space of all functions which satisfy f(kh) ~ kf(h), representation

define a ring

R(K) and it follows easily that

There is the construction eHK : R(K)

We

f : H

all k e K, h e H.

(K,V),

> V The induced

(H,W) is given by (~f)(h) = f(h~).

The

- 4

-

corresponding bordism construction EHK : ~ ( K ) is described as follows.

Given (M2n,K), let M2nx

quotient manifold obtained from M2nx (x,h) ~ ( x k , k - l k )

> ~(H)

where k g K.

H be the K H by the identification

There is an obvious right

action (M2nx

H,H) and this defines E . K HK that eF~< index = index EFK.

We can also show

In section 6 we restrict ourselves to H = Zp s, p an odd prime and s ~ i.

To obtain a workable invariant we form the

composition of index with trace : R(Zp s) Z(~)

>

Z(A)

where

denotes the integral linear combinations of the

pS-roots of unity.

This composition Tr : ~ 9 ~ ( Z p s)

is still a ring homomorphism.

> Z(A )

The fixed point theorem is

the contrapositive of (6.2)

Theorem:

If (M2n,zpS) has __oo fixed points then

Tr[M2n,zp s] = O. The critical point in the proof asserts that if (M2n,zpS) has no fixed points then [M2n,zp s] lies in the image of E :C~2n(ZpS-I ) result of [6, (7.8)].

> ~2n(ZpS),

which was the main

With this we need only show that the

composition

R(ZpS_I) is O.

e > R(Zp s) %.race > z(A )

This is merely an analysis of the behavior of eigen-

values and multiplicities when (ZpS-l,v) induces a representation (Zp s,W).

We are not able to prove (6.2)

immediately for a composite period by these methods because

-

5 -

the obvious generalization of [6~ (7.8)] is not valid. To complete the picture we should have a local invariant which computes Tr(M2n~zp s) in terms of the normal bundle to the fixed point set.

In section 7 we allow this a-%

local invariant to be forced on us.

Let~

denote the

bordism ring of orientation preserving actions (Bk,Zp s) where B k is a compact manifold and in ~ B k every isotropy group lies in Zp s-I (i.e. is a proper subgroup of ZpS). The action bounds if and only if there is a (wk+l~zp s) for which (Bk,ZpS) C

(~wk+l~zp s) as a compact invariant

oriented submanifold and for which every isotropy group in W k+l~ B k lies in Zp s-l. There is a naturalc/~ h ~ 1 7 6 1 7 6

: ~(Zps)

>~"

We show that if x ~///_ then there is an integer 0 < m < ~ 0 i n

and a Y2n ~ ~ 2 n (Zps) w i t h : ( Y 2 n ) = mX2n"

With

this

we

find

(7.2) Lc : ~

Theorem: > Q(~)

There exists a unique ring homomorphism for which

"-N is a commutative diagram. The fact that T r ( ~ ( Z p S ) ) ~

Z(~)C

Q(~)

interpret (7.2) as an integrability theorem.

allows us to

- 6 -

The obvious question is why we call this a local The answer lies in the ambivalent nature of

invariant.

~.

I t is important that the reader keep two isomorphic

interpretations of //( in mind.

The isomorphism was found

in [6, sec. 7]. Alternatively, p u t t i n g ~ = (pS _ 1)/2 we think o f ~ as the suitably defined bordism classes of objects (~i''''' ~

y

)

>

M2n' where ~ i ' ' ' ' ' ~

is an ordered

-tuple of complex vector bundles over a closed oriented

manifold.

We allow "0-bundles" which function as place

holders.

Addition i n ~ .

is by disjoint union and the ring

structure is [(

i'''"

) - - > M2n ][

' 1'''''

i'"''

)-->

)

> V2m ]

M2nx v 2m]

We can show that if P =~(Xl,X2~...,Xm,.'') graded polynomial ring o v e r ' w i t h 2m t h e n ~ ,

is the

generators xm of degree

is isomorphic to the y - f o l d

tensor product

P~_C/. "'" ~_C/_p"

The isomorphism sends I~ "'" ~ l ~ X m ~ l ~ ..'~i .th with xm in the 3 position into [0,'''0, ~,0,..-0) > CP(m - i)] t = X(j~m-

I) e ~ m "

The h o m o m o r p h i s m / : ~ / . ( Z p s) follows.

>~.

is described as

At each fixed point of (M2n~Zp s) we receive an

orthogonal representation, with no non-zero isotropy vector, of Zp s in the linear space of vectors normal to the fixed point

-

set at this point.

7

-

All along the component of the fixed

point set through this given point we can reduce the structure group of the normal bundle to a product U(nl)M . . . ~ U ( n y

).

In the resulting complex structure

on the normal bundle the representation a diagonal matrix wherein for i ~ j ~

.

~J

of Zp s appears as

occurs with multiplicity

nj

The orientation of M 2n together with the

orientation on the normal bundle induced by the complex structure determines an orientation on each component of the fixed point set and A~[M 2n ~Zp s ] e ~ all components of the resulting (3.. Now we see that Lc :vIf(.

is the sum over

elements in //12n" J %

>

Q(~)

can be intrepreted

as a function of the normal bundle to the fixed point set. By some elementary calculations

we are able to show that

the entire problem of understanding computing Lc([D2,ZpS]) by z

> z~

Lc can be reduced to

where Zp s acts on the closed 2-cell

, where ~ =

exp (2~i/pS).

In section 8, just for completeness,

we sketch out a

computation showing

Lc[ D 2 ,Zp s ]

=

_i

+~

- - O

l-A This is done by making use of the Hirzebruch-Riemann-Roch theorem for curves. $ =

Thus we define a curve S C

[Zl,Z2, Zs]IZ 1

pS

+ z2

pS

+ z8

pS

CP(2) by = 0

and note there is an action (S,Zp s) with [Zl,Z2, Z3]

I

>

[Zl, z2,~z3]

-8-

and fixed point set

~

I

[S,Zp s] = pS[D2~ZpS]

~ pS-i [-1 1 ~ J ~ o E

It turns out that

, thus Tr S~Zp s] = pSLc([D2,ZpS]).

To compute Tr[S,Zp s] we identify (ZpS,HI(S;R),J)

(Zp s,hO,l(s)), 1-forms.

with

where hO~ 1 is the space of anti-holomorphic

The problem becomes that of computing t h e % r a c e

of the representation

(ZpS~hO,l(s)).

We want to do this in

terms of the finite fixed point set so we shall have to localize.

Let~/----> S be the sheaf of local rings of holo-

morphic functions on S and let T : S

> S be the generator

of ZpS~ then by composition we obtain T

S and (ZpS,HI(s;~).~

>S

A simple modification

Theorem shows that ( Z p S ~ H I ( S ; ~ ) we have passed to a representation

~- (ZpS, hO'l(s)).

step.

Now computation of trace (Zp s, H 1 (S; ~ ) First it is s h o ~

quotient map ~ : S projective line.

Thus

of Zp s on a sheaf cohomology

group which is the desired localization

as follows.

of Dolbeault's

is carried out

that ~ Z p s = ~ ( i ) ~

thus the

> CP(1) is a ramified covering of the We denote b y ~

> CP(1) the sheaf which

is the direct image of ~-----> S under ~ .



t e n s o r p r o d u c t of the Hopf bundle

CP(1) with itself.

Clearly tr (ZpS, HI (CP (i) ; ~ ) =~

ps-IAJ 0

dim H I ( c P ( 1 ) ; ~ . J

tZ% ~j The dim HI(cP(1);~i) is read off directly from the RiemannRoch formula.

Since

Tr [S,Zp s] = t r

(Zp s, hO, 1 (S)) - t r

l+)

we can show Tr (S,Zp s) = _pS 1 - ~

(ZpS,hl'O(s))

and so compute Lc[D2,ZpS].

In view of t h e c o m p r e h e n s i v e t h e o r y of s u r f a c e s

and

r a m i f i e d c o v e r i n g s d e v e l o p e d in c o n n e c t i o n w i t h knot t h e o r y

it would seem reasonable to ask if there is an alternative computation of trace (ZpS~ H 1 (S;R)~J). In section 9 we present the Atiyah-Bott Fixed Point Formula as it applies to maps of odd prime power with a finite set of fixed points.

We do not indicate the remarkable

applications of this result as they may be read in [i~ p. 24, (3.33), etc.].

-

i0

-

This closes the discussion of Atiyah-Bott. With section i0 we begin the study o f involutions which Very briefly we have

preserve a weakly complex structure. an involution

(T,B n) on a compact manifold and we suppose

that on some (2k - n)R ~

>

B n there is a complex

structure which commutes with id ~

dT.

This idea is

suitably stable and it results in assigning naturally a weakly complex structure to the tangent bundle of the fixed point set and a complex structure to the normal bundle. the free case we get the~'~ U bordism module ~'~(Z2), no restriction on the fixed point set we g e t ~ ( Z ring of all weakly complex involutions third bordism r i n g ~ interpretation.

In

and with

2) the bordism

on closed manifolds.

A

U is introduced with again its ambivalent

From one point of view it is bordism classes

of pairs (T,B k) where T is a weakly complex involution on a compact manifold with no fixed points in ~ B k. mind we c a n d e f i n e ' ~ : tain

an e x a c t

"'"

~

think o f ~ [ ~

+l(Z2 )

and i n f a c t

we o b -

k+l

are naturally defined.

We may on the other hand

U as the bordism classes of complex bundles

> V 2 h I over closed weakly complex manifolds.

us that ~ U 2k+l = O of course. over ~

k(Z2)

sequence

>-O..-k+l(Z2 )

where ~

k+l

With this in

This shows

F u r t h e r m o r e ~ . U(Z 2 ) is generated

by the antipodal maps /[A,S2n+I]~O (not freely,). L 2k)C 2k_l(Z2) and with the aid of the

-

generating

ii

-

set9

(~U) =~ U (Z2). S i n c e ~ is a poly2k 2k+l nomial ring over Z with a generator in each even dimension it follows that d U (Z 2) = O. 2k+l We want to present an effective approach to the study of

~(Z

2) and especially to its ring structure.

impact of Boardman's work [

~

The decisive

] was our realization from

it that the systematic development of a suitable algebraic machine would bear fruit in the structure problem for

Section ii and especially

(11.5) will be immediately

recognized as a direct translation of Boardman to the weakly complex case.

Let D e u ~ U be the involution on the closed 2 2-cell given by z > -z. We introduce the ring F. as the quotient o f ~

U by the ideal which (i - D) generates.

We

W

find F. is graded by the dimension of the base space of [~

> V 2n ] and F. is a graded polynomial ring o v e r ~

generated by the conjugate Hopf bundles As a matter of f a c t




F.

as the composition

duIz21 .

.

>

F.

> F.

9

In (11.5) we find a sequence of involutions [(T~M 2k

2 closed algebraic varieties which have these properties

on

(i)

[M 2k] is indecomposable i n ~

(2)

every non-empty component of the fixed point set of

(3)

(T,M 2k ) has dimension ~ 2[k/2]

if n = [k/2] and v2n C

M 2k is the union of the

2n-dimensional components of the fixed point set then [V 2n] is indecomposable i n A

for k-odd and

decomposable for k even

(4)

if ~

> V 2n is the normal bundle then the

~[~

> v2n]~l with n = k/2 generate F~~4" as a

g r a d ed . A - m o d u l e 9

It is not difficult to see that all together generates F.

as a polynomial ring over Z 2.

The definitions of section 12 were arrived at largely ~U by trial and error. First we define in each~2n+l(Z2) a commutative ring structure with unit [A,S2n+I].

The definition

uses the Smith homomorphism & :~'~U(z 2) ~ > ~ U ( z 2 ) which was *~ U discussed in section i0. If x,y ~.~2n+l(Z2) we choose x''Y' e u ~ n + 2 xy

with~(x')

= x' 9 ( Y ' ) Y '=A

The product

----e~2n+l(Z2) is defined to be ~n+l~(x'y').

The product

-

13

-

is well defined, associative and commutative.

In addition~

~ ~ U (Z 2) is a ring homomorphism. A : (n+l)+l(Z2) > ~ 2n+] With this we can introduce the ring ~ Z 2 ) ~ an element of which i s a sequence

X2n+l

t!or which X2n+l ~ 2 n + l ( Z 2 )

and AX2(n+l)+l = X2n+l f o r a l l complete d e s c r i p t i o n f o r m a l power s e r i e s

element o f f [ t ]

n 2 0.

of~(Z2).

We can g i v e a q u i t e

Let.~.U[t]

be t h e r i n g of

over the graded ring.fLU;

has the form Z

ring homomorphism

[X2r]tr"

that

is~ an

We define a

>O (z 2) by assigning to

[x2r]t r the sequence ZO X2n+l =

~n

0

[A,S2(n_r)+l]Ex2 r

This ring homomorphism is onto.

~:

(Z2)"

The kernel of J2 is the

principal ideal generated by a certain power series 2 +

[L2r]t r. The coefficients L 2r can be explicitly 1 constructed. The Chern numbers of [L 2r] are all divisible by 2 and in fact 2 together with the [L 2k+I-2] generate the ideal I C ~ _

U consisting of those cobordism classes all of

whose Chern numbers are divisible by 2. J

Finally, there is a

_

decreasing filtration o f ~ Z 2 )

, f(k)9

F

(k + I ) D

...,

by ideals for which the associated graded algebra (k)/ _/[

f

(k + i) is naturally isomorphic to .

divisors.

The ring

(Z 2) has no torsion and no zero

A similar analysis o f ~ ( Z 2 ) / 2 ~ ( Z

2) is also made.

Now a stable boundary homomorphism ~ ' : i s d e f i n e d by a s s i g n i n g

to A2k ~

2k the sequence

----->

(Z 2)

-

14

-

A k ~ (X2k Dn+l ) E ~~! 2Un + l ( Z 2 )} 9 We show easily that ~ ' multiplicative. ~'(AD)

is

The stability refers to the fact that

= ~'(A).

We obtain thus an induced

,r.

The composite ring homomorphism ~

:~(Z

2) - - > ~ ( Z

may be thought of as a local invariant which is defined in terms of the normal bundle to the fixed point set.

It is

natural to ask for a corresponding global invariant. is done in section 13 by introducing structions

(Z 2) =

2n(Z2).

2n(Z2) is a sequence

~(X2r+ for a l l r _> n. P :

2n(Z2)

is Onto.

A construction in

-_ r

Obviously ~ n ( Z 2 ) .>

the ring of con-

a(~2r(Z2

2) = ~ ( X 2Ur ) D

This

satisfies

e ~ 2r+2 is a group.

2n(Z2) by P

We define

X2n and we show P

~ J ~ 2 n ( z 2 ) we 0 define a rather improbable looking product as follows. If {X2r ~ and jn

In the weak d i r e c t s u m ~ ( Z 2)

are

two constructions then t h e i r product

JZ2r ~ is given by L ~[n+m Z2r = X2nY2(r_n) + x2n+2 E(Y2(r_n_l) ) +'" "+ X2(r-m) e(y2m) for each r _> n + m.

Our product becomes more reasonable

when we observe E :

(Z 2) ~o

>~_U[t]

is a ring homo-

oo

By characterizing the

k

&. ~ n n ernel of P . ~ (Z2) - - > ~ ( Z

2) we can show that

ker (P)/~ ker (E) = O, and hence ~ ( Z 2) is in fact corn-

2)

-

15

-

mutative because XY - YX ~. ker ( P ) ~ ker (E). The commutative diagram U

really codifies into an algebraic framework all of the basic results about weakly complex involutions. The main result of section 14 is that ~ '

is a monomorphism.

The e s s e n t i a l

a

(Z 2) is

play.

monomorphism.

2)

= F/2F,

>

Z2)/2

It is here that (11.5) comes in-

In section 12 we filtered

~(k)D~(k

,.>~[Z

step in proving t h i s con-

sists of showing that the induced F, = ~

: F.

(Z2) by ideals

+ i)... for which the associated graded algebra

-'--~'~(k)/~(k + i) is naturally isomorphic to ,t/k, the 0 quotient of ~ by the ideal which 2 and [CP(1)] generate. Now consider

2k 2

~rom (11.5).

diagram we see t h a t ~ " Q ( ~ ?

~

U

) ~~ v ' 3

By use of the commutative

~ ) s

--L~

Furthermore v

,%

~(/~ ) ~{-) ~ is a set of generators of ,4 as a graded [ ~-" 2k -~" 2k)2 polynomial ring o v e r ~ o. Since F~ is generated as a polyO go oo nomial ring over Z o by ~ ( ~ 9 u ) ~ and ~ ~(k)/~(k + i)-'2"A it follows immediately %hat ~ " monomorphism. ~'

:

F~ - - - ~

E :~2n(Z2)

Since~(Z2)

: F.

-> ~

(Z2} is a

has no 2-torsion we see that

(Z2) is a monomorphism.

A corollary is that

>_c~U[t] is also a monomorphism, thus any con-

Z 2)

-

16

-

oO struction {X2r}n is uniquely determined by the sequence ( X2r n In section 15

we

prove a Boardman type result about

the dimension of the fixed point set.

If (T,M 2k) is a

weakly complex involution on a closed manifold for which every non-empty component of the fixed point set has dimension % 2n and if k > 3n then there are cobordism classes [A2k]~[B 2k-2] for which

[Z2,Z2][A2k ]

[T,M 2k] = modulo

+

[T,CP(1)][B 2(k-2)]

U

2(~'2k(Z2).

Here (Y,CP(1)) is [z,w]

,>

[-z,w].

In section 16 we come back to the Atiyah-Bott circle of ideas for weakly complex involutions.

Now if (T,V n)

is an analytic map of period 2 on an algebraic variety it would seem most natural to consider the global invariant n

Z

(-l)itr(T*~i). Unfortunately we cannot define 0 this for an arbitrary weakly complex involution. Instead

we define a local invariant Lc : gard

.

>

Z(i/2).

We re-

~ U as the polynomial ring over .~U generated by

conjugate Hopf bundles

and we define 0~ U Lc to be that unique ring homomorphism o f ~ . into Z(I/2) for which L c [ ~ Lc([v2m] [ ~

k>

> CP(n)

> CP(n)] = (I/2) n+l and for which M2n]) = td [v2m]Lc[ ~

denotes Todd genus. the image o 1 ~ . ~ U

[

> M 2n] where td

We show that Lc is integral valued on .(Z 2)

> ~ U ..

An interesting step in

the proof is the explicit construction of a set of homo-

-

17

-

geneous generators f o r ~ / ~ ( Z 2) as a graded algebra ~U.

over

We go on to present a formula computing Lc for

a complex k-plane bundle over a closed weakly complex manifold in terms of the characteristic classes of the bundle and the Chern classes of the base manifold. These notes are rough and contain only sketches for proofs.

They are intended as an introduction to the re-

markable ideas of Atiyah and Bott, and of Boardman as they apply to periodic maps.

Our treatment of Atiyah-Bott is

frankly scaled down and does not hint at the elegant techniques of associating analysis to topology in their paper [i

].

While we have generalized the Boardman work on involutions of unoriented manifolds to the weakly complex case it should be realized that Boardman developed his ideas in a much larger framework and the involution study was just one application.

In sections 12 and 13 we set up

the definitions so that analogous rings and~(Z

Zp),

(Zp)

2) can be defined for any cyclic group of prime

order in the weakly complex (oriented, unoriented) All that is needed is a Smith operator. for the ring of c o n s t r u c t i o n s ~ ( Z p ) .

case.

The same is true We do not know~

however, what form (11.5) takes in any but the Z 2 case (weakly complex, unoriented). worth

pursuit.

It would seem this matter is

-

18

-

We assume that the reader has studied [4] and [6]. Without comment we often use the fact t h a t ~ U ( z 2 analogous to the o r i e n t e d ~ ( Z p ) ~

p an odd prime.

) is

-

i.

19

-

The index In this section we develop the elementary linear

algebra requisite to our eventual consideration

of the

Atiyah-Bott Fixed Point Theorem as it ~pplies t_~o periodic maps of odd prime op_~r period.

We shall define an index,

as an element of the complex representation any complex representation

ring, R(H), for

of the finite group H which pre-

serves a non-singular inner product. Let (H,V) denote a finite dimensional, representation.

complex linear,

We shall say that two representations

(H,V) and (H,V') are equivalent if and only if there is a complex linear isomorphism L : (H,V) -~'(H,V') which is equivariant. We next impose some additional sentation

(HtV).

singular,

conjugate symmetric

We suppose that V is equipped with a non-

is preserved by H. (w,v) = ( ~ )

structure on the repre-

Thus

inner product,

(v)w)) which

(v,w) is complex linear in v,

and if v ~ O then for some w e V, (v,w) / O.

We also assume that for h e H~ (hv,hw) = (v,w).

It follows,

incidentally that (v,w) is conjugate linear in w. We denote this situation by (H,V,(.,.)). that (H~V~(.,.))

is equivalent to (H,V',(.,.)')

if there is a complex linear equivariant L : (H,V)

We now say

isomorphism

> (H,V') for which (L(v),L(w))'

can form sums and products as follows. set ((v' + v,~ + w')) = (v,w) + (v',w')'

if and only

m (v,w).

On V ~

We

V' we

and on V ~ V '

we

-

20

-

put ( ( v ( ~ v ' ~ w ~ w ' ) ) = ( v , w ) ( v ' , w ' ) ' .

Let P.I(H) be the

Grothendieck r i n g generated by complex r e p r e s e n t a t i o n s preserving a non-singular~ conjugate symmetric i n n e r product.

The index w i l l

be a c e r t a i n r i n g homomorphism

of RI(H ) i n t o R(H). Given (H~V~('~')) we s h a l l f i n d an e q u i v a r i a n t complex linear operator D : (H~V)

> (H~V) such that

(i)

02=

I

(2)

(v,D(v)) = (D(v),w)

(3)

(v,D(v)) > O i f

v / O.

We use D to s p l i t V i n t o a d i r e c t sum V+ ~

v_

vlD(v)

--

-

V_ where

.

Since D commutes w i t h the action of H we obtain representations (H,V+) and (H,V_).

We set ind ( H , V , ( ' , ' ) ) = (H,V+) - (H,V_) e R(H).

We must show that such an operator D does e x i s t .

It will

not in general be unique~ so we s h a l l have to show t h a t the

representation classes of (H~V+) and (H~V_) are unique. Let U I ( V ) C G L ( V )

be the subgroup of "unitary" operators

in the given inner product,

By assumption H C U I ( V ) .

CI(H)#~ UI(V) be the centralizer of H in UI(V). by.the

linear space of operators L : V

Let

Also denote

> V which are

"self-adjoint" in the sense that (v~L(w)) = (L(v)~w) and finally let ~ C ~ b e

those operators which are also "positive

definite" in that (v~D(v)) > 0 if v / O.

We are going to

- 21

prove that ~/~CI(H)

-

is non-empty and connected.

To show t h a t ~ I s

non-empty select an arbitrary

Hermitian inner product ~ v , w > o n V.

Since (v,w) is non-

singular we see by duality that for each w e V there is a unique Do(W) e V such that (V,Do(W)) =_for all v g V. Clearly D O is linear and (Do(v},w) = (~Do-~}) = < ~ > = 4 v , w >

(V,Do(V))

> o

v

= (V,Do(W))

o.

Using D o we set up a linear isomorphism of J w i t h space'of For L e J w e

the linear

all Hermitian operators with respect to~/v,w>. note that

= (v,L(w))= (L(v),w) = (w~L--~'Q")') = <W,Do e( l> = ~DolL (v),w> hence

D-1L e~. O

Furthermore~is

carried o n t o ~ + C

positive definite Hermitian operators. ~+

is a convex open cell i n ~ ,

It is well k n o ~ that

hence so also i s ~ i n ~

Now we can bring in the action of H. on

y conjugation~ preserving~,

The group acts

that is, L

> h

Lh.

i note that (v, h_lw ) = (h- lhv ~h-w) = (hv,w).

To 9 see that h-iLh ~ J LetJ(H)C~

, the

be the linear of operators fixed under this #

action of Hi t h u s ~ ( H )

is the space of equivariant linear

operators which are "self-adjoint." map r : J - - > ~ ( H )

A linear projection is given by r(L) = i/~ (H)g~EH g-iLg.

-

Since~s ~H)

22

-

invariant and convex this retracts ~ o n t o

=~/~H).

in~(H)~

We see t h a t C H )

is open and convex

hence it too is an open convex cell ( i n ~ ( H ) ) .

NOW we want the elements of order 2 in ~ H ) . D e ~H)

then D is certainly non-singular.

that D -I e ~ ( H ) ;

We assert

for (v,D- lw) = (DD-Iv, D-Iw) = (D -i v, w)

and (v,D-Iv) = (DD-Iv,D-Iv) Now D

If

= (D-Iv,D(D-Iv))

> O if v / O.

> D -I is a map of period 2 on an open convex cell

hence its fixed point set is connected and non-empty.

Ob-

viously D = D -I if and only if D 2 = I. (i.i)

Lemma:

Given

(H,V,(.,.)),

linear operators D : (H,V) (i)

D2=

(2)

(v,D(w)) ~ (D(v),w)

(3)

(v,D(v))

the set of all

> (H~V) for which

I

> O for v ~ O

forms a n o n - e m p t y c o n n e c t e d s e t in GL(V). The reader should show that this set is in fact /~CI(H). V = V+ ~

Thus we have an invariant splitting V_.

We must show that this does not depend in

any essential way on thechoice of D.

Let T C G L ( V )

(closed) subset of elements of order 2. signature of ~ i s Denote~ k.

T the

the dimension of its -1 eigenspace.

for O ~ k ~ dim V~ by T k the elements of signature

Since ~ C I ( H )

some k.

Forge

be the

is connected we have ~ C I ( H )

C T k for

-

23

-

Let Mk(V) be the Grassman manifold of k-planes in V and let ~

> Mk(V) be the associated k-plane bundle with

structure group GL(V).

There is the natural action of H

as a group of bundle maps ( H ~ )

~

(H~Mk(V)).

point of H in Mk(V) we obtain a representation fibre.

At a fixed

of H in the

The techniques of [6~ (2.2)] show that the equivalence

class of this representation

is constant on each component of

the fixed point set of (H~Mk(V)). Let (7-: T k ----> Mk(V) assign t o ' i t s is clearly onto~ but ~ ( ~ f ~ C I ( H ) )

-i eigenspace.

This

will lie in one component

of the fixed point set of (H,Mk(V)) ~ hence the representation class of (H~V_) is independent of D E ~ / I C I ( H ) . argument shows this is also true for (H,V+).

A similar

Thus

ind (H,V,(.,.)) = (H,V+) - (H,V_) ~ R(H) may be defined.

The

reader should prove that this induces a ring homomorphism ind : RI(H)

> R(H).

We shall next prove a standard

vanishing result. (1.2)

Lemma:

If for (H~V~('~.))

se__!~-annihilatin q subspace W ~ V ind ( H , V , ( ' , ' ) ) We write V = V+ 0

V_.

there is an invariant~

with 2 dim W = di____mV then = 0 ~: R ( H ) .

On V+ the inner product is

positive definite and on V_, negative definite, Wf%V+

= W/~V_

=

0 .

hence

Thus we see that the projections

(H,W) into (H~V+) and (H~V_) are both injections and from dimensional considerations (H,V+) -~" (H,W) -~'(H,V_),

it follows that

hence ind (H,V,(.,.)) = O.

of

-

24

-

We shall briefly comment on the conjugate skewsymmetric case.

In this case (v~w) = -(w~v)~ but otherwise the situation

is the same. ~v,w>=

We merely set ~ v ~ w > =

(v~iw) so that

-i(v,w) = (-iv,w) = -(w,-iv) = ( w - ~ a n d

we are back

in the conjugate symmetric case and we take ind (H,V,~','>). A remark about products is needed when V and V' both have conjugate skewsymmetric inner products. ((v ~ v ' , w ~ w ' ) ) of course. v,w>~,,

= ((v|

= -(v,w)(v',w')',

We take

which is conjugate symmetric

The sign choice is dictated by the observation that

w'>' = (v , iw)Cv' , iw')' = - ( C v ~

,w|

v' ,iw~Diw'))

w' )) = - ( v , w ) ( v ' , w ' ) ' .

Thus the product formula for index will be preserved.

-

25

-

Real representations

2.

Let us now consider a real linear representation on a finite dimensional triples

(H,V~(.,-))

symmetric~

real vector space.

where (v~w) is a non-singular,

real-valued

innerproduct.

section 1 we can define a ring ri(H) triples

(H~V~(.~.))

representation

We consider

By analogy with in terms of the real

as well as a ring r(H)~ the real

ring.

Now again by analogy we find a real

linear operator d : (H~V)

> (H,V) with

(i)

d2=

I

(2)

(v,dw)

(3)

(v,dv) > 0 if v ~ 0

= (dv,w)

and decompose V into the invariant eigenspaces V = V+

~

V_.

We set ind (H,V,(.,.))

There is the complexification follows.

On V ~ R

C let (H,V ~ R

of d r

= (H,V+) - (H,V_) ~ r(H).

ri(H)

> RI(H) given as

C) be

h(v + iv') = h(v) + ih(v') and (v + i v ' , w

+ iw')=

(v,w)

+ (v',w')

+ i((v',w)

This is easily seen to be conjugate symmetric, and H-invariant.

+ (dv',w')

= (v~dw) + ( v ' ~ d w ' )

+ i((dv',w)

-

(dv,w'))

(v,w'))

non-singular

Now let D(v + iv') = d(v) + id(v')~

(v + i v ' ~ d w + i d w ' ) = (dv,w)

-

+ i((v'~dw)

then -

= (dv + i d v ' , w

(v~dw')) + idw')

-

26

-

and (v + i v ' , d v = (v,dv)

+ idv')

+ (v',dv')

Clearly D 2 = I.

V_~

= (v,dv) > 0 if

+ (v',dv')

v + iv'

+ i((v',dv)

-

(v,dv')

/0.

The • eigenspaces of D are V + ~

R C and

C, therefore the diagram R

ind

ai(H)

~- a(H) >

commutes.

r(H)

The verticle arrows are complexifications.

The case of a real skewsymmetric innerproduct requires rather more attention. introduce on V ~ R

=

+ iw'~v + i v ' > =

(v',w) + (w',v) + i ( ( w , v )

(w',v) + (v',w) + i((v,w)

+ (v',w'))

= ( w ' , v ) + (v',w) - i ( ( w , v )

+ (w',v'))

=~v

+ iv',w +

~'>.

Thus we obtain a conjugate symmetric ( H , V ~ R shall compute ind ( H , V ~ R Let%be

+ (w',v'))

We should note t h a t

This is H-invariant. w

Given a skewsymmetric (H~V~ (',')) we

C,~.,.>).

We

C,~.,.~) as follows.

the family of real linear operators J : V

for which (i)

(v,J(w)) = - ( J ( v ) , w )

(2)

(v,J(v))

> o i f v / o.

>V

-

27

Obviously ~ is analogous t o Z ~ a n d same way. J e ~

Thus~and

~(H)are

-

in fact may be studied the

open convex cells.

(H), then J is non-singular and _j-i

e

(H)

If for

(v,-j-iw) = (Jj-Iv,-j-iw) = -(j-Iv,-jj-iw) =

- (-j-iv,w) and (v,-j-iv) = (jj-iv,_j-iv) = -(j-iv,-j(j-l(v)) = (j-iv,j(j-l(v))) Again J

> _j-i is a map of period 2 on the open convex

cell ~ ( H ) , nected.

> 0 if v / O.

thus its fixed point set is non-empty and con-

If J = _ j - I

then j2 = -I, so this time we are

finding complex structures on V~ which commute with the action of H. (2.1)

Lemma:

~kewsymmetric

If (H,V,(.,.)) preserves ~ 9on-singula~

innerproduct,

structures J : (H,V)

then the family o_ffcomplex

> (H,V) for which

(i)

(v,J(w)

= -(J(v),w)

= (w,J(v))

(2)

(v,J(v)) > O !! v ~ O

forms a non-empty connected subset of GL(V). The relation of these J to the computation of ind (H,V ~ R

C,=< D(v

) - iJ(w

-

(v',

J(w))

), w) - (J(v), + iv'),

w + iw'

w ))

>

and 0

J(v'))

is given J(v)

) and the

That

complex

- 1 eigenspace

by v

> J(v) + iJ(v)

- (v', J(v))

is

of D.

If

(

) + iJ(v)

- J(v)

+ iv

+ iv.

index

+ iv.

Note

t

9

A

that

= - v + iJ(v)

= i(J(v)

+ iv).

Similarly

is given

Thus

, .>~=

in this

representation

then

of (V, J) w i t h the + 1 eigenspace

of (V, -J) with the - 1 - e i g e n s p a c e

(H, V, < .

is, this

>

and the + 1 eigenspace

if v + iv' = - J(v

isomorphism

> - J(v) ind

then v = J(v')

linear

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

the • 1 eigenspaces

~ while

> j2(v)

J(v))

J(v'))

- iJ(v)

= (v, J(v'))

iS(v)

v) + i

complex

+ iv , J(v')

+ i((v,

J(v

v = - J(v

O~ thus W D J ( W )

annihilating.

)

= 0 =

0

since W is self-

Now 2 dim W = dim V~ thus V = W + J(W).

use this to define an isomorphism

We

(H~ V~ J) _o-(H~ V~ -J) as !

follows.

We send w + J(w')

J(w + J(w')) = -w Since ind (H, V ~ R the lemma follows.

-> w - J(w )~ then

+ J(w) - - > C,

) = (H, V, J) - (H, V, -J)

-

3.

30

-

Extension and reduction It will be useful for us to determine the relation of

index to two standard constructions Let K ~ H be a subgroup~

in representation

theory.

then there is the multiplicative

reduction homomorphism r (H, V, (. , .))

>

: R (H) > R (~) given by KH I I (K, V, (. ,. )) together with the

: R(H)

similar r

It is quite easy to verify that

>R(K).

KH R (H)

It

ind > R H) r

rKH

KH ind R (K) >' RI K) I is a commutative diagram. Let us next consider the induced representation. given (K, V, (. , 9 )). functions f : H

We are

Let W be the linear space of all

> V which satisfy f(kh) = kf(h)

for all k e K~ h e H. by (~f)(h) = f(h~).

R(H).

First we shall show that index is a bordism invariant. Suppose~

then, that there is an action

(B 2n+l, H) on a

compact oriented manifold for which (I~ B2n+l !

If

~"

in H --

>

(B2n+l 2n+l

o-

therefore

E.

annihilating.

be the image of i

Hn+I(B2n+I,M2n;R)

and observe that W is

!

v u

v

!

w )nO')

((i*(v') We w i s h

t

= (v U w )t%

~/ i * ( w ' ) ) / % o to

show t h a t

i.~r)

~*

W

= 0

~) = O, and W i s

self-

2 dim W = dim Hn(M2n;R).

!

Let

class then

For v , ~ w , e Hn( B2n+l ;R) we have

H-invariant. i~-(

;Z) is the orientation

i > Hn(M2n R ) ~

Hn(B2n+l R)

(

2n

Consider now

9

Let W C H n ( M 2 n ; R )

i.

, M

H) = (M 2n, H)

!

be a d i r e c t

summand o f W.

(v) ~ 0 e Hn+I(B 2n+l ,M 2n ;R).

If

v E W , then

By the Lefschetz Duality

.. .

-

Theorem there is a w

~. ((~*

,

n

e H (B

(v) u w ' ) ,..,~- ' )

=

2n

~.

38

-

~R) with

((v,.,

i

*(w' )),~o")

/o.

If follows now that W' is dually paired with W into R~ thus dim W = dim W' = 1/2 dim Hn(M2n~R)).

If n is odd

we apply (2.2) directly and if n is even W ~ R C

is an

invarian% self-annihilating subspace of Hn(M2n|c) and we can apply (1.2).

The reader may ascertain that reversing

the orientation of M 2n merely changes the sign of the index, thus ind (M2n~ H) only depends on the bordism class [M2n~ H].

This argument for showing index is a bordism

invariant goes back to Pontrjagin. Next~ let us point out why we only consider the middle dimension.

We could write

(H, H'x'(M2n~c)) = (H, Hn(M2n~c)) + Z o

n-1 (H, Hj (M2n ;C) (~ H.2n- j (M2n|c))

With the cup-product we obtain a non-singular H-invariant innerproduct on each term HJ(M2n;c) ~

H2n-j (M2n;c)

for 0 ~ j < n, but we can immediately apply either (1.2) or (2.2) to see that the resulting index is O. For a product (M 2 n x H (M

2n

X

V 2m, H) we recall that

~ M2n v2m;c) -~'H ( ;C) ~ H ~ ( V 2 m ; c )

product of graded algebras is formed. with (H~ Hn(M2n;c) ~ H m ( v 2 m ; c ) ) .

where the tensor We are only concerned

In particular if

-

!

v,

w ~Hn(M2n;c)

(V~

v )'{W@W

t

v

-

!

and v , w ) = (-I)

39

nm

6 Hm(v2mlc) t

t

(VW@V W ).

b o t h odd t h e n we see t h a t

then If

the innerproduct

n and m are for

M2n)( " V 2m

is still just the product of that for M 2n with that for V 2m in the sense we defined products in the skewsymmetric case.

Thus we a l w a y s have

(ind [M2n, H])(ind [V 2m, H]) = ind [M2n X

V 2m, HI.

We wish to relate index to the homomorphisms R

KH

.

.

We shall leave R

It


2n

-

~ pS

~2n+l(ZP s; ~" ~ ~' ~s )s-I

We now consider

)--> _

y has finite order.

(z 2n

)

;

pS/

We showed t h a t Tr a n n i h i l a t e s

2n

)

pS

t h e image of

pS

(7.2) Lc:~(Z *

2n Ib@orem:

;~s~#l

2n

)

p

pS J i s . l'_

-1 is

to.

on

> Q(~) for which

_

is a commutative diagram. If x c ~_2n.Zps;

) then by (7.1) we can choose ~ J

0 < m

V 2m]

!

!

[(#1''''' ~p)

The total degree of 2n + 2(dim ~ i

N 2n where # I , . . . , F ~

We allow "O-bundles" which is merely a

place holding device.

[(#1'''',~

-->

+ "'"

>

M2n]

)

> M2n.x v2m].

is

+ dim#F)"

For every pair of integers (i ~ j ~ # xCj,n + i) = [~,...,0, ~

and n ~ 0) let

,0,...,0) ---> CP(n)], which denotes

the c o n j u g a t e Hopf b u n d l e in t h e j t h

position.

To

~j = (i ~ i I ~...~ i ), a sequence of finite length, we associate r

X(~)

= X(j, il)...X(j , ir )"

d e s c r i b e d as f o l l o w s .

Y(cO l " ' "

An~-module

basis o f ~ .

can be

Let

0.,;~ ; Sz,...,s / ) s

=

x(~)(x(1,1)) i x(~2)(x(2,1) )s2 9 .. •

If some 4~) = ~ we just omit this term. J

Ic•

We require that each

-

s j -> O.

47

-

The total degree of this element is

2(n(O.~)

where n ( % )

+ ...

+ n(c,,J/j)

+ S

= il + ... + i.r

form a h o m o g e n e o u s ~ - m o d u l e

1

+

+

,..

S

)

The Y(6L)I,..., ~ '

Sl''''' s/

as t h e ~ - f o l d

of a graded polynomial ring wlth itself.

tensor product

Let P =~(Xl~X2,...~Xm~...)

polynomial ring on generators with deg (xm) = 2m.

An isomorphism

P~

P@... -0_

0

P ~ ~

-o_

is given by

l~p...

~l~D

x

~

l|

... ~)l

> X(j,m).

m

The h o m o m o r p h i s m ~ follows

:~(Zps

)

>~.

At each fixed point of (N2n Z

pS

is described as

) we receive an

orthogonal representation, with no non-trivial isotropy vector of Z

pS

in the linear space of vectors normal to the fixed point

set at this point.

)

basis of~-~ .

We can also i n t e r p r e t ~

be the g r a d e d

y

All along the component of the fixed point

set through this given point we can reduce the structure group of the normal bundle to a product U(n I) x...x U(n~).

In the

I

resulting complex structure on the normal bundle the representation of Z pS appears as a diagonal matrix wherein ~J occurs with multiplicity n. for 1 < j ~ / 9 The orientation 3 [

- 48 -

on M 2n

together with the orientation of the normal bundle

induced by the complex structure, defines an orientation on the component of the fixed point set being considered. this on each component of the fixed point set a n d ~ [ M

We do 2n,Zps]

is the sum over all these components of the resulting elements in ~

.

The reader may wish to formulate the definition of

2n

.

pS

.

s

s-i

So far we have no computation for Lc.

We see that Lc is

uniquely determined by its values on the X(j, n + i). (7.3)

Lemma:

For n > 0 and I ~ j ~ y

Lc(X(j, n + I)) = Lc(X(j, i))

n+l j n eve~

Lc(X(j, n + i)) = 1 - (Lc(X(j, I)) n+lj n o ~ d . We simply define an action (CP(n + i), Zps) by [Zl,... , z ] n+2 that ~ [ C P ( n in~2n+2.

. > [Zl,... , z

n+l

, z

n+l

~ J].

We assert

+ i), Zps] = X(j, n + i) + (-I) n+l CLc(X(j, i))) n+l The fixed point set of (CP(n + i), Zps) is the

disjoint union of CP(n) and [0,...0, i].

At the point we

introduce the local co-ordinates (z ,..., 1 so that

I

z

n+l

' Zn+l

)

> [il,... , i

) -->[~i

n+l

, I]

'''''Zn+l

[~i

n+l'

-

49

-

The orientation of the local coordinating agrees with the natural orientation of CP(n + i) if and only if n + 1 is even, hence the sign (-i) n+l.

On the other hand the normal

bundle to CP(n) is ~ > CP(n) and in each fibre the repret sentation of Z is multiplication by 2 . pS 9

Now surely Tr([CP(n + l)~Z

]) = ind [CP(n + i)] = 0 or 1 pS according to whether n is even or odd. Since Tr[CP(n + I),Z

pS

] = Lc(X(j, n + i)) + (-l)n+l(Lc(X(j, i))) n+l

the lemma follows. Finally we shall have to compute Lc(X(j, i)) s Q ~ ) 1 ~ j ~

=(pS-l)/2.

for

This will be done in the next section.

-

8.

50

-

Periodic maps on a Riemann surface We shall sketch a proof of the Atiyah-Bott Fixed Point

Formula for certain key periodic maps on Riemann surfaces. Our approach here is along the lines of the original Woods Hole Fixed Point Theorem.

We shall use sheaf cohomology

to establish relations between analysis and algebraic topology.

A general treatment of sheaf cohomology is found

in Bredon [2], and we shall refer to Gunning and Rossi for background in the sheaf theoretic approach to complex variables [9]. Let M 2 be a closed Riemann surface.

The real valued

harmonic 1-forms on M 2 uniquely represent HI(M2~R).

The

usual duality operator~ * , on 1-forms is locally represented by al(x,y)dx + a2(x,y)dy

>-a2(x,y)dx + al(x,y)dy and i f ~

is a harmonic 1-form then so is * 4 -

In terms of harmonic

forms the non-singular skewsymmetric innerproduct on HI(M2~R) is given by = / ( a l b 2 - a2bl)dXdy M2

Obviously. (~,

*~)

=-i

2 M

and since ~ / ~ . ~

=-(*~,~)

=-.~/A/~

and finally ( ~ ,

.~)

it follows that

> 0 if o ~

O.

Thus

the *-operator defines the complex structure (HI(M2~R),J). Consider on the other hand~ the complex cohomology HI(M2;C).

-

51

-

This is split into a direct sum hl,O ~ h O' i of complex subspaces where h I'O is the space of holomorphic 1-forms, and hO~I the space of anti-holomorphic 1-forms.

An

isomorphism of (HI(M2;R),J) with h0'I is defined as follows. If o *~6-

> o g(Fi)k

~ ~k(Ui).

For f e

k(Ui )

we observe that f/(Fi)k is regular since (Fi)k has a zero of order k at each fixed point~ but f also has a zero of order at least k.

Obviously this quotient lies in ~o (Ui).

In faet~ if U is any open set in CP(1) for which there is an F e fl(U)

having a simple zero at each fixed point

and vanishing nowhere else, then

~(U) ~ k ( U )

for 0 < k < n.

O

Thus the lemma is proved. (8.3)

Lemma:

For 0 < k < n the sheaf

> CP(1)

!~ naturally isomorphic to the sheaf of 9arms of local holomorphic sections of the k-fold tensor product of the H_Ho_pf bundle with itself. On

~)-l(u2/q U I) we have F 1 = GI2F2 where GI2 e ~o ( U 2 ~ U 1 )

may be regarded as Gl2([Zl~ (GI2)k : U 2[% U 1

z2]) = Zl/Z 2.

We shall prove that

> C* defines the k-fold tensor product of

the Hopf bundle with itself; ~ k W = C x C \ O,

and let C

t(z I, z 2 , ~ ) =

> CP(1).

Let

act on W )~ C by

(tz I, tz 2, t-k~),

then

p : (w

c)/c*

by ((Zl, z2,~))

-> cp(1)

is

k

,> c P ( 1 ) .

a vector in the total space.

Now

We denote

-

56

-

-1 and $ 2 ( [ Z l , z2],~ ) = ((zl, z 2 ' ~ / ( Z l ) k ) ) " On (U2/% U1)xC we have CP(1) over U is given by

[z I, z2]

> ((Zl, z2, gl/(z2)k )), U 1 z% U

[z I, z 2]

> ((zl, z2, gi(zl)k)), U2/~

U.

On the intersection U 2 ~ U 1/% U ((Zl' z2' g2/(zl)k))=

((Zl' z2' ((Zl/Z2)kgl)/(zk)k))

= ((Zl, z2, gl/(z2)k)) Thus the corresponding section is well defined. Conversely, a holomorphic section of ~ k over the open U is uniquely specified by a pair of holomorphic functions gi : U i ~

U

> C, i = I, 2 for which g2

=

(GI2)k gl on U 2 ~

UI/A

U.

-

Now gi ~ ~

57

-

( U i f 3 U) and we note that on ~ - I ( u 2 ~ U1 ~ U)

gl(F1)k = gl(G12)k(F2)k = g2s k so gl(Fl)k~

g2(F2)k define an element of F k ( U ) .

the proof of (8.3).

J

But now we can read o f / ~ d i m Riemann-Roeh

Formula~

[11~/5~].

dim H~

HI(cP(1);~k)

If c is the characteristic

from Hirzebruch's

First

- dim HI(cP(1);

ch(~ k) = 1 + kc.

This completes

) =

~ U(Z 2)

is an 1 3 _ U _

module homomorphism we have an exact sequence

(z) 2

(z) ~

~

2

>

0

-

64

-

then o~ IV2m ] = [Z2~Z 2 ~[ V 2m ] which is

If Iv2m] s ~ U 2m

just the permutation of two copies of V 2m. We have a natural U ~U ~ (Z )-module structure on ~ (Z ) given by 2

2

[TI,V2m][T~M2n+I] = [T 1 X in ~

~" U 2(m+n)+l

'~(

(Z). 2

(x)y) = x ~y

M 2n+l]

For x s

we have 2

e

(Z). .

~

Let e :

2

be t h e a u g m e n t a t i o n (10.1)

T, V 2m X

(Z) ~

- - > ~

U

2

U

Lemma:

Fo...~r x ~ ( ~

(Z) ~

x[A,S I]

=

2

s(x)[A,S I]

~.~U

in_.O_ (Z). 2

This is merely the analog of [4~ (35.2)]. argument applies here. homomorphism A : ~

The same

We can also introduce the Smith

U(Z2) -->_~._~(Z2) ~U ~ which is an ~-~U_

module homomorphism of degree -2.

Given (T,Mn) there is for

N large a unique equivariant homotopy class of equivariant maps f : (T~Mn)

> (A~S2~+3).

regular on the invariant S2~+i C f-l(S2A/+I) = vn-2 C

We may assume f is transverse S 2~/+3

then

Mn is a closed invariant submanifold

with a trivial complex normal bundle.

This normal bundle~

together with the weakly complex structure on M n ~ uniquely determines an invariant weakly complex structure on V n-2. We put ~[T,M n] = [T~V n-2] and by suitable transverse regularity

-

65

-

argumen~ it is shown that ~ is well defined [4~ p. 92]. ~-xr'~ U (10.2) Lemma: I_ffG[T~ M 2n+l ] = 0 for [T,M 2n+l] e --k/_2n+l(Z 2)__ then there is a [X 2n] e ~

U 2n

with ['I',M2n+l] = [x2n][A,S I].

The argument is like [4,(37.7)].

Let us next prove a

commutativity. (10.3)

Lemma:

The diagra m

~ 2 n+2

@

~ U > Q_2 +l(z )

2n

-

2n-i

2

is commutative. Suppose [T,B 2n] e

on (B2nX D 2) by (x,z)

2n' then [T,B2n]D is represented > (T(x), -z).

Note that

@(B2nX D2) = / ~ B 2nX D2 ~ B2n X ~ D 2

with (~B2nx

D2) ~ (B 2nx ~ D 2) = @B 2n • S1.

(A,S 2M/+3) as the j o i n

f : (T,

B2n )

(A~S2~+I)o(A,SI).

We regard

Suppose that

> (A, S2&/+l ) is an equivariant map~ then we

shall define an equivariant map F : (Z2,3(B2nX

D2)) -->(A~S 2A/+3)

by I(l-t)f(x) + tz, x&

B 2n , I zl = l, o _< t _< 1

F(x, t z ) = O+z, x ~ B 2n, Izl = i, t = 1

-

Now F-I(s 2n+l) is exactly A~[Z2,B2nX

66

-

~ B 2n X

0 , thus

D 2] = ~ [ T , B 2n] which is the assertion of

iemma.

The reader may verify the indicated transverse x. regularity. We shall now show that A is an~U(z2)-module....

t-'U( .(Z 2 ) and y ~ : _ ~ . Zp)

homomorphism.

(10.4)

Lemma :

For x ~:

A(xy) = xAy.

Let x = [T,V 2m] and y = [T, M 2n+l] then xy is represented by (T 1 x T,V 2mX f

:

(TI,M2n+I)

S 2n+l 9

M2n+l).

Let the equivariant map

> (A,S 2~+3) be transverse regular on

The map F : (T 1 X

T~V 2 m x

M 2n+l ) --> (A~ S2~,+3 )~

F(X~y) = f(y)~ is still transverse regular on S F-I(s2n+l) = v2m X f-l(s2n+l)

hence the lemma

As always we can interpret~-~/~U vector bundles; that is ~

2d+l and

U " ~ 2m

as bordism classes of U

Cb

__.~ m=n+k

CBUCk)). 2n

The isomorphism is described by assigning to (T~ B 2m ) the normal bundle to the fixed point set.

Recall that the fixed

point set receives a weakly complex structure and its normal bundle a complex structure.

Conversely~ given a k-plane

bundle over a closed weakly complex manifold~ V > v2n~ 2n we denote by D ( ~ ) > V the associated closed 2k-cell c

bundle.

The weakly complex structure on V 2n together with

-

the complex structure on ~

67-

determines a weakly complex

structure on the total space D(~), which is a compact 2(n+k)-manifold.

On D(~) we introduce the weakly complex

involution which sends each fibre into itself and which on each fibre agrees with-I. fixed points.

Thus [ ~

inverse isomorphism.

~o~uo~ ~ E ~

Clearly (T,~(D(~))) > V 2n]

has no

> [T,D(~)] is the

In terms of this representation the

> v~n~

__> v~Jl: C~,~f~

> v~

The element D is the trivial line bundle over a point. regard ~ U

~U

as the bordism classes of O-bundles.

know the structure of ~ U [ C~ the

--> CP(D)

quite well~ [9,(2.2)].

v~J~

We We

Let

denote the conjugate Hopf bundles, then U U is the graded polynomial algebra o v e r ~ / _ generated by ~

> GP(n)]

9

-

ii.

68

-

The ring F Let ~ c ~

U be the ideal generated by I-D~ then the

quotient algebra set

c'~ U /

~

will be denoted by F .

=

(BU(k))

2n I* so that

k=O

and ~o

for each integer n > 0

2n

= 9

=

2n~.

2n~.

l hus F~ is a graded algebra with F. =

n

n,* (ii.i)

We

2~7~ where

2n~*

Lemma:

The ideal ~ c o n t a i n s

no homogeneous >

onto. X~

An element A(I-D)

: d

)

>

lies in the image o__[f

~ if and only if there is an element

u s

(Z) ~

for

which

s(x)

s 2.

and

(x

= A.

2

Suppose B2n ~

2n and B2n = (I-D)A.

Since

is a

weak direct sum we can write A = 7 o N A2k ~ thus A2k = 0 for k < n and A2(n+m)

= B2nDm for m > O.

we conclude that B2n = 0 since

Since A2n+l = 0 = B2nDN-n+I has no zero divisors.

note explicitly the corollary that o

> 2k

is

exact.

z) 2k

2

> F *

We

-

69

-

~ U Next for any A2n~ n

2n we can surely find

[x2(n-r)] Dr so t h a t

~(A2n-

~

Dr ) = O,

n1[ x 2 ( n - r ) ]

1 which means that A2D - 7 n[X2(n-r)31 Dr ~

img)"

We then

observe that in F ) [x2(n-r~ = [X2(n-r)] Dr , but [X2(n-r)] ~ d U ( z ~U . thus (Z)

) as the trivial involution on X 2(n-r) 2 > F

is onto.

suppose @(A) = ~(AD) e ~ . U.(Z2 ). We write

Finally,

A ~(A2ND) = ~

(A ) = O, and thus we proceed to show that 2N

~ (A2k) = ~(A2(k_I)D) = 0 for 0 J k J N. write ~ ( x )

= A.

By (I0.I)

[A,S l]

x[A,S l] = but [4~

~(AD)

Thus we may

= 0 =

~

(xlD)

= x [A, S I] , hence e(x) e 2 _ ~

U

(37.6) by analogyJ. Since ~ U

is a graded polynomial algebra o v e r ~ U with

generators D and

[

> CP(n)

also a graded polynomial algebra in each ~ 2 n '

n > 09

it foltows that F

o v e r' .f-)_u

We note t h a t

..~U

is

~ith a generator C

by

F .)(-

[X2r]

> [X2r]'l. We are not able to work effectively with F

2 ~Uc~_U

denote

the

itself. Let

(prime) ideal of all those complex

-

70

-

cobordism classes which are divisible by 2.

A

=

We denote by

~-"~

the quotient ring • U / 2 nU. This 2n is a polynomial algebra over Z with a generator in each 2 even dimension [9 ]. We shall study F

..4.

.

- F ~ -

_/~ _Oo

It is clear from the structure of F. that ~

is still graded by

~

= zn

"

/2

= F./2F..

. Of course

This

F.

zn

is a graded polynomial algebra o v e r ~

with a generator in

each .,-z-'~~ n > i. 2n Such a generator is indecomposable with respect to the That

and 2n r~ then an element of F "~'. is indecomposable with

-algebra structure.

is, let F

=

*

1 respect

to the

-Z~-algebra

structure

if

~

1

and o n l y

if

it

does

not lie in the image of

To obtain a set of generators for F ~ algebra over A

as a graded polynomial

we choose in each ~/f 2n ~ n > i~ one element

which is -/~ -indecomposable. We may characterise ~ - i n d e c o m p o s a b i l i t y .

Let

be a k-plane bundle over a weakly complex manifold.

~

Express

the total Chern class of ~

in factored f o r m ~ i k ( l + t j )

for n > 0 let (7"2n[ ~

V 2n] g Z be the value of the

symmetric function Z

~>

> V 2n

then

Ik(tJ )n on the fundamental class of V 2n.

This is clearly a bordism invariant.

Since

-

C;-2n[ ~ ~

C

71

-

> ~ V2hI = C;'2n[~ --> V2n] ~. / obtain T w~ . . ;e_ .a/ r -

homomorphism C;- : ~ > Z, and a r 2n " 2n 2n 2n It is easily seen that (%- is 0 mod 2 on 2n ( i m ( g ~ ~ X ( ~ = / ~e F~ > F ~4~) FI

(11.2) Lemma:

/2~ zn

zn

>Z 9 2

An element e

e_//is indecomposable 2n 2n -algebra structure if and only if

with respect to the ~ (~2n(e2n) = 1 mod 2.

Equiva i ant ly, e2n = [ ~

> CP(n)] + ~ - d e c o m p o s a b l e

terms, if and only if ~2n(e2n) = 1 mod 2.

The proof is

immediate since ~ 2 n [
CP(n) = ~ i. Ff

and~

> [x 2 r ]. 1

gives us an embedding ~

C

polynomial ring over Z 2.

Thus we can also think of F ?

graded polynomial ring over Z 2. in

n ~ I.

as a

We shall require two generators

A set of generators over Z 2 is best oo

described as a sequence ~kJJE2 ~

2n

itself is a graded

with E

. We shall require that the

and E

i Now [

A~

:

both in

4n+2

generate

4n

graded polynomial algebra overJIL . certainly induces an augmentation e

4n

F~

-->

->~

as a

*

v2 ] --> [v2 ]

.

We must

= V 2n] e Z be described as follows.

Express

the total Chern class of the stable tangent bundle of V 2n in the factored form

(i+ ~.) and let S2n[ ~ > V 2n] be the 3 n v a l u e of the symmetric f u n c t i o n (~) on the fundamental 1

1

c l a s s of V2n.

J

Obviously the v a l u e of S2n only depends on

[V2n] e __Q U

A cobordism c l a s s [V2n] i s indecomposable in

if and only if S2n[V2n] # 0 mod 2, 2n / 2 j+l -2 S2n[V2n] / 0 mod 4, 2n = 2 j+l -2, j > 0

[9, ( 3 . 2 ) ] . 7

(11.3)

Theorem:

E4n , E4n+2 e ~ 2 n ,

Let --" ~E~. 6 zKj~ .be ..a . . sequence ~ --- f o r which then

polynomial ring over Z 2 if

E2k generates ~. -2

as a graded

-

73

-

i)

fo___rrn ~ i, gr" (E ) = 1 mod 2. 2n 4n

2)

for 2nm& 2J+i-2, S 2 n ( E 4 n + 2 )

/ 0 mod 2 and

S2n(E4n+2) ~ 0 m__ood4 for 2n = 2J+i-2. 3)

for

1, S2n(E4n)

n 2

= 0 mocl 2 ( o r 4 ) .

It is inconvenient to work with the grading on F. ~ thus

ZO2~ "~~

we introduce the f itration F2"~n =

(11.4)

Corollarl:

Let

2

be a sequence with flt

2

E4n ~ E4n+2 i_nnF2n.

If the components of E4n , E4n+2 in

s a t i s f y th___eehypothesis o f (11.3) then

2k

)

*

generates

.

2

as a polynomial algebra over Z 2. Let p : ~ U ( z

2n

> F ~4" be the composite ring homomorphism

2

*

d Iz > % ~ 2

*

>,

*

9> O.

*

The remainder of this section is devoted to a proof of (11.5)

Theorem: There is a sequence

2k

e

oO 2k

(

such t h a t

I)

e(

2)

p(v 4~_ j n ), p(

3)

-k ) e

is indecomposable 2k--


1 is odd, then for ~ 4 n

we

can

use the involution on CP(2n) given by T[Zl,''', Z2n+l] = [-Zl,... , -Zn+ I, Zn+ 2 ,''', Z2n+l]" The fixed point set is the disjoint union of CP(n) and CP(n-I), p~ hence p[T,CP(2n)] e 2n" The normal bundle to CP(n) is n ~-->

CP(n) and since n is odd, ~ n [ n

~

> CP(n)] = 1 sod 2.

S2n[CP(n)] = +(n + l)and n > 1 is odd we see that [CP(n)] is decomposable

in ~

even if n + 1 = 2J.

S4n[CP(2n)] = 1 mod 2. by [Zl, z2, z3]

Finally,

For n = 1 we can use (T,CP(2)) given

> [Zl~ z2, - z3] but we must use a distorted

weakly complex structure on CP(2); namely, than ~ ( ~ 7

, [5, p. 33

].

the weakly complex structure

~)7 % %

structure [CP(1)] = 0 in

U 2

~ ( ~ ~

rather

This imparts to CP(1)~CP(2) >CP(1) and in this

e

To continue we shall need the following construction.

Let

> CP(n) be the conjugate Hopf bundle and let jC

.> CP(n) be its sum with a trivial j-plane bundle.

Let p : CP(n,j)

> CP(n) be the associated projective space

bundle with fibre CP(j).

The total space is an algebraic

variety of real dimension 2(n+j).

In [9, (4.4)] we showed

that for j > 0 S

2(n+j)

[Cp(n

j)] = + ' -

n

(-i)

i( n+Ji

) + k)

-

We denote by

~

> CP(n~j)

75

-

the canonical line bundle~ a

point of which is a pair consisting of a line in a fibre of ~

j C

> CP(n)~ together with a vector in that line.

An elementary calculation

shows that

~(n+j)[

We now take up the construction of ~

~

> CP(n,j)] = • 1

with k = 4s.

We

"2 k

assert that

(l)

[CP(2,4s - 2 ) ]

(2)

[CP(1,2s - i)] is decomposable

(3)

4s there is a complex analytic involution on CP(2~4s - 2)

is indecomposable

in~

8s

in

with fixed point set the disjoint union of

CP(1,2s - 1 ) , CP(1) x C P ( 2 s - 2 ) , CP(2s - 1) and CP(2s - 2 ) . Now

(-1)

(

"

)

+ 4s

-

2 =

hence [CP(2,4s - 2)] is indecomposable~

(-1)

(

)

+ 4s

-

2 =

1 mod

but on the other hand

i 2s (-I) ( i ) + 2s - 1 = 0 mod 2 so [CP(l~2s - I)] is

~

decomposable

in ~

.

The involution must be explicitly

described.

Let U(1) act on S 5 by (Zl~ z2~ z3k = (zlt, z2t ~ z3t)

where

Izil

S5 X

= i t then CP(2~4s - 2) is the quotient of

CP(4s I- 2) with respect to

(x,

[Wl,...

, W4s_l])t

= (xt,

[wlt , w2,... , W4s_l]).

The involution on S 5 X CP(4s - 2) given by ~ ( x ,

= (Zl,

z 2, - z3~ [ - W l ~ ' ' ' ,

- W2s ~ W 2 s + l ' ' ' ' '

[w])

W4s_l])

2,

-

76

-

commutes with the action of U(1) and induces

(T,CP(2,4s

Note that f~ and t e U(1) can have a coincidence

- 2)).

if and only

if t = • 1 since the U(1) action is free. For t = i we find that the fixed point set of ~

is the

disjoint union of

S3 X CP(2s- 1)= ~((z[, z2, 0), [Wl,... , W2s, 0,... 0])) t and S 3 X

CP(2s - 2) = ~ ((zi, z2, 0), [0~..., O, W2s+l,...,

In CP(2~4s

- 2) these become CP(l~2s

- i) and CP(1) X

CP(2s - 2)

respectively. If t = - 1 then ~ X CP(2s-

I) =

and t have coincidences

at

((O,O,z3) , [Wl,O,...,O , W2s+l,...,W4s_l])

and slM

CP(2s-

2) = ~(O,O,z3),

[0 , w2,...,

W2s, 0,...0])~.

These become CP(2s - i) and CP(2s - 2) respectively CP(2~4s

- 2).

is as described. ing embedding

CP(1)

- 2))

Note that the normal bundle to the result-

of CP(l,2s

into a Whitney sum. >

in

Thus the fixed point set of (T~CP(2,4s

- l)cCP(2,4s

- 2) splits naturally

The first summand is p (q), where

is the normal bundle in CP(2) and the second

summand is (2s - I) ~

> CP(I~2s

:

- I).

i mod 2.

Now

w 4s-i

II}

-

We may t h e r e f o r e

P(~8

s

) ~ Fw{ 4s

take (T,CP(2,4s

77

-

- 2)) = ~------~8s and

as required.

Let us now consider CP(2n,2j + I).

Until we indicate

otherwise we assume 2n + 2j + 1 / 2 k - I.

Since


O, we take n = 2 r, then 2

= 0 mod 2 so that [CP(2 r+l ,2s - 2r+l+ i) ] is

indecomposable

in~

, as is [CP(2r,s - 2 r)].

We ,ee

an,

a

part of the fixed point set is indecomposable Finally we come to s = 2 r - i. (T,CP(2 r+l , 2 r+l - i)); that is n

=

in~

men,

on,,

.

If r > 1 we use 2r~ j

noted [CP(2r,2 r - i)] is indecomposable

=

2r

in ~

-

i.

As

we

and from

[9, (4.3), part iv] it follows that CP(2 r - i, 2r)] is decomposable

in~

.

Thus we consider

For s = 1 we take n = 1 and j = O.

(T,CP(2,1)).

different in this case. t=-

1 ~

X

CP(1)

The fixed point set is slightly

We take the coincidences

and we get S12~ CP(1) =

hence the fixed point set of (T,CP(2,1)) copies of CP(1)

and certainly 3[CP(1)]

Thus all the

P(
2 with

2n ' and a.~i we went along we checked ) ~ F''t

out the hypothesis

of (11.3) on the highest dimensional

of each fixed point set.

This part is decomposable

if k is even and indecomposable will follow with this selection.

if k is odd.

Thus

part

in (11.5)

-

12.

The ring~(Z

80

-

2)

We shall define a commutative ring structure with unit ~-- U in each bordism group ~ 2 n + l ( Z 2 ) for which ,~-U U 2n+l(Z2) is a ring homomorphism. A : ~2(n+l)+l(Z2) >~ Next we introduce the ring ~.U

2) = ninvlim -->~

~(Z

(~2n+l(Z2)

A

~(Z

2) is

introduced 9 In this section we shall regard

as the bordism

classes of pairs (T~B n) consisting of a weakly complex involution on a compact manifold with no fixed points in n ~ B . We recall again the exact sequence

>~_IU

0

>

~U If x e~2n+l(Z2)

U . ~

and /

e~

K = max(n~m) and k = min(n~m). is defined as follows. with ~ ( x ' )

xy

=

= x and

aK+l(

U(z21

>0.

U 2m+l(Z2) we set

The product x Y ~ 2 k + l ( Z

Choose x' e ~(y')

~

2n+2 and y' e

= y, then

(x'y')).

We must first show that xy is well defined.

Suppose that

2) 2m+2

-

81

-

U

~(X")

= X also,

then there is

Z e,~2n+2(Z 2)

Now

+

According

to (10.4), AK+I

Since K ~ m, ~(y")

= y, hence

examine

then A(xy)

xy

Lemma:

') + zAK+ I y.

= AK+I~(x'y

A similar argument

applies

= yx is well defined. of this product

to the Smith homomorphism. and if n ~ m

first the case n < m.

We choose

x' e

(10.3)

xy

it follows

(10.2)

Am+l("~(x'y'D)

: Am'~(x'y')

+

= Ay,

hence

A(~(y'D)

[X 2m] ei-LUOm~ with

+ [A,S I] Ix 2m] = y. =

(Z).

Thus ~ ( y ' D

+ ~(x'D

+ D [x2m]) = y and

[x2m]))

Am(x) [X 2m]

= xAy since m > n.

The case n > m is worked

out by considering

A(xy) = An+2('~(x'y'D + x'D[X2m]))

= An + l ' ~ ( x ' y ' )

+ An+l(x)[X 2m]

= xAy. An immediate

corollary

is

Z

2m-i

that A ~ ( y ' D )

to find

2n+2

(y') = &y E

with 2m

~(y'D)

if

Next we shall

If n < m then xy = xAy,

(x') = x and y' s

We apply

zy

= x&y.

Consider with

AK+I ~ ( x "zy ' )

y = O.

the relation

(12.1)

From

X" = x' + / ( z ) .

with

-

y)

=

O.

-

(12.2)

8 2

-

For e v e r y pa!K x , y

Lemma:

&(xy)

= &x&y.

Suppose n ~ m, then A(xy) = xAy, but then m - 1 < n, so xAy = AxAy.

The associativity of the product will be

proved with the aid of (12.1). (12.3)

~U If x~ y~ z e ~I 2-n +-

Lemma:

(Z) 2

then

(xy)z = x(yz). We choose x'

y' 9 z ' appropriately in

definition x(yz) = xAn+l~(y'z')~ xA

n+l~

(y'z')

in

=

2n+~

and ( 1 2 . 1 )

2n§215

By

2n+2 and by (12.1)~

(x'(y'z')).

Using t h e associativity

we h a v e

(y, z' ) ) -- "2n+2'8((•

9(x',,,')-: = (xy)z.

Thus the product in ~ U

2n+l(Z2) is associative9

The unit

element is [A, S 2n+l] for ~ D n+l = [A~S 2n+l ] and

~n+lg(x'Dn+1) = ~ (x'). Next we introduce the ring

z 2) -- ninvlim (fl 2n+l (z2) < --> ~ An element of X2n+l e ~

e,

Z 2) is a sequence

z~))

2 (n+l ~i (

f f Xon+1

where

U 2n+l(Z2) and AX2(n+l)+l = X2n+l for n > O.

-

83

-

Let ~LU[t]~ denote the ring of formal power series in one variable over the graded ring ~ L O ; that is~ an element of ~U

It] is a formal expression Z o [ x 2 r ] t r.

>~(Z 2) is

A homomorphism % ) : _ ~ U [ t ] follows.

rZfx2r [ ]t r we associate the sequence

To a given

~ 7 [ A , S 2(n-r)+l] [x2r].

X2n+l =

Clearly AX2(n+1)+l = X2n+l. Suppose t h a t induction

defined as

2n+l

e

Let us prove that ~

(Z 2) and suppose by way of

we have chosen [ x 2 r ] ~

X2k+l =

0 _< r < n so t h a t

k[A,s2(k-r)+l ] [X 2r]

7o

for all k < n.

is onto.

Now

A(X2n+l - Z T - I [ A , S

2(n-r)+l] [x2r]) = O,

so by (10.2) there is a [X 2n ] for which n

X2n+l = ~

[A,S

2(n-r)+l

] [X

2r

].

z-~o

In this way we may inductively find the coefficients of a power

series

which~

To show that

under

~

~

,

is

carried

into

c f X2n+l

is multiplicative we shall write out a

formula for the product in terms of the generating set {[A,S2n+l]I7

If

.

-

84

x = ~On[A,S 2(n-r)+l]

[X 2r]

y = Zon[A,S2(n-s)+l]

[y2S]

then we choose

x' =

Zo

n

D

-

n-r+l x2r z,n n-s+l [ ] and y' = D o

[y2S ]

so that

[X 2r]

x ,, y = zO2nD2n-t+l(Z

[y2S ])

r+s=t and A n+l~(x'y')

n 2 = 2 ( 9[A,S (n-t)+l] ( ~

[x2r] [y2S]

>

r+s=t

This formula for the coefficienl.of

(~

formula for that of t t in hence

~

[A~S2(n-t~ I] is also the

[x2r]t r

)(gj

)

[y2S]tS ,

is indeed multiplicative.

We want to characterize the kernel of

~

very precisely.

We shall geometrically construct the coefficients of a power series which generates the kernel of Let C numbers.

as a principal ideal.

denote the multiplicative group of non-zero complex L 2n We shall construct a sequence of actions (C*, )

for n ~ i.

We take L 2 = CP(1) and (C*~L 2) to be

t[z~w] = [z~tw].

Suppose

let W = C M C , -"~0,0~ from L 2 n x

~

(C*, L 2n )

and let L 2(n+l) be the space obtained

W by identifying

all t e C 9

has been defined, then

(x~(z~w)) with (tx~(tz~tw))

for

We shall denote a point in L 2(n+l) by ((x~(z~w))).

Of course L2(n+l) with fibre CP(1) 9

is an algebraic variety fibred over L 2n We define (C*, U 2(n+l) ) by

-

t((x,(z,w)))

85

=

-

((t-lx,(t-lz,.))).

We are really interested in the involutions

(T~L 2(n+l))

given by = ((x,(z,-w)))= Denote by Fn C

((-x,(-z,.))).

L 2n the fixed point set of (T~L 2n).

fixed point set Fn+ 1 is described as follows.

The

First F n c

by x

>

((x~(O~l)))~

but in Fn+ 1 there is also L 2 n C

by x

>

((x~(l~O))).

That is~ Fn+ 1 = F n L/ L 2n with

F n s L 2n = ~.

To compute ~

[T~ L 2(n+l) ] ~

L 2(n+l)

2(n+l) we

shall have to specify the normal bundle to En+ I. L2n c

Fn+ 1

The

Fn+ 1 has trivial normal bundle and in fact the

isomorphism of L 2 n ~

C onto this normal bundle is

(x,~)

.....> ((x,(l,~))).

x

((x,(O~l)))

>

NOw if L 2 n C

L 2(n+l) by

then the normal bundle to F n C L 2(n+l)

is the sum of the normal bundle of F n in L 2n with a trivial line bundle.

Inductively we see that Fn+ 1 = (2pts) together n

with a disjoint union ~ / L 2j. j=l a trivial

~

The normal bundle to L 2j is

(n-j+l)-plane bundle. 2(n+l)

[T,L

and since ~ = 2s2n+l

n+l ] = 2D

+Z

Now we can write

~n +

n-j+l I D

2j [e

O we arrive at

n [A,S 2(n-j)+l] 1

[L 2j] = O.

]

-

412.4)

Theorem:

86

-

The kernel of

~

: ~..U[t]

- - > ~ ~ ( Z 2)

is the principal ideal generated by 2 + 7~J[L2r]t r. ,

Let

a-J

--Zn [X2r]t r lie in this kernel.

1

Since

[A,S 1 ] [X ~ = 0 we can surely write [X~ : 2[Y ~] .

Suppose

we have found [yO]~...~[y2(k-l)] so that

[x 2n] : 2[u 2n] + for 0 < n < k.

,~on - l [ y 2 r ]

[L2(n-r)]

We can write

r+s:n

: [A'S1][x2k] + Z k-14 z o k - s - l [ A ~ ' s 2 ( k - s - r ) + l ] [ L 2 r ] ) [ y 2 S ] " Now ,~'7ok-s-l[A,s2(k-r-s)+l][L2r

] =-

[A,S1][L2(k-s)].

Thus we have [A,SI]([X 2k] - --~'Z;-I[y2s][L 24k-l-s) ]) = O. As in [4, 437.6)] there is a [y2k] e ~ U

[X 2k] = 2[y 2k] + ~

2k with

[y2s][L2r].

S +r= k Thus

we have shown inductively that

ZEx2r

[L2r])( ,~ [u

tr = c2 § 1

The following statement is the analog of [4, 446.3)] and is proved just the same way.

-

(12.5)

Theorem:

ideal) I C ~

87

-

The coefficients [L 2r] all lie in the

U, of those cobordism classes all of whose

Chern numbers are divisible b_y 2.

Furthermore I is generated

as a polynomial xi_9_q over Z b__y 2 and [L 2j+I-2] where j ~ i. Finally we shall define the stable ring homomorphism :

>

Z2).

If A r

we set k

,"O]nCA) = ~k .~(ADn+l) ~ . ~

U 2n+l

Clearly A ~n+l(A}

= ~

(Z). 2

CA) by definition) so we define n

:

>

(A

in

Z 2) by assigning to A the sequence Z ). 2

~'(AD) = ~ ' ( A ) .

The stability here refers to the fact We also observe that if n + 1 _< k then

~n(A) = Ak-n-Ic~A)) To see that ~ '

and in particular ~k_l(A) = ~ ( A ) . is multiplicative we note that in view

of the stability of ~ ' we need only consider a pair U A,B ~l')..2k and show that ~ k _ l (A)~) k-I (B) = ~ k-i ( ~ } .

We j u s t observed t h a t

definition

I~))k-iCA)~k-ICB) = (9)(A),~CB).

~ CA) 0CB) = ~ k6DCAB), but k = 2 k -

By

(k-l) - I,

thus ~ k ~(AB) =~k_I(AB). We shall also denote by 9 F, into ~ ( ~ ) . ~

(Z2 ) ~

' the induced homomorphism of

The composite homomorphism , ~>

(7:2) is a local invariant in the

-

sense

88

-

that it is really determined by the normal bundle to

the fixed point set of an element in d 1 , ( Z 2 ).

In the next

section we shall discuss the corresponding global invariant. We shall close this section by discussing the natural Jo filtration of ~ ( Z 2 ) . Note that for each k ~ 0 there is the ideal I ( k ) c ~

U [t] consisting of the power series of

=x3 the form ~ It]

and

p(k) C~(Z

k

[x2r]tr.

('1 I ( k ) k=O

This forms a decreasing filtration

= O.

Correspondingly

2) as follows.

An element %

we d e f i n e

ideals

E F(k)

if and

of

only if there is a power series fit) e I(k) for which

(fCt)) (12.0)

l_~fg(t) = Z o [ y 2 r ] t r is a power series

Lemma:

r-1

for which ~(g(t))

e I "(k) then there exist unigue cobordism

classes [Z~

2(k-l)] for which

[y2r] = ~ r[L2(r-s)][z2S] ifr O. [z2S]t s for which ~o o%~

. with degree +2 is defined

by C(x) = ? ( x - ~ ( x ) . l ) ) D

=~(x)D-~(x)D.

The useful feature of C is U (13.i) Lemma: ~F~ x s

, C(x) lies in the image

We wish to show that ~ ( x - e ( x ) . l ) D )

= oX[A'S1]e.~._.~ U[Z 2)e(x)[A'sl],

but this is (i0.I). This suggests the introduction~ for each n > O~ of the group ~2n(Z2)

X

2

which consists of all sequences

where x2r e (~U_zr(Z2 ) and r = FI

X2r+21ClXrlI*nlD*n+'Z*I*2ilDr+'i for r > n.

Let

-

Since C and~ a r e "

C (Z2) beUthe weak direct both ~ - m o d u l e

sum

2n ..IZ21. homomorphisms there is a

natural gradedY_(~U-module structure on ~ ( Z 2). onto homomorphism P : ~

(Z2)

>~(Z

A natural

2) is given by

-

P(X) = X2n for X ~ ~ 2 n(Z2). of degree O. (13.2)

93

-

This is an ~

U -module homomorphism

To introduce a product into ~(Z2) Lemma:

we shall need

For an_y ~ai___rx, y e ~ U ( z 2)

c(xy) = / ( x ) c ( y ) +

c(x)~(y)= c(x)/3(y) + ,(x)c(y). /

We simply write

c(•

: ~(•215

:~(•

/

+/~(•

2

= f~Cx) C(y) + CCx)sCy).

"If X r_. "'~"~2n (Z2)~ Y ~'~'2m ("Z2j ) we define X'Y = Z - c ~ 2 (n+m ) (Z 2) by Z2r = X2nY2(r_n) + X2(n+l)6(Y2(r_n_l)) +...+ X2(r_m)e(Y2m ) for each r > m + n.

We must show that~(Z2r+2)

Using (13.2) and the fact that C is an _ ~ - m o d u l e we

= C(Z2r). homomorphism

have

C(Z2r) -/~(X2n)C(Y2(r_n)) +

+ C(X2n)e(Y2(r_n))

C(X2(n§

=/(X2nY2(r+l_n)

+...+ C(x2(j_m))e(Y2m) + X2(n+l)e(Y2(r_n)) + X2(n+2)e(Y2(r+l_n))

+...+ X2(r+l_m)e(Y2m)) =~(Z2r+2)" The proof of the associativity of this product is a routine argument involving the rearranging of terms and it is left to

-

94

-

the reader. Since {j2(. z .n+m;. = XnYm we see that P :~

>~L.(z 2) is a ring homomorphism. Let us

(Z2)

define a ring homomorphism E

: ~ ( Z 2)

>~U[t]

by E(X)

=Z ~:(X2r)tr n

where X E G" (Z2). 2n (13.3)

We can now state the factoring result

Theorem: The diaq!am

U

~(z2)

~(z 2) ""/'~(~ u. (z2)~ is

commutative.

We recall that for X =

X2r n

/(~(X2r+2) =//~(X2n)Dr_n+ 1- ~

r g(x2i)Dr+l-i /W n

but ~(~(X2r+2) = O, therefore for r > n ~

(x2n)Dr-n+l) = Z

rn E(~i)[A'S2(r-i)+l]

-

95

-

~-U

in~2r+l(Z2). Now /~J/~(X2n)Dr-n+l) = ~rS/~(X2n )Dr-n+l and by stability we have ~r~(X2n

r r

)) = 2

)[A,S2(r-i) +I] 2i

n

for r > n~ but this is the assertion of the theorem. We may think

the =ompo,ition

m

global invariant corresponding to ~

I 21

/

,,

:~(Z

.

2

)

~

>~Z

2

).

We shall strain the reader's credulity by stating that~/~(Z2 ) is a commutative ring. We shall need some preliminary remarks before showing this. We recall the elements [T,L2n] of the last section.

We

let [T~L~ = [Z2~Z2] then we have the identity [T,L2n+2] = / [ T , We d e f i n e

a sequence

L 2n ]D + [L2n]D.

_v 2

by

[Z2,Z 2] = vo

[T,L2 ] = v 2 + ~(vo) [T,L2] r~LT,L2rj = v Thus V2r e

d

2r

+ 7~-i e(v2j) [T,L2(r-j)].

U

2r(Z2) is inductively defined.

Now/[T,L 2r+2] =?(V2r+2 ) + for e(v2j ~ [T 'e2(r+l-J) ] ?[T,L2r]D + [L2r]D ~(V2r)D + Z ; -I e(v2j?[T,L2(r-J)]D + [L2r]D

-

96

-

[T,L2(r-J)]D _ ~ [ T , L2(r+l-J) ] = - [L2(r-j)]D for

Since

0 ~ j ~ r-i

we have the formula

/ (v2r+2) = / ( V 2 r ) D - ~ r - i

e(v2j)[L2(r_J)]D- e(V2r/[T,L2] + [L2~D.

According to the defining equation r-i - [L2r]D + ~(V2r)D = - ~ o

e (v2i) [L2(r-J) ]D

therefore /

(v2r+2) ~

(v2r)D + e(V2r)D- e(V2r~[T, L2] "/(V2r }D - e(V2r)D

since

/~[T ~L2] = 2D.

f

%r

We have therefore shown that oo

,4")

We next assert that by E : aQ~-(Z2)

>~r~ [t] this

2

is carried into a generator of the kernel of ~ 9 This is seen as follows.

According to our definition e(v2j)tJ)(l + Z

or

1 oO e(v2j)tJ = (2 + Z 1

1

i

e~

oo

[L2r]tr)( 1 + Z 1 ~=

= 1 + (I + l c~

We shall use of P :~2n(Z2)

[L2r]tr)

[L2r]tr)-i 1

[L2r]tr)-i 1

v 2r ~ in the characterization of the kernel

--> ~ ( Z 2 ) .

Let I(n)c~/_U [t] be the ideal c~

of power series of the form S

[x2r]t r. n

-

(13.4)

Lemma:

kernel of ~ P l(n+l)CI(n)

97

-

There is an isomorphism of I(n) onto the

: ~2n(Z2)

-->

2n which when restricted to

is also _an isomorphism of l(n+l) with the kernel

> (~2n(Z2).

Of P : ~ 2 n ( Z 2 )

This isomorphism is described as follows. oo

2

To

x

{ [x2r]tr /we ass~ n

the sequence

2r

by n=r

X2n

= vo Ix 2n ]

X2n+2 = vo Ix 2n+2 ] + v2[X 2n]

X2r

=

Z O E'-n v2j [X2(r-J) /o

We must show that {X2rl ~ ~ (vo ) =

[Z2,Z 2] = 0 we have

]

2n(Z2).

Noting that

(X2r+2) =

(v2j)[X2

but ~ (v2")3 = C(v2j_2) for 1 _< j _< r+l-n~ hence

/ (X2r+2) = ~ 0 "-n C(v2i)[X 2 ( r - i ) ] By c o m p o s i n g t h i s we s e e i m m e d i a t e l y

: C(X2r).

homomorphism w i t h E : that

it

is

n(Z2)

a monomorphism.

that every element in the kernel o f ~

, >.~

[t]

We m u s t s e e

P has this form

~0

Suppose {Y2rl~lies

in the kernel ~

then

n

~

(Y2n ) = 0 and Y2n lies in the image of o~ :~-~2n'

>

n (z2);

],

-

98

-

U 2n with vo[X 2n] = Y2n"

that is, there is a unique IX2n] e ~

Suppose next that by way of induction we have found [X 2n ] . 9 9~ [X 2r ] with Y2r = Z o r - n

v2j IX2( r-j ) ] "

Consider/(Y2r+2

- ~

r+l-n v2j[X 2 ( r + l - j ) ])

1 =~(Y2r+2 ) - Er+l-n~(v2j)[X i

-~7

/

-n C(v2i)[X 2 ( r - ; ) ]

= C(Y2r)

U There is thus a unique IX 2r+2] ~-'~2r+2 Y2r+2 = vo[x2r+2] + ~ r + l - n J-J i l(n)

> ker(?P)

2(r+l-j)] = C(Y2r)

v2j

- C(Y2 r) with

[X 2(r+l-j)]

is also onto.

= O.

This proves

The second part of the lemma

is left for the reader. (13.5)

Theorem:

The r i n q ~ Z

Suppose X ~: ~2n(Z2), Y e r XY - YX , ker(P:~(n+m)(Z2) E(XY - YX) = 0 also.

2) is commutative.

then

> d u2(n+m) (Z2)) and

From (13.4) it follows that XY - ~

= O.

There is i n ~ ( Z 2) the deletion homomorphism, d, which is an ~ - m o d u l e

homomorphism of degree +2 given by

~

By definition p P d in ~ ( Z

Jn+l

= CP of course.

To relate d to the product

2) we define a ring homomorphism of degree 0

-

C

99

-

U

e :

(Z2) ~ >

.~

by e(

x2

) = e(X2n)

(13.6) d(X'Y)

Lemma:

E;..~2n.

_For X~ Y ~ ~

= X-d(Y)

+ d(X).e(Y)

(Z2)

= d(X).Y

+ e(X).d(Y).

xLO

Observe the similarity with (13.2), which will be the basis

of

Y =

2

our

argument.

Consider

first

X =

2r

and

then m

p

P(d(XY)

- X.d(Y)

- d(X)'e(Y))

= C(x2nY2n)

-/(X2n)C(Y2n)

- C(X2n)~:(Y2n)

= O.

We may thus apply the characterization of the kernel of p to the difference. E(d(XY)

- Xd(Y)

On the other hand we see easily that

- dCX)e(Y))

since

= 0 t h u s d(XY) = Xd(Y)

ker(E)~ker(/~P) = O. The f o r m u l a / f o l l o w s i m m e d i a t e l y from t h e b i l i n e a r i t y shall

use t h i s

characterize

P

deletion the kernel

for

+ d(X)e(Y)

a general

of the product.

homomorphism i n t h e n e x t o f E : ~ V ( Z 2) ~ > ~ U [ t ] .

section

pair We to

-

14.

Applications We denote by ~ "

induced by ~ '

: F,X

: F

> ~(Z W

> 2

i00

-

~

X

).

(Z2) the homomorphism From (13.3) we obtain

a commutative diagram

u[ G

(Z2)

{Z 2)

~

r(z2)

P

>

F~

According to (11.5) we can choose so t h a t the p(

k) e

.

with

4

k(Z2)

generates F~. as a polynomial

to

r i n g over Z2 generators

2k

and in a d d i t i o n { 6 ( ~ 2 k ) is a set Of \ 2 of / ~ . From the commutative diagram From (12.1l) we

conclude e a s i l y t h a t f ~ "

(14.l) F.

Theorem:

2)

.A. : F. ~ >

Z 2) is a monomorphism.

The stable boundary homomorphism is an embedding.

We just saw that ker(~') ~ 2F..

However,~Z

no 2-torsion, thus ker(~') = 2 ker(~'). graded polynomial algebra o v e r ~

2) has

Since F. is a

a non-trivial element in

F. cannot be infinitely divisible, therefore ker(~') = O.

-

(14.2)

I01

-

For each n ~ 0

Corqllary:

E:C2(Z)n

>.CLUE t ]

2

is a monomorphism. Suppose t h a t

X =

2

is

a construction

for

which

n

E(X)

= O.

By ( 1 3 . 3 ) ,

~'/~(X2n)

that~(X2n) lies in the / we showed t h a t ~ c o n t a i n s #P(x) tion

=p(X2n)

ideal

generated

E(X) = 0 i m p l i e s

of ~

X = O.

P found in

(13.4)

Thus we see t h a t

by

is

In

(ll.1)

therefore

we see t h a t a construction

(X2r)

in _ ~ .

The kernel of E : G ( Z 2

in

We

n homomorphism

shall use this~ along with the deletion Corollary:

by 1 - D.

If we turn tO the c h a r a c t e r i z a -

is uniquely determined

(14.3)

we see

no homogeneous elements~

= 0 already.

of the kernel

2n(Z2)

= O~ b u t by ( 1 4 . 2 )

to prove

)

the ideal

~-/=(X- dX,, X s g(Z2)and e(X)= 0 s ..~U). To see d(X.Y)

~is

an ideal we use (13.6) according

= XdY~ s i n c e

Suppose t h a t Y = Y(n)

+ Y(n+l)

e(X) = O~ t h e r e f o r e

E(Y) = O. +...+

0 ~ j ~ m.

By ( 1 4 . 2 ) ~

this

m.

length

We see t r i v i a l l y

Consider first that

E(Y(n)) = E(dY(n))~

X-Y - dX.Y = XY - d ( X . Y ) .

We w r i t e

Y(n+m) where Y ( n + j ) m ~ 1.

e(Y(n))

to which

The p r o o f m :

1.

is

by i n d u c t i o n

Thus Y : Y ( n )

= 0 ~ - ~ 2U n"

thus E(dY(n)

E~2(n+j)(Z

+ Y(n+l))

2) f o r on

+ Y(n+l).

But t h e n

= 0 and since

-

dY(n) + Y(n+l) ~

~2(n+l)(Z2)

102-

it follows that Y(n+l) = - dY(n)

as required. Now consider Y = Y(n) +...+ Y(n+m).

Again e(Y(n)) = 0

and we write

Y = Y(n)

- dY(n)

+ (dY(n)

+ Y(n+l))

+...+

Y(n+m).

Since E(Y(n) - dY(n)) = 0 we have E(dY(n) Thus Y is

+ Y(n+l)

expressed

as

+...+ the

Y ( n + m ) ) = O. sum o f an e l e m e n t

and an element in ker(E) of length m - I.

in

the

ideal

Thus inductively

every member of ker(E) has the form X - dX with e(X) = O. Certainly ~ C

ker(E).

-

15.

1 0 3 -

Dimension of the fixed point set The purpose of this section is a proof of (15.1)

Theorem:

~e_~% (T~M 2k) be a weakly complex

involution on a closed manifold for which every component of the fixed point set has dimension ~ 2n. ITeM 2k] lies~ modulo 2 ~ ( Z 2 )

If k > 3n~ then

~ in the ideal of

(Z 2)

which is 9enerated by [ ~ , Z 2] !~gether with [T,L2]. We shall need several lemmas of course. ~-

We begin with

U

(15.2)

Lemma: If x E ~ 2n+l(Z 2) has order 2 then there U is a [X 2n] E ~ 2 n for which x = [A~SI][X2n]. The proof is by induction over n. [X 2n-2] with Ax = [x2n-2][A~SI].

That is, we can find

But then

A(x - [x2n-2][A~S3]) = O r so by (IO.I) we can write x = [x2n-2][A, S3] + [y2n][A,$1].

Now

~

[T~ L4] = 0 implies ~hat 2[A,S 3] + [L2][A,S i] = 0

since~[T,L

4] = 2D2 + [L2]D and ~ =

O.

Now 0 = 2x = 2[x2n-2][A,S 3] = - [x2n-2][L2][A, Sl]. But [x2n-2][L 2] = 2[Z 2n] as in [4, (37.6) ], and since [L 2] = [CP(1)] is a generator of ~

we have [X 2n-2] = 2[Y 2n-2]

and x = [x2n-2][A,S3] + [y2n][A, SI] = ([y2n] _ [y2n-2][L2])[A, SI].

-

104

-

This is applied now to show (15.3)~ Lemma:

I f f ( Y 2 k ) = ~O~-I 2[y 2r]D k-r for

there are cobordism classes [A~k] , [B 2 k - 2 ]

Y2k e ~2k(Z2), U and an X2k e ~ 2 k

(Z2) for which

Y2k = [Z2'Z2][A2k] + [T'L2][B2k-2] + 2X2k

We

@I Z= k-l[y2r]Dk-r)

has order 2, thus by (15.2)

/~( ~ok-l[y2r]D k-r) = [B2k-2][A,S I] and there is X2k e ~(X2n)

k(Z2) with

= Z k-I [y2r]Dk-r _ [B2k-2]D.

Since~(T~L 2) = 2D we have

~

(2X2k + [B2k-2][T,L2]) = ; ( Y 2 k ).

Because we know the kernel of lexplicitly Y2k = 2X2k

we can write

+ [Z2'Z2][A2k] + [T'L2][B2k-2]"

We can now compute the kernel of O

2k (Z2

(

2k"

Suppose that

(Z2k) = 2A2k ,U

then there is a 2k such that

2k

(z), 2

together

with

[yO]~...,[y2k-2]

-

y

(a2k) = A2k -

105

-

Zo_l[y2r]Dk_ r

and ~(Y2k

) ~(Z2k)-

y(a2k

) = Zok-12[y2r]D k-r.

If we apply (15.3) we find I " %

(15.4)

Lemma:

I__f~(z2k) s

then there is

/ with cobordism classes [A2k]~ [B2k_2 ] X2k s ~ U 2k(Z2) %oge_~ther for which Z2k = 2X2k + [ A 2 k ] [ z 2 , z 2] + [ B 2 k - 2 ] [ T , L 2 ] .

We may regard F. /2 F.

as obtained from

.

as the quotient of the ideal generated by (I - D).

. Just

as in (ii.i) this ideal has no homogeneous elements~ so the kernel of 2~-" ~k(Z2

) p>

Fj-* coincides with the kernel

u

(z2)

/296

-->

u2k .

Next we need a more p r e c i s e

" :

F.It

(Z2) 9 We had filtered

filtration F 2n' "~" while on ~ ( Z filtration ~(k). (15.5)

statement

Lemma:

F2n /'~ (r

Fwt.

about

by the increasing

2 ) there is the decreasing

The two filtrations are related as follows. If k > 3n then

= O.

This is a Bondman type of result.

A Z2-base for F~2n

-

106

-

is given by the product terms 9

il ) p ( ~ 2 ) . . . p ( P(~--~4

. 3n ~ 4 n ) P ( ~ 6 31)...p(~) 4 n + 2 )

with wt = 2i I + 4i 2 +...2ni

If we apply ~ ~(wt

n

+ 23

1

+...+ 2n3

n

2n 9

" to this product the result lies in

+ Jl +'''+ Jn )' but not in ~ ( w t + Jl +'''+ Jn + i).

With n fixed we see that wt _< 2n implies 31 +...+ 3 n < n, hence wt + Jl +'" "+ 3n < 2n + n = 3n. is taken on by ~ 6

)n.

We can now prove (15.1). into

(k) and k > 3n.

We know that [T,M 2k] projects

But p[T,M 2k] e F2n , hence

/~"p([T,M2k]) = 0 and p[T,M 2k] = O. ~([T,M2k])~6~u /

This maximum value

This implies

k and we apply (15.4). 6

Corollary:

Modulo 2 1 ~ U the cobordism class [M2k] lies

in the ideal qenerated by [CP(1)].

-

16.

107

-

U A local invariant for .-~ ( Z 2 ) , In our discussion of the Atiyah-Bott fixed point theorem

we directly define the global Trace invariant and then found that a suitable local invariant was forced on us.

Now we shall

reverse orientation and for weakly complex involutions start with a local invariant. (16.1)

Inteqrability:

%~

If Lc :

> Z(i/2) is the

unique rinq homomorphism determined by th___eeconditions Lc[7

> CP(n)] = 1 -(I/2) n+l , n > O

%

(2)

~v2m.J

U

for [

~~

and 2m ---

LC([~

2s > M2r][V 2m] = (Lc[

> M2r])(td[V2m]) ~ U

then Lc is integral valued on the image of H:j

m>~

U ,.

.(Z 2)

To prove this result we shall construct a set of elements [T,V(n+l,k)], n ~ 0 r k ~ O r which will generate the image of /~as

a graded algebra over ~ u

Lc~[Tk,V(n+l,k)]

and for which

= i.

We fix n ~ 0 and construct a sequence of actions (C*,V(n+l,k)) as follows.

First let (C*,V(n+l,0)) be the

action of C* on CP(n+l) given by t([zl~...~Zn+2]) = [Zl~...,Zn+l, tZn+2].

Suppose now that

(C*,W(n,k)) has been inductively defined, then set W = C X C

% 90,0~

-

108

-

and introduce (C*,V(n+l,k)x W) by t(x,(z,w)) = (tx,(tz,tw)) and denote by V(n+l,k+l) the quotient manifold(variety) (V(n+l,k)X W)/C*. ((x,(z,w)))

If ((x,(z,w))) ~ V(n+l,k+l) then

> [z,w] is a fibring of V(n+l,k+l) over CP(1)

with fibre V(n+l~k).

t((x,(z,w)))

Finally we take (C*,V(n+l,k+l)) to be

= ((x,(z,tw)))

= ((t-lx,(t-lz,w))).

The involution (Tk+l,V(n+l,k+l)) will be ((x,(z,w)))

>

((x,(z, -w))) = (Tk(X) , (-z,w))).

analyse the fixed point set of Tk+ I.

Let us

Note that if w = 0

then we have a submanifold of fixed points

((x,(z,O)))

= ((xz -t , (1,o)))

: V(n+l,k)c V(n+l,k+l).

If w ~ O, however, then ((x,(z,w))) = ((x,(z, -w))) : ((Tk(X)l(-zlw))) if and only if z = 0 and x e F(Tk)CV(n+I,k).

Since

((x,(O,w))) = ((xw-l,(o,l))) we may identify this part of the fixed point set with F(Tk). union

of

To sum up, F(Tk+ I) is the disjoint

f F(TM) ~ ~ (x, (0, ll))~ C

V(n~l,k~l) with a copy

of

J

V(n+l,k)

{((x,(l,O)))~ c V(n+l,k+l).

There are, then, two

embeddings of V(n+l,k) X C into V(n+l,k+l) given by

(x,w)

> ((x, (1,w)))

(x,z)

> ((x,(z,1)))

The first shows that as a part of F(T

), V(n+l,k) is embedded k+l with a t~'ivial normal bundle, and the second shows us that the normal bundle of the F(Tk ) part of F(Tk+I) is the sum of the normal bundle to F(Tk)cV(n+l,k)

with a trivial line bundle.

-

109-

I f F = F(To) = CP(n) ~ ; ~ p t J = C P ( n + l )

= V(n+l,O),

then

inductively F(Tk+ 1) = F ~ (Uo k V ( n + l , j ) ) . Under ~

~

U : ~2(n+k+2)

(Z2)

[Tk+l,V(n+l,k+l)] = ([~

~ ">

U 2(n+k+2)

we

have

>CP(n)]+Dn+I)D k+l + Zo(Dk-j+l[v(n+l,j)].

Here we think of D as the line bundle over a point. Lc(D) = Lc[~

Since

>CP(o)] = 1/2 we have

Lc(~[Tk+l,V(n+l,k+l)]

= (i/2) k+l + Zo:(i/2)k-j+l(td[V(n+l,j)]).

/ If we can show that the Todd genus of [V(n+l,j)] = i then we will have k Lc~[Tk+l,V(n+l,k+l)]

= (i/2)k+l + Z

(i/2)i = 1 also. 1

To see that V(n+l,k) has Todd genus i we consider an alternative description of this algebraic variety.

Let

Z(n+l)1 be the complement of the origin in C n+2 and on Z(n+l) X W k let (C~) k+l act by

(tl,'", tk§

(Zl'w1)''" (Zk'Wk))

=( (tl~l'tl ~ ' ' ' ' ' t 2 t l ~ n + 2 (tk+IZk~tk+lWk))

)' (t2zl't2t3wl)' (t3z2' t3t4w2)'''"

then V(n+l~k) is the quotient variety

(Z(n+l) X wk)/(C~) k+l.

We can fibre V(n+l~k) over our old

-

ii0

-

friend L 2k from section 12 with fibre CP(n+l) (((~i'''''

~ n + 2 ) ' (Zl'Wl) ' " " " ' (Zk'Wk)))

In fact V(n+l~k)

bundle.

line bundle

~

>L 2k with a trivial

The cohomology ring H~(L2kIc)

22

(n+l)-plane

>L 2k for himself.

was extensively discussed in

and it is generated by forms of type (i~i).

a well known argument H*(CP((n+I)C@~);C)

By

is generated by

forms of type (i,i) also~ hence 1 = alg. genus genus

--> L 2k

associated to the sum of a certain

The reader may discover

[4~ (42.8)],

> ( ((Zl'Wl) ' " " " ' (Zk'Wk)))"

is the total space of p : CP((n+I)C~)%))

the bundle with fibre CP(n+l) canonical

by

(V(n+l~k)) = Todd

(V(n+l,k)). Now we must show that t h e / ~ E T k ~ V ( n + l ~ k ) ]

image o f / ~

as an algebra over

.

Let

generate the C

f

denote the algebra which is generated by all / ~ [T ,V(n+l~k)] F~uU / k As the polynomial ring generators of ~/~. we use D together with D(n+l) = ~ [ Y o ' V ( n + l , O ) ]

= [~

Next consider all sequences ~ ( @2)

=

r, the length of r

>CP(n)]

+ D n+l for n > O.

(J) = (I ~ i I ~ ... ~ i r) and set , and n(o2)

=

i I + .. .+ i r .

We

put D(~),s)

U = D(i ) ... D(i ) D s ~ ~/~( i r 2(n(~2 )+s)

and we understand that if (4) is empty we put D(@,s) = D s. Ranging over all ~0 and s these D( ~9 ~s) form a homogeneous

-

_~

-module base of

already.

iii

-

We see t h a t

t h e D(u;,O)

s~

Now

[Tk,V(ir,S)]

= D(ir)D s +

ZoS-IDS-J[v(ir,J)]

and so for s > 0

/ where LI.;= (1 < i 1 J . . . length

of cd u n t i l

combination

J it_l).

we f i n d

that,

modulo ~

over _ ~ U o f the D ( ~

D(C~s)

are an

applies

t o any element i n ~

element in

i)

-module base f o r

T h i s can be r e s t a t e d

over

We c o n t i n u e t o reduce t h e , D(~J,s)

= Dim 1 J i J s.

U .

as f o l l o w s .

by D c

As t h e

the same remark

Modulo~

lies in the graded polynomial generate

is a linear

2

9

every

subalgebra

In this subring~

however~

there are still some elements which lie in the image o f ~ . We can complete the proof that ~ that ( T k , V ( l , k ) ) (16.2)

:

Lemma:

(T, L 2(k+l))

= im(~)~

if we note

for k > 0 and show

l_ff x g 2n

(Z) 2

%s an element for ---

which

~

(x) = ~ - i

[x2r]D2(n_r)

then the____re . are cobordism _classes x = [Z2,Z2]Ey2n] +

.~f-1

[Y •] ~ .... [y2n] ___ for which [y2r][T,L2(n-r)].

-

Since ~ ( x )

= 0 we have

1 1 2 -

Zo

n-I

[x2r][A'S

2(n-r)-i

Applying A n-I to this equation we find that [X~

but then by [4, (37.6)], [X ~ = 2[Y~

- [Y~

=

Since

Z

n - l ( x 2r] - [Y~

--

1

Again 7 n-I ([X2r] - [Y~ 1 ~n-2 to see ([X2] - [Y~ Ix2] - [Y~

I] = O r

~ n-i [L2r]D2(n-r)

[T~L 2n] = 2D +

~x

] = O.

: 2[y2].

[T'L2n-2] = 2Dn-I + Z

S2(n-r)-l] = 0 so we apply = 0 and we write Since n-i [L2r]D2(n-r)-l] we have

2 - [y ][T,L 2n] - [y2][T,U2n-2]) = 7

n-I ([X2r] - [y~

] - [y2][L2r-2])D n-r.

2 We simply continue until we have [yO],...,[y2(n-l)]" with (x - 2

n-i [y2r ][T,L2(n-r) ]) = O.

As we know the kernel

O

of/~we

can write X = [Z2,Z2][Y 2hI + Z o n-I [y2r][T,L2(n-r)]. s

Thus any element of the form in the image of ~

~n-i

also lies in ~

,

[x2r]D2~n-r) which lies therefore ~

= imp)

as asserted. Actually we have really constructed a generating set for ~(Z

2) as an algebra over ~-~ U.

We need [Z2,Z2]~I and all

-

113

-

[Tk,V(n+l,k)] with n ~ O, k ~ O. We can also compute L c [ ~

> M 2n] for a complex k-plane

bundle over a closed weakly complex manifold in terms of the characteristic classes of the bundle and the Chern classes of M 2n.

To the conjugate bundle

{

L> N 2n we associate the

total K-theory Chern class [7~ (13.1)] expressed as C(~)

= 1 + ~l(~)t

2(~)

+...+ ~ k ( ~ ) t k.

i § ~l(~)t

§247

Let

~i(~)t i

§

denote the K-theory dual Chern class of { ; that is~ C_(~)D(~)

= i.

If ~

is a line bundle then

c_(~) = 1 § ( ~ -

1)t : i - (i - {

)t

thus ~o

for a line bundle.

If x e ~(M2n;z)

is the characteristic

cohomology characteristic class of the line bundle then oo

ch(D(~))

=

~

(i - e-X)it i = ~ O n ( l

- e-X)it i.

In general D( is now a k-plane bundle

we may express its total cohomology

Chern class in factored form b y ~)k ( l_ + X r 1 oO 1

and then

-

and ch(ds (~))

114

-

is the resulting coefficient of t s expressed

in terms of the cohomology invariants of ~ .

Note that

ch(ds( ~ )) e ~ n s H2r(M2n;Q)so that i f s > ~ then ch(ds(~))

= 0 and --S d (~)

is a torsion element in K(M 2n)

We also recall [7, p. 75-78] that

T-I(M2n) =

Yl . . . . Yn (l-e-Yl)...(l-e-Yn)

e H*(M2n;Q)

t?",-, n

where '~11 (l+Y r) is the factored form of the total Chern l

class of M 2n. (16.3)

Theorem:

I_ff ~ - - >

M 2n i__%sa complex k-plane

bundle over a closed weakly co__qm_plexmanifold,

Lc[~

> M2n] = 7 7 ( 1 / 2 ) k + i < r

We point out that

integer,

then

) ) T - l ( M 2 n ) ' ~ 2 n >"

~ c h ( d i ( ~ ))T-I(M2n), O-2n~ is an

thus we can i n t r o d u c e ~ c : ~o

> Z(1/2) by

We wish to show that O~c = Lc. the routine verification that ~ c

We leave to the reader

is multiplicative.

It is

then sufficient to show that C~C[ f

> CP(n)] = 1-(i/2) n+l.

If x e H2(CP(n);Z)

is the characteristic class of ? > CP(n) -x i -~ -x n+l then ch(_di(~)) = (I - e ) , and T(CP(n)) = (x/(l - e )) 9

-

115-

Thus ~h(d i(~)T(CP(n)) '

0 - 2 > = <xiCx/(l-e -x))n+l-i o- >= td CP(n-i) = 1 n ' 2n

for 0 ~ i ~ n, hence

~c[ ~

> CP(n)] = zon(I/2)

i+l

-- 1 -

(i/2) n+l

which establishes (16.3). It would seem most probable that if (T~Vn) is an analytic map of period 2 on a variety then Lc[T,V n] =Ion(-l)itr(T ,hO'i).

References

i.

M. F. Atiyah and R. Bott~ Notes on the Lefschetz Fixed Point Theorem for Elliptic Complexes, mimeographes notes, Harvard University, 94 pp. (1964).

.

G. E. Br@don, Sheaf Theory, McGraw Hill series in higher math. 272 pp. {1967)

.

J. M. Boardman, On Manifolds with Involution, Bull. Amer. Math. Soc. vol. 73, p. 136-138 (1967). (Also Unoriented bordism and cobordism which is to appear).

.

.

P. E. Conner and E. E. Floyd, Differentiable Periodic Maps, vol. 33~ Ergebnisse der Mathematik und ihre Grenzgebiete~ Springer-Verlag, 148 pp. ~964). , Torsion in SU-Bordism, Math. Soc. 72 pp. (1966).

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