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46 P. E. Conner University of Virginia, Charlottesville

Seminar on Periodic Maps

1967

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, microfllm and/or microcard) or by other procedure without written permission from Springer Verlag. © by Springer-Verlag Berlin· Heidelberg 1967. Library of Congress Catalog Card Number 67-31229 Printed in Germany. Tide No.736

Contents

.................................... 1 The index . · •• 19 Real representations ••••• . ..... ........ .. . 25 Extension and reduction •• . .. . ........ . ..... • ••• 30 The functor (J-"; (H;1;F ).. . ............ • • ••••• • • • 33Index as a bordism invariant ••••••••.••••••••••••••••• 36 The trace invariant •••••••••••••• ............ • •• 41 Introduction •••

1.

2.

3. 4.

5. 6.

7. The local invariant....

.

44

8. Periodic maps on a Riemann surfaces. •••• • •• 50 9. The Atiyah-Bott formula ••••.•••••••••••••••••••••••••• 60 10. Weakly complex involutions....... ••••••••• • ••••• 62 11. The ring F..... .............. . 68 12. The ring -{(,( Z2) • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• 80 13. The of constructions ••••••••••••••••••••••• 92 14. Applications ••••••••••••••••••••••• • .100 15. Dimension of the fixed point set. ..103 16. A local invariant for (Z2) ••••••.•• • •••••••• • • 107

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

References •••••••••••••••••••••

.................... • .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 (5((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 applied to

1n

f0!m to which it has £een

of 2dd Erime power

[1, 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)

=

(w,v).

Because we want our definition to resemble as closely as possible that of [1, p. 21, Ex. 6] we adopt the following approach.

We find that the set of linear transformations

D : (H,V)

(H,V), commuting with the action of Hand

satisfying (1)

(v,Dw) - (Dv,w)

(2)

(v,Dv)

(3)

2 D = Id

> 0, v 10

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+ (D

V_ into the ±l eigenspaces of

D and,so receive representations (H,V+) and (H,V_).

We let

index (H,V,(.,.)) b