Lectures on the Action of a Finite Group
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73 Pierre E. Conner University of Virginia, Charlottesville
1968
Lectures on the Action of a Finite Group
Springer-Verlag Berlin· Heidelberg· New York
Lecture Notes in Mathematics A collection of informal reports and seminars Edited by A. Dold, Heidelberg and B. Eckmann, ZUrich
73 Pierre E. Conner University of Virginia, Charlottesville
1968
Lectures on the Action of a Finite Group
Springer-Verlag Berlin· Heidelberg· New York
All rights reserved. No part of this book may be translated or reproduced in any form without written permission from Springer Verlag. © by Springer-Verlag Berlin' Heidelberg 1968 Library of Congress Catalog Card Number 68-57939 Printed in Germany. Title No. 3679
CONTENTS
Introduction........
.. .. ..
1
Chapter I Line Bundles with Operators •••••••••••••••• 1. A co-ordinate system with operators ••..•••••••• 2. Covariant stacks ••••.•••••••••••••••••••••••••• 3. Covariant stacks with operators ••••••••••••••••
19 19
29 32 35
4. An example
5. Two spectral sequences •.•.••••.••.•.••.••.••••• 43 6. Sheaves with operators ••••...•.•.•..•.•••.•..•. 7. Nerves of coverings ••••••...••.••.••••••••••••• 8. The sheaves p: (Q/iT) (lr;x) 9. Topological examples ••••.•••••••.••.••...•••••• 10. Holomorphic line bundles with operators •••••••• 11. Maps of prime period ••••••••.••••••••••••••••••
47
56 60
66
69 80
Chapter II Orientation Preserving Involutions ••••.••• 82 1. The bordism group An (2k) ••.••.••••.•.•••..•.••• 82
87 93
2. Self-intersection
3. The structure of
a
The ring *(Z2) •.•••••••••..•••.•..••••.•.••• 99 5. A trace invariant •••••••••••••.•••.•.•••••..••• 107
4.
6. Examples References
119 '
123
INTRODUCTION
These notes are based upon a series of lectures given, by the kind invitation of Professor Albrecht Dold, at the Mathematics Institute of the University of Heidelberg.
The
first chapter is aimed at a demonstration of the principle of borrowing ideas and techniques from the various branches of modern algebraic topology and using them to attack a problem in transformation groups.
As far as we know, this
principle was formally stated first by Borel, who followed it in a most elegant fashion. We simply contrive a problem and then set about it.
In
defense of the question we should point out that there has been recently a considerable interest, with profitable results, in the application of vector bundles with operators to the study of finite transformation groups. refer to
L:
together with a left action of
as a group of complex
linear bundle maps covering the action of'T'on X. tensor product these form an abelian is the action of
h (x
'J')
(hX'l!"
Via the The unit
on the product bundle given by
h
eW.
We ask how
l
(IT.'X)
is determined.
- 2 -
If we think of the case of line bundles alone, without operators, we immediately recall that such is determined uniquely by its Chern class in H2(X;Z).
Our first idea then
is to find a suitable replacement, a kind of equivariant cohomology group, in which we can associate to every line bundle
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