Equivariant Cohomology Theories
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Glen E. Bredon University of California, Berkeley
Equivariant Cohomology Theories 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 (photoatat, microfilm and/or microcard) or by other procedure without wrlttel:l permission from Springer Verlag. C by Springer-Verlag Berlin' Heidelberg 1967. Library of Congress Catalog Card Number 67 - 25284 Printed in Germany. Title No. 7354.
Preface These notes constitute the lecture notes to a series of lectures which the author gave at Berkeley in the spring of
1966. Our central objective is to provide machinery for the study of the set [[X1Y]] of equivariant homotopy classes of equivariant maps from the G-space X to the G-space Y (with base points fixed by G).
(For various reasons we restrict our atten-
tion to the case in which G is a finite group.)
An important
tool for this study is equivariant cohomology theory.
It is
immediately seen, however, that the classical equivariant cohomology theory is quite inadequate for the task. Our first object then is to develop an "equivariant classical cohomology theory" (as opposed to "classical equivariant cohomology theory") which is readily computable and which, for example, allows the development of an equivariant obstruction theory.
This is done in Chapter I and the obstruction
theory is considered in Chapter II.
Our cohomology theory
includes the classical theory as a special case. An approximation to [[X;Y]] lim[[SnX;Sn y ] ] which forms a group.
is the stable object If Y is a sphere, with a
given G-action, this leads to the stable equivariant cohomotopy groups of a G-space X.
These form an "equivariant generalized
cohomology theory" and such theories are considered briefly in Chapter IV and related to the equivariant classical cohomology. When X and Yare both spheres with (standard) involutions on them, the groups lim[[SnX;Sn y ] ]
are analogues of the stable
homotopy groups of spheres and constitute the case of greatest interest to us at present.
It is in fact this case which inspired the general theory expounded
in these notes.
Originally we intended to include a fifth chapter in these
notes which would apply the general theory to this special case.
However, the
special case has since expanded in length and in importance to the extent that we have decided to publish our results on this topic separately.
An
outline of these results has appeared in our research announcement "Equivariant stable stems" in Bull. Amer. Math. Soc. 73 (1967) 269-273. The main results in the present notes have been announced in "Equivariant cohomology theories," Bull. Amer. Math. Soc. 73 (1967) 266-268. Although we have restricted our attention, ,in these notes, to the case of finite groups it will be apparent that the theory goes through for cellular actions of discrete groups and this fact was incorporated in our research announcement (loc. cit.).
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