G-Complexes and Equivariant Cohomology

Abstract

Let G be a finite group and X a space on which G acts. In this chapter we will describe a cohomological analysis of X which involves H*(G; F _p ) in a fundamental way. First developed by Borei and then by Quillen, this approach is the natural generalization of classical Smith Theory. After reviewing the basic constructions and a few examples, we will apply these techniques to certain complexes defined from subgroups of a group G, first introduced by K. Brown and D. Quillen. Using results due to Brown and P. Webb we will show how these complexes provide a systematic method for approaching the cohomology of simple groups which will be discussed later on. One of the aims of this chapter is to expose the reader to the important part played by group cohomology in the theory of finite transformation groups. By no means is this a complete account; in addition it requires a different background than the previous chapters have. We recommend the texts [AP], [Brel] as excellent sources for those wanting to learn more about group actions.