Overview of group cohomology

A good introduction to the history of the cohomology of groups can be found in Mac Lane [171]. He traces the history back to the works of Hurewicz (1936) on aspherical spaces and Hopf (1942) on the relationship between the fundamental group and the second homology group of a space. We shall not dwell here on the historical development, but refer the reader to Mac Lane's article for further information and comments.

The purpose of this chapter is to give a survey of group cohomology and how it is connected to various other parts of mathematics, and in particular to topological and algebraic K-theory. In the first few sections, where we provide several definitions of group cohomology and show how they are related, we give fairly complete proofs. Later on, we lapse into description and give enough references so that the interested reader may chase up the proofs (we trust that the reader will also excuse some forward references to Chapter 3 on spectral sequences during the later sections of this chapter). We hope that this romp through large chunks of mathematics will be taken as a joy ride, and not as an indigestible pill.

The first approach to group cohomology, which we have already examined in some detail in Volume I, is the algebraic approach.