Chapter 18 - Cohomology of Groups

Abstract

This chapter discusses the cohomology of groups. The cohomology of groups is one of the crossroads of mathematics. It has its origins in the representation theory, class field theory, and algebraic topology. The theory of cohomology of groups in degrees higher than two really begins with a theorem in algebraic topology. The links between algebraic topology and the group theory lead naturally to the idea of cohomological finiteness conditions. It is useful to compare this with the idea of a finiteness condition in the abstract group theory. The latter notion, prevalent in the work of Philip Hall and other influential groups theorists of some 30 years ago, has had a powerful influence on the study of abstract infinite groups. Furthermore, one advantage of cohomology over homology is the existence of cup products and Yoneda products. Yoneda products are defined in terms of composition of maps.