The cohomology of groups

Abstract

This chapter discusses the cohomology of groups. The cohomology of groups is one of those branches of mathematics that is regarded as a tool for other areas of study. It has applications in homotopy theory, class field theory, representation theory, and K -theory. The clearest indication of the connection between algebraic topology and cohomology of groups is expressed in the theorem of Kan and Thurston [KaT]. For infinite groups, the cohomology theory is very much a part of group theory itself. Groups are often classified according to their homological properties, such as cohomological dimension. By contrast, the cohomology of finite groups is much more closely associated to modular and integral representation theory. The homology and cohomology of G are independent of the choice of the projective resolution. In addition, the homology and cohomology are functional in G, the group variable. However, the properties of this functionality are much more subtle. The restriction and inflation maps are also involved.