Introduction

Summary

If G is an arbitrary finite group (more generally a finitely presented possibly infinite group) it is an easy exercise in combinatorial topology to construct a finite 2-dimensional simplicial complex with fundamental group isomorphic to G. By attaching simplexes of dimension ≥ 3 in a systematic way it is possible to embed this 2-complex in a larger complex K(G, 1) without changing the fundamental group, but choosing K to have universal covering space homotopy equivalent to a point. One may then define the cohomology groups H^k(G, ℤ) of the discrete group G to be the cohomology groups of the space K. This definition is independent of the topological model chosen, and may indeed be copied algebraically with the coefficients ℤ replaced by some (left) ℤG-module A. It is clear that the graded ring {H^k(G, ℤ), k ≥ 0} is an important invariant of the group, indeed if the homomorphism ϕ: G₁ → G₂ of finite groups induces an isomorphism (ϕ^*: H^*(G₂, ℤ) → H^*(G₁, ℤ), then the groups G₁ and G₂ are isomorphic. However although the literature contains one or two striking applications of group cohomology, for example to the construction of infinite class field towers and to the study of outer automorphisms of p-groups, its systematic use as a tool has been held up by a lack of calculations for specific groups.