Lectures on the Cohomology of Groups

The cohomology theory of groups arose from both topological and algebraic sources. The starting point for the topological aspect of the theory was a 1936 paper by Hurewicz [7], in which he introduced aspherical spaces. These are spaces X such that πn(X) = 0 for n 6= 1. (Hurewicz had introduced higher homotopy groups just one year earlier, and he was now trying to understand the spaces with the simplest possible higher homotopy groups.) Hurewicz proved that such an X is determined up to homotopy equivalence by its fundamental group π := π1(X). Thus homotopy invariants of X can be thought of as invariants of the group π. Examples of such invariants include homology, cohomology, and the Euler characteristic. Thus we can define