An Introduction to the Cohomology of Groups (original) (raw)

H n (G, M) where n = 0, 1, 2, 3,. . ., called the nth homology and cohomology of G with coefficients in M. To understand this we need to know what a representation of G is. It is the same thing as ZG-module, but for this we need to know what the group ring ZG is, so some preparation is required. The homology and cohomology groups may be defined topologically and also algebraically. We will not do much with the topological definition, but to say something about it consider the following result: THEOREM (Hurewicz 1936). Let X be a path-connected space with π n X = 0 for all n ≥ 2 (such X is called 'aspherical'). Then X is determined up to homotopy by π 1 (x). If G = π 1 (X) for some aspherical space X we call X an Eilenberg-MacLane space K(G, 1), or (if the group is discrete) the classifying space BG. (It classifies principal G-bundles, whatever they are.) If an aspherical space X is locally path connected the universal cover˜X is contractible and X = ˜ X/G. Also H n (X) and H n (X) depend only on π 1 (X). If G = π 1 (X) we may thus define H n (G, Z) = H n (X) and H n (G, Z) = H n (X) and because X is determined up to homotopy equivalence the definition does not depend on X. As an example we could take X to be d loops joined together at a point. Then π 1 (X) = F d is free on d generators and π n (X) = 0 for n ≥ 2. Thus according to the above definition H n (F d , Z) = Z if n = 0 Z d if n = 1 0 otherwise. Also, the universal cover of X is the tree on which F d acts freely, and it is contractible. The theorem of Hurewicz tells us what the group cohomology is if there happens to be an aspherical space with the right fundamental group, but it does not say that there always is such a space.