Group Extensions and Cohomology for Locally Compact Groups. III

We shall define and develop the properties of cohomology groups Hⁿ(G,A) which can be associated to a pair (G, A) where G is a separable locally compact group operating as a topological transformation group of automorphisms on the polonais abelian group A. This work extends the results in [29] and [30], and these groups are to be viewed as analogues of the Eilenberg-Mac Lane groups for discrete G and A. Our cohomology groups in dimension one are classes of continuous crossed homomorphisms, and in dimension two classify topological group extensions of G by A. We characterize our cohomology groups in all dimensions axiomatically, and show that two different cochain complexes can be used to construct them. We define induced modules and prove a version of Shapiro’s lemma which includes as a special case the Mackey imprimitivity theorem. We show that the abelian groups Hⁿ(G,A) are themselves topological groups in a natural way and we investigate this additional structure.