Group extensions and cohomology for locally compact groups. III

Moore, Calvin C.

doi:10.1090/S0002-9947-1976-0414775-X

Group extensions and cohomology for locally compact groups. III
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Trans. Amer. Math. Soc. 221 (1976), 1-33 Request permission

Abstract:

We shall define and develop the properties of cohomology groups ${H^n}(G,A)$ 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 ${H^n}(G,A)$ are themselves topological groups in a natural way and we investigate this additional structure.

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