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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Group extensions and cohomology for locally compact groups. III

Author: Calvin C. Moore
Journal: Trans. Amer. Math. Soc. 221 (1976), 1-33
MSC: Primary 22D05; Secondary 22D10, 22D30
MathSciNet review: 0414775
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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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Article copyright: © Copyright 1976 American Mathematical Society