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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Group extensions and cohomology for locally compact groups. III
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by Calvin C. Moore PDF
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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Additional Information
  • © Copyright 1976 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 221 (1976), 1-33
  • MSC: Primary 22D05; Secondary 22D10, 22D30
  • DOI: https://doi.org/10.1090/S0002-9947-1976-0414775-X
  • MathSciNet review: 0414775