Symmetric continuous cohomology of topological groups

In this paper, we introduce a symmetric continuous cohomology of topological groups. This is obtained by topologizing a recent construction due to Staic, where a symmetric cohomology of abstract groups is constructed. We give a characterization of topological group extensions that correspond to elements of the second symmetric continuous cohomology. We also show that the symmetric continuous cohomology of a profinite group with coefficients in a discrete module is equal to the direct limit of the symmetric cohomology of finite groups. In the end, we also define symmetric smooth cohomology of Lie groups and prove similar results.

Keywords

continuous cohomology, group extension, Lie group, profinite group, symmetric cohomology, topological group

2010 Mathematics Subject Classification

Primary 20J06. Secondary 54H11, 57T10.

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Published 1 May 2013