Symmetric continuous cohomology of topological groups

TL;DR

This paper introduces a symmetric continuous cohomology theory for topological groups, extending previous algebraic constructions to topological and Lie group contexts, and characterizes related group extensions.

Contribution

It develops a topological version of symmetric cohomology, characterizes extensions, and connects cohomology of profinite groups with finite group cohomology, also defining symmetric smooth cohomology for Lie groups.

Findings

Symmetric continuous cohomology characterizes topological group extensions.

Cohomology of profinite groups equals the direct limit of finite group cohomologies.

Similar results hold for symmetric smooth cohomology of Lie groups.

Abstract

In this paper, we introduce a symmetric continuous cohomology of topological groups. This is obtained by topologizing a recent construction due to Staic (J. Algebra 322 (2009), 1360-1378), 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.