Topological Group Cohomology with Loop Contractible Coefficients

TL;DR

This paper establishes isomorphisms between different types of topological group cohomology under certain conditions and applies these results to finite-dimensional Lie groups, enhancing understanding of their cohomological properties.

Contribution

It proves isomorphisms between continuous and locally continuous cohomology for topological groups with loop contractible coefficients, extending to compactly generated and Lie groups.

Findings

Isomorphism between continuous and locally continuous cohomology groups.

Extension of results to compactly generated and Lie groups.

Application to various concepts of group cohomology for finite-dimensional Lie groups.

Abstract

We show that for topological groups and loop contractible coefficients the cohomology groups of continuous group cochains and of group cochains that are continuous on some identity neighbourhood are isomorphic. Moreover, we show a similar statement for compactly generated groups and Lie groups holds and apply our results to different concepts of group cohomology for finite-dimensional Lie groups.