A Cocycle Model for Topological and Lie Group Cohomology

TL;DR

This paper introduces a unified cocycle model for topological and Lie group cohomology, simplifying their study and revealing connections to Lie algebra cohomology and applications like the string group.

Contribution

It presents a new unified framework for topological and Lie group cohomology using cocycle models, bridging different constructions and applications.

Findings

Unified cocycle model for topological and Lie group cohomology

Connection established between group cohomology and Lie algebra cohomology

Application to constructing cohomology classes for the string group

Abstract

We propose a unified framework in which the different constructions of cohomology groups for topological and Lie groups can all be treated on equal footings. In particular, we show that the cohomology of "locally continuous" cochains (respectively "locally smooth" in the case of Lie groups) fits into this framework, which provides an easily accessible cocycle model for topological and Lie group cohomology. We illustrate the use of this unified framework and the relation between the different models in various applications. This includes the construction of cohomology classes characterizing the string group and a direct connection to Lie algebra cohomology.