A cocycle model for topological and Lie group cohomology

Abstract

We propose a unified framework in which the different constructions of cohomology groups for topological and Lie groups can all be treated on an equal footing. 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.