On Cohomology theory for topological groups

TL;DR

This paper introduces new cohomology theories for topological and Lie groups, focusing on measurable cochains with local continuity, and explores their properties and applications to group extensions.

Contribution

It develops novel cohomology frameworks based on locally continuous measurable cochains and links them to group extension classifications.

Findings

Second cohomology groups classify locally split extensions.

Cohomology theories are constructed for topological and Lie groups.

Basic properties of these cohomology theories are established.

Abstract

We construct some new cohomology theories for topological groups and Lie groups and study some of its basic properties. For example, we introduce a cohomology theory based on measurable cochains which are continuous in a neighbourhood of identity. We show that if G and A are locally compact and second countable, then the second cohomology group based on locally continuous measurable cochains as above parametrizes the collection of locally split extensions of G by A.