On discontinuities of cocycles in cohomology theories for topological groups

TL;DR

This paper explores the topological structure of cohomology classes in Moore's measurable cohomology for locally compact groups, establishing conditions under which it aligns with Segal's cohomology theory.

Contribution

It demonstrates that all classes in Moore's cohomology have representatives with enhanced topological properties and constructs an isomorphism with Segal's cohomology for specific modules.

Findings

All classes have representatives with additional topological structure.

A comparison map between Moore's and Segal's cohomology is constructed.

The comparison map is an isomorphism for certain modules.

Abstract

This paper studies Moore's measurable cohomology theory for locally compact groups and Polish modules. An elementary dimension-shifting argument is used to show that all classes in that theory have representatives with considerable extra topological structure beyond measurability. Using this, for certain target modules one can also construct a direct comparison map with a different cohomology theory for topological groups defined by Segal, and show that this map is an isomorphism.