Continuous Cocycles Endowed with Point-Open Topology

TL;DR

This paper studies the topology of continuous cocycles in topological groups, establishing conditions under which cohomology with inverse limit coefficients can be computed as inverse limits of cohomologies with simpler coefficients.

Contribution

It introduces a topological framework for cocycles and proves inverse limit theorems for nonabelian and abelian cohomology in this setting.

Findings

Identifies a natural isomorphism between cohomology of inverse limits and inverse limits of cohomologies for certain topological groups.

Establishes a topological structure on the set of cocycles that facilitates cohomological analysis.

Provides conditions under which cohomology groups commute with inverse limits in the context of compact and Hausdorff groups.

Abstract

Given a topological group $ G $ and a Hausdorff topological group $ A $ on which $ G $ acts continuously and compatibly with the group operation of $ A $, we study the set of continuous cocycles of $ G $ with value in $ A $. This set is a function space and can be endowed with several topologies. By imposing a suitable function space topology on the set of cocycles of $ G $ with value in $ A $, we propose a topological study of this set, and we prove, as our first main result, that if $ A $ is a compact group having a presentation as an inverse limit of compact and Hausdorff topological groups $ A_{r} $, for $ r $ in a directed poset $ R $, on which $ G $ acts continuously and compatibly with the group operation of $ A_{r} $ and equivariantly with respect to the transition maps, then one has a natural identification between the first nonabelian cohomology set of $ G $ with coefficients in the inverse limit $ A $ and the inverse limit of first nonabelian cohomology sets of $ G $ with coefficients in $ A_{r} $. Furthermore, we prove, as our second main result, that if $ G $ is compact and Hausdorff, and $ A $ is abelian - therefore one can define cohomology groups for all $ n\geq1 $ - then under a certain condition on $ A_{r} $ one has a natural identification between cohomology group with coefficients in the inverse limit $ A $ and the inverse limit of cohomology groups with coefficients in $ A_{r} $, for all $ n\geq1 $.