Continuity properties of measurable group cohomology

TL;DR

This paper investigates the continuity and regularization properties of measurable group cohomology for locally compact groups, establishing conditions under which it aligns with other cohomology theories and analyzing its topological features.

Contribution

It introduces a dimension-shifting technique to simplify the description of measurable cohomology and proves its equivalence with continuous cohomology for certain modules, also exploring topological properties.

Findings

Cohomology groups agree with continuous cocycles for Fréchet modules.

Cohomology vanishes in positive degrees for compact groups with Fréchet target modules.

Measurable-cochains cohomology groups are continuous under inverse and direct limits.

Abstract

A version of group cohomology for locally compact groups and Polish modules has previously been developed using a bar resolution restricted to measurable cochains. That theory was shown to enjoy analogs of most of the standard algebraic properties of group cohomology, but various analytic features of those cohomology groups were only partially understood. This paper re-examines some of those issues. At its heart is a simple dimension-shifting argument which enables one to `regularize' measurable cocycles, leading to some simplifications in the description of the cohomology groups. A range of consequences are then derived from this argument. First, we prove that for target modules that are Fr\'echet spaces, the cohomology groups agree with those defined using continuous cocycles, and hence they vanish in positive degrees when the acting group is compact. Using this, we then show that for Fr\'echet, discrete or toral modules the cohomology groups are continuous under forming inverse limits of compact base groups, and also under forming direct limits of discrete target modules. Lastly, these results together enable us to establish various circumstances under which the measurable-cochains cohomology groups coincide with others defined using sheaves on a semi-simplicial space associated to the underlying group, or sheaves on a classifying space for that group. We also prove in some cases that the natural quotient topologies on the measurable-cochains cohomology groups are Hausdorff.