Configured locally smooth cohomology and $\mathbb{Q}/\mathbb{Z}$-torsion in $H_3$ of diffeomorphism groups

TL;DR

This paper introduces configured group cohomology, a new geometric framework for constructing explicit cocycles on diffeomorphism groups, revealing the presence of $ ext{Q}/ ext{Z}$-torsion in their third homology groups.

Contribution

It develops configured group cohomology from geometric fillings, providing explicit cocycles and demonstrating $ ext{Q}/ ext{Z}$-torsion in the third homology of certain diffeomorphism groups.

Findings

Explicit locally smooth $ ext{R}/ ext{Z}$-valued 3-cocycles constructed

Third homology groups contain $ ext{Q}/ ext{Z}$-torsion subgroup

Framework applicable to groups preserving geometric structures

Abstract

We introduce configured group cohomology, a variant of locally smooth cohomology built from well-configured tuples and geometric fillings. This framework yields explicit locally smooth $\R/\Z$-valued $3$-cocycles of Chern--Simons type on diffeomorphism groups preserving geometric structures. As an application we show that, for several such groups, the third group homology contains a subgroup isomorphic to $\Q/\Z$.