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

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 $\Q/\Z$.