Chern--Simons-type $3$-cocycles and $\mathbb{Q}/\mathbb{Z}$-torsion in the third homology of discrete diffeomorphism groups

Abstract

We prove that the third integral group homology of both the volume-preserving diffeomorphism group and the strict contactomorphism group of the standard \(3\)-sphere contains \((\mathbb Q/\mathbb Z)^2\). The proof constructs two independent locally smooth Chern--Simons-type \(3\)-classes from the volume form and the standard framing of \(S^3\). A torsion-controlled local integration method and comparison with the primitive Cheeger--Chern--Simons class also detect \(\mathbb Q/\mathbb Z\) for several spherical space forms and symplectic projective manifolds.