Group cocycles on the volume-preserving diffeomorphism group

TL;DR

This paper constructs two types of group cocycles on the volume-preserving diffeomorphism group, linking one to the Euler class of flat sphere bundles, advancing understanding of geometric structures in infinite-dimensional groups.

Contribution

It introduces new group cocycles on the volume-preserving diffeomorphism group and connects one to the Euler class of flat sphere bundles, revealing novel geometric insights.

Findings

Construction of two group cocycles on the diffeomorphism group

Identification of one cocycle with the Euler class of flat sphere bundles

Enhanced understanding of geometric structures in infinite-dimensional groups

Abstract

We construct two kinds of group cocycles on the volume-preserving diffeomorphism group. We show that, for the volume-preserving diffeomorphism group of the sphere, one of the cocycles gives the Euler class of flat sphere bundles.