McDuff's secondary class and the Euler class of foliated sphere bundles

TL;DR

This paper constructs a cohomology class related to McDuff's secondary class in volume-preserving diffeomorphisms, demonstrating its transgression to the Euler class of foliated sphere bundles, extending known results to higher dimensions.

Contribution

It introduces a new cohomology class for volume-preserving diffeomorphisms of acyclic manifolds with sphere boundary and links it to the Euler class of foliated sphere bundles.

Findings

Constructed a cohomology class related to McDuff's secondary class.

Proved the class transgresses to the Euler class of foliated sphere bundles.

Extended the analogy of the Calabi invariant to higher dimensions.

Abstract

Tsuboi proved that the Calabi invariant of the closed disk transgresses to the Euler class of foliated circle bundles and suggested looking for its higher-dimensional analog. In this paper, we construct a cohomology class of the group of volume-preserving diffeomorphisms of a real-cohomologically acyclic manifold with sphere boundary, which is closely related to McDuff's secondary class, and prove that this cohomology class transgresses to the Euler class of foliated sphere bundles.