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

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.