Affine transverse foliations in sphere bundles

TL;DR

This paper proves that the Euler number of an oriented sphere bundle over a closed manifold with an amenable fundamental group is zero if the bundle admits a smooth transverse affine foliation, using an elementary proof.

Contribution

It establishes a new elementary proof linking the existence of a smooth transverse affine foliation to the vanishing of the Euler number in sphere bundles.

Findings

Euler number of the bundle vanishes under the given conditions

Existence of smooth transverse affine foliation implies topological constraints

Provides an elementary proof method for this vanishing result

Abstract

Let~$S^{n-1}\rightarrow E \rightarrow M^n$ be an oriented sphere bundle over an oriented closed manifold with amenable fundamental group. We provide an elementary proof that the Euler number of the bundle vanishes whenever the bundle admits a smooth transverse affine foliation.