Affine transverse foliations in sphere bundles

Abstract

Let~$S^{n-1}\to E\to M^n$ be an oriented sphere bundle supporting a {smooth} affine transverse foliation. We give a new and elementary proof of the following fact: if the fundamental group {${\pi }_1(M^n)$} is amenable, then the Euler number of the bundle vanishes. As a by-product, we give an upper bound for the Euler number of the bundle in the general case, that is, for non-amenable {${\pi }_1(M^n)$}.