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 an upper bound for the Euler number of the bundle. We also give a new and elementary proof of the following fact: if the fundamental group $\ddot{I}{€}_1(M^n)$ is amenable, then the Euler number of the bundle vanishes.