Diffeomorphism of total spaces and equivalence of bundles

Abstract

Let E 1 and E 2 be the total spaces of smooth, oriented vector bundles of rank k over the n-sphere. We show that if E 1 and E 2 are diffeomorphic, with orientation preserved, then the bundles are smoothly equivalent up to orientation of the base whenever k>[( n+1)/2]+1. With an additional hypothesis, the same conclusion holds when the base is an arbitrary closed, oriented n-manifold. Furthermore, if the base manifold is a homotopy n-sphere and if one of the bundles has a nowhere-zero cross-section, then the oriented bundles are smoothly equivalent up to orientation of the base in the case where k=[( n+1)/2]+1 as well. The latter statement is false if k<[( n+1)/2]+1, as several counterexamples illustrate. We show that each of these examples is an open manifold E admitting a complete metric of nonnegative sectional curvature for which the zero section of the nontrivial vector bundle, a standard sphere, is not the image of a soul in the sense of Cheeger and Gromoll under any diffeomorphism of E.