Pinching, Pontrjagin classes, and negatively curved vector bundles

Abstract

We prove several finiteness results for the class script M signa,b,π,n of n-manifolds that have fundamental groups isomorphic to π and that can be given complete Riemannian metrics of sectional curvatures within [a, b] where a ≤ b < 0. In particular, if M is a closed negatively curved manifold of dimension at least three, then only finitely many manifolds in the class script M signa,b,πa(M),n are total spaces of vector bundles over M. Furthermore, given a word-hyperbolic group π and an integer n there exists a positive ε = ε(n, π) such that the tangent bundle of any manifold in the class script M sign-1-ε,-1,π,n has zero rational Pontrjagin classes. © Springer-Verlag 2001.