Pinching, Pontrjagin classes, and negatively curved vector bundles

TL;DR

This paper establishes finiteness results for negatively curved manifolds with fixed fundamental groups and explores properties of their vector bundle structures, including Pontrjagin classes, in the context of differential geometry.

Contribution

It proves finiteness of certain negatively curved manifolds as total spaces of vector bundles and shows conditions under which their tangent bundles have trivial Pontrjagin classes.

Findings

Finitely many manifolds are vector bundle total spaces over a given negatively curved manifold.

Existence of a positive epsilon ensuring zero rational Pontrjagin classes for tangent bundles.

Results apply to manifolds with hyperbolic fundamental groups.

Abstract

We prove several finiteness results for the class $M_{a,b,G,n}$ of $n$-manifolds that have fundamental groups isomorphic to $G$ and that can be given complete Riemannian metrics of sectional curvatures within $[a,b]$ where $a\le b<0$. In particular, if $M$ is a closed negatively curved manifold of dimension at least three, then only finitely many manifolds in the class $M_{a,b,\pi_1(M), n}$ are total spaces of vector bundles over $M$. Furthermore, given a word-hyperbolic group $G$ and an integer $n$ there exists a positive $\epsilon=\epsilon(n,G)$ such that the tangent bundle of any manifold in the class $M_{-1-\epsilon, -1, G, n}$ has zero rational Pontrjagin classes.