Counting negatively curved manifolds
by
Igor Belegradek
University of Maryland, College Park

We prove various finiteness results for the class M_{a, b, \pi, n} of n-manifolds with fundamental groups isomorphic to \pi that can be given complete Riemannian metrics of sectional curvatures within [a, b] where a <= b < 0. Here are sample results.

• If M_{a, b, \pi, n} contains a closed manifold of dimension >= 3, the class M_{a, b, \pi, n} is precompact in the Lipschitz topology, in particular M_{a, b, \pi, n} falls into finitely many diffeomorphism classes.
• If \pi is a word-hyperbolic group, the class M_{a, b, \pi, n} breaks into finitely many tangential homotopy types.
• If \pi is a word-hyperbolic group, then for any n there exists \epsilon = \epsilon(n, \pi) > 0 such that the tangent bundle of any manifold in the class M_{-1-\epsilon, -1, \pi, n} has zero rational Pontrjagin classes.
• If M is a closed negatively curved manifold of dimension >= 3, then, up to diffeomorphism, only finitely many manifolds from the class M_{a, b, \pi1(M), n} are total spaces of vector bundles over M. (Due to a theorem of M. Anderson, the total space of any vector bundle over M admits a complete metric of pinched negative sectional curvature.)

Date received: March 29, 1998