Intersection Pairing in Hyperbolic Manifolds, Vector Bundles and Characteristic Classes

We present some constraints on the intersection pairing in hyperbolic manifolds. We also show that, given a closed negatively curved manifold M of dimension at least three, only nitely many rank k oriented vector bundles over M can admit complete hyperbolic metrics. x0. Introduction Hyperbolic manifolds are locally symmetric Riemannian manifolds of negative sectional curvature. Any complete hyperbolic manifold is a quotient of a rank one symmetric space by a discrete groups of isometries that acts freely. The main result is the following 0.1. Theorem. Let Y be a nite connected CW-complex such that 1 (Y) is torsion free group which is not virtually nilpotent. Assume that 1 (Y) has no nontrivial decomposition as an amalgamated product or an HNN extension over a virtually nilpotent group. Then, given a nonnegative integer n and homology classes ] 2 H m (Y) and ] 2 H n?m (Y), there exists K > 0 such that, (1) for any continuous map f : Y ! N of Y into an oriented complete hyperbolic n-manifold N that induces an isomorphism of fundamental groups, and (2) for any embedding f : Y ! N of Y into an oriented complete hyperbolic n-manifold N that induces a monomorphism of fundamental groups. the intersection number of the cycles f and f in N satisses jhf ; f ij K. The assumptions on 1 (Y) are used to invoke a compactness theorem due to Rips that the space of conjugacy classes of faithful discrete representations of 1 (Y) into the group of isometries of a rank one symmetric space is compact. For example, these assumptions hold if Y is a closed aspherical manifold of dimension at least three where 1 (Y) is word-hyperbolic. In particular, we get the following 0.2. Corrolary. Let M be a closed oriented K(?; 1) manifold of dimension n 3 such that ? is word-hyperbolic. Then there is a nite subset F H k (M) that contains the Euler class of any 1-incompressible topological embedding of M into a complete oriented hyperbolic (n + k)-manifold. Thus, F contains the Euler class of any oriented vector bundle R k ! E ! M if E admits a complete F-hyperbolic structure. The corollary generalizes a result of Kapovich Kap2] who proved that, given a closed oriented negatively curved manifold M of dimension n 3, there is a nite subset F H n (M) that contains the …