Euler characteristic of closed hyperbolic 4-manifolds.

By the Gauss-Bonnet theorem, the volume of a hyperbolic 4-manifold is proportional to its Euler characteristic. There are examples, constructed explicitly in [1] of hyperbolic 4-manifolds with every positive integer as their Euler characteristic. These examples are non-compact (with five or six cusps, I believe). But [2] observes that there are restrictions on the Euler characteristic that a closed hyperbolic 4-manifold may possess. In particular, it is shown in [3] that the Pontrjagin numbers of a hyperbolic manifold vanish. But the signature is a rational linear combination of those Pontrjagin numbers, so . And by Poincare duality, , so is even. A natural question to ask is: does there exist a hyperbolic 4-manifold with ? Now if such an also had , we would know the volume spectrum of closed hyperbolic 4-manifolds.

This certainly seems to parallel the case for 2-manifolds: all negative integers are the Euler characteristic of a hyperbolic 2-manifold, and all even negative integers are the Euler characteristic of a closed hyperbolic 2-manifold.

The vanishing of Pontrjagin numbers for hyperbolic manifolds also holds for pinched negative curvature under some conditions [1].

It is also a fact that the Stiefel-Whitney numbers vanish for a closed hyperbolic manifold (and the vanishing of the top Stiefel-Whitney class is the same thing as having even Euler characteristic).