Euler characteristic of closed hyperbolic 4-manifolds.
September 22, 2006 mathematics
By the Gauss-Bonnet theorem, the volume of a hyperbolic 4-manifold is proportional to its Euler characteristic. There are examples, constructed explicitly in  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  observes that there are restrictions on the Euler characteristic that a closed hyperbolic 4-manifold may possess. In particular, it is shown in  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 .
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).
 J.G. Ratcliffe, S.T. Tschantz, The volume spectrum of hyperbolic 4-manifolds, Experiment. Math. 9 (2000) 101–125. http://projecteuclid.org/getRecord?id=euclid.em/1046889595.
 J.G. Ratcliffe, The geometry of hyperbolic manifolds of dimension at least 4, in: Non-Euclidean Geometries, Springer, New York, 2006: pp. 269–286.
 S.-s. Chern, On curvature and characteristic classes of a Riemann manifold, Abh. Math. Sem. Univ. Hamburg. 20 (1955) 117–126.