Euler characteristic of closed hyperbolic 4-manifolds.
2006-09-22 20:18:50 +0000 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
MR1758804
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
MR2191252
observes that there are restrictions on the Euler characteristic that a closed hyperbolic 4-manifold may possess. In particular, it is shown in
MR0075647
that the Pontrjagin numbers of a hyperbolic manifold $M$ vanish. But the signature $\sigma(M)$ is a rational linear combination of those Pontrjagin numbers, so $\sigma(M) = 0$. And by Poincare duality, $\chi(M) \equiv \sigma(M) \pmod 2$, so $\chi(M)$ is even. A natural question to ask is: does there exist a hyperbolic 4-manifold $M$ with $\chi(M) = 2$? Now if such an $M$ also had $H_1(M) \neq 0$, 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:
MR1758804
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).