Euler characteristic of closed hyperbolic 4-manifolds.

 September 22, 2006 mathematics

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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 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 [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).

[1] J.G. Ratcliffe, S.T. Tschantz, The volume spectrum of hyperbolic 4-manifolds, Experiment. Math. 9 (2000) 101–125.

[2] J.G. Ratcliffe, The geometry of hyperbolic manifolds of dimension at least 4, in: Non-Euclidean Geometries, Springer, New York, 2006: pp. 269–286.

[3] S.-s. Chern, On curvature and characteristic classes of a Riemann manifold, Abh. Math. Sem. Univ. Hamburg. 20 (1955) 117–126.