On the topology of stationary black hole event horizons in higher dimensions

In four dimensions the topology of the event horizon of an asymptotically flat stationary black hole is uniquely determined to be the two-sphere S ². We consider the topology of event horizons in higher dimensions. First, we reconsider Hawking's theorem and show that the integrated Ricci scalar curvature with respect to the induced metric on the event horizon is positive also in higher dimensions. Using this and Thurston's geometric types classification of three-manifolds, we find that the only possible geometric types of event horizons in five dimensions are S³ and S² × S¹. In six dimensions we use the requirement that the horizon is cobordant to a four-sphere (topological censorship), Friedman's classification of topological four-manifolds and Donaldson's results on smooth four-manifolds, and show that simply connected event horizons are homeomorphic to S⁴ or S² × S². We show that the non-simply connected event horizons S³ × S¹ and S² × Σ_g and event horizons with finite non-abelian first homotopy group whose universal cover is S⁴, are possible. Finally, we discuss the classification in dimensions higher than six.