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

In four dimensions the topology of the event horizon of an asymptoticallyflat stationary black hole is uniquely determined to be the two-sphere $S^2$.We consider the topology of event horizons in higher dimensions. First, wereconsider Hawking's theorem and show that the integrated Ricci scalarcurvature with respect to the induced metric on the event horizon is positivealso in higher dimensions. Using this and Thurston's geometric typesclassification of three-manifolds, we find that the only possible geometrictypes of event horizons in five dimensions are $S^3$ and $S^2 \times S^1$. Insix dimensions we use the requirement that the horizon is cobordant to afour-sphere (topological censorship), Friedman's classification of topologicalfour-manifolds and Donaldson's results on smooth four-manifolds, and show thatsimply connected event horizons are homeomorphic to $S^4$ or $S^2\times S^2$.We find allowed non-simply connected event horizons $S^3\times S^1$ and$S^2\times \Sigma_g$, and event horizons with finite non-abelian first homotopygroup, whose universal cover is $S^4$. Finally, following Smale's results wediscuss the classification in dimensions higher than six.Comment: 12 pages, minor edits 27/09/0