On the Topology of Black Hole Event Horizons in Higher Dimensions

TL;DR

Using the extension of Hawking's theorem, the paper shows $\int_{M_H} \hat{\mathcal{R}}\, d\hat{S} > 0$ on the horizon under the dominant energy condition, and couples this with topological censorship to constrain horizon topology in higher dimensions. In five dimensions, horizons are limited to topology $S^3$ or $S^2\times S^1$; in six dimensions, simply connected horizons are $S^4$ or $S^2\times S^2$, with several non-simply connected possibilities. These conclusions are derived from combining Thurston geometry classifications with Friedman's and Donaldson's results on four-manifolds, and from cobordism considerations (h-cobordism/spin cobordism) in higher dimensions. Overall, the work narrows the landscape of higher-dimensional black-hole horizons and ties gravitational topology to global geometric constraints.

Abstract

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 $S^3$ and $S^2 \times S^1$. 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 $S^4$ or $S^2\times S^2$. We find allowed non-simply connected event horizons $S^3\times S^1$ and $S^2\times Σ_g$, and event horizons with finite non-abelian first homotopy group, whose universal cover is $S^4$. Finally, following Smale's results we discuss the classification in dimensions higher than six.