A generalization of Hawking's black hole topology theorem to higher dimensions
Gregory J. Galloway, Richard Schoen
TL;DR
This paper addresses the higher-dimensional topology of black hole horizons by generalizing Hawking's result to show that horizon cross sections in higher dimensions are of positive Yamabe type, not necessarily spherical. It develops a stability-operator/ conformation-deformation framework under the dominant energy condition, deriving a nonnegative scalar curvature on horizon cross sections without relying on the Jang equation. In five dimensions the results imply strong topological restrictions: horizon cross sections must be built from spherical spaces and $S^2 \times S^1$, ruling out $K(\pi,1)$ factors. Overall, the work provides a PSC-based obstruction perspective that aligns with known 5D horizon examples and extends topology constraints for higher-dimensional black holes.
Abstract
Hawking's theorem on the topology of black holes asserts that cross sections of the event horizon in 4-dimensional asymptotically flat stationary black hole spacetimes obeying the dominant energy condition are topologically 2-spheres. This conclusion extends to outer apparent horizons in spacetimes that are not necessarily stationary. In this paper we obtain a natural generalization of Hawking's results to higher dimensions by showing that cross sections of the event horizon (in the stationary case) and outer apparent horizons (in the general case) are of positive Yamabe type, i.e., admit metrics of positive scalar curvature. This implies many well-known restrictions on the topology, and is consistent with recent examples of five dimensional stationary black hole spacetimes with horizon topology $S^2 \times S^1$. The proof is inspired by previous work of Schoen and Yau on the existence of solutions to the Jang equation (but does not make direct use of that equation).