On the topology of stationary black holes
P. T. Chrusciel, R. M. Wald
TL;DR
The paper addresses whether stationary black holes must have spherical horizon topology. It leverages the Friedman-Schleich-Witt topological censorship theorem under the null energy condition to show that the domain of outer communication $\mathcal{J}$ is simply connected by constructing a global time function and identifying $\mathcal{J}\cong \mathbb{R}\times {\cal C'}$. With additional hypotheses on a compact horizon cross-section $K$ and a compactness condition on ${\cal C'}\setminus {\cal C}_{\text{ext}}$, it further proves that each component of $K$ is homeomorphic to a sphere, implying spherical topology of horizon cross-sections under the stated conditions. These results strengthen the black hole uniqueness program, remove reliance on horizon analyticity, and support harmonic-map approaches to rotation problems through topological constraints on $\mathcal{J}$ and horizon cross-sections.
Abstract
We prove that the domain of outer communication of a stationary, globally hyperbolic spacetime satisfying the null energy condition must be simply connected. Under suitable additional hypotheses, this implies, in particular, that each connected component of a cross-section of the event horizon of a stationary black hole must have spherical topology.