### Abstract

We prove that, under certain conditions, the topology of the event horizon of a four-dimensional asymptotically flat black-hole spacetime must be a 2-sphere. No stationarity assumption is made. However, in order for the theorem to apply, the horizon topology must be unchanging for long enough to admit a certain kind of cross section. We expect this condition is generically satisfied if the topology is unchanging for much longer than the light-crossing time of the black hole. More precisely, let M be a four-dimensional asymptotically flat spacetime satisfying the averaged null energy condition, and suppose that the domain of outer communication to the future of a cut K of is globally hyperbolic. Suppose further that a Cauchy surface for is a topological 3-manifold with compact boundary in M, and is a compact submanifold of with spherical boundary in (and possibly other boundary components in ). Then we prove that the homology group must be finite. This implies that either consists of a disjoint union of 2-spheres, or is non-orientable and contains a projective plane. Furthermore, , and will be a cross section of the horizon as long as no generator of becomes a generator of . In this case, if is orientable, the horizon cross section must consist of a disjoint union of 2-spheres.

Original language | English (US) |
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Article number | 012 |

Pages (from-to) | 1055-1061 |

Number of pages | 7 |

Journal | Classical and Quantum Gravity |

Volume | 12 |

Issue number | 4 |

DOIs | |

State | Published - 1995 |

### ASJC Scopus subject areas

- Physics and Astronomy (miscellaneous)

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## Cite this

*Classical and Quantum Gravity*,

*12*(4), 1055-1061. [012]. https://doi.org/10.1088/0264-9381/12/4/012