TOPOLOGY OF EVENT HORIZONS AND TOPOLOGICAL CENSORSHIP

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 C-K to the future of a cut K of I- is globally hyperbolic. Suppose further that a Cauchy surface Sigma for C-K is a topological 3-manifold with compact boundary partial derivative Sigma in M, and Sigma' is a compact submanifold of <(Sigma)over tilde> with spherical boundary in Sigma (and possibly other boundary components in M\Sigma). Then we prove that the homology group H-1(Sigma', Z) must be finite. This implies that either partial derivative Sigma' consists of a disjoint union of 2-spheres, or Sigma' is non-orientable and partial derivative Sigma' contains a projective plane. Furthermore, partial derivative Sigma = partial derivative I+[K]boolean AND partial derivative I-[I+], and partial derivative Sigma will be a cross section of the horizon as long as no generator of partial derivative I+[K] becomes a generator of partial derivative I-[I+]. In this case, if Sigma is orientable, the horizon cross section must consist of a disjoint union of 2-spheres.