TOPOLOGY OF EVENT HORIZONS AND TOPOLOGICAL CENSORSHIP

We prove that, under certain conditions, the topology of the event hor izon of a four-dimensional asymptotically flat black-hole spacetime mu st be a 2-sphere. No stationarity assumption is made. However, in orde r for the theorem to apply, the horizon topology must be unchanging fo r 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 preci sely, let M be a four-dimensional asymptotically flat spacetime satisf ying the averaged null energy condition, and suppose that the domain o f outer communication C-K to the future of a cut K of I- is globally h yperbolic. Suppose further that a Cauchy surface Sigma for C-K is a to pological 3-manifold with compact boundary partial derivative Sigma in M, and Sigma' is a compact submanifold of <(Sigma)over tilde> with sp herical 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 partia l derivative Sigma' contains a projective plane. Furthermore, partial derivative Sigma = partial derivative I+[K]boolean AND partial derivat ive I-[I+], and partial derivative Sigma will be a cross section of th e 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 or ientable, the horizon cross section must consist of a disjoint union o f 2-spheres.