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.
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