  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 $\Sm$ is
globally hyperbolic. Suppose further that a Cauchy surface $\Sigma$ for $\C_K$
is a topological 3-manifold with compact boundary $\partial\S$ in $M$, and
$\S'$ is a compact submanifold of $\bS$ with spherical boundary in $\S$ (and
possibly other boundary components in $M/\S$). Then we prove that the homology
group $H_1(\Sigma',Z)$ must be finite. This implies that either $\partial\S'$
consists of a disjoint union of 2-spheres, or $\S'$ is nonorientable and
$\partial\S'$ contains a projective plane. Further,
$\partial\S=\partial\Ip[K]\cap\partial\Im[\Sp]$, and $\partial \Sigma$ will be
a cross section of the horizon as long as no generator of $\partial\Ip[K]$
becomes a generator of $\partial\Im[\Sp]$. In this case, if $\S$ is orientable,
the horizon cross section must consist of a disjoint union of 2-spheres.}
