  Hawking's theorem on the topology of black holes asserts that cross sections
of the event horizon in 4-dimensional asymptotically flat stationary black hole
spacetimes obeying the dominant energy condition are topologically 2-spheres.
This conclusion extends to outer apparent horizons in spacetimes that are not
necessarily stationary. In this paper we obtain a natural generalization of
Hawking's results to higher dimensions by showing that cross sections of the
event horizon (in the stationary case) and outer apparent horizons (in the
general case) are of positive Yamabe type, i.e., admit metrics of positive
scalar curvature. This implies many well-known restrictions on the topology,
and is consistent with recent examples of five dimensional stationary black
hole spacetimes with horizon topology $S^2 \times S^1$. The proof is inspired
by previous work of Schoen and Yau on the existence of solutions to the Jang
equation (but does not make direct use of that equation).
