In a talk given here last fall I presented joint work with Rick Schoen, in which we obtained a generalization to higher dimensions of a classical result of Hawking concerning the topology of black holes. We proved, for example, that, apart from certain exceptional circumstances, cross sections of the event horizon in stationary black hole spacetimes obeying a standard energy condition are of positive Yamabe type. 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 $S^2 \times S^1$. In this talk I show how to rule out in this setting the possibility of any such exceptional circumstances (which might have permitted, e.g., toroidal cross sections). This follows from the main result to be discussed, which is a rigidity result for suitably outermost marginally outer trapped surfaces that are not of positive Yamabe type.