In spacetime dimension four, Hawking showed that the cross sections of an event horizon of asymptotically flat stationary black holes must have spherical topology. In higher dimensions such a simple characterization does not hold as illustrated by "black ring" solutions in spacetime dimension five. A natural question then arises: what are the possible horizon topologies in higher dimensions? In this talk, I'll discuss bounds on the first Betti number and structure results for the fundamental group of horizon cross-sections for extreme stationary vacuum black holes in arbitrary dimension, without additional symmetry hypotheses. The main element of the proof is a generalization of the Cheeger-Gromoll splitting theorem from Riemannian geometry that holds for the universal cover of the near-horizon geometry. This is joint work with Marcus Khuri from Stony Brook and Eric Woolgar from Univ. of Alberta.