1 Constraints on the topology of higher dimensional black holes

As discussed in the first chapter, black holes in four dimensions satisfy remarkable uniqueness properties. Of fundamental importance is the classical result of Carter, Hawking and Robinson that the Kerr solution, which is characterized by its mass M and angular momentum J , is the unique four dimensional asymptotically flat stationary (i.e., steady state) solution to the vacuum Einstein equations.1 A basic step in the proof is Hawking’s theorem on the topology of black holes [25], which asserts that for such black hole spacetimes, cross sections of the event horizon are necessarily spherical, i.e., are topologically 2-spheres.2 In short, for conventional black holes in four dimensions, horizon topology is spherical. However, as seen in the previous chapter, in higher dimensions, black hole horizons need not have spherical topology. With the remarkable discovery by Emparan and Reall [13] of the black ring solution, with its S1 × S2 horizon topology, the question naturally arose as to what, if any, are the restrictions on horizon topology in higher dimensional black holes. This issue was addressed in a paper of the author and Rick Schoen [19], in which we obtained a generalization of Hawking’s theorem to higher dimensions. This generalization is discussed in Sections 1.4 and 1.5. In preparation for that, we review Hawking’s black hole topology theorem in Section 1.2 and introduce some basic background material on marginally trapped surfaces in Section 1.3. Theorem 1.4.1 in Section 1.4 leaves open the possibility of horizons with, for example, toroidal topology in vacuum black hole space-