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 × 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).

We first show that the intrinsic, geometrical structure of a dynamical horizon (DH) is unique. A number of physically interesting constraints are then established on the location of trapped and marginally trapped surfaces in the vicinity of any DH. These restrictions are used to prove several uniqueness theorems for DH. Ramifications of some of these results to numerical simulations of black hole spacetimes are discussed. Finally, several expectations on the interplay between isometries and DHs are shown to be borne out.e-print archive: http://lanl.arXiv.org/abs/gr-qc/0503109 2 ABHAY ASHTEKAR AND GREGORY J. GALLOWAY

Motivated by recent interest in black holes whose asymptotic geometry approaches that of anti-de Sitter spacetime, we give a proof of topological censorship applicable to spacetimes with such asymptotic behavior. Employing a useful rephrasing of topological censorship as a property of homotopies of arbitrary loops, we then explore the consequences of topological censorship for the horizon topology of black holes. We find that the genera of horizons are controled by the genus of the space at infinity. Our results make it clear that there is no conflict between topological censorship and the nonspherical horizon topologies of locally anti-de Sitter black holes. More specifically, let D be the domain of outer communications of a boundary at infinity ''scri.'' We show that the principle of topological censorship ͑PTC͒, which is that every causal curve in D having end points on scri can be deformed to scri, holds under reasonable conditions for timelike scri, as it is known to do for a simply connected null scri. We then show that the PTC implies that the fundamental group of scri maps, via inclusion, onto the fundamental group of D: i.e., every loop in D is homotopic to a loop in scri. We use this to determine the integral homology of preferred spacelike hypersurfaces ͑Cauchy surfaces or analogues thereof͒ in the domain of outer communications of any four-dimensional spacetime obeying the PTC. From this, we establish that the sum of the genera of the cross sections in which such a hypersurface meets black hole horizons is bounded above by the genus of the cut of infinity defined by the hypersurface. Our results generalize familiar theorems valid for asymptotically flat spacetimes requiring simple connectivity of the domain of outer communications and spherical topology for stationary and evolving black holes.

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Over the past decade there has been an increasing interest in the study of black holes, and related objects, in higher (and lower) dimensions, motivated to a large extent by developments in string theory. The aim of the present paper is to obtain higher-dimensional analogues of some well known results for black holes in 3 + 1 dimensions. More precisely, we obtain extensions to higher dimensions of Hawking's black hole topology theorem for asymptotically flat ( = 0) black hole spacetimes, and Gibbons' and Woolgar's genus-dependent, lower entropy bound for topological black holes in asymptotically locally antide Sitter ( < 0) spacetimes. In higher dimensions the genus is replaced by the so-called σ -constant, or Yamabe invariant, which is a fundamental topological invariant of smooth compact manifolds.