  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
horizon topology of black holes. We find that the genera of horizons are
controlled by the genus of the space at infinity. Our results make it clear
that there is no conflict between topological censorship and the non-spherical
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 (PTC), that every causal
curve in D having endpoints 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
thereof) in the domain of outer communications of any 4-dimensional spacetime
obeying 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 slowly evolving black holes.
