TITLE: Topology of black holes
QUESTION [10 upvotes]: I've asked this question of some physicist friends of mine and I've never gotten a satisfactory answer: What is topologically possible for a neighborhood of a black hole? To clarify, I'm curious about the topology as a 4-manifold, although I'd also be interested to hear about time-like and space-like slices as well. I've heard that a time-like slice of the event horizon can be a torus or sphere, but this isn't really what I'd like to know, although I imagine that there is a close connection between the topology of the event horizon (as a 3-manifold) and the topology of a neighborhood of the black hole. Please ask if any clarifications are needed.

REPLY [15 votes]: So far J Verma and RBega provided two succint descriptions on topology of the apparent horizon itself (for any space-time admitted trapped regions), and so by association, the topology of the event horizon in a stationary asymptotically flat black-hole space-time. 
I'll try to provide an answer to an interpretation of the question you asked for: namely that of the topology of a neighborhood of the event horizon. That I have to interpret the question is because you have not actually provide a description of what you mean by a neighborhood. Using the Hawking topology theorem you can easily manage that the topology of the event horizon is $\mathbb{S}^2\times \mathbb{R}$, and is an embedded null hypersurface. So a tubular neighborhood of it necessarily has the topology $\mathbb{S}^2\times \mathbb{R}^2$, which is, for one thing, simply connected. But there is absolutely nothing to prevent you from choosing a neighborhood to be some arbitrary open set containing the event horizon with complicated topology. Indeed, you can easily imagine removing a four dimensional tube disjoint from the event horizon from the $\mathbb{S}^2\times\mathbb{R}^2$ set to get something that is not simply connected. 
Because of this freedom to choose subsets, the naive reading of you question leads to the answer that "pretty much as bad as you want". 
But that answer is rather physically useless: it doesn't capture anything essential about black holes. In fact, the answer given above is identical to the answer to the following question: let $U$ be an open connected subset of $\mathbb{R}^4$, what kinds of topology can $U$ admit? 
A more useful question to ask is: given an isolated gravitating body (such as a black hole), what is the topology of the space-time outside of it? And that question is one admitting a good answer. The content is the topological censorship theorem. In physicist's language, to quote Friedman, Schleich, and Witt, 

general relativity does not allow an observer to probe the topology of spacetime: Any topological structure collapses too quickly to allow light to traverse it. 

An early version of this is due to Gannon, who showed that Cauchy hypersurface with non-trivial topology will necessarily generate a development which is null geodesically incomplete. The FSW paper showed that under some restrictions, all the non-trivial topology must be hidden behind the event horizon. 
A stronger generalisation of the topological censorship theorem is due to Galloway, who later, with Schleich, Witt, and Woolgar, extended the result from asymptotically flat space-times to also asymptotically anti-deSitter ones. One interesting crucial assumption of these theorems is the requirement for null or time-like "Scri". 
A somewhat related paper is this one by Schleich and Witt which I didn't read in detail so cannot say more about.