I just read a little about Euler characteristics of surfaces and now I wonder:
Are black holes related to the hole of the torus?
Since black holes are things in space-time and the torus' hole is something that is related to a surface, I'm not sure if the concept of Euler characteristics can be carried over to black holes. Can it?
To put it another way: What the Euler characteristic analogon of space-time with one black hole?
I'm definitely not an expert, but here is my understanding. Mass (or energy) causes a change in the curvature tensor (not quite sure how this works). But the key is that this is a local property of the manifold. So imagine your torus is a ballon. Now poke the balloon. It will bend in, this changes the curvature close to where you push in. The amount that it curves is determined by how hard you push (which corresponds to the amount of mass) and the area over which you push in, this is how pressure works.
Now for a black hole, the mass is finite but the volume is 0, again this is all hypothetical, so this means that at the point of the mass the curvature is $\infty$, this is why it's called a singularity.
In the analogy of the balloon, it would be like pinching the balloon so that it is no longer smooth, but so that you get a "corner"
The key of all this is that it is essentially local and involves geometry.
Now if you talk about the hole of a torus, it is global and topological, not geometric. In fact a torus can be made to be as flat as a plane, locally at every point simultaneously (notice that this cannot be done for a sphere or a higher genus surface). So it is really of a different nature.
Now there was a comment about a wormhole. Well the idea there is that wormholes will change the topology of space-time to look like a torus, and there is a hypothesis that this can happen inside a black hole. But really this is jumping the gun, possibly to the point of being nonsensical. Before I get comments about my last statement, let me explain.
Because there is currently no physical theory that sufficiently explains what happens in a black hole, then in some sense there can be no wrong answers. So you can come up with all kinds of theories about what kinds of things can happen.