As I've stated in a prior question of mine, I am a mathematician with very little knowledge of Physics, and I ask here things I'm curious about/things that will help me learn.

This falls into the category of things I'm curious about. Have people considered whether spacetime is [simply connected](http://mathworld.wolfram.com/SimplyConnected.html)? Similarly, one can ask if it contractible, what its Betti numbers are, its Euler characteristic and so forth. What would be the physical significance of it being non-simply-connected?