general relativity - What is known about the topological structure of spacetime?

General relativity says that spacetime is a Lorentzian 4-manifold $M$ whose metric satisfies Einstein's field equations. I have two questions:

1. 
   

   What topological restrictions do Einstein's equations put on the manifold? For instance, the existence of a Lorentz metric implies some topological things, like the Euler characteristic vanishing.

   
   
   


2. 
   

   Are there any experiments being done or even any hypothetical experiments that can give information on the topology? E.g. is there a group of graduate students out there trying to contract loops to discover the fundamental group of the universe?