Originally posted by Soothfast
Very interesting. But, would it be fair to say that the temporal cross-section of the universe that constitutes "now" is a 3-sphere -- or at least is homeomorphic to one, modulo local anomalies such as black hole singularities?
The accelerating expansion of the universe seems to point to hyperbolic space, which would have such a cross-section homeomorphic to a 3-plane or whatever the three dimensional equivalent of a plane is (I think). But, the lambda-CDM model has the universe not having an accelerating expansion in the distant past, but after the inflationary era, and in those eras it would be a 3-sphere. I don't see how that patches together. Also one has to be a little cautious about making assumptions about the topology - why not a torus or some more exotic topology?
What is possible (in other words not ruled out by observation) is a situation similar to a torus. On a two dimensional surface the integral of the curvature is the Euler characteristic. For a sphere, or any surface homeomorphic to a sphere, that comes to 2, and the sphere has constant positive curvature everywhere. For a torus it comes to 0. The torus has a region with positive curvature, on the outer rim, and a region of negative curvature on the inner rim, and they cancel out. Moving up a dimension, the integral of the curvature is no longer a topological invariant, but something like that might fix up this interpolation problem. Regions of intense positive curvature (black holes?) fixing up the otherwise hyperbolic geometry to allow the whole thing to be homeomorphic to a 3-sphere. A note of caution, this is my speculation, I don't know that this is even correct so be cautious about repeating it except as a question...
In short that's still open. My
guess is it's homeomorphic to a 3-sphere (modulo microscopic wormholes and so forth) but I think that question's open.