Attraction from outside our universe:

Originally posted by Soothfast
The actual universe, as I understand it, may be thought of as a 3-sphere (a three-dimensional sphere).

Yes, my post acknowledged that possibility, but I wanted it clearly spelt out as there is often confusion between that an an actually spherical universe with edges.
Further, as Deepthought notes, it is not a known fact about the universe but rather one of three possibilities and we currently have no evidence pointing to one of them. All current measurements find the universe to be close to flat, but are not accurate enough to determine whether it is exactly flat or curved one way or the other (hyperbolic).

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?

It is unknown whether or not space is finite. People frequently assume that it is, but not because of evidence but rather a dislike of infinities. Also common is confusing 'the observable universe' which is a finite sphere of approximately known size with 'the universe' which is thought to be much larger than the observable universe and may be infinite. We had a discussion in the past about whether or not the actual universe could be smaller than the apparent observable universe, and I do not recall whether strong evidence was presented either way.

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.