I've heard it speculated that the spatial dimensions of the universe is a 3-sphere. Or a 3-torus. But usually, I guess, it's assumed that the "time" dimension just has its own geometry, like a line, in Cartesian product with the geometry of the spatial dimensions.
I don't know much about topology, nor the constraints on topology placed by geometry. So I don't know if the shape of the manifold "cares" that the geometry treats one of those dimensions differently (particularly because the dimensions are symmetric in a sphere). Basically, can a manifold whose metric has the Lorentz signature $-+++$ be a 4-sphere? Or more generally, can a manifold with a $(1,n-1)$ signature metric be an $n$-sphere?
Also, let me know if I'm using imprecise, bad language here.