Can a “time dimension” be part of a spherical topology?

Question

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.

Answer 1

No, Lorentzian manifolds can be spheres only in odd dimensions. This is because the Euler characteristic of compact Lorentzian manifolds must vanish, which it doesn't for $S^n$ for even $n$, cf. e.g. this MathOverflow question.