# Expansion of Universe: Spacetime metric and Topological Variations? [duplicate]

There are many posts, questions, comments and excellent answers on this website pertaining to cosmology, expansion, universe, spacetime et cetera. Yet, the following question was not asked or answered, so here it goes.

This question deals with - trying to 'force' an analogy between spacetime metrics and the underlying manifold and it's topological properties. (Due to my lack of mathematical knowledge, I will shamelessly abuse some language and notation here.)

Premise

We know the following very well:

1. The dynamics of spacetime metric is excellently described by Einstein's Field eqns: $$G_{\mu\nu} = k\ T_{\mu \nu}$$

2. The expansion of spacetime metric on a collosal scale is described by: $$ds^2 = dt^2 - a^2(t) \gamma_{i j} dx^i dx^j$$ The rate of change of this scale factor $a(t)$, in the form of the Hubble parameter, then characterizes expansion of spacetime metric as: $$H \equiv \frac{\dot{a}}{a}$$ Experiments have measured this parameter and roughly we have [1]: $$H \approx 67.80\pm0.77 \ \frac{Km}{s \ Mpc}$$

Given the above two information and the central role that the metric plays in Riemann Geometry and GR, a colossal amount of information can be extracted about our cosmos. NOTE - metric tensor, like any other tensor on a manifold, is a bi-linear map from a certain space to $\mathbb{R}$. It is also an added physical structure on the Riemannian Manifold, which relates our cherished notions from Geometry to the real world (metric defines physical distances).

Question

There is so much wonderful cosmology that has been done and continues to be studied using the metric. But due to their inherent nature, these studies are limited to the geometrical aspects only. My questions are:

1. What subject, if any, studies the topological aspect of spacetime (topological cosmology, for example)?
2. What is our current understanding from such studies? Any references or links will be appreciated.
3. Since only the physical space-time metric undergoes physical processes, is the topology of the underlying manifold an unchanging static constant?
4. More specifically: For the last 13.8 billions of years, the physical distance between two objects has increased (from the metric related physics) which is the expansion of universe. Has the coordinate distance between two objects on an appropriate scale changed? Is it even measurable? Does this question make any scientific sense? (appropriate meaning the same scale on which the universe is homogeneous and isotropic)

Finally, any general perspective on such a topic is welcome. Thanks in advance!

Reference

[1] arXiv-1303.5062