Sure, as far as finding solutions that is fine. But mathematically the primitive piece is the pseudo-Lorentzian manifold. That is the mathematical foundation on which everything else is built. Of course you can infer the foundation from measurements or start with the foundation. But from a mathematical construction, the manifold is the basis.PeterDonis said:I would say that GR, considered simply as a geometric theory of gravity, can be even more general than this: you can start with a solution on a local patch and ask what its maximal analytic extension is, and then see what topological manifold that maximal analytic extension has.
I could easily be wrong, but I think that conformal changes are not topological changes. I.e. the fact that they are not conformally the same does not imply that they are topologically different.PeterDonis said:we do have evidence that the physical spacetime of the universe is not conformally the same as Minkowski spacetime (since any FRW spacetime other than the edge case of the empty Milne universe is not); it doesn't have the same structure at infinity.
Of course, curvature singularities make topological defects, but we expect GR to break down there, so I wouldn’t count the existence of singularities in GR as evidence of non-trivial physical topology, even where GR is known to accurately describe the physical spacetime far from the singularity.