Spin connection for a paralellization takes more general forms than $SO(3,1)$ in different spacetime topologies?

I'm interested in a frame bundle over spacetimes with different topologies. In the trivial case of Minkowskian space ($$\mathbb{R}^{3,1}$$), a frame (or tangent space) at one point is going to be related to a frame (or tangent space) at another point via an $$SO(3,1)$$ rotation. Hence we get a spin connection valued in that group. In fact we can always reduce the structure group locally to this for a Lorentzian manifold (ie spacetime). I'm more interested in global structures on spacetimes.

Let's consider a spatial slice in a closed Friedman-Lemaitre-Robertson-Walker type spacetime, which has a spatial $$S^{3}$$ topology. A frame (or the tangent space) at an arbitrary point on the space is related to a frame (or the tangent space) at another point via an $$SO(4)$$ rotation, and any spatial triad (or tangent spaces) could be thus related.

Considering a full spacetime then with topology $$S^{3}\times\mathbb{R}$$ , wouldn't the structure group for a parallelization necessarily be $$SO(4,1)$$? In other words wouldn't the covariant derivative in an oriented orthonormal frame (the parallelization) necessarily have an $$SO(4,1)$$ spin connection? Which itself would reduce to the standard $$SO(3,1)$$ spin connection in the neighborhood of a point? If you want to formulate your answer in terms of fiber bundles that's totally fine! (:

Are you trying to embed the de Sitter group $$SO(4,1)$$ into your theory one way or another? If so, you may refer to the MacDowell–Mansouri version of gravity here, where the expanded spin connection on the de Sitter group $$SO(4,1)$$ encompasses both the original $$SO(3,1)$$ spin connection and the coset $$SO(4,1)/SO(3,1)$$ tetrad.
• +1 thank you! I'm actually not going quite that route, I'm actually just wondering if different topologies in general affect the spin connection when it's over a finite area. For example a closed-cyclic FLRW universe would have something like an $S^{3}xS^{1}$ topology which then by my reasoning would have an $SO(4,2)$ connection which upon reduction locally to the Lorentz group would have to bear traces of that former symmetry (since the Lorentz group locally would be made of broken generators from the initial group. I am trying to stay within General Relativity though, if possible. Commented Feb 8, 2021 at 2:22
• "would have something like an $S^3 x S^1$ topology which then by my reasoning would have an $SO(4,2)$ connection", I am not sure about the $S^1$ part, since you may prefer not going down the closed-time-loops rabbit hole. As for $SO(4,2)$ conformal gravity, there are some attempts, see e.g. : journals.aps.org/prd/abstract/10.1103/PhysRevD.51.1674 Commented Feb 8, 2021 at 17:17