Is Modern Physics Stagnant? A Deep Dive into Topology

The simple idea of the universe being an R^3 space with a time dimension, typically represented by these people as the Minkowski R^{3,1}, is naive and ignorant given the problems with modern physics, bugging every field imaginable.
Firstly, we know (?) the universe should be finite, and expanding [whether the Hubble constant is in shambles due to the incompetence of cosmology is a different matter], thus, cosmologists love to create a hypothetical boundary pushed by dark energy leading to expansion. Why, firstly, is there even a need to construct an imaginary boundary in order to restrict the R^3 of space, when topological structures with such properties, including finiteness, are easily available ?
What's comical is that if the universe were to be of a richer topology, say S^3, or S^4, etc, trying to model it using three spatial dimensions is foolishness and would lead to unavoidable singularities !
Cosmologists love the dots-on-a-balloon analogy of the expansion of the universe, but the moment it comes to simulating the universe and developing theories and experimentation, they run back to the flat comfort of the R^{3,1}. Why, then, do you understand that analogy in the first place, and yet not try to implement it ?
The standard model already uses gauge groups to make sense of the fundamental particles and to explain their different symmetries, those are not flat, simple groups. Perhaps try that with the universe itself ?
Have they tried to formulate the universe with a richer topology that allows particles to exist and guarantees their topological existence ? That shall lead to solitons, and non linearity, which Physicists [mainstream] are for some reason hesitant to accept as a reality of this universe. If Physicists do not embrace non linearity, Physics shall continue to stay stagnant.