I came across this paper recently called The Small Scale Structure of Spacetime and the following idea occured to me:
To uninformed humans the universe appears Euclidean but we know from GR that on a larger scale it behaves like a Riemannian manifold. Now the paper claims that on a smaller scale, things are 2-dimensional. However, the underlying space is still the same, i.e. that of a manifold.
I have two questions:
Why do we always deal with manifolds. Perhaps there is a better topological space that explains what happens on all scales. Does anyone know of any competing theory to GR say that does not use manifolds? Are there any theories that make use of the idea of a topologically stratified space?
Have their been attempts at a unified theory where the geometry of the universe varies at different scales, thus affecting the physics of things at different scales? For example, think of a toy universe that is a sphere. However, this sphere is not made of a smooth material but rather out of a fine mesh of wire. There are thus three scales here: on a larger scale, the universe is curved; on a medium scale, the universe looks flat and 2D; and on a smaller scale, the universe looks 1D almost everywhere.