Let's say General Relativity is the study of the Einstein equation on smooth Lorentzian manifolds, i.e. pseudo-Riemannian manifolds of signature $(n-1,1)$.

I've heard more than once people say that the reason why there are many non-diffeomorphic smooth structures on $\mathbb{R}^n$ only for $n=4$ would be somehow "related to the fact that the spacetime in which we live is $4$-dimensional". Or, on the other hand, that there are exotic smooth structures only on $\mathbb{R}^4$ because "that's the dimension of spacetime".

So, first of all

- Are there properties or phenomena of general relativity that are "specific" to $\mathbb{R}^{3,1}$, that is, the equivalent version on $\mathbb{R}^{n-1,1}$ does not hold for $n\neq 4$, or more generally on any other Lorentzian manifold homeomorphic to $\mathbb{R}^4$?

I know string theorists proved that if you cross $\mathbb{R}^{n-1,1}$ by a Calabi-Yau manifold then $n=4$ would result in a "critical dimension" for some properties (supersymmetry?) of string theory on the whole $\mathbb{R}^{3,1}\times\mathrm{CY}$, but I'm wondering if something special happens already at the level of GR.

And

- Has anybody seriously considered the relationship between "the spacetime being $4$-dimensional" (i.e. $n=4$ being a "critical dimension" for GR/ string theory on $\mathbb{R}^{n-1,1}$) and the existence of exotic $\mathbb{R}^n$'s only for $n=4$? To which conclusions did they arrive? Which explanations did they provide?