# Categories (other than TOP) to rule out exotic $\mathbb{R}^4$?

I'm a non-mathematician who is interested in differential topology.

If I understand correctly, the existence of exotic $$\mathbb{R}^4$$ is directly linked to the failure of smooth h-cobordism theorem between 4-manifolds. Exotic $$\mathbb{R}^4$$s also exist in PL since PL is isomorphic to DIFF in dimension 4, but in TOP they are homeomorphic to the standard $$\mathbb{R}^4$$.

My question is: could there be a category of manifolds, other than TOP, that can rule out exotic $$\mathbb{R}^4$$?

For example, we might define an "X-morphism" less strict than diffeomorphism (but stricter than homeomorphism) so that all the exotic $$\mathbb{R}^4$$s are "X-morphic" to the standard $$\mathbb{R}^4$$. Or we might have a category of "nice" manifolds, excluding manifolds with fractal structures. Would any of these be a real (and useful) thing in mathematics?

• Not quite what you're looking for, but there's a result that any $C^p$ manifold for $p \geq 1$ can be given a unique $C^\infty$ structure. Aside from that, there are more exotic things like diffeological spaces, orbifolds, absolute neighborhood retracts, etc. But I don't think they're really what you're looking for either, and I'm not familiar with the state of the question exotic manifolds outside the usual continuous, PL, or smooth categories. Apr 5 at 3:26
• There is also notion of Lipshitz homeomorphism which I believe has been studied in this context. But I am no expert, please consult the literature about it. Apr 5 at 11:28