Closed 4-manifolds with uncountably many differentiable structures

Question

I know that $\mathbb{R}^4$ admits uncountably many differentiable structures and I was wandering what happen if we consider closed (or just compact) 4-manifolds. Are there any closed (or compact) 4-manifolds with uncountably many differentiable structures? And with countably many?

Answer 1

This is a summary of comments.

Your question was answered here in a comment by Ryan Budney. In the nutshell: The categories DIFF and PL are equivalent in dimension 4 (this is due to Kirby and Siebenmann, but one can easily spend a year trying to understand their proof). A PL structure on an n-manifold is equivalent to a "combinatorial triangulation", i.e., a triangulation where the link of each simplex is combinatorially homeomorphic to the closed n-ball. Since there are only countably many finite simplicial complexes, it follows that there are only countably many closed differentiable 4-manifolds.

Furthermore, I think, it is also true in higher dimensions that there are only countably many smooth closed manifolds (even though, DIFF $\ne$ PL), but I would have to check this.

Lastly, according to Qiaochu Yuan's comment, Dolgachev surface (which is a closed 4-manifold) admits countable infinitely many smooth structures.