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

No, there are only countably-many diffeomorphism types of compact 4-manifolds, since each one has a finite PL-compatible triangulation, and that determines the diffeomorphism type uniquely. Moreover, there's an outstanding conjecture in 4-manifold theory: every compact smoothable 4-manifold admits countably-infinite distinct smooth structures.
– Ryan BudneyJun 16 '14 at 22:10

2

I suppose this conjecture is due to Fintushel-Stern, but is there actually some reference for its first occurence?
– ThiKuJun 17 '14 at 3:22

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The first occurrence of it I remember was at Ron Stern's talk at the last Topology Festival. I think that was at Cornell in the summer of 2012. Fintushel and Stern would likely know of any earlier occurrence.
– Ryan BudneyJul 29 '14 at 16:57