I know that $\mathbb{R}^4$ admits uncountably many differentiable structures. So, I wonder what happens if we consider closed 4-manifolds. Are there any closed 4-manifolds with uncountably many differentiable structures?
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15$\begingroup$ 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. $\endgroup$– Ryan BudneyCommented Jun 16, 2014 at 22:10
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3$\begingroup$ I suppose this conjecture is due to Fintushel-Stern, but is there actually some reference for its first occurence? $\endgroup$– ThiKuCommented Jun 17, 2014 at 3:22
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2$\begingroup$ 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. $\endgroup$– Ryan BudneyCommented Jul 29, 2014 at 16:57
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3$\begingroup$ For the record, there's no need to pass to PL structures, and one can just work with handle decompositions instead (which uses Morse theory and not the Hauptvermutung). $\endgroup$– Marco GollaCommented May 8 at 21:39
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