3
$\begingroup$

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?

$\endgroup$
3
  • 11
    $\begingroup$ 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. $\endgroup$ Jun 16, 2014 at 22:10
  • 2
    $\begingroup$ I suppose this conjecture is due to Fintushel-Stern, but is there actually some reference for its first occurence? $\endgroup$
    – ThiKu
    Jun 17, 2014 at 3:22
  • 1
    $\begingroup$ 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. $\endgroup$ Jul 29, 2014 at 16:57

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.