46
$\begingroup$

Is there a smooth compact 4-manifold whose universal covering is an exotic $R^4$, i.e. is homeomorphic but not diffeomorphic to $R^4$?

Remark. I am aware of examples (due to Mike Davis) of compact $n$-manifolds whose universal covering spaces are fake $R^n$'s i.e. are contractile but not homeomorphic to $R^n$.

Improve this question
$\endgroup$
4
  • $\begingroup$ Do you have a reference for the examples by Mike Davis? Do you know if they are smooth? $\endgroup$
    – Michael Albanese
    Commented May 15, 2019 at 14:00
  • $\begingroup$ @MichaelAlbanese: M.Davis, Groups Generated by reflections and aspherical manifolds not covered by Euclidean space, Annals of Math, 1983. $\endgroup$
    – Moishe Kohan
    Commented May 15, 2019 at 14:05
  • $\begingroup$ Thanks. Do you know if these examples are smooth? The paper is quite long, and I could only find a discussion about smoothness at the end, but it is unclear to me whether the results there suffice to deduce the compact manifolds constructed are smooth. $\endgroup$
    – Michael Albanese
    Commented May 15, 2019 at 14:17
  • $\begingroup$ I would need to check the higher-dimensional case but they are definitely PL and in dimensions $< 7$, DIFF=PL (Kirby-Siebenmann). Hence, you have smooth examples at least in dimensions 4, 5 and 6. $\endgroup$
    – Moishe Kohan
    Commented May 15, 2019 at 14:28

1 Answer 1

Reset to default
41
$\begingroup$

This is problem 4.79(A) of Kirby's 1995 problem list, contributed by Gompf. It is my impression that it is still open.

There is some small progress: Remark 7.2 in this article observes that their constructions imply that a specific countable set of examples of exotic $\Bbb R^4$s cannot possibly cover a closed manifold. This is not a huge reduction, as only countably many exotic $\Bbb R^4$s could possibly be covers of the countable set of closed smooth 4-manifolds, anyway. But at least the examples are somewhat explicit.

I couldn't find any other references to this problem in the literature, but that doesn't mean there aren't any.

UPDATE: I emailed Bob Gompf; he is not aware of any recent progress.

Improve this answer
$\endgroup$
2
  • $\begingroup$ Ah, you are right I should have checked Kirby's list first, before asking! Incidentally, the right link is: arxiv.org/pdf/math/9712292.pdf $\endgroup$
    – Moishe Kohan
    Commented Oct 13, 2018 at 14:56
  • $\begingroup$ @Moishe Thanks, I fixed that and added the official word. $\endgroup$
    – mme
    Commented Oct 13, 2018 at 18:04

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .