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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 $R^n$.

Moishe Kohan
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  • Do you have a reference for the examples by Mike Davis? Do you know if they are smooth? – Michael Albanese May 15 '19 at 14:00
  • @MichaelAlbanese: M.Davis, Groups Generated by reflections and aspherical manifolds not covered by Euclidean space, Annals of Math, 1983. – Moishe Kohan May 15 '19 at 14:05
  • 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. – Michael Albanese May 15 '19 at 14:17
  • 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. – Moishe Kohan May 15 '19 at 14:28

1 Answers1

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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.

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