40
$\begingroup$

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 $R^n$.

$\endgroup$
  • $\begingroup$ Do you have a reference for the examples by Mike Davis? Do you know if they are smooth? $\endgroup$ – Michael Albanese May 15 '19 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 May 15 '19 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 May 15 '19 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 May 15 '19 at 14:28
35
$\begingroup$

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.

| cite | improve this answer | |
$\endgroup$
  • $\begingroup$ 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 Oct 13 '18 at 14:56
  • $\begingroup$ @Moishe Thanks, I fixed that and added the official word. $\endgroup$ – Mike Miller Oct 13 '18 at 18:04

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.