Exotic $R^4$ as the universal covering space

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 $$R^n$$.

• Do you have a reference for the examples by Mike Davis? Do you know if they are smooth? 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. 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. 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. May 15 '19 at 14:28

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.