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Is there any classification for all topological spaces that have $\mathbb{R}^n$ as their universal cover?

123...
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    You're pretty much asking for a classification of groups acting freely, properly discontinuously on $\mathbb{R}^n$. – Najib Idrissi Oct 01 '15 at 15:34
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    And the answer is "no, absolutely not". Every hyperbolic manifold is of this form, and the structure of those is very complicated. –  Oct 01 '15 at 16:16

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A more restrictive question has a more reasonable answer, namely a classification of all Riemannian manifolds that have $\mathbb{R}^n$, with the usual metric, as their universal cover. I think that these are precisely the complete flat manifolds. Bieberbach classified all the closed flat manifolds: in particular, they all admit finite covers by flat tori $\mathbb{R}^n/\Gamma$ (where $\Gamma$ is a lattice). Their fundamental groups are space groups.

As Mike Miller says in the comments, without this restriction the problem is at least as complicated as the problem of classifying hyperbolic manifolds (since hyperbolic space is homeomorphic and even diffeomorphic to Euclidean space), which is probably hopeless.

Qiaochu Yuan
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  • One thought: classifying oriented closed manifolds with $\Bbb R^n$ as their universal cover shouldn't be far off from the classification of $K(G,1)$s with an oriented closed manifold model (by the Borel conjecture, which is probably true), and these have a nice conjectural classification: the $G$ are precisely finitely presented Poincare duality groups. Other contractible manifolds without boundary should be too "asymmetrical" to usually admit compact quotients; I don't know any actual conjectures or theorems that make this idea precise. (There should be a non-orientable version too.) –  Oct 02 '15 at 15:53
  • Thank you very much for your helps. Actually, I want to consider this problem for those spaces that are free $\mathbb{Z}_2$-spaces. (a free $\mathbb{Z}_2$-space, is a topological space $X$ with fixed point free involution over it). – 123... Oct 03 '15 at 09:59
  • @123 I really don't think that helps. I can only imagine the added complication making a classification worse. –  Oct 03 '15 at 15:48