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Question. Is there a $3$-dimensional integer homology sphere whose universal cover is $\mathbb{R}^3$?

I believe the answer is in the positive and I am looking for (precise) references. If not in dimension $3$, I would be happy with higher dimensional examples.

This question is related to my post: Linking topological spheres and also to the post Have examples of non-simple connected higher-dimensions integer homology sphere? (This link was provided by Ian Agol).

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    $\begingroup$ Yes: an integral homology 3-sphere that admits a hyperbolic metric. These are fairly easy to construct, but I'm on a train right now, so I can't post a precise reference to one of the constructions (this is why this is a comment rather than an answer). $\endgroup$ – Andy Putman Jan 24 '19 at 19:54
  • $\begingroup$ Related question in higher dimensional case: mathoverflow.net/q/247502/1345 $\endgroup$ – Ian Agol Feb 1 '19 at 18:59
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In a sense, most $3$-manifolds have universal cover $R^3$. In particular, this is the case for hyperbolic $3$-manifolds. And there do exist integer homology spheres which are hyperbolic. Two explicit examples I found by googling: Auckly: Surgery numbers of 3-manifolds: a hyperbolic example and Hom, Lidman: Surgery obstructions and hyperbolic integer homology spheres. It should be possible to get more examples from SnapPy.

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    $\begingroup$ If you tell me your name (you can e-mail me) I will put you in the acknowledgements since I will use this result in a paper. $\endgroup$ – Piotr Hajlasz Jan 25 '19 at 1:57
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In fact, this is true for any closed connected 3-manifold with aspherical fundamental group. More generally, the universal cover of a closed 3-manifold is $S^3$ punctured at $0, 1, 2$ points or a tame Cantor set.

Aspherical homology spheres abound, for example $1/n$ surgery on a knot for $n$ sufficiently large. Other examples are Brieskorn spheres $\Sigma(p,q,r)$ with $p,q,r$ pairwise relatively prime and $1/p+1/q+1/r \leq 1$.

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    $\begingroup$ Your first link seems to take me to a dropbox page with no associated pdf. $\endgroup$ – Mike Miller Jan 24 '19 at 21:18
  • $\begingroup$ @MikeMiller there should be a gif, it’s a slide from a talk I gave. Are you logged in to dropbox? $\endgroup$ – Ian Agol Jan 24 '19 at 21:57
  • $\begingroup$ I get the notice: "The folder ‘/Public/public_html/cover’ doesn’t exist." It seems that your link doesn't include any user data so I am not sure it should link to one of your files. (I am mostly just curious about the argument!) $\endgroup$ – Mike Miller Jan 24 '19 at 22:57
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    $\begingroup$ Sorry for the many comments. I found a link to the slide here following through your webpage. $\endgroup$ – Mike Miller Jan 25 '19 at 0:34
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There are several good answers but I thought I'd chuck in one more. Prior to all of the important results on geometrization mentioned above, Waldhausen had shown that any Haken manifold (any 2-sphere bounds a ball and the manifold contains an incompressible surface) has universal cover $\mathbb{R}^3$. Simple examples of Haken homology spheres are obtained by gluing two non-trivial knot complements in such a way that meridian and longitude are interchanged.

Piotr also asked about higher dimensional examples. These are harder to come by. In dimension $4$ there is "Some examples of aspherical 4-manifolds that are homology 4-spheres" by Ratcliffe and Tschantz, Topology Volume 44, Issue 2, March 2005, Pages 341-350. I don't know about dimensions higher than that.

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Here are two references which also might be of help:

The introduction of

Saveliev, Nikolai, Invariants for Homology 3-spheres, Encyclopaedia of Mathematical Sciences, 140. Springer (2002).

has a description and construction of Seifert Fibered Homology Spheres with infinite fundamental group, $\S$1.1.4 in the 2002 edition. (I have yet to read the 2013 edition.) Also, Peter Scott's paper:

Scott, Peter, The geometries of 3-manifolds, Bull. Lond. Math. Soc. 15, 401-487 (1983). ZBL0561.57001.

could then serve as a reference for the fact that the universal cover of a Seifert fibered homology sphere with infinite fundamental group has $R^3$ as a universal cover. Here I am relying on the complete classification of orientable $S^2 \times \mathbb{R}$ manifolds (see p. 457-459 of Scott's paper).

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