Homology sphere with $\mathbb{R}^3$ as the universal cover 
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).
 A: 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). 
A: 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.
A: 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 $1/p+1/q+1/r \leq 1$. 
A: 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.
