TITLE: Homology sphere with $\mathbb{R}^3$ as the universal cover
QUESTION [9 upvotes]: 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).

REPLY [10 votes]: 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.