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> **Question.** *Is there a $3$-dimensional integer homology sphere whose universal cover is $\mathbb{R}^3$?* 
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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](https://mathoverflow.net/q/294953/121665) and also to the post [Have examples of non-simple connected higher-dimensions integer homology sphere?](https://mathoverflow.net/q/247502/121665) (This link was provided by Ian Agol).



Homology sphere with $\mathbb{R}^3$ as the universal cover
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](https://www.math.ksu.edu/~dav/surno.pdf) and
[Hom, Lidman: Surgery obstructions and hyperbolic integer homology spheres](http://www.ams.org/journals/proc/2018-146-03/S0002-9939-2017-13925-2/home.html).
It should be possible to get more examples from [SnapPy](https://www.math.uic.edu/t3m/SnapPy/#).


