Let $M^n$ be a smooth manifold whose universal cover is homeomorphic $\mathbb{S}^n$, are there examples where $M^n$ is not homeomorphic to a space form ? 


The answer may vary if you replace homeomorphic by diffeomorphic, and I'm also interested in this question under this restriction.


I came to this question while reading surveys about sphere theorems, where the non simply-connected case is harder to get (while you already know that the universal cover is a sphere, you need more work to show it is actually a space-form).



Manifold whose universal covering is a sphere but which is not a space form? 
There are lots of fake lens spaces, and fake spherical space forms (search on these keywords). In particular, a construction of fake lens spaces is in chapter 12 of Milnor's  ["Whitehead torsion"](http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.bams/1183527946). Here are details: 
start with any lens space $L$ with fundamental group of order different from 2,3,4, 6, so that its Whitehead group is infinite. Then there are infinitely many distict manifolds that are h-cobordant to $L$. On the other hand, Corollary 12.13 implies that any h-cobordism between lens spaces is a product. (Of course, I assume dimension $\ge 5$ here).


