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).
Best Answer
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". 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).