TITLE: Manifold whose universal covering is a sphere but which is not a space form? 
QUESTION [12 upvotes]: 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).

REPLY [8 votes]: The geometrization theorem implies that 3-dimensional manifolds covered by $S^3$ are diffeomorphic to space forms. 
There are examples of Cappell-Shaneson and Fintushel-Stern
of fake $\mathbb{RP}^4$s (in the Fintushel-Stern case, they actually
show there is a smooth exotic free involution of $S^4$).