TITLE: Is every closed simply connected manifold a non-trivial covering space?
QUESTION [25 upvotes]: We know that every universal covering space is simply connected. The converse is trivially true, every simply connected space is a covering space of itself. But I'm wondering what simply connected spaces, specifically manifolds, are covering spaces of another (connected) space other than itself.
I found this paper that shows that there exist certain contractible open simply connected manifolds that aren't non-trivial covering spaces of manifolds. If we restrict to closed simply connected manifolds, can we also find such examples, or can there always be a different manifold that it covers?

REPLY [19 votes]: Below, I'll give an example which, as a topological manifold can non-trivially cover, but as a smooth manifold it cannot.
Recall that in each dimension, the set of diffeomorphism types of $S^n$ forms an abelian group under the connect sum operation, where the inverse to an element is given by switching the orientation.
Elements of order $2$ in this group are precisely the exotic spheres which admit an orientation reversing diffeomorphism.
In dimension $10$, the group of exotic spheres has order $6$.  By Cauchy's theorem, there is an element $\Sigma$ of order $3$.  In particular, such a $\Sigma$ does not admit an orientation reversing diffeomorphism.
It follows from an easy application of the Lefschetz fixed point formula that any orientation preserving diffeomorphism $f:\Sigma\rightarrow \Sigma$ must have a fixed point.  Thus, $\Sigma$ cannot smoothly cover anything.
On the other hand, $\Sigma$, being homeomorphic to $S^{10}$ can topologically cover $\mathbb{R}P^{10}$.