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?
Best Answer
The manifold $\mathbb{CP}^2$ is not the universal covering space of any manifold other than itself.
One way to see this is to note that if $M \to N$ is a $d$-sheeted covering of closed manifolds, then $\chi(M) = d\chi(N)$. As $\chi(\mathbb{CP}^2) = 3$, we see that $d = 1$, in which case $N = \mathbb{CP}^2$, or $d = 3$. If a three-sheeted covering were to exist, the manifold $N$ would satisfy $\pi_1(N) \cong \mathbb{Z}_3$ (as this is the only group of order $3$). As $\mathbb{Z}_3$ has no index two subgroups, the manifold $N$ is orientable. But then the signature of $N$ satisfies $1 = \sigma(\mathbb{CP}^2) = 3\sigma(N)$ which is impossible, so the only manifold which is covered by $\mathbb{CP}^2$ is itself.
More generally, the connected sum of $k$ copies of $\mathbb{CP}^2$ does not cover any manifold other than itself. The argument above for orientability of the quotient doesn't apply when $k$ is even, instead you can use the argument in this answer (which works for any $k$).