Why isn't $\mathbb{CP}^2$ a covering space for any other manifold? This is one of those perhaps rare occasions when someone takes the advice of the FAQ and asks a question to which they already know the answer.  This puzzle took me a while, but I found it both simple and satisfying.  It's also great because the proof doesn't use anything fancy at all but it's still a very nice little result.

Euler characteristic is multiplicative, so (since $\chi(P^2)=3$ is a prime number) if $P^2\to X$ is a cover, $\chi(X)=1$ and $\pi_1(X)=\mathbb Z/3\mathbb Z$ (in particular, X is orientable). But in this case $H_1(X)$ is torsion, so (using Poincare duality) $\chi(X)=1+\dim H_2(X)+1>1$.
