Let $X_k$ the connected sum of k projective planes. I wonder about necessary and sufficient conditions to know wheter there exists a covering $\pi: X_{k'} \to X_k$ if k and k' are integers.
A necessary condition is that the euler characteristic of $X_{k'}$ has to be multiple of the euler characteristic of $X_k$, so is the sufficient condition what it's more difficult.
The case of orientable surfaces is easy, but this is more difficult I think, and I can't find it anywhere.
I would appreciate any hint about this.
Edit: when I wrote $X'_k$ I meant $X_{k'}$, so $X_{k'}$ is another connected sum of projective planes.