TITLE: Sufficient condition for coverings between non-orientable surfaces
QUESTION [5 upvotes]: Let $X_k$ be the connected sum of $k$ projective planes. I am interested in necessary and sufficient conditions for the existence of a covering $\pi: X_{k'} \to X_k$, where $k$
and $k'$ are integers.
A necessary condition is that the Euler characteristic of $X_{k'}$ is a multiple
of the Euler characteristic of $X_k$. Though obtaining a sufficient condition
seems more difficult.
The case of orientable surfaces is easy, but this is more difficult than I think,
and I can't find it anywhere.
I would appreciate any hint about this.

REPLY [8 votes]: Your necessary condition, rephrased slightly, is sufficient. It should say that the Euler characteristic of $X_{k'}$ is a positive multiple of $\chi(X_k)$. This modification is to take care of the possibility that $X_k$ is $RP^2$, which is not covered by any other non-orientable surface.  If $\chi(X_k)= \chi(X_{k'}) = 0$, then both are Klein bottles, and certainly there is a covering.  Finally if $\chi(X_k)<0$, then it is a connected sum of a torus with some number of projective planes.  So it has non-orientable covers of all degrees; these are determined by their Euler characteristics.  So, given $X_{k'}$ with $\chi(X_{k'}) = n \chi(X_k)<0$, choose a degree n cover of $X_{k}$; it will be homeomorphic to $X_{k'}$.  
The case of orientable surfaces follows by the same argument.