How to exhibit an étale cover of a surfaces.

Question

I have this statement about complex algebraic projective surfaces.
Let $S$ a surface such that the Euler characteristic of $S$ is negative. $X_{top}(S)<0$. Then there exist an étale cover $S^{'} \rightarrow S$ such that this inequality holds
$p_{g} (S^{'}) \le 2q(S^{'})-4 $.
The inequality is not difficult to verify but, first, in the proof there is this assertion that I cannot check:
due to the hypothesis we get $q(S)\ge 1$. Hence there are connected étale covers of any degree $m$. So what is the reason of the existence of those étale covering? Any suggestion?

Answer 1

I think, you need smoothness here. Then $\chi(S)<0$ together with Poincare duality implies that $b_1(S)>0$. By Hurewicz theorem, it follows that $\pi_1(S)$ admits an epimorphism to $Z$, hence, to $Z_m$. The kernel of this homomorphism defines a degree $m$ regular etale cover of degree $m$ over $S$.