Genus of a surface Let $S$ be a genus 3 closed orientable surface. Let $R\rightarrow S$ be a degree 2 covering map. What is the genus of $S$ ? Do I have to use the Euler characteristic of a surface presentation which is a topological invariant? I guess the question should be what is the question of R?
Thanks
 A: Let us assume that $R$ and $S$ have triangulations, and further that the covering map $\pi\colon R\to S$ of degree two respects the triangulation on $S$ (that is, $\pi$ is a cellular-map). The Betti numbers of $R$ (resp. $S$) are $1,2g_R,1,0,\ldots$ (resp. $1,2g_S,1,0,\ldots$) which can be found using standard techniques. So, the Eular-characteristics are given by $$\chi(R)=1+2g_R-1=2g_R\\ \chi(S)=1+2g_S-1=2g_S.$$
Now, we can also give the Euler characteristic of a surface in terms of the vertices, edges, and faces of a triangulation, and so we also have $$2-2g_R=\chi(R)=V_R+E_R-F_R\\ 2-2g_S=\chi(S)=V_S+E_S-F_S.$$
Given that $\pi$ is assumed to take vertices to vertices, etc, and is of degree two, so the preimage of every $k$-cell is a disjoint union of two  $k$-cells, we should have $$V_R=2V_S,\: E_R=2E_S,\:F_R=2F_S$$ and so $2-2g_R=V_R+E_R-F_R=2(V_S+E_S-F_S)=2(2-2g_S)=2(-4)=-8\implies g_R=5$.

It wasn't clear from your question if you wanted to find $g_R$ from $g_S$ or vice-versa, but from the above you can see that $g_R=2g_S-1$ in all cases.
There is a small concern in the above that the covering map $\pi$ given may not in fact be a cellular map. This concern is justified, and I should have really shown that we can take a suitably fine triangulation of $S$ which properly lifts up to a triangulation on $R$ via $\pi$. The easiest way to do this is to put a triangulation on $R$, consider the induced partition by polygons of $S$ by $\pi$ (which may not be a triangulation) that we get by mapping the $1$-skeleton of $R$ to $S$. We can then refine this partition so that it really is a triangulation, and then lift this new triangulation back up to $S$ via $\pi$. This new triangulation on $R$ will then, by construction, respect the cellular structure of $S$ with respect to $\pi$.
