Think of the surface of genus $k$ as a sphere with $k$ tubes sewn in. Calculate its Euler characteristic by trangulating.
I know that I need to make the genus covered by infinitely many triangle then use the formula
$$\chi=V-E+F$$
where $V$ is the number of vertices on the tessellated surface, $E$ is the number of edges on the tessellated surface and $F$ is number of polygonal faces.
My professor told me to remove $2k$ non-adjacent faces from $S^2$, then triangulate a closed cylinder $S^1 \times [0,1]$.
Why do I need to remove $2k$ non-adjacent faces from $S^2$? I know that $\chi(S^2)=2$, but I can't see what it has to do with this move.
I also know that the surface of genus $k$ admits a Lefschetz map homotopic to the identity with 1 source, 1 sink and $2k$ saddles. This implies that $\chi =2-2k$. So I guess this is the answer for this problem. The only thing I need to do is using triangulating to get to that result.
Starting from a genus $k$ surface, delete two triangles, then sow in one tube made from two triangles. What happens to the number of faces, vertices, and edges?