Let $M$ be a orientable 2-closed surface,prove $H^1(M)$ is direct sum of an even number of $\Bbb Z$

Question

Let $M$ be a orientable 2-closed surface,prove $H^1(M)$ is direct sum of an even number of $\Bbb Z$

Could anyone give some hints?

Answer 1

Hint: Think about how you would triangulate such a surface. Think about how you can triangulate a $1$-holed torus, $2$-holed torus, $\dots$ , $n$-holed torus. This of course relies on what Georges mentioned in the comments; the fact that $M$ is just a sphere with $n$ handles.

Then think about what this triangulation is homotopy equivalent to, and what the first cohomology of this homotopy equivalent object is. Here you'll need that $H^1(S^1) \cong \Bbb{Z}$.