I'm attempting to compute the Betti numbers for the closed orientable genus 2 surface, and I'm realizing I'm completely lost. I guess to begin, I need to compute the simplicial chain complexes, then derive the homologies from there, and then I can use those two to compute the Betti numbers. The thing is, I really have no clue where to begin.
I understand the definition of the simplicial chain complexes: we take the free abelian group over the k-th simplices of our surface. For my given surface, I assume that means we take the free abelian groups with respectively: 1 generator (0-simplices), 9 generators (1-simplices), and 6 generators (2-simplices). So calling our free abelian groups a product of $\mathbb{Z}$'s, I think we'd get a chain as follows:
$\mathbb{Z} \rightarrow \mathbb{Z}^9 \rightarrow \mathbb{Z}^6$.
I don't know what the next step from here is though, and computing the Homologies (and then ranks for Betti numbers) has been thoroughly confusing me. Any help would be welcome!