Homology groups of genus $g$ Riemann surface with $n$ punctures.

Question

I want to know whether one can compute $H_k(\Sigma_{g,n})$ with elementary methods like singular chain complex? By $\Sigma_{g,n}$ I mean a genus $g$ surface with $n$ circular holes. If not, I'm not familiar with the standard method, which is Mayer-Vietoris sequence. Could you please introduce me to a lecture or note that help me work this out? Or any link to the exact solution to this problem would be very helpful.