the fundamental group of punctured surface

Question

Let $S_{g,m}$ be a surface of genus $g$ with $m$ punctured, we know the fundamental group of $S_{g,0}$ is $$ \pi_1(S_{g,0}) = \left\langle a_1, b_1, \dots, a_g, b_g {~\large\mid~} [a_1, b_1] \dots [a_g,b_g] = 1 \right\rangle, $$ then what is the fundamental group of $S_{g,m}$? Thanks in advance.

Answer 1

Hint: Show that $S_{g,m} \simeq \bigvee_{i=1}^{m+2g-1}S^1$ for $m>0$. To do this, use the fundamental polygon of $S_g$; this should get easier as you add more holes!