# the fundamental group of punctured surface

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.

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!
• In particular, you probably showed this for $m=1$ when originally calculating the fundamental groups of $S_{g,0}$ via Van Kampen's theorem. – Dan Rust Jun 10 '14 at 13:27
• So $\pi_1(S_{g,m})=\mathbb{Z}^{m+2g-1}$, $m>0$? – user151938 Jun 11 '14 at 5:40
• @user151938 Careful: what you wrote generally refers to the free abelian group. $\pi_1(S_{g,m})$ is the free group on $m+2g-1$ generators. – user98602 Jun 11 '14 at 5:47