# Fundamental group of the n-holed torus

I am trying to determine the fundamental group of the n-holed torus

I know that the fundamental group of the torus is $\mathbb{Z} \times \mathbb{Z}$

The n-holed torus deformation retracts onto n copies of the circle linked together (i.e. an extended version of the figure 9 loop - with n loops)

What would its fundamental group be?

I am guessing something of the form $\mathbb{Z} \times \mathbb{Z} \times .... \mathbb{Z}$

• I believe it is not even abelian, you can see this intuitively by noticing one can embed the n-fold figure eight(that is, a space with the same fundamental group as $\mathbb{R}^2-{x_1,..,x_n}$) in the n-fold torus. I believe (but am not sure) this induces an injection of fundamental groups, so the fundamental group of the n-fold torus has a non-abelian subgroup, hence it is not abelian. May 17, 2016 at 20:01

The $n$-holed torus has as fundamental group the group presented as

$$\langle a_1, b_1, \ldots, a_n, b_n \mid [a_1,b_1]\cdots[a_n,b_n] = 0\rangle$$

where $[a, b] = aba^{-1}b^{-1}$.

As an example, consider this octagon: Identify all corners, then identify the edges as labeled, and you get a 2-holed torus. The sequence $a_1b_1a_1^{-1}b_1^{-1}a_2b_2a_2^{-1}b_2^{-1}$ of edge labelings immediately gives you the generating relation for the fundamental group.

• For $n=2$, is that supposed to be $[a_1,b_1][a_2,b_2] = 0$ or $[a_1,b_1] = [a_2,b_2] = 0$?
– user14972
May 17, 2016 at 20:10
• It's supposed to be $[a_1,b_1][a_2,b_2] = 0.$
– Anon
May 17, 2016 at 20:12
• Can you explain how you got this please? May 17, 2016 at 20:18
• Are $a_i$ and $b_i$ loops? Did you use the triangulation on simplicial complex to get this? May 17, 2016 at 20:19