## Fundamental Group of Torus (van Kampen method)

There are various methods of computing fundamental groups, for example one method using maximal trees of a simplicial complex (considered a slow method). There is one “trick” using van Kampen’s Theorem that makes it relatively fast to compute the fundamental group.

This “trick” doesn’t seem to be explicitly written in books, I had to search online to learn about it.

# Fundamental Group of Torus

First we let $U$ and $V$ be open subsets of the torus  (denoted as $X$)as shown in the diagram below. $U$ is an open disk, while $V$ is the entire space with a small punctured hole. We are using the fundamental polygon representation of the torus. This trick can work for many spaces, not just the torus. $U$ is contractible, thus $\pi_1(U)=0$. $U\cap V$ has $S^1$ as a deformation retract, thus $\pi_1(U\cap V)=\mathbb{Z}$. We note that $X=U\cup V$ and $U\cap V$ is path-connected. These are the necessary conditions to apply van Kampen’s Theorem.

Then, by Seifert-van Kampen Theorem, $\displaystyle\boxed{\pi_1(X)=\pi_1(U)\coprod_{\pi_1(U\cap V)}\pi_1(V)}$, the free product of $\pi_1(U)$ and $\pi_1(V)$ with amalgamation.

Let $h$ be the generator in $U\cap V$. We have $j_{1*}(h)=1$ and $j_{2*}(h)=aba^{-1}b^{-1}$. ( $j_1:U\cap V\to U$ and $j_2:U\cap V\to V$ are the inclusions. )

Therefore \begin{aligned} \pi_1(X)&=\langle a,b\mid aba^{-1}b^{-1}=1\rangle\\ &=\langle a,b\mid ab=ba\rangle\\ &\cong\mathbb{Z}\times\mathbb{Z} \end{aligned} Free Math Newsletter