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

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

vankampen_torus

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2 Responses to Fundamental Group of Torus (van Kampen method)

mathwizzardblog says:

October 21, 2018 at 6:57 pm

Hey, I’m afraid your method is not rigorous. You have to make sure that base point is in every open set in your decomposition. And you didn’t mention about base point in this solution. There is analogous method with decomposition using analogous sets, but they have to be “hooked” in one chosen corner of the fundamental polygon of the torus.

Best!

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- mathtuition88 says:
  
  October 21, 2018 at 7:04 pm
  
  Hi, thanks for your comment. For path-connected spaces, fundamental group is independent of choice of basepoint.
  
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