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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