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


Free Math Newsletter

 

Advertisements

About mathtuition88

http://mathtuition88.com
This entry was posted in topology and tagged , , . Bookmark the permalink.

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

  1. 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!

    Like

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Google+ photo

You are commenting using your Google+ account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s

This site uses Akismet to reduce spam. Learn how your comment data is processed.