Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

In Hatcher on page 84 there is the following proposition: For a connected graph $X$ with maximal tree $T$, $\pi_1 (X)$ is a free group with basis the classes $[f_\alpha]$ corresponding to the edges $e_\alpha$ of $X - T$.

I tried to apply this to the torus $T^2$ with the two edges $e_a, e_b$ and the maximal tree $T = \{ x_0\}$ where $x_0$ is the point connecting the two edges.

The problem is that then I get the free group $F(a,b)$ instead of $\mathbb{Z} \oplus \mathbb{Z}$.

Where am I making the mistake? Many thanks for your help!

share|cite|improve this question
Hmm. How is the torus a graph (= 1-dim CW-complex)? In fact, you're not that far off, the relation $[a,b] = aba^{-1}b^{-1}$ from the two-cell gives you the desired $\mathbb{Z}^2$. – t.b. Jul 7 '11 at 13:02
well, I thought I could just use the $1$-skeleton to get the fundamental group of the whole complex. Need to work out how to put things together correctly I think... – Rudy the Reindeer Jul 7 '11 at 13:46
up vote 3 down vote accepted

In this proposition a graph is a regular, 1-dimensional CW-complex.

In fact you computed the fundamental group of the 1-Skeleton of the Torus, which is a bouquet of $2$ $S^1$ and has therefore $\mathbb Z * \mathbb Z$ as fundamental group.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.