# Computing the fundamental group of a cell complex

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!

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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

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.