# The Covering Transformation Theorem - Fundamental Group of the Torus

Recall from The Covering Transformation Theorem for Finding Fundamental Groups page that if $X$ is a topological space and $(\tilde{X}, p)$ is a universal cover of $X$ then:

(1)(Where $A(\tilde{X})$ denotes the set of covering transformations of $\tilde{X}$.)

We will now look at applying this result to find the fundamental group of the torus, $T^2$.

Recall that if $X = T^2$ then $\tilde{X} = \mathbb{R}^2$ is the universal cover of the torus with the map. We define $p : \mathbb{R}^2 \to T^2$, the covering map, by first rolling $\mathbb{R}^2$ into an infinite length cylinder, and then coiling the cylinder onto itself to form the torus.

For each $n \in \mathbb{N}$ let $s_n : \mathbb{R}^2 \to \mathbb{R}^2$ and $t_n : \mathbb{R}^2 \to \mathbb{R}^2$ be defined for all $(x, y) \in \mathbb{R}^2$ by:

(2)Then the set of covering transformations of $\mathbb{R}^2$ (with respect to $p$) is generated by $s_1$ and $t_1$. We also have that $s_mt_n = t_ns_m$ for all $m, n \in \mathbb{N}$ and so:

(3)So the fundamental group of the torus is $\mathbb{Z} \times \mathbb{Z}$.