The Covering Transformation Theorem - Fund. Group of the Torus

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)

\begin{align} \quad A(\tilde{X}) \cong \pi_1(X, x) \end{align}

(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, $T^2$.

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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)

\begin{align} \quad s_n(x, y) &= (x + n, y) \\ \quad t_n(x, y) &= (x, y + n) \end{align}

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)

\begin{align} \quad A(\mathbb{R}^2) = \langle s_1, t_1 : s_1t_1 = t_1s_1 \rangle \cong \mathbb{Z} \times \mathbb{Z} \cong \pi_1(X, x) = \pi_1(S^1, x) \end{align}

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