Definition: By a closed curve in a surface $S$ we will mean a continuous map $S^1 \to S$. We will usually identify a closed curve with its image in $S$. A closed curve is called essential if it is not homotopic to a point, a puncture, or a boundary component.
I want to prove the following lemma:
Lemma. Let $\alpha$ and $\beta$ be two essential simple closed curves in a torus $T$. Then $\alpha$ is isotopic to $\beta$ if and only if $\alpha$ is homotopic to $\beta$.
Proof. One direction is vacuous since an isotopy is a homotopy. So suppose that $\alpha$ is homotopic to $\beta$. We immediately have that $i(\alpha, \beta)=0$.
Also we have:
Theorem: The nontrivial homotopy classes of oriented simple closed curves in a torus $T$ are in bijective correspondence with the set of primitive elements of $\pi_1(T) ≈ \mathbb{Z}^2$. An element (p, q) of $\mathbb{Z}^2$ is primitive if and only if $(p, q) = (0, \pm1)$, $(p, q)=(\pm1, 0)$, or $\gcd(p, q)=1$.
