# Closed path on torus

How to proof that closed path $l$ (see picture) on 2-dimensional torus don't homotopic to trivial path, using definition torus as CW-complex? (closed path on surface $S$ is a map $f:[0, 1]\to S$, such that $f(0)=f(1)=x_0$) Thanks.

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It is difficult to give an answer that will be useful to you, without knowing your mathematical background. It would also be helpful to know where the question comes from. Is this for a class? Self-study? Do you have someone to talk to? What approaches have you tried? – Sam Nead Sep 15 '11 at 21:35

Four proof sketches. $\newcommand{\ZZ}{\mathbb Z} \newcommand{\RR}{\mathbb R}$
2. The fundamental group of the torus is $\ZZ \times \ZZ$. The vertical and horizontal loops give the generators and so neither is trivial.
3. Since the torus is the quotient $\RR^2/\ZZ^2$ we can take the preimage of the vertical loop in $\RR^2$ and get a collection of vertical lines. By homotopy lifting, a trivial loop in the torus lifts to an infinite collection of trivial loops in $\RR^2$.