How to rigorously prove that simple closed curves on a surface are primitive closed curves ?

2011-11-03T14:42:56Z

Let me first state the definitions :

A not-nullhomotopic closed curve / loop $c$ on an orientable surface $X,c:[0,1]\to X$ is called simple closed curve is $c|[0,1)$ is injective and [ $c(0)=c(1) ] ; $ A closed curve / loop $c$ is called primitive if in the fundamental group $\pi_1(X,c(1)),$ the homotopy class $[c]$ can NOT be written as $[c]= [\gamma]^n$ for some closed curve $\gamma$ with $\gamma(0)=\gamma(1)=c(0)=c(1) $ and for some $n\ge 2$.

My question is : rigorously prove that simple closed curves are primitive .

It is visually pretty clear , but I have difficulty proving it. Thanks !

Answer by Sam Nead for How to rigorously prove that simple closed curves on a surface are primitive closed curves ?

2011-11-03T22:24:44Z

Suppose that $c = \gamma^n$ in $\pi_1(X)$. Note that, as $\pi_1(X)$ is torsion free and $c$ is assumed to be non-trivial, the element $\gamma$ generates an infinite cyclic subgroup $\langle \gamma \rangle < \pi_1(X)$. Let $A = X^\gamma$ be the cover of $X$ corresponding to the subgroup $\langle \gamma \rangle$. So $\pi_1(A)$ is also infinite cyclic. Since $X$ is orientable, so is $A$. It follows from the classification of surfaces $A$ is a (non-compact) annulus.

Note that $\gamma$ can be lifted to $A$ and this lift, $\gamma'$, is homotopic to the core curve of $A$. Likewise $c$ lifts to a curve $c'$ and we have $c' = (\gamma')^n$ in $\pi_1(A)$. Since $c$ is simple in $X$ the lift $c'$ is simple in $A$. By the intermediate value theorem (sort of!) the only simple curves in $A$ are isotopic to the trivial curve and to the core curve. Thus $n = \pm 1$ and we are done.

How to rigorously prove that simple closed curves on a surface are primitive closed curves ? - MathOverflow

How to rigorously prove that simple closed curves on a surface are primitive closed curves ?

Answer by Sam Nead for How to rigorously prove that simple closed curves on a surface are primitive closed curves ?