Let $T$ be a compact oriented torus, let $p\in T$ be a point, and let $T^*$ be $T - \{p\}$.

In Farb-Margalit's Primer on mapping class groups, in the discussion after Proposition 1.5 they say that "the homotopy classes of simple closed curves in $T^*$ are in bijective correspondence with those in $T$", but they give no hint of a proof.

I presume they left out the word "essential"? Even if we're talking about essential simple closed curves, what is this bijection? The only natural thing is to look at the map induced by the inclusion $T^*\hookrightarrow T$, but then this would mean that in each coset of the commutator subgroup of $\pi_1(T^*,q)$ (for some base point $q\in T^*$), there is a distinguished class which can be represented by an essential simple closed curve?

My questions are: What is the right statement (and how do you prove it)? Also, given an element $w\in\pi_1(T^*,q)$ written as a word in $x,y$ for some nice choice of basis $x,y\in\pi_1(T^*,q)$, is it possible to recognize when $w$ has a representative which is a simple closed curve? an essential simple closed curve?

I'm a novice in this area, so references would be appreciated as well!


There is exactly one simple closed curve in each primitive homology class in $T^\ast$ and that curve is the shortest one in the homology class. Conversely, the simple closed curves on $T$ correspond exactly to the primitive homology classes (primitive = class $(a, b)$ where gcd of $a, b$ (sadly, also denoted by $(a, b)$ usually) equals $1$). For references, see the paper

McShane, Greg; Rivin, Igor, A norm on homology of surfaces and counting simple geodesics, Int. Math. Res. Not. 1995, No. 2, 61-69 (1995). ZBL0828.30023.

and references therein.

(the existence of a bijective correspondence in and of itself is not interesting since both are countable sets).

  • $\begingroup$ No, $W_{m, n}$ are the so called Sturmian words. Draw the line from $0$ to $(m, n)$ and look at the sequence of intersection with horizontal and vertical grid lines... $\endgroup$ – Igor Rivin Jan 27 at 15:50

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