Consider a compact Riemann surface of genus $g\geq2$. An **admissible system** of Jordan curves is a finite collection of Jordan curves $\{\gamma\_1,\cdots,\gamma\_n\}$ such that


1. they are nonintersecting with each other;
2. no two are freely homotopic;
3. none is homotopic to 0


This concept is mentioned in Masur's *on a class of geodesic in teichmuller space* and in this paper the author mentions that an admissible system has at most $3g-3$ Jordan curves. I am wondering why and how to prove it.


My attempt would be to consider a fundamental polygon with $4g$ sides, consecutively labeled by
$$a\_1,b\_2,a\_1^{-1},b\_1^{-1},\cdots,b\_g^{-1}$$
Now choose a point $p$ on the edge $a\_1$, then $p$ appears on $a\_1^{-1}$ as well. Connecting them gives a closed curve. We cannot do the same for $b\_1$, but we can do this for $a\_2,a\_3,\cdots,a\_g$. This gives $g$ curves. Finally an edge is also a closed curve. Now any curve not null-homotopic should cross an edge of the fundamental polygon, but I cannot seem to find another one without intersecting the existing ones. So I can find at most $g+1$ curves. Where do the $3g-3$ curves come from?



Number of curves in an admissible system of Jordan curves on a surface
It comes from the so called "pants decomposition". A "pair of pants" (or simply pants) is a sphere with $3$ holes. Every compact surface of genus $g\geq 2$ can be decomposed into such pants. The $3g-3$ curves are the pants boundaries.


To visualize, represent your
surface in the following way. Consider the sphere with $g+1$ holes, and your surface
is obtained by gluing this to its double (mirror image). Now decompose the
sphere with $g+1$ holes into pants. To do this you draw $g-2$ disjoint closed curves.
The mirror image will contain the same number of them. So the total is
$$(g+1)+2(g-2)=3g-3.$$


