Filling systems on surfaces

Abstract

Let $F_g$ be a closed orientable surface of genus $g$. A set $\mathrm{\Omega }=\{{\gamma }_1,\dots ,{\gamma }_s\}$ of pairwise non-homotopic simple closed curves on $F_g$ is called a filling system or simply be filling of $F_g$, if $F_g\setminus \mathrm{\Omega }$ is a disjoint union of $b$ topological discs for some $b\ge 1$. A filling system is called minimally intersecting, if the total number of intersection points of the curves is minimum, or equivalently $b=1$. The size of a filling system is defined as the number of its elements. We prove that the maximum possible size of a minimally intersecting filling system is $2g$. Next, we show that for $g\ge 2\text{ and }2\le s\le 2g$ with $(g,s)\ne (2,2)$, there exists a minimally intersecting filling system on $F_g$ of size $s$. Furthermore, we study geometric intersection number of curves in a minimally intersecting filling system. For $g\ge 2$, we show that for a minimally intersecting filling system $\mathrm{\Omega }$ of size $s$, the geometric intersection numbers satisfy $max\{i({\gamma }_i,{\gamma }_j)|i\ne j\}\le 2g-s+1$, and for each such $s$ there exists a minimally intersecting filling system $\mathrm{\Omega }=\{{\gamma }_1,\dots ,{\gamma }_s\}$ such that $max\{i({\gamma }_i,{\gamma }_j)|i\ne j\}=2g-s+1$.