Filling systems on surfaces

Abstract

Let $F_g$ be a closed orientable surface of genus $g$. A set $\Omega =\{{\gamma }_1,\dots ,{\gamma }_s\}$ of pairwise non-homotopic simple closed curves on $F_g$ is called a \emph{filling system} or simply a \emph{filling} of $F_g$, if $F_g\setminus \Omega$ is a union of $b$ topological discs for some $b\ge 1$. A filling system is called \emph{minimal}, if $b=1$. The \emph{size} of a filling is defined as the number of its elements. We prove that the maximum size of a filling of $F_g$ with $b$ complementary discs is $2g+b-1$. Next, we show that for $g\ge 2,b\ge 1\text{ with }(g,b)\ne (2,1)$ (resp. $(g,b)=(2,1)$) and for each $2\le s\le 2g+b-1$ (resp. $3\le s\le 2g+b-1$), there exists a filling of $F_g$ of size $s$ with $b$ complementary discs. Furthermore, we study geometric intersection number of curves in a minimal filling. For $g\ge 2$, we show that for a minimal filling $\Omega$ of size $s$, the \emph{geometric intersection numbers} satisfy $\max \left\{i({\gamma }_i,{\gamma }_j)|i\ne j\right\}\le 2g-s+1$, and for each such $s$ there exists a minimal filling $\Omega =\left\{{\gamma }_1,\dots ,{\gamma }_s\right\}$ such that $\max \left\{i({\gamma }_i,{\gamma }_j)|i\ne j\right\}=2g-s+1$.

arXiv commentPrevious title has changed, Theorem 1.1 and Theorem 1.2 in the previous version are generalized, 17 pages, 8 figures