Filling systems on surfaces

TL;DR

This paper investigates the maximum size of filling systems on surfaces, constructs fillings of all sizes within certain bounds, and analyzes the geometric intersection numbers of curves in minimal fillings.

Contribution

It establishes the maximum size of fillings with a given number of complementary discs and constructs fillings of all sizes within these bounds, also analyzing intersection numbers.

Findings

Maximum size of a filling with b discs is 2g+b-1.

Existence of fillings for all sizes between minimal and maximum.

Bounds on intersection numbers in minimal fillings.

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 $b\geq 1$. 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 $g\geq 2, b\geq 1\text{ with }(g,b)\neq (2,1)$ (resp. $(g,b)=(2,1)$) and for each $2\leq s\leq 2g+b-1$ (resp. $3\leq s\leq 2g+b-1$), 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 $g\geq 2$, we show that for a minimal filling $\Omega$ of size $s$, the \emph{geometric intersection numbers} satisfy $\max \left\lbrace i(\gamma_i, \gamma_j)| i\neq j\right\rbrace\leq 2g-s+1$, and for each such $s$ there exists a minimal filling $\Omega=\left\lbrace \gamma_1, \dots, \gamma_s \right\rbrace$ such that $\max\left\lbrace i(\gamma_i, \gamma_j) | i\neq j\right\rbrace = 2g-s+1$.