Filling of closed Surfaces

TL;DR

This paper classifies minimal fillings of genus 2 surfaces with a single disc complement, proves the minimal number of discs for filling pairs, and constructs fillings with specified disc counts for higher genus surfaces.

Contribution

It provides a complete classification of minimal fillings for genus 2 surfaces and constructs fillings with arbitrary disc counts for higher genera, advancing understanding of surface fillings and mapping class group actions.

Findings

Minimal fillings of $F_2$ have 3 or 4 curves.

Unique minimal fillings up to mapping class group action.

Constructed fillings with specified disc counts for higher genus.

Abstract

Let $F_g$ denote a closed oriented surface of genus $g$. A set of simple closed curves is called a filling of $F_g$ if its complement is a disjoint union of discs. The mapping class group $\text{Mod}(F_g)$ of genus $g$ acts on the set of fillings of $F_g$. The union of the curves in a filling forms a graph on the surface which is a so-called decorated fat graph. It is a fact that two fillings of $F_g$ are in the same $\text{Mod}(F_g)$-orbit if and only if the corresponding fat graphs are isomorphic. We prove that any filling of $F_2$ whose complement is a single disc (i.e., a so-called minimal filling) has either three or four closed curves and in each of these two cases, there is a unique such filling up to the action of $\text{Mod}(F_2)$. We provide a constructive proof to show that the minimum number of discs in the complement of a filling pair of $F_2$ is two. Finally, given positive integers $g$ and $k$ with $(g, k)\neq (2, 1)$, we construct a filling pair of $F_g$ such that the complement is a union of $k$ topological discs.