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| Main Authors: | , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2503.04116 |
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| _version_ | 1866916645149081600 |
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| author | Kumar, Rakesh Parsad, Shiv |
| author_facet | Kumar, Rakesh Parsad, Shiv |
| contents | Let $S_g$ be a closed orientable surface of genus $g\geq 2$. A collection $Ω= \{ γ_1, \dots, γ_s\}$ of pairwise non-homotopic simple closed curves on $S_g$ such that $γ_i$ and $γ_j$ are in minimal position, is called a \emph{filling system} or a \emph{filling} of $S_g$ if the complement $S_g\setminus Ω$ is a disjoint union of $b$ topological discs for some $b\geq 1$. The \emph{size} of a filling system is defined as the number of its elements. We prove that the maximum size of a filling system on $S_g$ with $ 1 \leq b \leq 2g-2$ boundary components is $2g+b-1$. Furthermore, we give a lower bound on mapping class group orbits of filling systems of maximum size with $ 1 \leq b \leq g-2$ boundary components. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_04116 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Filling systems of maximum size Kumar, Rakesh Parsad, Shiv Geometric Topology 57M15, 05C10 Let $S_g$ be a closed orientable surface of genus $g\geq 2$. A collection $Ω= \{ γ_1, \dots, γ_s\}$ of pairwise non-homotopic simple closed curves on $S_g$ such that $γ_i$ and $γ_j$ are in minimal position, is called a \emph{filling system} or a \emph{filling} of $S_g$ if the complement $S_g\setminus Ω$ is a disjoint union of $b$ topological discs for some $b\geq 1$. The \emph{size} of a filling system is defined as the number of its elements. We prove that the maximum size of a filling system on $S_g$ with $ 1 \leq b \leq 2g-2$ boundary components is $2g+b-1$. Furthermore, we give a lower bound on mapping class group orbits of filling systems of maximum size with $ 1 \leq b \leq g-2$ boundary components. |
| title | Filling systems of maximum size |
| topic | Geometric Topology 57M15, 05C10 |
| url | https://arxiv.org/abs/2503.04116 |