Liftable braids and the coloured braid groupoid
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866918116667162624 |
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| author | Licata, Joan Vértesi, Vera |
| author_facet | Licata, Joan Vértesi, Vera |
| contents | When $π:\widetildeΣ\rightarrow D^2$ is a cover of the disc branched over $n$ marked points, the braid group $B_n$ acts on the disc by homeomorphisms fixing the marked points setwise. A braid $β$ \textit{lifts} if there is a homeomorphism $\widetildeβ\in \textit{Mod}(\widetildeΣ)$ such that $β\circ π=π\circ \widetildeβ$. For arbitrary covers, the \textit{lifting homomorphism} taking $β$ to $\widetildeβ$ is only defined on a proper subgroup of the braid group. This paper extends the lifting homomorphism to a map from a coloured braid groupoid to a mapping class groupoid for all simple covers of the disc. We characterise the lift of every coloured braid, recovering the classical lifting homomorphism on the liftable braid group. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_05146 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Liftable braids and the coloured braid groupoid Licata, Joan Vértesi, Vera Geometric Topology Combinatorics Group Theory Representation Theory 57K10, 57M12, 20F36, 57M07, 57K20, 20F65 When $π:\widetildeΣ\rightarrow D^2$ is a cover of the disc branched over $n$ marked points, the braid group $B_n$ acts on the disc by homeomorphisms fixing the marked points setwise. A braid $β$ \textit{lifts} if there is a homeomorphism $\widetildeβ\in \textit{Mod}(\widetildeΣ)$ such that $β\circ π=π\circ \widetildeβ$. For arbitrary covers, the \textit{lifting homomorphism} taking $β$ to $\widetildeβ$ is only defined on a proper subgroup of the braid group. This paper extends the lifting homomorphism to a map from a coloured braid groupoid to a mapping class groupoid for all simple covers of the disc. We characterise the lift of every coloured braid, recovering the classical lifting homomorphism on the liftable braid group. |
| title | Liftable braids and the coloured braid groupoid |
| topic | Geometric Topology Combinatorics Group Theory Representation Theory 57K10, 57M12, 20F36, 57M07, 57K20, 20F65 |
| url | https://arxiv.org/abs/2508.05146 |