Year: 2023
Type: preprint
Abstract: The main result of this article is that pure orbifold braid groups fit into an exact sequence $1\rightarrow K\rightarrowπ_1^{orb}(Σ_Γ(n-1+L))\xrightarrow{ι_{\textrm{PZ}_n}}\textrm{PZ}_n(Σ_Γ(L))\xrightarrow{π_{\textrm{PZ}_n}}\textrm{PZ}_{n-1}(Σ_Γ(L))\rightarrow1.$ In particular, we observe that the kernel $K$ of $ι_{\textrm{PZ}_n}$ is non-trivial. This corrects Theorem 2.14 in [12](arXiv:2006.07106). Moreover, we use the presentation of the pure orbifold mapping class group $\textrm{PMap}^{\textrm{id},orb}_n(Σ_Γ(L))$ from [8] to determine $K$. Comparing these orbifold mapping class groups with the orbifold braid groups, reveals a surprising behavior: in contrast to the classical case, the orbifold braid group is a proper quotient of the orbifold mapping class group. This yields a presentation of the pure orbifold braid group which allows us to read off the kernel $K$.
Source: arXiv (Cornell University)
Author: Flechsig, Jonas
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funder: Deutsche Forschungsgemeinschaft
OpenAlex ID: W4375958466