Braid groups and mapping class groups for 2-orbifolds

Abstract

The main result of this article is that pure orbifold braid groups fit into an exact sequence $1\rightarrow K\rightarrow\pi_1^{orb}(\Sigma_\Gamma(n-1+L))\xrightarrow{\iota_{\textrm{PZ}_n}}\textrm{PZ}_n(\Sigma_\Gamma(L))\xrightarrow{\pi_{\textrm{PZ}_n}}\textrm{PZ}_{n-1}(\Sigma_\Gamma(L))\rightarrow1.$ In particular, we observe that the kernel $K$ of $\iota_{\textrm{PZ}_n}$ 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 $\textrm{PMap}^{\textrm{id},orb}_n(\Sigma_\Gamma(L))$ 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 $K$.