Braid groups and mapping class groups for 2-orbifolds

The main result of this article is that pure orbifold braid groups fit into an exact sequence $1\to K\to {\pi }_1^{orb}({\mathrm{\Sigma }}_{\mathrm{\Gamma }}(n-1+L))\stackrel{{\iota }_{{\text{PZ}}_n}}{\to }{\text{PZ}}_n({\mathrm{\Sigma }}_{\mathrm{\Gamma }}(L))\stackrel{{\pi }_{{\text{PZ}}_n}}{\to }{\text{PZ}}_{n-1}({\mathrm{\Sigma }}_{\mathrm{\Gamma }}(L))\to 1.$ In particular, we observe that the kernel $K$ of ${\iota }_{{\text{PZ}}_n}$ is non-trivial. This corrects Theorem 2.14 in 12. Moreover, we use the presentation of the pure orbifold mapping class group ${\text{PMap}}_n^{\text{id},orb}({\mathrm{\Sigma }}_{\mathrm{\Gamma }}(L))$ 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 $K$.