Liftable braids and the coloured braid groupoid

TL;DR

This paper extends the classical lifting homomorphism from braid groups to a broader coloured braid groupoid for simple covers of the disc, providing a comprehensive characterization of liftable braids.

Contribution

It introduces a generalized lifting homomorphism from a coloured braid groupoid to a mapping class groupoid applicable to all simple disc covers.

Findings

Extended the lifting homomorphism to coloured braid groupoids.

Characterized the lift of every coloured braid.

Recovered the classical lifting homomorphism on liftable braids.

Abstract

When $\pi:\widetilde{\Sigma}\rightarrow D^2$ 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 $\beta$ \textit{lifts} if there is a homeomorphism $\widetilde{\beta}\in \textit{Mod}(\widetilde{\Sigma})$ such that $\beta\circ \pi=\pi\circ \widetilde{\beta}$. For arbitrary covers, the \textit{lifting homomorphism} taking $\beta$ to $\widetilde{\beta}$ 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.