Liftable braids and the coloured braid groupoid

Abstract

When $\ddot{I}€:\tilde{\hat{I}}\pounds \to D^2$ 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 ${\hat{I}}^2$ \textit{lifts} if there is a homeomorphism ${\tilde{\hat{I}}}^2\in \text{Mod}(\tilde{\hat{I}}\pounds )$ such that ${\hat{I}}^2\circ \ddot{I}€=\ddot{I}€\circ {\tilde{\hat{I}}}^2$. For arbitrary covers, the \textit{lifting homomorphism} taking ${\hat{I}}^2$ to ${\tilde{\hat{I}}}^2$ 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.

22 pages, 13 figures