Liftable braids and the coloured braid groupoid
Joan Licata, Vera V\'ertesi

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 is a cover of the disc branched over marked points, the braid group acts on the disc by homeomorphisms fixing the marked points setwise. A braid \textit{lifts} if there is a homeomorphism such that . For arbitrary covers, the \textit{lifting homomorphism} taking to 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.
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