Lifting of homeomorphisms to branched coverings of a disk
The aim of the paper under review is to provide explicit sets of generators for special subgroups of the braid group \(B_n\) which are of geometric interest. Here the braid group \(B_n\) is considered as the group of isotopy classes of homeomorphisms of the \(2\)-disc which fix point-wise the boundary and permute a finite set of \(n\) marked points.NEWLINENEWLINEConsider \(d\)-sheeted, not necessarily connected, branched coverings of the \(2\)-disc which are \textsl{simple}, that is their fibres are either generic and consist of \(d\) points or contain precisely \(d-1\) points (one of them being a \textsl{branch point}). There is only a finite number of non-generic fibres which are assumed to be precisely the fibres above the \(n\) marked points of the disc.NEWLINENEWLINEFor each such branched covering one can consider the subgroup of \(B_n\) whose elements correspond to homeomorphisms of the disc which lift to the covering in such a way that a fibre over a fixed base-point is fixed point-wise. Note that, if a homeomorphism lifts to the covering, so do all the homeomorphisms in its isotopy class. These are the special subgroups considered in the paper.NEWLINENEWLINEThe authors prove that these groups are generated by powers of half-twists along well-specified arcs, by considering two cases: The case where the covering is connected is obtained by induction on the order of the covering, while the general case is reduced to the connected one.NEWLINENEWLINEThis result generalises work by \textit{J. S. Birman} and \textit{B. Wajnryb} [Geometry and topology, Proc. Spec. Year, College Park/Md. 1983-84, Lect. Notes Math. 1167, 24--46 (1985; Zbl 0589.57009)] and by the two authors [Topology Appl. 155, 1820--1839 (2008; Zbl 1154.57002)] on connected coverings of degrees \(3\) and \(4\), and work by \textit{F. Catanese} and \textit{B. Wajnryb} [Topology 30, 641--651 (1991; Zbl 0755.57001)] and by \textit{M. Mulazzani} and \textit{R. Piergallini} [Rend. Ist. Mat. Univ. Trieste 32, Suppl. 1, 193--219 (2001; Zbl 1001.57004)] on connected coverings of degree \(n+1\).
scientific article; zbMATH DE number 6035687
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