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2020 Boundary braids
Michael Dougherty, Jon McCammond, Stefan Witzel
Algebr. Geom. Topol. 20(7): 3505-3560 (2020). DOI: 10.2140/agt.2020.20.3505


The n–strand braid group can be defined as the fundamental group of the configuration space of n unlabeled points in a closed disk based at a configuration where all n points lie in the boundary of the disk. Using this definition, the subset of braids that have a representative where a specified subset of these points remain pointwise fixed forms a subgroup isomorphic to a braid group with fewer strands. We generalize this phenomenon by introducing the notion of boundary braids. A boundary braid is a braid that has a representative where some specified subset of the points remains in the boundary cycle of the disk. Although boundary braids merely form a subgroupoid rather than a subgroup, they play an interesting geometric role in the piecewise Euclidean dual braid complex defined by Tom Brady and the second author. We prove several theorems in this setting, including the fact that the subcomplex of the dual braid complex determined by a specified set of boundary braids metrically splits as the direct metric product of a Euclidean polyhedron and a dual braid complex of smaller rank.


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Michael Dougherty. Jon McCammond. Stefan Witzel. "Boundary braids." Algebr. Geom. Topol. 20 (7) 3505 - 3560, 2020.


Received: 23 January 2019; Revised: 3 February 2020; Accepted: 24 February 2020; Published: 2020
First available in Project Euclid: 5 January 2021

MathSciNet: MR4194287
Digital Object Identifier: 10.2140/agt.2020.20.3505

Primary: 20F36
Secondary: 20F65

Keywords: braid groups

Rights: Copyright © 2020 Mathematical Sciences Publishers


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Vol.20 • No. 7 • 2020
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