Volume 20, issue 7 (2020)

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Boundary braids

Michael Dougherty, Jon McCammond and Stefan Witzel

Algebraic & Geometric Topology 20 (2020) 3505–3560

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.

braid groups
Mathematical Subject Classification 2010
Primary: 20F36
Secondary: 20F65
Received: 23 January 2019
Revised: 3 February 2020
Accepted: 24 February 2020
Published: 29 December 2020
Michael Dougherty
Department of Mathematics and Statistics
Swarthmore College
Swarthmore, PA
United States
Jon McCammond
Department of Mathematics
University of California, Santa Barbara
Santa Barbara, CA
United States
Stefan Witzel
Mathematical Institute
Gießen University