A presentation for Hilden's subgroup of the braid group
Stephen Tawn
TL;DR
This paper provides a finite presentation for the mapping class group of a 3D ball with n unknotted arcs, using its action on a simply-connected complex to analyze its structure.
Contribution
It introduces a new finite presentation for the subgroup of the braid group related to Hilden's subgroup, based on topological and combinatorial methods.
Findings
Finite presentation of the mapping class group established.
Utilizes the action on a simply-connected complex for derivation.
Provides tools for understanding the structure of braid-related subgroups.
Abstract
Consider the unit ball, B = D x [0,1], containing n unknotted arcs a_1, ... a_n such that the boundary of each a_i lies in D x {0}. We give a finite presentation for the mapping class group of B fixing the arcs {a_1, ..., a_n} setwise and fixing D x {1} pointwise. This presentation is calculated using the action of this group on a simply-connected complex.
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Taxonomy
TopicsGeometric and Algebraic Topology · Homotopy and Cohomology in Algebraic Topology · Algebraic Geometry and Number Theory