Homotopy braid groups are torsion-free

TL;DR

This paper proves that the group of braids up to link-homotopy is torsion-free for any number of components, extending previous results and providing new insights into braid group properties.

Contribution

It generalizes Humphries' torsion-free result to all components and offers an explicit solution to the existence of non-abelian torsion-free quotients of braid groups.

Findings

Link-homotopy braid groups are torsion-free for any number of components

Provides an alternative proof that braid groups are torsion-free

Answers a question on non-abelian torsion-free quotients of braid groups

Abstract

We show that, for any number of components, the group of braids up to link-homotopy is torsion-free. This generalizes a result of Humphries up to six components, and provides an explicit solution to a question posed by Lin and addressed by Linell and Schick regarding the existence of non-abelian torsion-free quotients of the braid group. The proof relies on the diagrammatic theory of welded braids and uses the Artin representation. As a corollary, we obtain yet another proof that braid groups themselves are torsion-free.