Milnor invariants of braids and welded braids up to homotopy

Darné, Jacques

doi:10.2140/agt.2024.24.1277

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Milnor invariants of braids and welded braids up to homotopy

Jacques Darné

Algebraic & Geometric Topology 24 (2024) 1277–1320

DOI: 10.2140/agt.2024.24.1277

Abstract

We consider the group of pure welded braids (also known as loop braids) up to (link-)homotopy. The pure welded braid group classically identifies, via the Artin action, with the group of basis-conjugating automorphisms of the free group, also known as the McCool group $P Σ_{n}$ . It has been shown recently that its quotient by the homotopy relation identifies with the group $h P Σ_{n}$ of basis-conjugating automorphisms of the reduced free group. We describe a decomposition of this quotient as an iterated semidirect product which allows us to solve the Andreadakis problem for this group, and to give a presentation by generators and relations. The Andreadakis equality can be understood, in this context, as a statement about Milnor invariants; a discussion of this question for classical braids up to homotopy is also included.

Keywords

braids, welded braids, link homotopy, Milnor invariants, central filtrations, Lie algebras, Johnson morphisms

Mathematical Subject Classification

Primary: 20F12, 20F14, 20F18, 20F28, 20F36

Secondary: 16W25, 16W70, 20F05

References

Publication

Received: 1 April 2020

Revised: 1 July 2022

Accepted: 2 January 2023

Published: 28 June 2024

Authors

Jacques Darné
LAMFA Université de Picardie Jules Verne Amiens France
https://sites.google.com/view/jacques-darne/

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Volume 24, issue 3 (2024)