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Milnor invariants of braids and welded braids up to homotopy

Jacques Darné

Algebraic & Geometric Topology 24 (2024) 1277–1320

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 PΣn. It has been shown recently that its quotient by the homotopy relation identifies with the group hPΣn 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.

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
Received: 1 April 2020
Revised: 1 July 2022
Accepted: 2 January 2023
Published: 28 June 2024
Jacques Darné
Université de Picardie Jules Verne

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