Milnor invariants of braids and welded braids up to homotopy

Invariants de Milnor des tresses et des tresses soudées à homotopie près

Jacques Darné (1, 2)

1 IF - Institut Fourier
2 IMB - Institut de Mathématiques de Bourgogne [Dijon]

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 hP Σ n of basis-conjugating automorphisms of the reduced free group. In the present paper, we describe a decomposition of this quotient as an iterated semi-direct 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.

Domaines: Topologie géométrique [math.GT]; Topologie algébrique [math.AT]; Théorie des groupes [math.GR]

Jacques Darné. Milnor invariants of braids and welded braids up to homotopy. 2020. ⟨hal-02105277v2⟩