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Milnor invariants of braids and welded braids up to homotopy

Jacques Darné

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 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.

Milnor invariants of braids and welded braids up to homotopy

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

Paper Structure

This paper contains 36 sections, 52 theorems, 73 equations.

Table of Contents

  1. Acknowledgements:
  2. Reminders: strongly central series and Lie rings
  3. A very short introduction to the Andreadakis problem
  4. Lazard's theorem Lazard (see also Darne1):
  5. Semi-direct product decompositions Darne2:
  6. The Johnson morphism Darne1:
  7. The case of the free group
  8. Magnus expansions:
  9. Magnus' theorem:
  10. The Andreadakis problem and the Johnson morphism:
  11. The reduced free group and its Lie algebra
  12. The reduced free group
  13. The reduced free algebra
  14. Centralizers
  15. Derivations and the Johnson morphism
  16. ...and 21 more sections

Key Result

Proposition 1.3

For any integer $n$, the group $RF_n$ is $n$-nilpotent. For any set $X$, the group $RF[X]$ is residually nilpotent.

Theorems & Definitions (120)

  • Definition 1
  • Definition 3
  • Definition 1.2
  • Proposition 1.3
  • proof
  • Definition 1.4
  • proof
  • Definition 1.6
  • Proposition 1.7
  • proof
  • ...and 110 more