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| Format: | Preprint |
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2019
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| Online Access: | https://arxiv.org/abs/1904.10677 |
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| _version_ | 1866913421965918208 |
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| author | Darné, Jacques |
| author_facet | Darné, Jacques |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1904_10677 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Milnor invariants of braids and welded braids up to homotopy Darné, Jacques Algebraic Topology Group Theory 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. |
| title | Milnor invariants of braids and welded braids up to homotopy |
| topic | Algebraic Topology Group Theory |
| url | https://arxiv.org/abs/1904.10677 |