Brunnian braids on surfaces.
type: Item
instance of: scholarly article
title: Brunnian braids on surfaces.
zbMATH Open document ID: 1270.20036
author: Roman Mikhailov
author: Vladimir Vershinin
author: Jie Wu
author: Valeriy G. Bardakov
published in: Algebraic \& Geometric Topology
publication date: 2012-08-20T00:00:00Z
full work available at URL: https://arxiv.org/abs/0909.3387
review text: A Brunnian braid is a braid that becomes trivial after removing any one of its strands, and let \(\text{Brun}_n(M)\) be the subgroup of the \(n\)-strand Brunnian braids on a connected 2-manifold \(M\). In the paper under review, the authors investigate the relationships between \(\text{Brun}_n(M)\) and the symmetric commutator subgroup \(R_n(M)\) generated by normal closures of standard generators \(A_{i,n}\) of pure braid group \(P_n(D^2)\). That is, \(\text{Brun}_n(M)\) is isomorphic to \(R_n(M)\) when \(M\) is different from a sphere and real projective space; and it is a group extension of \(\pi_{n-1}(S^2)\) by \(R_n(M)\) otherwise. Moreover, they prove the finite presentability for \(P_n(M)/\text{Brun}_n(M)\) and \(B_n(M)/\text{Brun}_n(M)\) for \(n\geq3\).
Mathematics Subject Classification ID: 20F36
Mathematics Subject Classification ID: 57M07
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 55Q40
zbMATH DE Number: 6068608
zbMATH Keywords: pure braid groups
zbMATH Keywords: Brunnian braids
zbMATH Keywords: homotopy groups
zbMATH Keywords: symmetric commutator subgroup
zbMATH Keywords: finitely presented groups
reviewed by: Byunghee An
MaRDI profile type: Publication
Wikidata QID: Q116180360
OpenAlex ID: W2093474576
arXiv ID: 0909.3387
cites work: On presentations of surface braid groups.
cites work: Configurations, braids, and homotopy groups
cites work: Coproducts of crossed P-modules: applications to second homotopy groups and to the homology of groups
cites work: Van Kampen theorems for diagrams of spaces
cites work: A colimit of classifying spaces
cites work: Configuration Spaces.
cites work: The braid groups of \(E^ 2\) and \(S^ 2\)
cites work: The Word Problem and Consequences for the Braid Groups and Mapping Class Groups of the 2-Sphere
cites work: The braid groups of the projective plane.
cites work: Towards a characterization of smooth braids
cites work: Q4344080
cites work: Decomposable Braids and Linkages
cites work: Decomposable Braids as Subgroups of Braid Groups
cites work: Artin braid groups and homotopy groups
cites work: On symmetric commutator subgroups, braids, links and homotopy groups
cites work: Q5513066
cites work: Symmetric ideals in group rings and simplicial homotopy.
cites work: Lie algebras and Lie groups. 1964 lectures, given at Harvard University.
cites work: Braid Groups of Compact 2-Manifolds with Elements of Finite Order
cites work: Combinatorial descriptions of homotopy groups of certain spaces
DOI: 10.2140/AGT.2012.12.1607
description: scientific article; zbMATH DE number 6068608