Homotopy invariants of links
type: Item
instance of: scholarly article
title: Homotopy invariants of links
zbMATH Open document ID: 0668.57014
DOI: 10.1007/BF01393902
author: Kent E. Orr
published in: Inventiones Mathematicae
publication date: 1989-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143657
review text: In this paper the author introduces a concordance invariant of links, \(\theta_ k\), which is defined whenever all the Milnor \({\bar \mu}\) invariants of length \(\leq k\) vanish. Equivalently, \(\theta_ k\) is defined whenever the lower central series quotients of the fundamental group up to order k are indistinguishable from those of a trivial link. Various facts about \(\theta_ k\) make it possible to give a precise formula for the number of independent Milnor numbers of length \(k+1.\) The details are as follows: Let L be a link with m components in \(S^ 3\). Let \(\pi\) denote the fundamental group of the complement, F denote the free group on m letters, and \(\pi /\pi^{(k)}\), \(F/F^{(k)}\) denote the lower central series quotients for \(k=1,2,...,\omega\). Let \(\{\bigvee^{m}_{i=1}S^ 1\subset S^ 3-L\},\) denoted by \(\tau\), be a choice of meridians for L. This induces a homomorphism \(F\to \pi\). If all Milnor numbers of length \(\leq k\) vanish, this induces an isomorphism \(F/F^{(k)}\cong \pi /\pi^{(k)}\). Let K be the Eilenberg-MacLane space \(K(F/F^{(k)},1)\) obtained from \(\bigvee^{m}_{i=1}S^ 1\) by adding cells of dimension \(\geq 2\) and let \(K_ k=K/\bigvee^{m}_{i=1}S^ 1\). Extension over meridianal disks gives a canonical map \(\rho\) : \(S^ 3\to K_ k\). Define \(\theta_ k(L,\tau)=[\rho]\in \pi_ 3(K_ k)\). The following properties of \(\theta_ k\) hold: Theorem: 1. \(\theta_ k(L,\tau)=0\) if L is a boundary link. 2. If \(k>\ell\) there is a homomorphism \(\psi_{k,\ell}=\pi_ 3(K_{\ell})\to \pi_ 3(K_{\ell})\) such that \(\psi_{k,\ell}(\theta_ k(L,\tau))=\theta_{\ell}(L,\tau)\). 3. If \(\tau_ 1\), \(\tau_ 2\) are two choices of meridians for L then there is an automorphism \(\alpha\) of \(\pi_ 3(K_ k)\) such that \(\alpha (\tau_ 1)=\tau_ 2\). 4. \(\theta_ k(L,\tau)\) is additive under connected sum of based links. 5. If \(x\in \pi_ 3(K_ k)\) then there is a link L such that \(\theta_ k(L,\tau)\) is defined and equals x. If \(L_ 1\) and \(L_ 2\) are t…
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 57M05
zbMATH DE Number: 4094402
zbMATH Keywords: Milnor invariants
zbMATH Keywords: concordance invariant of links
zbMATH Keywords: lower central series quotients of the fundamental group
zbMATH Keywords: boundary link
zbMATH Keywords: choices of meridians
zbMATH Keywords: connected sum of based links
MaRDI profile type: Publication
cites work: Homological localization towers for groups and Π-modules
cites work: Homotopy limits, completions and localizations
cites work: Link cobordism
cites work: Isotopy invariants of links
cites work: Link concordance invariants and homotopy theory
cites work: Homology, Massey products and maps between groups
cites work: A new technique for the link slice problem
cites work: Cohomological topics in group theory.
cites work: Q4135772
cites work: Link groups
cites work: Higher Order Massey Products and Links
cites work: New link invariants and applications
cites work: Milnor's & &nbsp; 956; -Invariants and Massey Products
cites work: Homology and central series of groups
cites work: Milnor's Invariants and the Completions of Link Modules
cites work: Q4125490
OpenAlex ID: W1989031560
description: scientific article; zbMATH DE number 4094402