Link concordance and algebraic closure. II
type: Item
instance of: scholarly article
title: Link concordance and algebraic closure. II
zbMATH Open document ID: 0692.57010
DOI: 10.1007/BF01393697
published in: Inventiones Mathematicae
publication date: 1989-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143688
review text: This is the sequel to Part I reviewed above (see Zbl 0692.57009). The author defines a group G to be algebraically closed (AC) if each finite system of equations \(x_ i=w_ i\in G*F(x_ 1,...,x_ n)\), where each \(w_ i\) is a product of conjugates of elements of G, has a unique solution in G. For each group G he proves the existence of an algebraic closure \(G\to \hat G\) satisfying the universal property (in the usual sense) with respect to homomorphisms into AC-groups. Then he establishes the following ``localization'' properties: 1) \(G\to \hat G\) is normally surjective and homologically 2-connected. 2) For any normally surjective homomorphism \(\phi\) : \(G\to H\) (G finitely generated, H finitely presented), the induced homomorphism \(G\to \hat H\) is homologically 2- connected. The algebraic closure is the algebraic localization functor corresponding to the (topological) Vogel localization, in fact: if X is a finite complex with fundamental group G, then \(\hat G\) is the fundamental group of the Vogel localization of X. In the second part the author applies the concept of algebraic closure to link theory in \(S^ 3\). He defines the notion of \(\hat F-\)link (satisfying a condition ``ostensibly'' stronger than vanishing \({\bar \mu}\)-invariants; ``..'' is related to the canonical map \(\hat F\to \tilde F\) \(=\) nilpotent completion of the free group F). For those links a (based) concordance invariant \(\theta (L)\in H_ 3(\hat F)\) is defined. The author proves that for an \(\hat F-\)link L, \(\theta (L)=0\) if and only if L is concordant to a sublink of a homology boundary link. Moreover, each element in \(H_ 3\hat F\) is realized as \(\theta\) (L) for some \(\hat F-\)link L. In an appendix the author mentions several open problems relating interesting questions concerning the algebraic closure to important questions in link concordance.
Mathematics Subject Classification ID: 57Q45
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 57M05
zbMATH DE Number: 4133129
zbMATH Keywords: algebraically closed group
zbMATH Keywords: nilpotent completion of free group
zbMATH Keywords: algebraic closure
zbMATH Keywords: algebraic localization functor
zbMATH Keywords: Vogel localization
zbMATH Keywords: fundamental group of the Vogel localization
zbMATH Keywords: \^F-link
zbMATH Keywords: homology boundary link
zbMATH Keywords: link concordance
author: Jerome Levine
reviewed by: Uwe Kaiser
MaRDI profile type: Publication
cites work: Homological localization towers for groups and Π-modules
cites work: Link concordance invariants and homotopy theory
cites work: The codimension two placement problem and homology equivalent manifolds
cites work: Link cobordism
cites work: Q3697912
cites work: Concordance and homotopy. I: Fundamental group
cites work: Unlinking up to cobordism
cites work: Trivializing ribbon links by Kirby moves
cites work: Cobordism of Knots and Blanchfield Duality
cites work: Les nœuds de dimensions supérieures
cites work: A calculus for framed links in \(S^3\)
cites work: Seifert Matrices and Boundary Link Cobordisms
cites work: Link concordance and algebraic closure of groups
cites work: Knot cobordism groups in codimension two
cites work: Q3247621
cites work: New link invariants and applications
cites work: Cobordisms of semi-boundary links
cites work: Q5555545
cites work: Homology and central series of groups
cites work: Q3769080
cites work: A Note on Algebraically Closed Groups
cites work: The topology of discrete groups
OpenAlex ID: W2063261219
description: scientific article; zbMATH DE number 4133129