Forme de Blanchfield et cobordisme d'entrelacs bord
type: Item
instance of: scholarly article
title: Forme de Blanchfield et cobordisme d'entrelacs bord
zbMATH Open document ID: 0632.57014
DOI: 10.1007/BF02621935
author: Julien Duval
published in: Commentarii Mathematici Helvetici
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/140071
review text: In this paper much of the algebraic structure and properties of knot modules are shown to extend to ``boundary link modules''. A boundary link is a collection of m disjoint n-spheres in \((n+2)\)-space (n\(\geq 1)\) whose components bound disjoint submanifolds. There is a canonical epimorphism \(\pi \to F_ m\), where \(\pi\) is the fundamental group of the complement and \(F_ m\) the free group of rank m. The homology of the associated free covering is a collection of modules \(\{H_ i\}\) over A, the group ring of \(F_ m\). It is proved here that they satisfy duality properties: \(\overline{t(H_ i)}\approx e^ 2(t(H_{n-i}))\), \(\overline{f(H_ i)}\approx e^ 1(f(H_{n+1-i}))\), where t, f refer to the Z-torsion submodule and Z-torsion-free quotient, respectively. Furthermore, when n is odd, the middle-dimensional duality on \(f(H_ i)\) is induced by a nonsingular \(\pm\)-Hermitian pairing generalizing the Blanchfield pairing for knots. This pairing takes values in the quotient \(\Lambda\) /A, where \(\Lambda\) is a suitable localization of A. Two theorems concerning this pairing extend analogous theorems for knots: (1) Any algebraically permissible pairing can be realized by a boundary link; (2) the group of concordance classes of boundary links is isomorphic to a Witt group of pairings.
Mathematics Subject Classification ID: 57Q45
Mathematics Subject Classification ID: 11E16
Mathematics Subject Classification ID: 57R90
zbMATH DE Number: 4028429
zbMATH Keywords: boundary link modules
zbMATH Keywords: fundamental group of the complement
zbMATH Keywords: associated free covering
zbMATH Keywords: Hermitian pairing
zbMATH Keywords: Blanchfield pairing
zbMATH Keywords: concordance classes of boundary links
zbMATH Keywords: Witt group
reviewed by: Jerome Levine
MaRDI profile type: Publication
OpenAlex ID: W2054329125
description: scientific article; zbMATH DE number 4028429