Forme de Blanchfield et cobordisme d'entrelacs bord
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.
scientific article; zbMATH DE number 4028429