Homology boundary links and Blanchfield forms: Concordance classification and new tangle-theoretic constructions
The main motivating open problem is the classification of the set of concordance or cobordism classes of links (a link is a collection of a finite number \(m\) of nicely embedded \(n\)-spheres in the \((n + 2)\)- sphere). This classification is known for knots (the case \(m = 1)\) in dimensions \(n > 1\). Also, the classification of boundary links (i.e. admitting \(m\) disjoint Seifert surfaces) modulo boundary link cobordism is known. However, not every link is concordant to a boundary link; in fact, some years ago the present authors defined new invariants which showed that many homology boundary links are not concordant to any boundary link (it is unknown whether or not every link is concordant to a homology boundary link). The two ingredients of these invariants were a complexity associated to a ``pattern'' \(P\) and a function of the universal Blanchfield form \(B\) of the homology boundary link, which may also be viewed in terms of cobordism classes of Seifert matrices. ``The primary aim of the present paper is to show that any such pair \((P,B)\) may be realized by an explicit geometric construction. The main theorem is a corollary of a new and interesting construction for links that is useful in creating homology boundary links with prescribed properties. An important secondary goal is to classify homology boundary links modulo a suitable cobordism relation.'' For definitions and more details we recommend the extensive and carefully written introduction of the paper.
scientific article; zbMATH DE number 633645