Belts and \(\kappa{}\)-invariants of link maps in spheres
A link map \(f_ 1 + \dots + f_ r: S^{p_ 1} + \dots + S^{p_ r} \to S^ m\) is a collection of maps from several spheres into a single sphere such that the \(f_ i(S^{p_ i})\) are disjoint but each \(f_ i\) may be singular. Link homotopy is a homotopy of such maps. One can also consider link maps into \(R^ m\) -- these two types of link maps are equivalent as long as each \(p_ i \leq m - 2\), but otherwise they will be different. The inclusion \(R^ m \subseteq S^ m\) induces a function between these link homotopy sets and, if we choose a point in the complement of the image of a link map into \(S^ m\), then stereographic projection produces a link map into \(R^ m\). The present work studies these functions between the two link homotopy sets. In particular, the link homotopy invariants (\(\kappa\) and \(\kappa'\) invariants) obtained by considering the natural maps of \(S^{p_ 1} \times \dots \times S^{p_ r}\) into the configuration spaces of \(r\) distinct points in \(R^ m\) and \(S^ m\) are studied and compared.
scientific article; zbMATH DE number 219900