Link homotopy in \(\mathbb{R}{}^ 3\) and \(S^ 3\) (Q1183093)

A link map in \(\mathbb{R}^ 3\) (or \(S^ 3\)) is a map from several disjoint spheres into \(\mathbb{R}^ 3\) (or \(S^ 3\)) such that the images of different spheres are disjoint. A link homotopy is just a homotopy which is a link map at each stage. The case in which each sphere is a circle (and so we may restrict attention to imbedded links except that self-intersections occur within any component during a link homotopy) was first studied by Milnor and the classification completed by Habegger-Lin. In this paper the case when at least one of the spheres has dimension \(>1\) is considered, and a complete classification is obtained for the case of 2- component links, and for 3-component links in \(S^ 3\) (except partial results are obtained when only one of the spheres has dimension \(>1\)). For example, 2-component link maps in \(S^ 3\) are all homotopically trivial but in \(\mathbb{R}^ 3\) the presence of \(S^ 2\) at \(\infty\) permits non-trivial link maps.