Link homotopy with one codimension two component

Abstract

Link maps with one codimension two component are studied and an invariant of link maps modulo link homotopy is constructed using ideas from knot theory and immersion theory. This invariant is used to give examples of nontrivial link homotopy classes and to show that there are infinitely many distinct link homotopy classes in many dimensions. A link map with the codimension two component embedded is shown to be nullhomotopic. These ideas are applied to the special case of 2 2 -spheres in S 4 {S^4} to give simple examples of the failure of the Whitney trick in dimension 4 4 .