Kirk's invariant may have mixed terms

Abstract

A link map is a map from a union of spheres into another sphere with pairwise disjoint images. Two link maps are link homotopic if one can be deformed into the other by a homotopy through link maps. This rather crude equivalence relation was introduced in 1954 by Milnor[13] for classical links and an algorithm for the classification problem in this setting was given in 1990 by Habegger and Lin[4]. In the past years, link homotopy in higher dimensions has also attracted much renewed interest. For link maps in a large metastable range, the corresponding classification problem has been reduced to standard questions in homotopy theory by Koschorke[9, 10].

In this paper we consider link maps of 2-spheres into the 4-sphere. This is another interesting case which differs considerably from the two cases above because of the failure of Whitney's trick in dimension four. Here the first contribution was made in 1984 by Fenn and Rolfsen[2] who established the first non-trivial link map of two 2- spheres into the 4-sphere. Their idea has been generalized by Kirk[6] to define his σ-invariant that distinguishes infinitely many different link homotopy classes. For several years, Kirk's invariant remained the strongest invariant for link maps into the 4-sphere. Only recently, it became known that Kirk's invariant suffices to classify two-component link maps in dimension four [12].