The triviality of a certain secondary invariant of link homotopy in dimension 4

A link map is a map of spheres into another sphere with pairwise disjoint images, and a link homotopy is a homotopy through link maps. In this talk I will discuss the problem of classifying, up to link homotopy, two-component link maps of two-spheres in the four-sphere. This setting is particularly interesting because, as usual, four-dimensional topology presents unique difficulties. It is conjectured that such link maps are classified by an invariant due to Kirk, and a “secondary” invariant has subsequently been proposed by Li to detect counterexamples to this conjecture. After giving a brief history of the subject, in this talk I will discuss the (very geometric) constructions of these invariants, and outline a proof that Li's invariant cannot detect such examples; indeed, it is a strictly weaker invariant.