Link Maps in the 4-sphere

Ash Lightfoot (Indiana)

Abstract: To a link map $f$, a map of two 2-spheres in the 4-sphere with disjoint images, P. Kirk associated a pair of polynomials $\sigma(f)$ that is invariant under link homotopy and obstructs link homotoping $f$ to an embedding. (A link homotopy is a homotopy through link maps.) It is an open question whether $\sigma$ is a complete obstruction; that is, whether $\sigma(f)=(0,0)$ implies that $f$ is link homotopic to an embedding. So as to find a counterexample, G-S. Li introduced a ``secondary'' invariant $\omega(f)$, which assumes the ``primary'' obstruction $\sigma(f)$ vanishes, and measures a further obstruction. In this talk I will discuss the motivation behind and the (very geometric) constructions of these invariants, and outline a proof that $\omega$ cannot detect counterexamples.