An unlinking theorem for link maps in the 4-sphere

In this talk, having placed one component $f_1$ of a link map $f_1\sqcup f_2: S^2_1\sqcup S^2_2\to S^4$ into a standard form, we construct 2-spheres representing generators for the second homotopy group of its complement $S^4\setminus f_1$. These 2-spheres are constructed using accessory disks and Whitney disks for the immersion $f_1$. Our first application of this construction is to compute the image of Kirk's invariant (which was first proved by Kirk in his foundational work). We then establish criteria, in terms of Wall intersections, for a link map to be link homotopically trivial. These criteria will be seen to be relatively weak; indeed, the proof of their sufficiency will require an application of Freedman's embedding theorem.
This is the fourth in a series of talks in which we give a careful exposition of a recent ground-breaking paper of Rob Schneiderman and Peter Teichner, The Group of Disjoint 2-Spheres in 4-Space. arXiv:1708.00358.
A link map $f_1\sqcup f_2:S^2_1\sqcup S^2_2\to S^4$ is a map of two 2-spheres into the 4-sphere such that $f(S^2_1)\cap f(S^2_2)=\emptyset$, and a link homotopy is a homotopy through link maps. That is, throughout the homotopy each component may self-intersect, but the two components must stay disjoint. Schneiderman and Teichner resolved a long-standing problem by proving that such link maps, modulo link homotopy, are classified by a certain invariant due to Paul Kirk. (This is a higher-dimensional analogue of the classical result in knot theory that the linking number classifies links $S^1\sqcup S^1\to S^3$ up to link homotopy.) The goal of these talks is to obtain a complete understanding of the proof of this result.