Classifying link maps in the four-sphere (II)

This is the second 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 $f:S^2_1\sqcup S^2_2\to S^4$ 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.
Previously we discussed immersions of surfaces in four-manifolds, finger moves, Whitney disks and Wall's algebraic intersection numbers. In this second talk, we apply these techniques in the setting of link maps, and our goal is two-fold. We first introduce Kirk's link homotopy invariant and investigate its properties. We then introduce a standard (or “unknotted”) form of an immersed 2-sphere in 4-space, and show that any link map may be arranged so that one component is standard.