Introduction to Kirby diagrams of four-manifolds

A Kirby diagram is a convenient way to describe a handle decomposition of a four-manifold. It consists of a link in 3-space, where each component is either equipped with an integer, or the component is an unknot with a “dot”. In this talk we will explain the meaning of such a diagram and how to use them to illustrate certain diffeomorphisms between four-manifolds. In relation to the series of talks described below, our particular aim is to exhibit a Kirby diagram for the complement in the 4-sphere of an immersed 2-sphere (or, rather, a regular neighborhood thereof), and use the Kirby diagram to construct certain surfaces in the four-manifold.
This is the third 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.