Constructing locally flat surfaces in 4-manifolds

Abstract

There are two main approaches to building locally flat embedded surfaces in 4-manifolds: direct methods which geometrically manipulate a given map of a surface, and more indirect methods using surgery theory. Both rely on Freedman-Quinn's disc embedding theorem. In this expository article, we give an introduction to these methods by sketching proofs of the following results: every primitive second homology class in a closed, simply connected 4-manifold is represented by a locally flat embedded torus (Lee-Wilczynski); and every Alexander polynomial one knot in $S^3$ is topologically slice (Freedman-Quinn).

34 pages, 18 figures. In v2, we have shortened the title, added a few more figures, streamlined the exposition, and corrected some typos, following a referee report