Constructing locally flat surfaces in 4-manifolds

TL;DR

This paper introduces methods for constructing locally flat surfaces in 4-manifolds, highlighting direct geometric manipulations and surgery theory, with applications to representing homology classes and knot sliceness.

Contribution

It provides an accessible overview of techniques for building locally flat surfaces, including proofs of key results in 4-manifold topology and knot theory.

Findings

Primitive second homology classes represented by locally flat tori

Alexander polynomial one knots are topologically slice

Both methods rely on Freedman-Quinn's disc embedding theorem

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).