The relative Whitney trick and its applications

TL;DR

This paper introduces the relative Whitney trick, a geometric operation that simplifies immersed surfaces in 4-manifolds, leading to new results on link homotopies, concordances, and Gordian distances in topology.

Contribution

The paper develops the relative Whitney trick and applies it to prove new results on link homotopies, concordances, and Gordian distances in 4-manifold topology.

Findings

Every link in a homology sphere is homotopic to a topologically slice link in a contractible 4-manifold.

Any link in a homology sphere is order k Whitney tower concordant to a link in S^3 for all k.

Explicit bounds on Gordian distances for 3-component links in homology spheres.

Abstract

We introduce a geometric operation, which we call the relative Whitney trick, that removes a single double point between properly immersed surfaces in a $4$-manifold with boundary. Using the relative Whitney trick we prove that every link in a homology sphere is homotopic to a link that is topologically slice in a contractible topological $4$-manifold. We further prove that any link in a homology sphere is order $k$ Whitney tower concordant to a link in $S^3$ for all $k$. Finally, we explore the minimum Gordian distance from a link in $S^3$ to a homotopically trivial link. Extending this notion to links in homology spheres, we use the relative Whitney trick to make explicit computations for 3-component links and establish bounds in general.