The relative Whitney trick and its applications

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 links in homology spheres are equivalent to links in S 3 up to any level of the Whitney tower filtration that approximates link concordance. 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.