The relative Whitney trick and its applications

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.