  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.
