A relative Whitney trick and some applications

In a 4-manifold, the Whitney trick seeks to remove a pair of oppositely signed intersection points between immersed surfaces. I will discuss recent joint work with Chris Davis and JungHwan Park where we describe a relative Whitney trick. The relative Whitney trick seeks to remove a single double point in a properly immersed surface, at the expense of changing the boundary of the surface by a homotopy. Our main application is to prove that any link in a homology 3-sphere is homotopic to a link that bounds a collection of locally flatly embedded discs in a contractible topological 4-manifold. In other words, every link in a homology sphere is homotopic to a topologically slice link.