Generalizing the notion of sliceness for links in , a link in a homology sphere is called slice if it bounds a disjoint union of locally flat embedded disks in a contractible 4-manifold. It is trivial to see that any link in can be changed by a homotopy to a slice link, indeed any link is homotopic to the unlink. We prove that the same is true for links in homology spheres. Our argument passes through a novel geometric construction which we call the relative Whitney trick. If time permits we will explore an application of the relative Whitney trick to the existence of Whitney towers.
