Tim Cochran , Stefan Friedl And

which restricts to the given links at the ends. A link is called (topologically) slice if it is concordant to the trivial m–component link or, equivalently, if it bounds a flat embedding of m disjoint slice disks D2 ∐ · · · ∐D2 ֒→ D4. In the special case m = 1 we refer to the link as a knot. If the embeddings above are required to be C, or smooth, then these notions are called smoothly concordant and smoothly slice. The study of link concordance was initiated by Fox and Milnor in the early 1960s arising from their study of isolated singularities of 2-spheres in 4-manifolds. It is now known that specific questions about link concordance are equivalent to whether or not the surgery and s-cobordism theorems (that hold true in higher dimensions) hold true for topological 4-manifolds. Moreover, the difference between a link being topologically slice and being smoothly slice can be viewed as “atomic” for the existence of multiple differential structures on a fixed topological 4-manifold. There is only one known way to construct a smoothly slice link, namely as the boundary of a set of ribbon disks [Ro90]. The known constructions of (topologically) slice links are also fairly limited. In 1982 Michael Freedman proved that any knot with Alexander polynomial 1 is slice [F85]. It is known that some of these knots cannot be smoothly slice and hence cannot arise from the ribbon construction. Freedman [F85, F88] and later Freedman and Teichner [FT] gave other techniques showing that the Whitehead doubles of various links are slice. The 4-dimensional surgery and s-cobordism theorems (for all fundamental groups) are in fact equivalent to the free sliceness of Whitehead doubles of all links with vanishing linking numbers, see [FQ90]. Here a link is freely slice if the