Robust 4−manifolds and Robust Embeddings

A link in the 3−sphere is homotopically trivial, according to Milnor, if its components bound disjoint maps of disks in the 4−ball. This paper concerns the question of what spaces give rise to the same class of homotopically trivial links when used in place of disks in the analogous definition. We show that there are 4−manifolds for which this property depends on their embedding in the 4−ball. The main motivation for this work is the A-B slice problem, a reformulation of the 4−dimensional surgery conjecture. As a corollary, this provides a new, secondary, obstruction in the A-B slice problem for a certain class of decompositions of D .