ROBUST FOUR-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 is concerned with the question of what spaces, when used in place of disks in an analogous definition, give rise to the same class of homotopically trivial links. We show that there are 4-manifolds for which this property depends on their embedding in the 4-ball. This work is motivated by the A-B slice problem, a reformulation of the 4-dimensional topological 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(4).