On the relative slice problem and four dimensional topological surgery (Q1818752)

A pair of disjoint links \((L,H)\) in \(S^3\) is called relatively slice if the components of \(L\) bound disjoint embedded (topologically flat) discs in the handlebody obtained from the 4-ball by attaching 2-handles along the components of \(H\) with zero framings. The main result of this paper is that a restricted class of link pairs is not relatively slice. This is related to open questions in four-dimensional topological manifolds. The four-dimensional surgery conjecture fails for manifolds with fundamental group a free group if and only if link pairs in a certain infinite family are not relatively slice. Thus, the paper describes approaches which may be useful in the search for an obstruction to 4-dimensional surgery.