Open Problems

Abstract

Open problems in the study of topological 4-manifolds are explained in detail. An important open problem is to determine whether the disc embedding theorem and its antecedents hold for all groups; in other words, whether all groups are good. The disc embedding conjecture and the surgery conjecture are stated. The relationships between these conjectures and their various reformulations are explained. Of particular interest are the reformulations in terms of freely slicing certain infinite families of links. In particular, the surgery conjecture is true if and only if all good boundary links are freely slice. Good boundary links are the many-component analogues of Alexander polynomial one knots.