A new technique for the link slice problem (Q1060741)

A link in \(S^ 3\) is called slice if it is the boundary of a collection of disjoint, topologically flat discs in \(D^ 4\). The problem of whether certain classes of links are slice arises as an obstruction to removing restrictions on \(\pi_ 1\) in the 4-dimensional surgery theorem and the 5-dimensional s-cobordism theorem. In this paper, a technique for slicing links is introduced, and the following theorem proved: any link which is an untwisted Whitehead double of a tame boundary link is slice; moreover the complement in \(D^ 4\) of the slice discs is homotopy equivalent to a wedge of circles, with \(\pi_ 1\) freely generated by meridinal loops. The method of proof, roughly, is to construct the slice complement as an abstract 4-manifold using handles. Attaching further 2-handles (corresponding to replacing the slice discs) gives a contractible 4- manifold bounded by \(S^ 3:\) the topological Poincaré conjecture then says that this is \(D^ 4\), as desired.