Link compositions and the topological slice problem (Q2367220)

The disc embedding theorems needed to establish the \(s\)-cobordism theorem and exactness of the surgery sequence for topological 4-manifolds are at present known to hold only when the relevant fundamental groups are elementary amenable [the author and \textit{F. S. Quinn}, Topology of 4- manifolds (1990; Zbl 0705.57001)]. Casson and the author showed that exactness of the surgery sequence in general could be reduced to the study of certain ``atomic'' surgery problems, and that these problems could be solved if and only if links in certain families are ``free flat slice'' [\textit{A. Casson} and the author, in ``Four-manifold theory'', Proc. AMS-IMS-SIAM Joint Summer Res. Conf., Durham/N.H. 1982, Contemp. Math. 35, 181-199 (1984; Zbl 0559.57008)]. A \(\mu\)-component link \(L=\bigcup L_ i\) in \(S^ 3\) is free flat slice if there are \(\mu\) disjoint 2-discs \(\bigcup D_ i\) properly and flatly embedded in \(D^ 4\) such that \(\partial D_ i=L_ i\) and \(\pi_ i(D^ 4\setminus \bigcup D_ i)\) is freely generated by meridians. One such atomic family consists of the ``good'' boundary links. A \(\mu\)-component boundary link \(L\) is good if, roughly speaking, it admits a system of \(\mu\) disjoint Seifert surfaces which do not link each other homologically. If we require also that there be simple closed curves on the Seifert surfaces which represent a symplectic half-basis (Lagrangian submodule) for the intersection forms on these surfaces, and which themselves together form a boundary link, we obtain the more restricted class of \(\partial^ 2\)- links. Here it is shown that links in the latter class are free flat slice. A new atomic family of links is described, and it is shown that good boundary links are flat slice in simply connected 4-manifolds with the integral homology of \(S^ 3\times [0,\infty)\). The arguments use the language of the Kirby calculus and the notion of ``grope'', and are very condensed.