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Link compositions and the topological slice problem. (English) Zbl 0782.57010

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 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” [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.

MSC:

57N13 Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010)
57M25 Knots and links in the \(3\)-sphere (MSC2010)
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