Link compositions and the topological slice problem

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THE four-dimensional topological surgery “theorem” is still an open question for most fundamental groups. (The class of groups for which the theorem is known coincides with the elementary amenable groups [3], [S].) The general case, is equivalent (Ch. 12, [9]) to the existence of free,$ flat slices for certain class of links which we have called atomic [2]. There is some choice about the set of atomic links but they are all manifestly iterated compositions. In fact, part of the motivation for searching for new atomic families is the possibility that they will meet a known sufficient condition for the existence of free, flat, slices. Consider the following list of increasing conditions on smooth links in S3, the last of which