Freedman, Michael H.; Lin, Xiao-Song On the \((A,B)\)-slice problem. (English) Zbl 0845.57016 Topology 28, No. 1, 91-110 (1989). The remaining questions regarding 4-dimensional topological surgery seem to be well expressed in terms of the link-slice problem for certain links \(L'\) which are Whitehead doubles, \(L' = \text{Wh} (L)\). The first author introduced in [A geometric reformulation of four dimensional surgery, Topology Appl. 24, 133-141 (1986); ibid., 143-145 (1986; Zbl 0627.57004)] a method involving the theory of decomposition spaces for “undoubling” the problem and expressing surgery in terms of a condition, “\((A,B)\)-slice”, on \(L\). The full four-dimensional (topological) surgery conjecture is equivalent to ‘all generalized Borromean rings are \((A,B)\)-slice’. We begin the analysis of the \((A,B)\)-slice problem using the methods of: handlebody theory, combinatorial group theory (the “Magnus expansion”), and secondary operations (in the form of lower central series and Massey product calculations). We find a constraint on the \((A,B)\)-decompositions that can arise in an \((A,B)\)-slicing of any homotopically essential link \(L\). This result applies to the case of most interest since every generalized Borromean ring is homotopically essential. We are unable to go beyond this to show that homotopically essential links are not \((A,B)\)-slice, in fact, we still do not know if the Borromean rings are \((A,B)\)-slice. The ultimate goal is to develop a method of analyzing relative imbedding problems for all possible \((A,B)\) decompositions. This is not achieved. We reduce the general case to the case that \(A\) and \(B\) each have (relative) handle decompositions containing handles of indices one and two and our methods determine obstructions to (weak) \((A,B)\)-slicing when either side (say the \(A\)-side) lacks 1-handles. We are able to formulate an obstruction in the case of general decompositions but do not know how to evaluate it. Thus the results of this paper are consistent with the surgery conjecture although the program has been to find a (nontrivial) obstruction – specifically an obstruction to the Borromean rings being \((A,B)\)-slice. Since the outcome is not definitive, perhaps the main interest for the reader lies in seeing the classical techniques of link theory (relying on Milnor, Stallings, and Massey) arrayed against a novel problem. Cited in 2 ReviewsCited in 9 Documents MSC: 57N13 Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010) Keywords:topological surgery; link-slice problem; handlebody; imbedding; \((A,B)\) decompositions Citations:Zbl 0627.57004 PDFBibTeX XMLCite \textit{M. H. Freedman} and \textit{X.-S. Lin}, Topology 28, No. 1, 91--110 (1989; Zbl 0845.57016) Full Text: DOI