Freedman, Michael H. Are the Borromean rings A-B-slice. (English) Zbl 0627.57004 Topology Appl. 24, 143-145 (1986). Let \(L=L_ 1\cup...\cup L_ n\) be a tame n-component link in \(S^ 3\). Let \(L_{n+1}\cup...\cup L_{2n}\) be an untwisted parallel to L. The author says L is A-B-slice if there exist pairwise disjoint 4-dimensional compact submanifolds \(A_ 1,...,A_ n\), \(B_ 1,...,B_ n\) of the 4- disc D and self-homeomorphisms \(f_ 1,...,f_ n\), \(g_ 1,...,g_ n\) of D such that for \(i=1,...,n\) the intersections \(A_ i\cap \partial D\) and \(B_ i\cap \partial D\) are tubular neighborhoods respectively of \(L_ i\) and of its parallel \(L_{i+n}\), and \(D=f_ i(A_ i)\cup g_ i(B_ i)\) is a smooth Heegaard-type decomposition of D which extends the standard genus 1 Heegaard decomposition of \(S^ 3=\partial D.\) Two simple observations justify the notion: If L is slice then L is A-B- slice; if L is A-B-slice then \(Link(L_ i,L_ j)=0\) for all \(i\neq j\). The main result asserts that the 4-dimensional topological surgery “theorem”, if true, implies that the Borromean rings are A-B-slice. The author remarks that “it appears to be a routine application of Donaldson’s theory to show that the Borromean rings do not satisfy the smooth category analogy of A-B-slice condition”. An example of a topologically nonslice but A-B-slice link is provided by the elementary Whitehead link. Reviewer: V.Turaev Cited in 1 ReviewCited in 4 Documents MSC: 57M25 Knots and links in the \(3\)-sphere (MSC2010) 57N13 Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010) 57S17 Finite transformation groups Keywords:tame n-component link; untwisted parallel; slice; A-B-slice; Borromean rings; topologically nonslice; Whitehead link PDFBibTeX XMLCite \textit{M. H. Freedman}, Topology Appl. 24, 143--145 (1986; Zbl 0627.57004) Full Text: DOI References: [1] Freedman, Michael, A geometric reformulation of four dimensional surgery, Topology, 24, 135-143 (1985), (this issue) [3] Quinn, Frank, Ends of Maps, III; Dimension 4 and 5, Diff. Geom., 81, 503-521 (1982) · Zbl 0533.57009 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.