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Are the Borromean rings A-B-slice. (English) Zbl 0627.57004

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

MSC:

57M25 Knots and links in the \(3\)-sphere (MSC2010)
57N13 Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010)
57S17 Finite transformation groups
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[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
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