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Topological concordance and F-isotopy. (English) Zbl 0569.57003

The author proves that every 2-component link in \(S^ 3\) with Alexander polynomial 1 is topologically concordant to the Hopf link. The proof uses a result of M. H. Freedman [Proc. Int. Congr. Math., Warszawa 1983, Vol. 2, 647-663 (1984)] that unobstructed 4-dimensional surgery problems over groups with polynomial growth are solvable. This applies to the group of the Hopf link, which is abelian. The rest of the proof consists of the construction of a certain degree 1 map between 4-manifolds, and the explicit calculation of its surgery obstruction.
F-isotropy is the equivalence relation on links generated by the move of replacing one component by a knot homologous to it in a regular neighbourhood. The above method also shows that a certain restricted class of F-isotopies can be realized by topological concordances.
Reviewer: J.Howie

MSC:

57M25 Knots and links in the \(3\)-sphere (MSC2010)
57N13 Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010)
57R67 Surgery obstructions, Wall groups
57R90 Other types of cobordism
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References:

[1] Freedman, J. Diff. Geometry 17 pp 357– (1982)
[2] DOI: 10.1090/S0273-0979-1983-15090-5 · Zbl 0519.57012 · doi:10.1090/S0273-0979-1983-15090-5
[3] Cohen, A Course in Simple-Homotopy Theory 10 (1973) · Zbl 0261.57009 · doi:10.1007/978-1-4684-9372-6
[4] Browder, Surgery on simply-connected manifolds 65 (1972) · Zbl 0239.57016 · doi:10.1007/978-3-642-50020-6
[5] Gromov, Publ. Math. I.H.E.S. 53 pp 53– (1981) · Zbl 0474.20018 · doi:10.1007/BF02698687
[6] DOI: 10.2307/1970726 · Zbl 0182.57303 · doi:10.2307/1970726
[7] HILlman, Alexander Ideals of Links 895 (1981) · Zbl 0491.57001 · doi:10.1007/BFb0091682
[8] DOI: 10.1112/plms/s3-27.1.126 · doi:10.1112/plms/s3-27.1.126
[9] Wall, Surgery on Compact Manifolds (1970)
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