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A note on knot concordance. (English) Zbl 1398.57027
This paper first proves that for any nontrivial element of the fundamental group of a closed 3-manifold other than \(S^3\) there are infinitely many distinct smooth almost-concordance classes in the free homotopy class of the unknot. Then, considering such classes on the boundary of a Mazur manifold, it shows that none of them bounds a PL disc in the manifold, although all are topologically slice. It also proves that all knots in the free homotopy class of \(S^1\times \text{pt}\) in \(S^1\times S^2\) are smoothly concordant (Theorem 2), noting that a similar result appears in [C. W. Davis et al., J. Lond. Math. Soc., II. Ser. 98, No. 1, 59–84 (2018; Zbl 06929431)]. The author’s technique is based on the intersection number of C. T. C. Wall [Surgery on compact manifolds. 2nd ed. Providence, RI: American Mathematical Society (1999; Zbl 0935.57003)] as applied to knot concordance by [R. Schneiderman, Algebr. Geom. Topol. 3, 921–968 (2003; Zbl 1039.57003)].

MSC:
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
57Q60 Cobordism and concordance in PL-topology
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