zbMATH — the first resource for mathematics

Cobordisms of semi-boundary links. (English) Zbl 0561.57012
Let \(L=(M,N)\) be a link consisting of two disjoint, closed, oriented, connected n-dimensional submanifolds of \(S^{n+2}\). L is said to be a semi-boundary link if M bounds a Seifert surface V disjoint from N, and N bounds a Seifert surface W disjoint from M. In this paper the ”Sato- Levine invariant” \(\beta (L)\in \pi_{n+2}(S^ 2)\) of a semi-boundary link is introduced; it is the homotopy class corresponding (via the Thom- Pontryagin construction) to the intersection of V and W. An equivalence relation between semi-boundary links, called ”\(\beta\)-equivalence”, is defined and shown to have these properties: concordant links are \(\beta\)- equivalent; the set of \(\beta\)-equivalence classes forms an abelian group under connected sum; and the invariant \(\beta\) defines an isomorphism between this abelian group and \(\pi_{n+2}(S^ 2)\). The Sato-Levine invariant and its generalization to links of more than two components have attracted a great deal of attention since the appearance of this paper. In the classical case \((n=1)\), \(\beta\) (L) is defined for links of two or three components in which all linking numbers are zero; T. D. Cochran [Concordance invariance of coefficients of Conway’s link polynomial, Invent. Math., to appear] has shown that \(\beta\) (L) is then related to the coefficient \(a_ 2\) in the Conway polynomial \(\nabla (z)=z^{m-1}\cdot (a_ 0+a_ 2z^ 2+...)\). (Here m is the number of components of L.) In higher dimensions (n\(\geq 2)\), K. E. Orr [Ph. D. dissertation, Rutgers University, 1985] has shown that \(\beta\) vanishes for all links of four or more components, and also vanishes for links of two or three simply-connected components.
Reviewer: L.Traldi

57Q45 Knots and links in high dimensions (PL-topology) (MSC2010)
57M25 Knots and links in the \(3\)-sphere (MSC2010)
Full Text: DOI
[1] Cappell, S.; Shaneson, J., Link cobordism, Comment. math. helv., 55, 20-49, (1980) · Zbl 0444.57008
[2] Cochran, T., On an invariant of link cobordism in dimension, 4, (1983), MIT, pre-print
[3] Levine, J., Knot cobordism in codimension two, Comment. math. helv., 44, 229-244, (1969) · Zbl 0176.22101
[4] Levine, J., Invariants of knot cobordism, Invent. math., 8, 98-110, (1969) · Zbl 0179.52401
[5] Problem list compiled by C. McA. Gordon, in Knot Theory, LNM #685, Springer-Verlag, Berlin.
[6] D. Ruberman, Concordance of links in S4, to appear in: Proceedings of 1982 research conference at Durham, New Hampshire, Proceedings of Symposia in Pure Mathematics.
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.