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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

##### MSC:
 57Q45 Knots and links in high dimensions (PL-topology) (MSC2010) 57M25 Knots and links in the $$3$$-sphere (MSC2010)
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##### References:
 [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.
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