Cobordisms of semi-boundary links
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; \textit{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\(\geq 2)\), \textit{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.
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