On an invariant of link cobordism in dimension four
It is not known in general whether a 2-dimensional link L (i.e., collection of disjoint imbedded 2-spheres in \(S^ 4)\) is slice (i.e., bounds disjoint 3-disks in \(D^ 5)\). A possible obstruction (in \({\mathbb{Z}}/2\) for a 2-component link) is the Sato-Levine invariant \(\beta\) defined by applying the Pontryagin construction to the intersection of Seifert ''surfaces'' for the individual components of L. This paper is devoted to showing \(\beta =0\) in many special cases. (In fact, it is a much more recent result of Kent Orr that \(\beta\) always vanishes.) The general technique used in the present paper is to reinterpret \(\beta\) as the bordism class of the spin manifold X, obtained by surgering \(S^ 4\) along L, and natural map \(X\to S^ 1\times S^ 1\). Among the conditions which are shown to imply \(\beta =0\) are: (1) one component of L is unknotted, (2) the group \(\pi\) of L maps into another group P which has the same low-dimensional \({\mathbb{Z}}/2\)-homology as the free group of rank 2. This latter case includes boundary links, which we know to be slice.
scientific article; zbMATH DE number 3907387
scientific article