On an invariant of link cobordism in dimension four
type: Item
instance of: scholarly article
title: On an invariant of link cobordism in dimension four
zbMATH Open document ID: 0568.57015
DOI: 10.1016/0166-8641(84)90002-6
author: Tim D. Cochran
published in: Topology and its Applications
publication date: 1984-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 57Q45
zbMATH DE Number: 3907387
zbMATH Keywords: Seifert surfaces
zbMATH Keywords: slice link
zbMATH Keywords: 2-dimensional link
zbMATH Keywords: Sato-Levine invariant
zbMATH Keywords: boundary links
reviewed by: Jerome Levine
MaRDI profile type: Publication
full work available at URL: 0166-8641(84)90002-6
OpenAlex ID: W2058033529
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description: scientific article; zbMATH DE number 3907387