Localization and finiteness in link concordance
type: Item
instance of: scholarly article
title: Localization and finiteness in link concordance
zbMATH Open document ID: 0695.57014
DOI: 10.1016/0166-8641(89)90049-7
published in: Topology and its Applications
publication date: 1989-01-01T00:00:00Z
review text: This article contains a very nice survey of recent results in link concordance theory (except for the recent examples of Cochran-Orr of links not concordant to boundary links) along with two new contributions of the author. The first of these is the construction of a concordance invariant of 2-dimensional links in 4-space whose vanishing is necessary and sufficient for the link to be slice. This invariant lives in the spin bordism group \(\Omega_ 4^{spin}(K(\hat F))\), where \(\hat F\) is the ``algebraic closure'' of the free group F and \(K(\hat F)\) is the mapping cone of the map of Eilenberg-MacLane spaces K(F,1)\(\to K(\hat F,1)\), a construction whose prototype is due to Kent Orr. Another related result is a non-triviality condition on \(H_*(\hat F)\) which will imply the existence of a 3-dimensional link in 5-space not concordant to a sublink of a homology boundary link. The second new result is a geometric condition on a link to insure that it is a sublink of a homology boundary link. This condition involves the existence of a finite system of surfaces indexed by formal brackets which begins with Seifert surfaces for the link components and proceeds by adding surfaces whose boundary is a component of the intersection of previous surfaces in the system. The main point is that these intersections should eventually coincide with boundary components of surfaces already in the system, a property the author calls ``self-referencing''.
Mathematics Subject Classification ID: 57Q45
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 57Q60
Mathematics Subject Classification ID: 20F14
zbMATH DE Number: 4138791
zbMATH Keywords: slice link
zbMATH Keywords: link concordance
zbMATH Keywords: concordance invariant of 2-dimensional links in 4-space
zbMATH Keywords: spin bordism group
zbMATH Keywords: 3-dimensional link in 5-space
zbMATH Keywords: homology boundary link
zbMATH Keywords: Seifert surfaces
author: Tim D. Cochran
reviewed by: Jerome Levine
MaRDI profile type: Publication
full work available at URL: 0166-8641(89)90049-7
OpenAlex ID: W1976235113
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description: scientific article; zbMATH DE number 4138791