Localization and finiteness in link concordance
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''.
concordance invariant of 2-dimensional links in 4-space
scientific article; zbMATH DE number 4138791
scientific article