Volume 10, issue 3 (2006)

 Recent Issues
 The Journal About the Journal Editorial Board Editorial Interests Editorial Procedure Submission Guidelines Submission Page Subscriptions Author Index To Appear Contacts ISSN (electronic): 1364-0380 ISSN (print): 1465-3060
Blanchfield and Seifert algebra in high-dimensional boundary link theory I: Algebraic $K$–theory

Andrew Ranicki and Desmond Sheiham

Geometry & Topology 10 (2006) 1761–1853
 arXiv: math.AT/0508405
Abstract

The classification of high-dimensional $\mu$–component boundary links motivates decomposition theorems for the algebraic $K$–groups of the group ring $A\left[{F}_{\mu }\right]$ and the noncommutative Cohn localization ${\Sigma }^{-1}A\left[{F}_{\mu }\right]$, for any $\mu \ge 1$ and an arbitrary ring $A$, with ${F}_{\mu }$ the free group on $\mu$ generators and $\Sigma$ the set of matrices over $A\left[{F}_{\mu }\right]$ which become invertible over $A$ under the augmentation $A\left[{F}_{\mu }\right]\to A$. Blanchfield $A\left[{F}_{\mu }\right]$–modules and Seifert $A$–modules are abstract algebraic analogues of the exteriors and Seifert surfaces of boundary links. Algebraic transversality for $A\left[{F}_{\mu }\right]$–module chain complexes is used to establish a long exact sequence relating the algebraic $K$–groups of the Blanchfield and Seifert modules, and to obtain the decompositions of ${K}_{\ast }\left(A\left[{F}_{\mu }\right]\right)$ and ${K}_{\ast }\left({\Sigma }^{-1}A\left[{F}_{\mu }\right]\right)$ subject to a stable flatness condition on ${\Sigma }^{-1}A\left[{F}_{\mu }\right]$ for the higher $K$–groups.

 Desmond Sheiham died 25 March 2005. This paper is dedicated to the memory of Paul Cohn and Jerry Levine.
Keywords
Boundary link, algebraic $K$–theory, Blanchfield module, Seifert module
Mathematical Subject Classification 2000
Primary: 19D50, 57Q45
Secondary: 20E05