Blanchfield and Seifert algebra in high-dimensional boundary link theory I: Algebraic K–theory

Ranicki, Andrew; Sheiham, Desmond

doi:10.2140/gt.2006.10.1761

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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

DOI: 10.2140/gt.2006.10.1761

arXiv: math.AT/0508405

Abstract

The classification of high-dimensional $μ$ –component boundary links motivates decomposition theorems for the algebraic $K$ –groups of the group ring $A [F_{μ}]$ and the noncommutative Cohn localization $Σ^{- 1} A [F_{μ}]$ , for any $μ \geq 1$ and an arbitrary ring $A$ , with $F_{μ}$ the free group on $μ$ generators and $Σ$ the set of matrices over $A [F_{μ}]$ which become invertible over $A$ under the augmentation $A [F_{μ}] \to A$ . Blanchfield $A [F_{μ}]$ –modules and Seifert $A$ –modules are abstract algebraic analogues of the exteriors and Seifert surfaces of boundary links. Algebraic transversality for $A [F_{μ}]$ –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 $K_{*} (A [F_{μ}])$ and $K_{*} (Σ^{- 1} A [F_{μ}])$ subject to a stable flatness condition on $Σ^{- 1} A [F_{μ}]$ 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

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Publication

Received: 6 October 2005

Revised: 14 July 2006

Accepted: 2 September 2006

Published: 2 November 2006

Proposed: Wolfgang Lück

Seconded: Peter Teichner, Steve Ferry

Authors

Andrew Ranicki
School of Mathematics University of Edinburgh Edinburgh EH9 3JZ Scotland, UK

Desmond Sheiham

Volume 10, issue 3 (2006)