Volume 10, issue 3 (2006)

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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 μ–component boundary links motivates decomposition theorems for the algebraic K–groups of the group ring A[Fμ] and the noncommutative Cohn localization Σ1A[Fμ], for any μ 1 and an arbitrary ring A, with Fμ the free group on μ generators and Σ the set of matrices over A[Fμ] which become invertible over A under the augmentation A[Fμ] A. Blanchfield A[Fμ]–modules and Seifert A–modules are abstract algebraic analogues of the exteriors and Seifert surfaces of boundary links. Algebraic transversality for A[Fμ]–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[Fμ]) and K(Σ1A[Fμ]) subject to a stable flatness condition on Σ1A[Fμ] 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
References
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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