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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

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[Fμ] and the noncommutative Cohn localization Σ-1A[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*-1A[Fμ]) subject to a stable flatness condition on Σ-1A[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

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