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Blanchfield and Seifert algebra in high-dimensional
boundary link theory I: Algebraic K–theory
Andrew Ranicki and Desmond Sheiham
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Geometry & Topology 10 (2006)
1761–1853
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Abstract
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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*(Σ-1A[Fμ]) subject to a
stable flatness condition on Σ-1A[Fμ]
for the higher K–groups.
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Desmond Sheiham died 25 March 2005.
This paper is dedicated to the memory of Paul Cohn and Jerry
Levine.
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Keywords
Boundary link, algebraic K–theory,
Blanchfield module, Seifert module
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Mathematical Subject Classification
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
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