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

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*(Σ-1A[Fμ]) subject to a stable flatness condition on Σ-1A[Fμ] for the higher K–groups.