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

Abstract

The classification of high-dimensional mu-component boundary links motivates decomposition theorems for the algebraic K-groups of the group ring A[F_mu] and the noncommutative Cohn localization Sigma^{-1}A[F_mu], for any mu>0 and an arbitrary ring A, with F_mu the free group on mu generators and Sigma the set of matrices over A[F_mu] which become invertible over A under the augmentation A[F_mu] to A. Blanchfield A[F_mu]-modules and Seifert A-modules are abstract algebraic analogues of the exteriors and Seifert surfaces of boundary links. Algebraic transversality for A[F_mu]-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_mu]) and K_*(Sigma^{-1}A[F_mu]) subject to a stable flatness condition on Sigma^{-1}A[F_mu] for the higher K-groups.