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 $A\left[F_{\mu }\right]$ and the noncommutative Cohn localization ${\Sigma }^{-1}A\left[F_{\mu }\right]$, for any $\mu \ge 1$ and an arbitrary ring $A$, with $F_{\mu }$ the free group on $\mu$ generators and $\Sigma$ the set of matrices over $A\left[F_{\mu }\right]$ which become invertible over $A$ under the augmentation $A\left[F_{\mu }\right]\to A$. Blanchfield $A\left[F_{\mu }\right]$–modules and Seifert $A$–modules are abstract algebraic analogues of the exteriors and Seifert surfaces of boundary links. Algebraic transversality for $A\left[F_{\mu }\right]$–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 $K_{\ast }\left(A\left[F_{\mu }\right]\right)$ and $K_{\ast }\left({\Sigma }^{-1}A\left[F_{\mu }\right]\right)$ subject to a stable flatness condition on ${\Sigma }^{-1}A\left[F_{\mu }\right]$ for the higher $K$–groups.