Celebratio Mathematica — Ranicki

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Works connected to Jerome Paul Levine

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A. Ranicki and D. Sheiham: “Blanchfield and Seifert algebra in high-dimensional boundary link theory, I: Algebraic \( K \)-theory,” Geom. Topol. 10 : 3 (2006), pp. 1761–1853. Desmond Sheiham died 25 March 2005. This paper is dedicated to the memory of Paul Cohn and Jerry Levine. MR 2284050 Zbl 1125.19003 ArXiv math/0508405 article

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 \( \mu \geq 1 \) 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.

Desmond Sheiham	Related
Paul Moritz Cohn	Related
Jerome Paul Levine	Related