Algebraic K-theory of group rings and the cyclotomic trace map

TL;DR

This paper proves the rational injectivity of the Farrell-Jones assembly map for algebraic K-theory under certain conditions, linking number theory and topology to describe parts of K-groups of group rings.

Contribution

It generalizes previous results by connecting algebraic K-theory, number theory conjectures, and topological cyclic homology, providing new insights into the structure of K-groups.

Findings

Rational injectivity of the assembly map under mild conditions

Description of a large summand of K-groups in terms of group homology

Applicability to Whitehead groups in many cases

Abstract

We prove that the Farrell-Jones assembly map for connective algebraic K-theory is rationally injective, under mild homological finiteness conditions on the group and assuming that a weak version of the Leopoldt-Schneider conjecture holds for cyclotomic fields. This generalizes a result of B\"okstedt, Hsiang, and Madsen, and leads to a concrete description of a large direct summand of $K_n(\mathbb{Z}[G])\otimes_{\mathbb{Z}}\mathbb{Q}$ in terms of group homology. In many cases the number theoretic conjectures are true, so we obtain rational injectivity results about assembly maps, in particular for Whitehead groups, under homological finiteness assumptions on the group only. The proof uses the cyclotomic trace map to topological cyclic homology, B\"okstedt-Hsiang-Madsen's functor C, and new general isomorphism and injectivity results about the assembly maps for topological Hochschild homology and C.