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

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 Bökstedt, 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, Bökstedt-Hsiang-Madsen's functor C, and new general isomorphism and injectivity results about the assembly maps for topological Hochschild homology and C.

To appear in Advances in Mathematics. 77 pages

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Cited by in corpus (8)

• Algebraic $K$-theory, assembly maps, controlled algebra, and trace methods
• Purity in chromatically localized algebraic $K$-theory
• Comparing cyclotomic structures on different models for topological Hochschild homology
• K-theory and actions on Euclidean retracts
• Equivariant Morse theory on Vietoris-Rips complexes & universal spaces for proper actions
• Assembly maps for topological cyclic homology of group algebras
• Towards computing the rational homology and assembly maps of generalised Thompson groups
• On the Farrell-Jones conjecture for localising invariants