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Assembly maps for topological cyclic homology of group algebras

  • Wolfgang Lück EMAIL logo , Holger Reich , John Rognes und Marco Varisco
Veröffentlicht/Copyright: 2. Juni 2017

Abstract

We use assembly maps to study 𝐓𝐂(𝔸[G];p), the topological cyclic homology at a prime p of the group algebra of a discrete group G with coefficients in a connective ring spectrum 𝔸. For any finite group, we prove that the assembly map for the family of cyclic subgroups is an isomorphism on homotopy groups. For infinite groups, we establish pro-isomorphism, (split) injectivity, and rational injectivity results, as well as counterexamples to injectivity and surjectivity. In particular, for hyperbolic groups and for virtually finitely generated abelian groups, we show that the assembly map for the family of virtually cyclic subgroups is injective but in general not surjective.

Award Identifier / Grant number: Leibniz Award

Funding source: Simons Foundation

Award Identifier / Grant number: #419561

Funding statement: We have been financially supported by the first author’s Leibniz Award, granted by the Deutsche Forschungsgemeinschaft, and by his European Research Council Advanced Grant “KL2MG-interactions” (#662400); by the Collaborative Research Center 647 in Berlin; and by a grant from the Simons Foundation (#419561, Marco Varisco).

Acknowledgements

We thank the referee for a detailed and thoughtful report. We also thank the Hausdorff Research Institute for Mathematics in Bonn, where parts of this work were completed during the 2015 Trimester Program on “Homotopy theory, manifolds, and field theories.”

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Received: 2016-07-14
Revised: 2017-04-18
Published Online: 2017-06-02
Published in Print: 2019-10-01

© 2019 Walter de Gruyter GmbH, Berlin/Boston

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