Abstract
The goal of the paper is to define Hochschild and cyclic homology for bornological coarse spaces, i.e., lax symmetric monoidal functors XHHG and XHCG from the category GBornCoarse of equivariant bornological coarse spaces to the cocomplete stable ∞-category Ch∞ of chain complexes reminiscent of the classical Hochschild and cyclic homology. We investigate relations to coarse algebraic K-theory XKG and to coarse ordinary homology XHG by constructing a trace-like natural transformation XKG→XHG that factors through coarse Hochschild (and cyclic) homology. We further compare the forget-control map for XHHG with the associated generalized assembly map.
Original language | English |
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Pages (from-to) | 463-493 |
Number of pages | 31 |
Journal | Journal of Homotopy and Related Structures |
Volume | 15 |
Early online date | 24 Jul 2020 |
DOIs | |
Publication status | Published - Dec 2020 |
Bibliographical note
Funding Information:Open Access funding provided by Projekt DEAL. This work formed part of the author’s PhD thesis at Regensburg University. It is a pleasure to again acknowledge Ulrich Bunke, this work would not exist without him. The author also thanks Clara Löh, Denis-Charles Cisinski and Alexander Engel for helpful discussions, and the anonymous referees for constructive comments and recommendations. The author has been supported by the DFG Research Training Group GRK 1692 “Curvature, Cycles, and Cohomology” and by the DFG SFB 1085 “Higher Invariants”.
Publisher Copyright:
© 2020, The Author(s).
Copyright:
Copyright 2020 Elsevier B.V., All rights reserved.
Keywords
- Algebraic Topology
- Coarse Geometry
- K-theory and homology