Cyclic homology for bornological coarse spaces

Luigi Caputi

doi:10.1007/s40062-020-00263-3

Cyclic homology for bornological coarse spaces

Luigi Caputi^*

^*Corresponding author for this work

Mathematical Science

Research output: Contribution to journal › Article › peer-review

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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
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	https://doi.org/10.1007/s40062-020-00263-3
Publication status	Published - Dec 2020

Keywords

Algebraic Topology
Coarse Geometry
K-theory and homology

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title = "Cyclic homology for bornological coarse spaces",

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.",

keywords = "Algebraic Topology, Coarse Geometry, K-theory and homology",

author = "Luigi Caputi",

note = "Funding Information: Open Access funding provided by Projekt DEAL. This work formed part of the author{\textquoteright}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 L{\"o}h, 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: {\textcopyright} 2020, The Author(s). Copyright: Copyright 2020 Elsevier B.V., All rights reserved.",

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doi = "10.1007/s40062-020-00263-3",

language = "English",

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N1 - 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.

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N2 - 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.

AB - 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.

KW - Algebraic Topology

KW - Coarse Geometry

KW - K-theory and homology

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EP - 493

JO - Journal of Homotopy and Related Structures

JF - Journal of Homotopy and Related Structures

SN - 1512-2891

ER -

Cyclic homology for bornological coarse spaces

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