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.
|Number of pages||31|
|Journal||Journal of Homotopy and Related Structures|
|Early online date||24 Jul 2020|
|Publication status||Published - Dec 2020|
- Algebraic Topology
- Coarse Geometry
- K-theory and homology