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Quantitative K-homology of uniform Roe algebras

  • Hao Guo
  • Chenxi Wang
  • Guoliang Yu
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Abstract

We introduce quantitative K-homology for uniform Roe algebras of discrete proper metric spaces with bounded geometry, based on the controlled KK-theory of Willett and Yu. We construct a comparison map from Lipschitz K-theory and prove that, for chain geodesic spaces, it is an isomorphism in a quantitative sense; in particular, quantitative K-homology is isomorphic to the inverse limit of the Lipschitz K-theory groups as the Lipschitz constant tends to zero. We also establish a pairing between quantitative K-homology and quantitative K-theory.

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Cite

BibTeXRIS
BibTeX
@misc{guo2026quantitative,
  title = {{Quantitative $K$-homology of uniform Roe algebras}},
  author = {Hao Guo and Chenxi Wang and Guoliang Yu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.34050},
  eprint = {2610.34050},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.34050},
}
RIS
TY  - UNPB
TI  - Quantitative K-homology of uniform Roe algebras
AU  - Hao Guo
AU  - Chenxi Wang
AU  - Guoliang Yu
PY  - 2026
DA  - 2026/10/27
PB  - arXiv
DO  - 10.48550/arxiv.2610.34050
UR  - https://arxiv.org/abs/2610.34050
AB  - We introduce quantitative K-homology for uniform Roe algebras of discrete proper metric spaces with bounded geometry, based on the controlled KK-theory of Willett and Yu. We construct a comparison map from Lipschitz K-theory and prove that, for chain geodesic spaces, it is an isomorphism in a quantitative sense; in particular, quantitative K-homology is isomorphic to the inverse limit of the Lipschitz K-theory groups as the Lipschitz constant tends to zero. We also establish a pairing between quantitative K-homology and quantitative K-theory.
ER  -