A geometric representative for the fundamental class in KK-duality of Smale spaces
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866909229042892800 |
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| author | Gerontogiannis, D. M. Whittaker, Michael F. Zacharias, Joachim |
| author_facet | Gerontogiannis, D. M. Whittaker, Michael F. Zacharias, Joachim |
| contents | A fundamental ingredient in the noncommutative geometry program is the notion of KK-duality, often called K-theoretic Poincaré duality, that generalises Spanier-Whitehead duality. In this paper we construct a $θ$-summable Fredholm module that represents the fundamental class in KK-duality between the stable and unstable Ruelle algebras of a Smale space. To find such a representative, we construct dynamical partitions of unity on the Smale space with highly controlled Lipschitz constants. This requires a generalisation of Bowen's Markov partitions. Along with an aperiodic point-sampling technique we produce a noncommutative analogue of Whitney's embedding theorem, leading to the Fredholm module. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2205_13395 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | A geometric representative for the fundamental class in KK-duality of Smale spaces Gerontogiannis, D. M. Whittaker, Michael F. Zacharias, Joachim K-Theory and Homology Dynamical Systems Operator Algebras 37D20, 19K33, 58B34 (primary), 37B10 (secondary) A fundamental ingredient in the noncommutative geometry program is the notion of KK-duality, often called K-theoretic Poincaré duality, that generalises Spanier-Whitehead duality. In this paper we construct a $θ$-summable Fredholm module that represents the fundamental class in KK-duality between the stable and unstable Ruelle algebras of a Smale space. To find such a representative, we construct dynamical partitions of unity on the Smale space with highly controlled Lipschitz constants. This requires a generalisation of Bowen's Markov partitions. Along with an aperiodic point-sampling technique we produce a noncommutative analogue of Whitney's embedding theorem, leading to the Fredholm module. |
| title | A geometric representative for the fundamental class in KK-duality of Smale spaces |
| topic | K-Theory and Homology Dynamical Systems Operator Algebras 37D20, 19K33, 58B34 (primary), 37B10 (secondary) |
| url | https://arxiv.org/abs/2205.13395 |