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arXiv · 2205.13395

A geometric representative for the fundamental class in KK-duality of Smale spaces

D. M. Gerontogiannis·Michael F. Whittaker·Joachim Zacharias
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Abstract

A fundamental ingredient in the noncommutative geometry program is the notion of KK-duality, often called K-theoretic Poincar\'{e} duality, that generalises Spanier-Whitehead duality. In this paper we construct a $\theta$-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.

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D. M. Gerontogiannis, Michael F. Whittaker, Joachim Zacharias. 2022-05-26. A geometric representative for the fundamental class in KK-duality of Smale spaces. https://doi.org/10.1016/j.jfa.2024.110455

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