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

TL;DR

This paper constructs a geometric representative of the fundamental class in KK-duality for Smale spaces, extending noncommutative geometry tools to dynamical systems with controlled partitions.

Contribution

It introduces a new Fredholm module representing the fundamental class in KK-duality for Smale spaces, utilizing generalized Bowen partitions and a noncommutative Whitney embedding.

Findings

Constructed a $ heta$-summable Fredholm module for KK-duality

Developed dynamical partitions with controlled Lipschitz constants

Established a noncommutative Whitney embedding theorem

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.