Homology and K-theory of dynamical systems. II. Smale spaces with totally disconnected transversal

Proietti, Valerio; Yamashita, Makoto

doi:10.48550/arxiv.2104.10938

Cited by 4 publications

(5 citation statements)

References 22 publications

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“…This is based on two sets of results. The first are those of Proietti and Yamashita [21,22], who show that the Smale space homology here agrees with the groupoid homology as studied by Matui. Further, Matui has recently adapted the results which are used in the next section for the K-theory of the C * -algebras to the case of groupoid homology: see [19].…”

Section: Homologysupporting

confidence: 80%

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Binary factors of shifts of finite type

Putnam

2023

Ergod. Th. Dynam. Sys.

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We construct two new classes of topological dynamical systems; one is a factor of a one-sided shift of finite type while the second is a factor of the two-sided shift. The data are a finite graph which presents the shift of finite type, a second finite directed graph and a pair of embeddings of it into the first, satisfying certain conditions. The factor is then obtained from a simple idea based on binary expansion of real numbers. In both cases, we construct natural metrics on the factors and, in the second case, this makes the system a Smale space, in the sense of Ruelle. We compute various algebraic invariants for these systems, including the homology for Smale space developed by the author and the K-theory of various $C^{*}$ -algebras associated to them, in terms of the pair of original graphs.

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Section: Homologysupporting

confidence: 80%

“…It is worth noting that recent work of Proietti and Yamashita [21,22] shows that there are very close relations between the K-theory of the C * -algebras and the Smale space homology, as well the cohomology of certain groupoids.…”

Section: Introductionmentioning

confidence: 94%

Binary factors of shifts of finite type

Putnam

2023

Ergod. Th. Dynam. Sys.

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“…Examples of recent research in this direction are the HK conjecture of Matui [5], or the relation between the homology theory of Smale spaces and the K-theory of their corresponding C * -algebras [8]. In this latter example, a special case of the methods developed here (i.e., when the groupoid is torsion-free and ample) has already been applied with great success and lead to many interesting results in topological dynamics, as is demonstrated by the papers [9,10,11].…”

Section: Introduction and Main Resultsmentioning

confidence: 81%

Categorical approach to the Baum-Connes conjecture for étale groupoids

Bönicke¹,

Proietti²

2022

Preprint

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We consider the equivariant Kasparov category associated to an étale groupoid, and by leveraging its triangulated structure we study its localization at the "weakly contractible" objects, extending previous work by R. Meyer and R. Nest. We prove the subcategory of weakly contractible objects is complementary to the localizing subcategory of projective objects, which are defined in terms of "compactly induced" algebras with respect to certain proper subgroupoids related to isotropy. The resulting "strong" Baum-Connes conjecture implies the classical one, and its formulation clarifies several permanence properties and other functorial statements. We present multiple applications, including consequences for the Universal Coefficient Theorem, a generalized "Going-Down" principle, injectivity results for groupoids that are amenable at infinity, the Baum-Connes conjecture for group bundles, and a result about the invariance of K-groups of twisted groupoid C * -algebras under homotopy of twists.

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“…For now, it seems appropriated to mention that the counterexample in the present paper can be used to show that there is a counterexample to the HK-conjecture in the class of groupoids obtained from the unstable relation of Smale spaces with totally disconnected stable sets. This is of interest in light of recent results of Proietti and Yamashita [18] connecting the homology of étale groupoids to Putnam's homology theory for Smale spaces [20].…”

Section: Introductionmentioning

confidence: 95%

A counterexample to the HK-conjecture that is principal

Deeley

2022

Ergod. Th. Dynam. Sys.

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Scarparo has constructed counterexamples to Matui’s HK-conjecture. These counterexamples and other known counterexamples are essentially principal but not principal. In the present paper, a counterexample to the HK-conjecture that is principal is given. Like Scarparo’s original counterexample, our counterexample is the transformation groupoid associated to a particular odometer. However, the relevant group is the fundamental group of a flat manifold (and hence is torsion-free) and the associated odometer action is free. The examples discussed here do satisfy the rational version of the HK-conjecture.

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Homology and K-theory of dynamical systems. II. Smale spaces with totally disconnected transversal

Cited by 4 publications

References 22 publications

Binary factors of shifts of finite type

Binary factors of shifts of finite type

Categorical approach to the Baum-Connes conjecture for étale groupoids

A counterexample to the HK-conjecture that is principal

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