{"title":"Homology and $K$-theory of dynamical systems II. Smale spaces with totally disconnected transversal","authors":"Valerio Proietti, Makoto Yamashita","doi":"10.4171/jncg/494","DOIUrl":null,"url":null,"abstract":"We apply our previous work on the relation between groupoid homology and $K$-theory to Smale spaces. More precisely, we consider the unstable equivalence relation of a Smale space with totally disconnected stable sets and prove that the associated spectral sequence shows Putnam's stable homology groups on the second sheet. Moreover, this homology is in fact isomorphic to the groupoid homology of the unstable equivalence relation.","PeriodicalId":54780,"journal":{"name":"Journal of Noncommutative Geometry","volume":"8 1","pages":"0"},"PeriodicalIF":0.7000,"publicationDate":"2023-05-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Noncommutative Geometry","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.4171/jncg/494","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
We apply our previous work on the relation between groupoid homology and $K$-theory to Smale spaces. More precisely, we consider the unstable equivalence relation of a Smale space with totally disconnected stable sets and prove that the associated spectral sequence shows Putnam's stable homology groups on the second sheet. Moreover, this homology is in fact isomorphic to the groupoid homology of the unstable equivalence relation.
期刊介绍：
The Journal of Noncommutative Geometry covers the noncommutative world in all its aspects. It is devoted to publication of research articles which represent major advances in the area of noncommutative geometry and its applications to other fields of mathematics and theoretical physics. Topics covered include in particular:
Hochschild and cyclic cohomology
K-theory and index theory
Measure theory and topology of noncommutative spaces, operator algebras
Spectral geometry of noncommutative spaces
Noncommutative algebraic geometry
Hopf algebras and quantum groups
Foliations, groupoids, stacks, gerbes
Deformations and quantization
Noncommutative spaces in number theory and arithmetic geometry
Noncommutative geometry in physics: QFT, renormalization, gauge theory, string theory, gravity, mirror symmetry, solid state physics, statistical mechanics.
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