{"title":"Persistent homology with non-contractible preimages","authors":"K. Mischaikow, C. Weibel","doi":"10.4310/hha.2022.v24.n2.a16","DOIUrl":null,"url":null,"abstract":". For a fixed N , we analyze the space of all sequences z = ( z 1 , . . . , z N ), approximating a continuous function on the circle, with a given persistence diagram P , and show that the typical components of this space are homotopy equivalent to S 1 . We also consider the space of functions on Y -shaped (resp., star-shaped) trees with a 2-point persistence diagram, and show that this space is homotopy equivalent to S 1 (resp., to a bouquet of circles).","PeriodicalId":55050,"journal":{"name":"Homology Homotopy and Applications","volume":null,"pages":null},"PeriodicalIF":0.8000,"publicationDate":"2021-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"4","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Homology Homotopy and Applications","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.4310/hha.2022.v24.n2.a16","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 4
Abstract
. For a fixed N , we analyze the space of all sequences z = ( z 1 , . . . , z N ), approximating a continuous function on the circle, with a given persistence diagram P , and show that the typical components of this space are homotopy equivalent to S 1 . We also consider the space of functions on Y -shaped (resp., star-shaped) trees with a 2-point persistence diagram, and show that this space is homotopy equivalent to S 1 (resp., to a bouquet of circles).
期刊介绍：
Homology, Homotopy and Applications is a refereed journal which publishes high-quality papers in the general area of homotopy theory and algebraic topology, as well as applications of the ideas and results in this area. This means applications in the broadest possible sense, i.e. applications to other parts of mathematics such as number theory and algebraic geometry, as well as to areas outside of mathematics, such as computer science, physics, and statistics. Homotopy theory is also intended to be interpreted broadly, including algebraic K-theory, model categories, homotopy theory of varieties, etc. We particularly encourage innovative papers which point the way toward new applications of the subject.
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