{"title":"Magnitude homology and homotopy type of metric fibrations","authors":"Yasuhiko Asao, Yu Tajima, Masahiko Yoshinaga","doi":"arxiv-2409.03278","DOIUrl":null,"url":null,"abstract":"In this article, we show that each two metric fibrations with a common base\nand a common fiber have isomorphic magnitude homology, and even more, the same\nmagnitude homotopy type. That can be considered as a generalization of a fact\nproved by T. Leinster that the magnitude of a metric fibration with finitely\nmany points is a product of those of the base and the fiber. We also show that\nthe definition of the magnitude homotopy type due to the second and the third\nauthors is equivalent to the geometric realization of Hepworth and Willerton's\npointed simplicial set.","PeriodicalId":501119,"journal":{"name":"arXiv - MATH - Algebraic Topology","volume":"87 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Algebraic Topology","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.03278","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
In this article, we show that each two metric fibrations with a common base
and a common fiber have isomorphic magnitude homology, and even more, the same
magnitude homotopy type. That can be considered as a generalization of a fact
proved by T. Leinster that the magnitude of a metric fibration with finitely
many points is a product of those of the base and the fiber. We also show that
the definition of the magnitude homotopy type due to the second and the third
authors is equivalent to the geometric realization of Hepworth and Willerton's
pointed simplicial set.
在本文中,我们证明了具有共同基点和共同纤维的两个度量纤度具有同构的幅同调,甚至具有相同的幅同调类型。这可以看作是 T. Leinster 所证明的一个事实的一般化,即具有有限多点的度量纤度的幅是基点和纤维的幅的乘积。我们还证明了第二位和第三位作者关于幅同调类型的定义等同于赫普沃思和威勒顿的点简集的几何实现。