{"title":"论大小（共）同源中的有限生成及其扭转","authors":"Luigi Caputi, Carlo Collari","doi":"10.1112/blms.13143","DOIUrl":null,"url":null,"abstract":"<p>The aim of this paper is to apply the framework developed by Sam and Snowden to study structural properties of graph homologies, in the spirit of Ramos, Miyata and Proudfoot. Our main results concern the magnitude homology of graphs introduced by Hepworth and Willerton, and we prove that it is a finitely generated functor (on graphs of bounded genus). More precisely, for graphs of bounded genus, we prove that magnitude cohomology, in each homological degree, has rank which grows at most polynomially in the number of vertices, and that its torsion is bounded. As a consequence, we obtain analogous results for path homology of (undirected) graphs.</p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"56 11","pages":"3434-3451"},"PeriodicalIF":0.8000,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/blms.13143","citationCount":"0","resultStr":"{\"title\":\"On finite generation in magnitude (co)homology and its torsion\",\"authors\":\"Luigi Caputi, Carlo Collari\",\"doi\":\"10.1112/blms.13143\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>The aim of this paper is to apply the framework developed by Sam and Snowden to study structural properties of graph homologies, in the spirit of Ramos, Miyata and Proudfoot. Our main results concern the magnitude homology of graphs introduced by Hepworth and Willerton, and we prove that it is a finitely generated functor (on graphs of bounded genus). More precisely, for graphs of bounded genus, we prove that magnitude cohomology, in each homological degree, has rank which grows at most polynomially in the number of vertices, and that its torsion is bounded. As a consequence, we obtain analogous results for path homology of (undirected) graphs.</p>\",\"PeriodicalId\":55298,\"journal\":{\"name\":\"Bulletin of the London Mathematical Society\",\"volume\":\"56 11\",\"pages\":\"3434-3451\"},\"PeriodicalIF\":0.8000,\"publicationDate\":\"2024-09-10\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://onlinelibrary.wiley.com/doi/epdf/10.1112/blms.13143\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Bulletin of the London Mathematical Society\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1112/blms.13143\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Bulletin of the London Mathematical Society","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1112/blms.13143","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
On finite generation in magnitude (co)homology and its torsion
The aim of this paper is to apply the framework developed by Sam and Snowden to study structural properties of graph homologies, in the spirit of Ramos, Miyata and Proudfoot. Our main results concern the magnitude homology of graphs introduced by Hepworth and Willerton, and we prove that it is a finitely generated functor (on graphs of bounded genus). More precisely, for graphs of bounded genus, we prove that magnitude cohomology, in each homological degree, has rank which grows at most polynomially in the number of vertices, and that its torsion is bounded. As a consequence, we obtain analogous results for path homology of (undirected) graphs.
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