{"title":"图的有限覆盖同调中的子表示","authors":"Xenia Flamm","doi":"10.1017/S0017089523000150","DOIUrl":null,"url":null,"abstract":"Abstract Let \n$p \\;:\\; Y \\to X$\n be a finite, regular cover of finite graphs with associated deck group \n$G$\n , and consider the first homology \n$H_1(Y;\\;{\\mathbb{C}})$\n of the cover as a \n$G$\n -representation. The main contribution of this article is to broaden the correspondence and dictionary between the representation theory of the deck group \n$G$\n on the one hand and topological properties of homology classes in \n$H_1(Y;\\;{\\mathbb{C}})$\n on the other hand. We do so by studying certain subrepresentations in the \n$G$\n -representation \n$H_1(Y;\\;{\\mathbb{C}})$\n . The homology class of a lift of a primitive element in \n$\\pi _1(X)$\n spans an induced subrepresentation in \n$H_1(Y;\\;{\\mathbb{C}})$\n , and we show that this property is never sufficient to characterize such homology classes if \n$G$\n is Abelian. We study \n$H_1^{\\textrm{comm}}(Y;\\;{\\mathbb{C}}) \\leq H_1(Y;\\;{\\mathbb{C}})$\n —the subrepresentation spanned by homology classes of lifts of commutators of primitive elements in \n$\\pi _1(X)$\n . Concretely, we prove that the span of such a homology class is isomorphic to the quotient of two induced representations. Furthermore, we construct examples of finite covers with \n$H_1^{\\textrm{comm}}(Y;\\;{\\mathbb{C}}) \\neq \\ker\\!(p_*)$\n .","PeriodicalId":50417,"journal":{"name":"Glasgow Mathematical Journal","volume":"65 1","pages":"582 - 594"},"PeriodicalIF":0.5000,"publicationDate":"2021-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":"{\"title\":\"Subrepresentations in the homology of finite covers of graphs\",\"authors\":\"Xenia Flamm\",\"doi\":\"10.1017/S0017089523000150\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Abstract Let \\n$p \\\\;:\\\\; Y \\\\to X$\\n be a finite, regular cover of finite graphs with associated deck group \\n$G$\\n , and consider the first homology \\n$H_1(Y;\\\\;{\\\\mathbb{C}})$\\n of the cover as a \\n$G$\\n -representation. The main contribution of this article is to broaden the correspondence and dictionary between the representation theory of the deck group \\n$G$\\n on the one hand and topological properties of homology classes in \\n$H_1(Y;\\\\;{\\\\mathbb{C}})$\\n on the other hand. We do so by studying certain subrepresentations in the \\n$G$\\n -representation \\n$H_1(Y;\\\\;{\\\\mathbb{C}})$\\n . The homology class of a lift of a primitive element in \\n$\\\\pi _1(X)$\\n spans an induced subrepresentation in \\n$H_1(Y;\\\\;{\\\\mathbb{C}})$\\n , and we show that this property is never sufficient to characterize such homology classes if \\n$G$\\n is Abelian. We study \\n$H_1^{\\\\textrm{comm}}(Y;\\\\;{\\\\mathbb{C}}) \\\\leq H_1(Y;\\\\;{\\\\mathbb{C}})$\\n —the subrepresentation spanned by homology classes of lifts of commutators of primitive elements in \\n$\\\\pi _1(X)$\\n . Concretely, we prove that the span of such a homology class is isomorphic to the quotient of two induced representations. Furthermore, we construct examples of finite covers with \\n$H_1^{\\\\textrm{comm}}(Y;\\\\;{\\\\mathbb{C}}) \\\\neq \\\\ker\\\\!(p_*)$\\n .\",\"PeriodicalId\":50417,\"journal\":{\"name\":\"Glasgow Mathematical Journal\",\"volume\":\"65 1\",\"pages\":\"582 - 594\"},\"PeriodicalIF\":0.5000,\"publicationDate\":\"2021-07-15\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Glasgow Mathematical Journal\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1017/S0017089523000150\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Glasgow Mathematical Journal","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1017/S0017089523000150","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
Subrepresentations in the homology of finite covers of graphs
Abstract Let
$p \;:\; Y \to X$
be a finite, regular cover of finite graphs with associated deck group
$G$
, and consider the first homology
$H_1(Y;\;{\mathbb{C}})$
of the cover as a
$G$
-representation. The main contribution of this article is to broaden the correspondence and dictionary between the representation theory of the deck group
$G$
on the one hand and topological properties of homology classes in
$H_1(Y;\;{\mathbb{C}})$
on the other hand. We do so by studying certain subrepresentations in the
$G$
-representation
$H_1(Y;\;{\mathbb{C}})$
. The homology class of a lift of a primitive element in
$\pi _1(X)$
spans an induced subrepresentation in
$H_1(Y;\;{\mathbb{C}})$
, and we show that this property is never sufficient to characterize such homology classes if
$G$
is Abelian. We study
$H_1^{\textrm{comm}}(Y;\;{\mathbb{C}}) \leq H_1(Y;\;{\mathbb{C}})$
—the subrepresentation spanned by homology classes of lifts of commutators of primitive elements in
$\pi _1(X)$
. Concretely, we prove that the span of such a homology class is isomorphic to the quotient of two induced representations. Furthermore, we construct examples of finite covers with
$H_1^{\textrm{comm}}(Y;\;{\mathbb{C}}) \neq \ker\!(p_*)$
.
期刊介绍:
Glasgow Mathematical Journal publishes original research papers in any branch of pure and applied mathematics. An international journal, its policy is to feature a wide variety of research areas, which in recent issues have included ring theory, group theory, functional analysis, combinatorics, differential equations, differential geometry, number theory, algebraic topology, and the application of such methods in applied mathematics.
The journal has a web-based submission system for articles. For details of how to to upload your paper see GMJ - Online Submission Guidelines or go directly to the submission site.