{"title":"Topological symmetries of simply connected 4-manifolds and actions of automorphism groups of free groups","authors":"Shengkui Ye","doi":"10.1093/qmath/haac042","DOIUrl":null,"url":null,"abstract":"Abstract Let M be a simply connected closed 4-manifold. It is proved that any (possibly finite) compact Lie group acting effectively and homologically trivially on M by homeomorphisms is an abelian group of rank at most two, when $b_{2}(M)\\gt2$. As applications, let $\\mathrm{Aut}(F_{n})$ be the automorphism group of the free group of rank $n.$ We prove that any group action of $\\mathrm{Aut}(F_{n})$ and $\\mathrm{GL}_{n}\\mathbb{Z}$, n > = 4, on $M\\neq S^{4}$ factors through $\\mathbb{Z}/2$, if the group action is by homologically trivial homeomorphisms.","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":"26 1","pages":"0"},"PeriodicalIF":0.6000,"publicationDate":"2023-01-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Quarterly Journal of Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1093/qmath/haac042","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Abstract Let M be a simply connected closed 4-manifold. It is proved that any (possibly finite) compact Lie group acting effectively and homologically trivially on M by homeomorphisms is an abelian group of rank at most two, when $b_{2}(M)\gt2$. As applications, let $\mathrm{Aut}(F_{n})$ be the automorphism group of the free group of rank $n.$ We prove that any group action of $\mathrm{Aut}(F_{n})$ and $\mathrm{GL}_{n}\mathbb{Z}$, n > = 4, on $M\neq S^{4}$ factors through $\mathbb{Z}/2$, if the group action is by homologically trivial homeomorphisms.
期刊介绍:
The Quarterly Journal of Mathematics publishes original contributions to pure mathematics. All major areas of pure mathematics are represented on the editorial board.