{"title":"Open book decompositions versus prime factorizations of closed, oriented 3-manifolds","authors":"P. Ghiggini, P. Lisca","doi":"10.2140/GTM.2015.19.145","DOIUrl":null,"url":null,"abstract":"Let M be a closed, oriented, connected 3–manifold and .B; / an open book decomposition on M with page † and monodromy ' . It is easy to see that the first Betti number of † is bounded below by the number of S S –factors in the prime factorization of M . Our main result is that equality is realized if and only if ' is trivial and M is a connected sum of copies of S S . We also give some applications of our main result, such as a new proof of the fact that if the closure of a braid with n strands is the unlink with n components then the braid is trivial.","PeriodicalId":115248,"journal":{"name":"Geometry and Topology Monographs","volume":"29 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2014-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Geometry and Topology Monographs","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.2140/GTM.2015.19.145","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 2
Abstract
Let M be a closed, oriented, connected 3–manifold and .B; / an open book decomposition on M with page † and monodromy ' . It is easy to see that the first Betti number of † is bounded below by the number of S S –factors in the prime factorization of M . Our main result is that equality is realized if and only if ' is trivial and M is a connected sum of copies of S S . We also give some applications of our main result, such as a new proof of the fact that if the closure of a braid with n strands is the unlink with n components then the braid is trivial.