{"title":"完全标记曲面定理","authors":"David Gabai, Mehdi Yazdi","doi":"10.4310/ACTA.2020.V225.N2.A4","DOIUrl":null,"url":null,"abstract":"In his seminal 1976 paper Bill Thurston observed that a closed leaf S of a foliation has Euler characteristic equal, up to sign, to the Euler class of the foliation evaluated on [S], the homology class represented by S. The main result of this paper is a converse for taut foliations: if the Euler class of a taut foliation $\\mathcal{F}$ evaluated on [S] equals up to sign the Euler characteristic of S and the underlying manifold is hyperbolic, then there exists another taut foliation $\\mathcal{F'}$ such that $S$ is homologous to a union of leaves and such that the plane field of $\\mathcal{F'}$ is homotopic to that of $\\mathcal{F}$. In particular, $\\mathcal{F}$ and $\\mathcal{F'}$ have the same Euler class. \nIn the same paper Thurston proved that taut foliations on closed hyperbolic 3-manifolds have Euler class of norm at most one, and conjectured that, conversely, any integral cohomology class with norm equal to one is the Euler class of a taut foliation. This is the second of two papers that together give a negative answer to Thurston's conjecture. In the first paper, counterexamples were constructed assuming the main result of this paper.","PeriodicalId":4,"journal":{"name":"ACS Applied Energy Materials","volume":null,"pages":null},"PeriodicalIF":5.4000,"publicationDate":"2020-08-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"5","resultStr":"{\"title\":\"The fully marked surface theorem\",\"authors\":\"David Gabai, Mehdi Yazdi\",\"doi\":\"10.4310/ACTA.2020.V225.N2.A4\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"In his seminal 1976 paper Bill Thurston observed that a closed leaf S of a foliation has Euler characteristic equal, up to sign, to the Euler class of the foliation evaluated on [S], the homology class represented by S. The main result of this paper is a converse for taut foliations: if the Euler class of a taut foliation $\\\\mathcal{F}$ evaluated on [S] equals up to sign the Euler characteristic of S and the underlying manifold is hyperbolic, then there exists another taut foliation $\\\\mathcal{F'}$ such that $S$ is homologous to a union of leaves and such that the plane field of $\\\\mathcal{F'}$ is homotopic to that of $\\\\mathcal{F}$. In particular, $\\\\mathcal{F}$ and $\\\\mathcal{F'}$ have the same Euler class. \\nIn the same paper Thurston proved that taut foliations on closed hyperbolic 3-manifolds have Euler class of norm at most one, and conjectured that, conversely, any integral cohomology class with norm equal to one is the Euler class of a taut foliation. This is the second of two papers that together give a negative answer to Thurston's conjecture. In the first paper, counterexamples were constructed assuming the main result of this paper.\",\"PeriodicalId\":4,\"journal\":{\"name\":\"ACS Applied Energy Materials\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":5.4000,\"publicationDate\":\"2020-08-17\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"5\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"ACS Applied Energy Materials\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.4310/ACTA.2020.V225.N2.A4\",\"RegionNum\":3,\"RegionCategory\":\"材料科学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"CHEMISTRY, PHYSICAL\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"ACS Applied Energy Materials","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.4310/ACTA.2020.V225.N2.A4","RegionNum":3,"RegionCategory":"材料科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"CHEMISTRY, PHYSICAL","Score":null,"Total":0}
引用次数: 5
摘要
Bill Thurston在其1976年的开创性论文中观察到,叶理的闭叶S的欧拉特征与[S]上评估的叶理的欧拉类(由S表示的同源类)一致。本文的主要结果是张叶理的逆:如果在[S]上评估的张叶理$\mathcal{F}$的欧拉类等于S的欧拉特征,并且下面的流形是双曲的,则存在另一张紧叶理$\mathcal{F’}$,使得$S$与叶的并集同源。特别是,$\mathcal{F}$和$\mathical{F’}$具有相同的Euler类。在同一篇论文中,Thurston证明了闭双曲3-流形上的张叶理具有最多为1的欧拉范数类,并推测反过来,任何范数等于1的积分上同调类都是张叶理的欧拉类。这是两篇论文中的第二篇,这两篇论文共同对瑟斯顿猜想给出了否定的答案。在第一篇论文中,假设本文的主要结果,构造了反例。
In his seminal 1976 paper Bill Thurston observed that a closed leaf S of a foliation has Euler characteristic equal, up to sign, to the Euler class of the foliation evaluated on [S], the homology class represented by S. The main result of this paper is a converse for taut foliations: if the Euler class of a taut foliation $\mathcal{F}$ evaluated on [S] equals up to sign the Euler characteristic of S and the underlying manifold is hyperbolic, then there exists another taut foliation $\mathcal{F'}$ such that $S$ is homologous to a union of leaves and such that the plane field of $\mathcal{F'}$ is homotopic to that of $\mathcal{F}$. In particular, $\mathcal{F}$ and $\mathcal{F'}$ have the same Euler class.
In the same paper Thurston proved that taut foliations on closed hyperbolic 3-manifolds have Euler class of norm at most one, and conjectured that, conversely, any integral cohomology class with norm equal to one is the Euler class of a taut foliation. This is the second of two papers that together give a negative answer to Thurston's conjecture. In the first paper, counterexamples were constructed assuming the main result of this paper.
期刊介绍:
ACS Applied Energy Materials is an interdisciplinary journal publishing original research covering all aspects of materials, engineering, chemistry, physics and biology relevant to energy conversion and storage. The journal is devoted to reports of new and original experimental and theoretical research of an applied nature that integrate knowledge in the areas of materials, engineering, physics, bioscience, and chemistry into important energy applications.