Contact topology of 3-manifolds

3 流道的接触拓扑

• 批准号: 0410066
• 负责人: Gordana Matic
• 金额: --
• 依托单位: University of Georgia Research Foundation Inc
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-08-01 至 2008-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0410066&HistoricalAwards=false
• 关键词: Contact; topology; manifolds

Tight contact structures are closely related to the topology of theunderlying 3-manifold. They provide bounds on the genus of the surfacerepresenting a homology class, are related to taut foliations on3-manifolds and to the topology of symplectic 4-manifolds, toSeiberg-Witten theory and Floer homology. Matic proposes to furtheranalyze contact manifolds using cut-and-paste techniques developed incollaboration with Honda and Kazez. The main part of cut-and-pastecontact topology are gluing theorems. Matic proposes to keep studyingthe gluing techniques and to apply them to various open questions, likethe existence of tight contact structures on Haken homology spheres.Here the techniques from foliation theory fall short. There are infact recent examples of Haken homology spheres that do not carry tautfoliations. As a result of her investigations she hopes to be able tobetter understand a basic question: what information about the topologyof the 3-manifold can we get from existence of a tight contactstructure. Three-dimensional manifolds are modeled on the three dimensional spacewe live in. Questions in symplectic and contact geometry wereoriginally motivated by dynamical questions in physics and contactstructures arise naturally in hydrodynamics. The proposed research hasthe potential to produce new results applicable to these questions.

紧接触结构与底层三流形的拓扑结构密切相关。 它们提供了代表同调类的表面属的界限，与 3-流形上的张紧叶化和辛 4-流形的拓扑、Seiberg-Witten 理论和 Floer 同调有关。 Matic 建议使用与 Honda 和 Kazez 合作开发的剪切粘贴技术进一步分析接触流形。 剪切粘贴接触拓扑的主要部分是粘合定理。 马蒂奇建议继续研究粘合技术，并将其应用于各种开放性问题，例如哈肯同源球上紧密接触结构的存在。在这里，叶状结构理论的技术存在不足。 事实上，最近有一些哈肯同源球的例子不带有互叶化。 通过她的研究，她希望能够更好地理解一个基本问题：我们可以从紧密接触结构的存在中获得有关三流形拓扑的哪些信息。三维流形是在我们生活的三维空间上建模的。辛和接触几何中的问题最初是由物理学中的动力学问题引发的，而接触结构自然地出现在流体动力学中。 拟议的研究有可能产生适用于这些问题的新结果。