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.

紧密的接触结构与执行3型manifold的拓扑密切相关。 它们提供了表面列出同源类别的属的界限，与3-manifolds的绷紧叶子以及Symbletic 4-manifolds，Toseiberg-Witten理论和浮点同源性有关。 Matic提议使用切割和粘贴技术与本田和Kazez不合转。 切割和pastecontact拓扑的主要部分是粘合定理。 Matic提议继续研究胶合技术，并将其应用于各种开放问题，例如在Haken同源性领域上存在紧密的接触结构。叶面理论的技术不足。 实际上，有一些没有含有粘胶的Haken同源球体的例子。 由于她的调查，她希望能够理解一个基本问题：关于三个manifold拓扑的信息我们可以从紧密的联系结构中获得。三维歧管是在我们生活的三维空间上建模的。符合性和接触几何形状的问题是由物理学和触点结构中的动态性问题自然出现在流体动力学中的动态问题。 拟议的研究有可能产生适用于这些问题的新结果。