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.

紧切触结构与底层三维流形的拓扑结构密切相关。 它们提供了一个同调类的曲面的亏格上的界，与3-流形上的紧叶理和辛4-流形的拓扑有关，与Seiberg-Witten理论和Floer同调有关。马蒂奇建议进一步分析接触歧管使用剪切和粘贴技术开发incolonywith本田和Kazez。 剪切粘贴接触拓扑的主要部分是胶合定理。 Matic建议继续研究胶合技术，并将其应用于各种开放的问题，如Haken同调球面上紧接触结构的存在。 事实上，最近有一些哈肯同调球的例子，它们不带重瓣。 作为她的调查结果，她希望能够更好地理解一个基本问题：什么样的信息的拓扑3流形，我们可以得到从存在的紧接触结构.三维流形是在我们生活的三维空间上建立的。 辛几何和接触几何中的问题最初是由物理学中的动力学问题激发的，而接触结构自然地出现在流体力学中。 拟议的研究有可能产生适用于这些问题的新结果。