Contact topology and automorphisms of surfaces

接触拓扑和表面自同构

• 批准号: 0711341
• 负责人: Gordana Matic
• 金额: $ 24.4万
• 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: 2007
• 资助国家: 美国
• 起止时间: 2007-07-01 至 2012-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0711341&HistoricalAwards=false
• 关键词: Contact; topology; automorphisms; surfaces

Fundamental work of Giroux established a one-to-one correspondence between contact structures on closed three-manifolds and automorphisms of surfaces up to stabilization via compatible open book decompositions. It is this correspondence between two classically studied, fundamentally important, objects that the PIs propose to study. Since positive stabilization of surface automorphisms generates an almost intractable equivalence relation, it is important to discern the properties of a contact structure from just a single representative automorphism. An example of such a result is the fact that automorphisms which are compositions of positive Dehn twists induce contact structures that are Stein fillable, i.e. that arise as natural boundaries of Stein manifolds. We are working to understand questions like: what property of an automorphism guarantees symplectic fillability, what property implies existence of Giroux torsion. Investigating these questions will have applications to the study of contact invariants in Heegaard-Floer homology theory in both the bounded and unbounded cases.Contact topology or geometry and its even dimensional counterparts, symplectic topology or geometry, were born out of the study of questions arising in classical mechanics and thermodynamics. Three-dimensional manifolds are mathematical objects modeled on the space in which we live. Contact structures on such spaces arise naturally in the study of fluid flows as the family of planes perpendicular to the flow. A familiar example of a contact structure occurs in the design of DLP front projection televisions where they dictate the use of literally millions of tiny mirrors rather than one large curved mirror. Considerable progress has been made in the last several decades on the three-dimensional contact topology and four-dimensional symplectic topology. Recent progress has allowed researchers to apply two-dimensional techniques to the inherently three-dimensional study of contact topology.

Giroux 的基础工作在封闭三流形上的接触结构和表面自同构之间建立了一对一的对应关系，直到通过兼容的开书分解达到稳定。 PI 提议研究的正是两个经典研究的、极其重要的对象之间的对应关系。 由于表面自同构的正稳定性产生几乎难以处理的等价关系，因此从单个代表性自同构中辨别接触结构的性质非常重要。 这种结果的一个例子是这样的事实：作为正 Dehn 扭曲的组合的自同构引起可斯坦因填充的接触结构，即作为斯坦因流形的自然边界而出现。我们正在努力理解这样的问题：自同构的什么属性保证了辛可填充性，什么属性意味着吉鲁挠率的存在。研究这些问题将应用于有界和无界情况下 Heegaard-Floer 同调理论中接触不变量的研究。接触拓扑或几何及其偶维对应物，辛拓扑或几何，诞生于对所出现问题的研究在经典力学和热力学中。三维流形是根据我们居住的空间建模的数学对象。 这种空间上的接触结构在流体流动研究中自然出现，作为垂直于流动的平面族。接触结构的一个熟悉的例子出现在 DLP 正投电视的设计中，它们要求使用数百万个微小的镜子，而不是一个大的曲面镜。 在过去的几十年里，三维接触拓扑和四维辛拓扑已经取得了相当大的进展。 最近的进展使研究人员能够将二维技术应用于接触拓扑的固有三维研究。