Contact Geometry, Contact Homology and Open Book Decompositions

摘要奖：DMS-0804820首席研究员：约翰·B·埃特尼尔这项建议的重点是更好地理解所有(奇)维接触结构的有趣性质，特别注意三维，并将接触几何技术应用于拓扑学中的问题。这项研究的第一个主要主题是开卷分解。接触几何和这些特殊的纤维连接提供了大量关于接触结构和低维拓扑的信息。作为这一提议的一部分，这一基本联系将被用来定义和研究接触结构的新的不变量和其中的特殊结--具体地说，是传奇结和横结点。从前人的工作中可以清楚地看出，这些不变量与三维中著名的紧与过扭二分以及接触结构的辛填充的性质有着微妙的联系，这反过来又对拓扑学的应用是至关重要的。该项目的这一部分预期的一些主要成果是对紧密和扭曲的接触结构中的Legendrian和横向结的各种分类结果，以及对接触外科的基本过程的更好理解。该项目还将研究联系结构和高维度的开放书籍之间的联系，目的是阐明高维度的存在问题，并试图在这里推广未来的概念。类似于与开卷的联系，接触结构也与叶理有关。这种联系在接触几何在低维拓扑中的应用中起到了重要作用。关于这种联系的许多悬而未决的问题将被探讨。提案的第二个主要主题是发展和分析更高维度的传奇接触同源。在这一理论中有许多美丽的解析和代数结构，其中许多结构还有待发现。此外，通过优雅的余法构造，人们可以用它来定义流形及其在欧氏空间中的嵌入的不变量。这些不变量似乎非常强大，将被彻底研究。流形上的接触结构是非常自然的对象，大约在两个世纪前，作为几何光学、热力学和经典力学的自然语言。每天，当平行停车、滑冰或观看一杯水中的棱镜运动时，人们都会遇到接触结构。接触结构已经被许多数学家研究过，似乎触及了数学和物理的不同领域，但直到最近才进入数学的前台。这是由于接触拓扑学取得了许多显著的突破，形成了丰富而美丽的理论，在三维和四维拓扑(即空间结构和时空结构)、弦理论和现代物理中的大对偶以及流体力学等领域都有许多深远的应用。主要调查者将研究接触几何和各种拓扑对象(如开卷分解和分叶)之间的关系。此外，他还将开发用于研究高维接触结构的解析和代数工具。总体而言，预计在本项目结束时，我们将对所有维度的接触结构有更好的理解，并看到对低维流形的更多令人惊讶的含义。

AbstractAward: DMS-0804820Principal Investigator: John B. EtnyreThe focus of this proposal is to better understand the intriguingnature of contact structures in all (odd) dimensions, withspecial attention given to dimension three and to apply contactgeometric techniques to questions in topology. The first maintheme of the proposed research is open book decompositions. Theconnection of contact geometry and these specially fibered linkshave provided a fount of information concerning contactstructures and low-dimensional topology. As part of this proposalthis fundamental connection will be exploited to define and studynew invariants of contact structures and special knots in them -specifically, Legendrian and transversal knots. From prior workit is clear these invariants have subtle connections to the famedtight vs. overtwisted dichotomy in dimension three and propertiesof symplectic fillings of contact structures, which in turn arecrucial to applications to topology. Some of the major outcomesexpected from this part of the project are various classificationresults for Legendrian and transversal knots in tight andovertwisted contact structures and a better understanding of thefundamental process of contact surgery. The project will alsostudy the connection between contact structures and open books inhigh dimensions with the goal of illuminating existence questionsin high dimensions and trying to generalize the notion ofovertwistedness here as well. Similar to the connection with openbooks, contact structures are also related to foliations. Thisconnection has been instrumental in applications of contactgeometry to low-dimensional topology. Many of the outstandingquestions concerning this connection will be explored. A secondmain theme of the proposal is the development and analysis ofLegendrian contact homology in higher dimensions. There is agreat deal of beautiful analytic and algebraic structure in thistheory, much of which is yet to be discovered. Moreover, throughthe elegant conormal construction one can then use this to defineinvariants of manifolds and of their embeddings in Euclideanspace. These invariants appear to be extremely powerful and willbe thoroughly investigated.Contact structures on manifolds are very natural objects, bornover two centuries ago, as a natural language for geometricoptics, thermodynamics and classical mechanics. Everyday, oneencounters contact structures when parallel parking a car,skating, or watching the prismatic play of light in a glass ofwater. Contact structures have been studied by manymathematicians and seem to touch on diverse areas of mathematicsand physics, but only recently have they moved into theforeground of mathematics. This is due to the many remarkablebreakthroughs in contact topology, resulting in a rich andbeautiful theory with many far reaching applications to areassuch as three and four dimensional topology (that is thestructure of space and space- time), string theory and large Ndualities in modern physics, and fluid dynamics. The PrincipalInvestigator will study the relation between contact geometry andvarious topological objects (such as open book decompositions andfoliations). In addition he will develop analytic and algebraictools for studying contact structures in high dimensions. As awhole it is expected that at the conclusion of this project wewill have a much better understanding of contact structures inall dimensions and see many more surprising implications for low-dimensional manifolds.