Contact topology of 3-manifolds

紧密接触结构与底层3流形的拓扑结构密切相关。它们提供了表示同调类的曲面的属上的界，与3流形上的紧叶理和辛4流形的拓扑、seiberg - witten理论和flower同调有关。Matic建议使用与本田和Kazez合作开发的剪切粘贴技术进一步分析接触歧管。剪切粘贴接触拓扑的主要部分是粘接定理。马蒂奇建议继续研究粘合技术，并将其应用于各种悬而未决的问题，比如哈肯同调球上紧密接触结构的存在。在这里，叶理理论的技术是不够的。事实上，最近有一些哈肯同调球的例子，它们不携带重叶。作为她的调查结果，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得到关于3流形拓扑的哪些信息。三维流形以我们生活的三维空间为模型。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引起的，接触结构在流体动力学中自然出现。拟议的研究有可能产生适用于这些问题的新结果。

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.