Tight Contact Structures and 3-dimensional Topology

摘要研究人员提出，基于对各种三维流形上紧密接触结构进行分类的新技术，通过接触结构探索三维拓扑结构。我们的主要目标是将涉及凸面和“旁路”的三维剪切和粘贴技术发展为一种很大程度上的组合技术。研究者建议从叶状和层压理论中引入思想和结构（与J. Etnyre和W. Kazez合作）。凸表面和旁路有助于将紧密接触流形分解成球，类似于Gabai的“缝合流形分解”。岩屑发生的表面上的分度曲线决定了紧密接触结构。目前正在进行的一个项目是仔细地遵循Gabai的缝合流形分解，在大多数3-流形上构建紧叶理，并通过与Gabai的构造相同的粘合方式构建紧密接触结构。我们希望为紧密接触结构提供一个有效的粘接定理。另一个研究方向是legendrian knot theory。Etnyre和Honda利用实体环面上紧密接触结构的分类，提出了Legendrian环面结和Legendrian 8字形结的分类。研究人员提议对三维空间进行研究。我们研究的三维空间局部与标准欧几里得三维空间相似。这些物体在全局上可能非常复杂，但局部观察者无法分辨其中的区别，就像蚂蚁无法分辨它是坐在一个平面上还是一个非常大的球体上一样。“有限的”二维空间已经被分类和理解了很长时间——它们是二维球体、甜甜圈、有2个洞的甜甜圈、有3个洞的甜甜圈等等，并通过洞的数量来区分。然而，尽管本世纪许多数学家做了很多工作，但三维空间的完整分类还远远没有被理解。在我们的工作中，我们试图通过强加一个额外的结构来更好地理解三维空间，这个结构被称为接触结构，非常宽松地说，相当于在三维空间的每个点上选择一个特定的方向（或一个旋转轴）。接触结构与四维几何、量子物理和动力学（如流体动力学）有着密切的联系，我们希望通过接触结构更好地理解三维空间。

Proposal: DMS-0072853AbstractThe investigators propose to explore 3-dimensional topology viacontact structures, based on new techniques in the classificationof tight contact structures on various 3-dimensional manifolds. Ourmain goal is to develop the 3-dimensional cut-and-paste techniquesinvolving convex surfaces and ``bypasses" into a largelycombinatorial one. The investigators propose to import ideas andconstructions from the theory of foliations and laminations (injoint work with J. Etnyre and W. Kazez). Convex surfaces andbypasses aid the decomposition of a tight contact manifold(eventually) into balls, similar to the `` sutured manifolddecomposition", due to Gabai. The ``dividing curves on thesurfaces along which the cuttings take place determine the tightcontact structure. A project which is currently under way is tocarefully follow the sutured manifold decompositions of Gabai inconstructing taut foliations on most 3-manifolds, and to constructtight contact structures by gluing in much the same way as Gabai'sconstruction. We hope to produce an effective gluing theorem fortight contact structures. Another direction of research isLegendrian knot theory. Using the classification of tight contactstructures on solid tori, Etnyre and Honda propose to classifyLegendrian torus knots and Legendrian figure eight knots. The investigators propose a study of 3-dimensional spaces. The3-dimensional spaces we study will locally be similar to thestandard Euclidean 3-dimensional space. These objects may be verycomplicated globally, but a local observer cannot tell thedifference, just as an ant cannot tell whether it is sitting on aflat plane or a very large sphere. `Finite' 2-dimensional spaceshave been classified and understood for a long time - they are the2-dimensional sphere, the doughnut, the doughnut with 2 holes, the doughnutwith 3 holes, etc., and are distinguished by the number of holes. However, in spite of work by numerous mathematicians this century,a complete classification of 3-dimensional spaces is far fromunderstood. In our work we seek to better understand 3-dimensionalspaces by imposing an additional structure, called a contactstructure, which, very loosely speaking, amounts to choosing apreferred direction (or a spinning axis) at every point in the3-dimensional space. Contact structures have intimate connectionswith 4-dimensional geometry, quantum physics, and dynamics (such asfluid dynamics), and we hope to gain better understanding of3-dimensional spaces through contact structures.