Tight Contact Structures and 3-dimensional Topology

提案：DMS-0072853摘要研究者们提出，基于各种三维流形上紧接触结构分类的新技术，通过接触结构探索三维拓扑。 我们的主要目标是将涉及凸面和"旁路”的三维剪切粘贴技术发展成一种大规模的组合技术。研究者们建议从叶理和层理理论中引进一些想法和构造（与J. Etnyre和W. Kazez）。凸面和旁路有助于将紧接触流形（最终）分解为球，类似于Gabai提出的“缝合流形分解”。 岩屑产生的表面沿着的“分界曲线”决定了紧密接触构造。 目前正在进行的一个项目是仔细遵循缝合歧管分解的Gabai在构建拉紧叶理上的大多数3-流形，并构建紧接触结构胶合在很大程度上相同的方式Gabai的建设。 我们希望得到一个有效的紧接触结构的胶合定理。 另一个研究方向是勒让德纽结理论。Etnyre和本田利用固体环面上紧接触结构的分类方法，提出了Legendrian环面纽结和Legendrian 8字形纽结的分类方法。研究人员提出了一个三维空间的研究。 我们研究的三维空间在局部上类似于标准的欧几里德三维空间。 这些物体在整体上可能非常复杂，但局部观察者无法区分，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。 “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、有两个孔的甜甜圈、有三个孔的甜甜圈，等等，并且通过孔的数量来区分。然而，尽管这个世纪许多数学家做了大量的工作，三维空间的完整分类还远未被理解。 在我们的工作中，我们试图通过施加一种额外的结构来更好地理解三维空间，这种结构被称为接触结构，非常松散地说，相当于在三维空间中的每一点上选择一个参考方向（或旋转轴）。 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.