Tight Contact Structures and 3-dimensional Topology

研究人员基于对各种三维流形上紧密接触结构分类的新技术，提出了探索三维拓扑接触结构的建议。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成一个很大的组合技术。研究人员建议从叶层和层片理论中引入思想和结构(与J.Etnyre和W.Kazez共同工作)。由于Gabai，凸面和旁路有助于将紧密接触流形(最终)分解成球，类似于“缝合流形分解”。切割发生的表面上的分割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遵循Gabai的缝合流形分解，在大多数3-流形上构建拉紧的叶状结构，并通过与Gabai的构建大致相同的方式粘合来构建紧密的接触结构。我们希望给出一个关于紧接触结构的有效粘合定理。另一个研究方向是勒让德里亚的纽带理论。利用实心环上紧接触结构的分类，Etnyre和Honda提出了Legendrian环结和Legendrian字形八结的分类。研究人员提出了一项关于三维空间的研究。我们研究的三维空间将局部地类似于标准的欧几里得三维空间。这些物体可能在全球范围内是非常复杂的，但局部观察者无法区分，就像蚂蚁无法分辨它是坐在空中平面上还是坐在非常大的球体上一样。长期以来，人们对二维空间进行了分类和理解--它们是二维球体、甜甜圈、有两个洞的甜甜圈、有三个洞的甜甜圈等等，并根据洞的数量来区分。然而，尽管本世纪有许多数学家进行了工作，但对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个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，非常宽泛地说，这相当于在三维空间中的每一点选择参考方向(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对三维空间有更好的了解。

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.