Tight Contact Structures and 3-dimensional Topology

紧接触结构和 3 维拓扑

• 批准号: 0072853
• 负责人: Gordana Matic
• 金额: $ 13.43万
• 依托单位: University of Georgia Research Foundation Inc
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-07-15 至 2005-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0072853&HistoricalAwards=false
• 关键词: Tight; Contact; Structures; dimensional; Topology

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.

提案：DMS-0072853摘要研究人员建议基于各种3维流形上紧密接触结构分类的新技术，通过接触结构探索3维拓扑。 我们的主要目标是将涉及凸面和“旁路”的 3 维剪切和粘贴技术发展为一种很大程度上组合的技术。研究人员建议从叶状和层状理论中引入思想和结构（与 J. Etnyre 和 W 共同工作） .Kazez）。凸面和旁路有助于将紧密接触歧管（最终）分解成球，类似于“缝合”。流形分解”，归因于 Gabai。 沿切割发生的表面上的分割曲线决定了紧密的接触结构。 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 流形上构造拉紧的叶状结构，并通过与 Gabai 的构造大致相同的方式粘合来构造紧密的接触结构。 我们希望针对紧密接触结构提出有效的粘合定理。 另一个研究方向是勒让德结理论。 Etnyre 和 Honda 使用实体环面上的紧密接触结构分类，提出对勒让德圆环结和勒让德八字结进行分类。研究人员提出了一项 3 维空间的研究。 我们研究的三维空间局部类似于标准欧几里得三维空间。 这些物体在全球范围内可能非常复杂，但本地观察者无法区分其中的差异，就像蚂蚁无法区分它是坐在平面上还是位于非常大的球体上一样。 “有限”二维空间长期以来一直被分类和理解——它们是二维球体、甜甜圈、2个孔的甜甜圈、3个孔的甜甜圈等，并通过孔的数量来区分。然而，尽管本世纪许多数学家做了很多工作，但对 3 维空间的完整分类仍远未被理解。 在我们的工作中，我们试图通过施加一个额外的结构（称为接触结构）来更好地理解 3 维空间，非常宽松地说，这相当于在 3 维空间中的每个点选择一个首选方向（或旋转轴）。 接触结构与4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3维空间。