Essential laminations and immersed essential surfaces in 3-manifolds

3 歧管中的基本叠片和浸入式基本表面

• 批准号: 0220439
• 负责人: Tao Li
• 金额: $ 6.78万
• 依托单位: Oklahoma State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-08-31 至 2004-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0220439&HistoricalAwards=false
• 关键词: Essential; laminations; immersed; essential; surfaces

AbstractAward: DMS-0102316Principal Investigator: Tao LiEssential laminations and immersed essential surfaces are twoimportant objects in 3-manifolds. They generalize embeddedincompressible surfaces, and are remarkably useful in obtainingtopological and geometric information of 3-manifolds. The maingoal of this project is to explore the relationships between thetopology of 3-manifolds and these two objects. The tools thatthe investigator will use include branched surfaces and immersedbranched surfaces that have been proved to be extremely useful inthe investigator's previous work. The investigator intends tocontinue his research on essential laminations and immersedsurfaces with the following goals. (1) To construct essentiallaminations in 3-manifolds obtained from Dehn surgery onhyperbolic knots. The techniques to be developed in thisresearch could potentially have great impact on some famousconjectures in knot theory, e.g., Property P for knots and thecabling conjecture. (2) To find an algorithm to decide whether a3-manifold is a Seifert fiber space. The investigator intends touse immersed branched surfaces and normal surface theory to finda practical algorithm. (3) To show that two homotopy equivalent3-manifolds, which contain essential laminations, arehomeomorphic. (4) To find an algorithm to decide whether a3-manifold contains an essential lamination.Three-manifolds are objects modeled on the 3-dimensional spacethat we are living in. These objects can be found in many othersciences, such as physics, biology, and chemistry. A geometricway of studying 3-manifolds, which is extremely fruitful, is toview a 3-manifold as a collection of 3-dimensional pieces gluedtogether along 2-dimensional surfaces. The investigator plans tostudy the structure of 3-manifolds using such 2-dimensionalsurfaces, which are called essential laminations and immersedessential surfaces. The research in this project is related toknot theory, which has helped to understand the structure of DNA;it is also related to hyperbolic geometry, which has been used byphysicists to understand the universe.

Abstractaward：DMS-0102316原理研究者：道liessential层压和沉浸的必需表面是3个manifolds中的两个重要物体。 它们概括了嵌入不可压缩的表面，并且在获取3个manifolds的文化和几何信息方面非常有用。 该项目的少女是探索三曼群和这两个对象的thetopology之间的关系。 研究人员将使用的工具包括分支表面和沉浸式表面，这些表面已被证明在研究者以前的工作中非常有用。 调查人员打算以以下目标对基本层压和沉浸式曲折的研究进行研究。 （1）在从dehn手术的精细结节中获得的三个manifolds中构建本质层接。 在本研究中要开发的技术可能会对结理论中的某些著名概念产生重大影响，例如，用于结和提出猜想的财产P。 （2）找到一种算法来决定A3-Manifold是否是Seifert纤维空间。 研究人员打算将塔斯浸入分支表面和正常的表面理论中，以发现实用算法。 （3）表明两个同型等效3个manifolds，其中包含必需的层压效果，是塑形的。 （4）找到一种算法来决定A3-manifold是否包含必不可少的层压板。三个manifolds是对我们所生活的三维间距建模的对象。这些物体可以在许多其他知识中找到，例如物理，生物学和化学。 研究3个manifolds的几何学是非常富有成果的，它是一个3曼if虫，作为沿二维表面粘合的三维碎片的集合。 研究人员计划使用这样的2二维曲面来计划3个manifolds的结构，这些结构称为必不可少的层压和沉浸式表面。 该项目的研究与结论有关，该理论有助于理解DNA的结构；它也与双曲线几何形状有关，双曲线几何形状已被用来理解宇宙。