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.

摘要奖：DMS-0102316 首席研究员：李涛 基本叠层和浸没基本表面是三流形中的两个重要对象。 它们推广了嵌入式不可压缩表面，对于获取 3 流形的拓扑和几何信息非常有用。 该项目的主要目标是探索三流形拓扑与这两个对象之间的关系。 调查员将使用的工具包括分支表面和浸入分支表面，这些工具已被证明在调查员之前的工作中非常有用。 研究人员打算继续对基本叠片和浸没表面进行研究，以实现以下目标。 (1) 在双曲结上通过 Dehn 手术获得的 3 流形中构造基本叠层。 这项研究中开发的技术可能会对结理论中的一些著名猜想产生重大影响，例如结的属性 P 和布线猜想。 (2)寻找判断a3-流形是否为Seifert纤维空间的算法。 研究人员打算利用浸入分支曲面和法向曲面理论来寻找实用的算法。 (3) 证明两个同伦等价3-流形（包含必要叠层）是同胚的。 (4) 找到一种算法来确定 a3 流形是否包含必要的叠层。三流形是根据我们所生活的 3 维空间建模的对象。这些对象可以在许多其他科学中找到，例如物理学、生物学、和化学。 研究 3 流形的一种非常富有成效的几何方法是将 3 流形视为沿着 2 维表面粘合在一起的 3 维片段的集合。 研究人员计划使用这种二维表面来研究三流形的结构，这些表面被称为本质叠层和浸没本质表面。 该项目的研究与结理论有关，该理论有助于理解DNA的结构；它也与双曲几何有关，该理论已被物理学家用来理解宇宙。