Essential laminations and immersed essential surfaces in 3-manifolds

摘要奖：DMS-0102316主要研究人员：陶力本质层和浸没本质面是三维流形中的两个重要对象。它们推广了嵌入到可压缩曲面上，并且在获得三维流形的拓扑和几何信息方面非常有用。这个项目的主要目的是探索三维流形的拓扑学与这两个对象之间的关系。调查人员将使用的工具包括分支表面和浸入分支表面，这些表面在调查人员以前的工作中被证明是非常有用的。这位研究人员打算继续他对基本叠层和浸没表面的研究，目标如下。(1)在双曲纽结的Dehn手术得到的3-流形上构造本质层。这项研究中将要发展的技术可能会对纽结理论中的一些著名猜想产生巨大的影响，例如纽结的性质P和令人信服的猜想。(2)寻找判定3-流形是否为Seifert纤维空间的算法。研究人员试图利用浸入分枝曲面和法面理论来寻找一种实用的算法。(3)证明了包含本质叠层的两个同伦等价3-流形是同胚的。(4)寻找一种算法来判定一个3-流形是否包含本质层。3-流形是在我们所居住的3维空间上模拟的对象。这些天体可以在许多其他科学中找到，例如物理、生物和化学。研究三维流形的一种几何方法是将三维流形视为沿着二维表面粘合在一起的三维碎片的集合，这是非常有成效的。这位研究人员计划使用这样的二维表面来研究三维流形的结构，这种表面被称为本质层和浸没本质表面。该项目的研究与令牌论有关，有助于理解DNA的结构；也与双曲几何有关，物理学家已利用双曲几何来理解宇宙。

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.