Essential laminations and immersed essential surfaces in 3-manifolds

AbstractAward：DMS-0102316主要研究者：李涛本质层和浸入本质面是三维流形中的两个重要对象。 它们推广了嵌入不可压缩曲面，在获取三维流形的拓扑和几何信息方面非常有用。 本项目的主要目的是探索三维流形的拓扑结构与这两个对象之间的关系。 研究人员将使用的工具包括分支表面和浸入分支表面，这些表面在研究人员以前的工作中已经被证明是非常有用的。 研究人员打算继续他的研究基本层压和浸没表面与以下目标。 (1)在由双曲纽结的Dehn手术得到的三维流形中构造本质分层。 在这项研究中开发的技术可能会对纽结理论中的一些著名理论产生重大影响，例如，纽结的性质P与作图猜想 (2)寻找判定三维流形是否为塞弗特纤维空间的算法。 研究者试图利用浸入分支曲面和法向曲面理论来寻找一种实用的算法。 (3)证明了两个含有本质层的同伦等价3-流形是同胚的。 (4)寻找一个算法来判断三维流形是否包含本质层。三维流形是在我们所生活的三维空间中建模的物体。 这些对象可以在许多其他科学中找到，如物理学，生物学和化学。 研究三维流形的一种几何方法是将三维流形看作是沿沿着二维曲面粘合在一起的三维块的集合，这是一种非常有成效的方法。 研究者计划用这种二维曲面来研究三维流形的结构，这种二维曲面被称为本质叠层和浸入本质曲面。 这个项目的研究与纽结理论有关，它有助于理解DNA的结构;它也与双曲几何有关，物理学家用它来理解宇宙。

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.