Essential laminations and immersed essential surfaces in 3-manifolds

摘要/ abstract摘要：[dms -0102316]作者：李涛。本质片和浸没本质面是三维流形中的两个重要对象。它们推广了嵌入不可压缩曲面，在获取3流形的拓扑和几何信息方面非常有用。这个项目的主要目标是探索3-流形的拓扑结构和这两个对象之间的关系。研究者将使用的工具包括分支表面和浸入式分支表面，这在研究者以前的工作中被证明是非常有用的。研究者打算继续他的研究基本层压和浸没表面与以下目标。(1)构造双曲结的Dehn手术得到的3-流形的基本构型。在本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，结的性质P和电缆猜想。(2)寻找一种判定a3流形是否为Seifert光纤空间的算法。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)证明了包含基本片的两个同伦等价3-流形是同胚的。(4)寻找一种判定三维流形是否包含必要层合的算法。三维流形是以我们生活的三维空间为模型的物体。这些对象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非常富有成效的几何方法是将三维流形视为沿二维表面粘合在一起的三维块的集合。研究者计划使用这种二维表面来研究3-流形的结构，这些二维表面被称为基本层合和浸没基本表面。该项目的研究与结理论有关，结理论有助于理解DNA的结构；它也与双曲几何有关，双曲几何被物理学家用来理解宇宙。

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.