Heegaard splitting and surgery of 3-manifolds

Heegaard 分裂和三歧管手术

基本信息

批准号：
1005556
负责人：
Tao Li
金额：
$ 16.01万
依托单位：
Boston College
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2010
资助国家：
美国
起止时间：
2010-07-01 至 2014-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1005556&HistoricalAwards=false
关键词：
Heegaard splitting surgery manifolds

项目摘要

The PI plans to study the topology of 3-manifolds, especially Heegaard splitting of 3-manifolds, using methods of amalgamation, surgery, branched surface and lamination. The first goal of the project is to construct a hyperbolic counterexample to the Rank Conjecture. The second part of the proposed research is to use branched surfaces and laminations to study Dehn surgery on knots and links in reducible 3-manifolds. One goal is to prove a major part of the Cabling Conjecture. The third part of the project is to study various aspects of Heegaard splitting. There are many interesting connections between the proposed research and several active programs of other researchers. The PI plans to explore these connections and develop new tools to achieve these goals.Three-manifolds are objects modeled on the 3-dimensional space that we are living in. A donut and the spatial universe are both examples of 3-manifolds. These objects arise naturally in many contexts in physical and other natural sciences and model many interesting phenomena. A geometric way of studying 3-manifolds is to cut a complicated 3-manifold into a collection of simpler 3-dimensional pieces along 2-dimensional surfaces. For example a Heegaard splitting is such a decomposition. Using this idea, one can also construct useful new 3-manifolds by doing surgery on well-understood ones. The PI plans to study 3-manifolds using Heegaard splitting and surgery. The research targets several central questions in low-dimensional topology and knot theory, which has potential impact on other areas of scientific investigations, such as the topological structures of DNA.

PI计划使用合并，手术，分支表面和层压方法研究3个manifolds的拓扑，尤其是3个manifolds的Heegaard分裂。该项目的第一个目标是构建一个夸张的反例。拟议的研究的第二部分是使用分支表面和层压板来研究Dehn手术在可简化的3个manifolds中的结和链接上。一个目标是证明布线猜想的主要部分。该项目的第三部分是研究Heegaard分裂的各个方面。拟议的研究与其他研究人员的几个活跃计划之间有许多有趣的联系。 PI计划探索这些连接并开发新工具以实现这些目标。三个manifolds是在我们所居住的三维空间上建模的对象。甜甜圈和空间宇宙都是3个manifolds的例子。这些物体在物理和其他自然科学的许多情况下自然出现，并模拟许多有趣的现象。研究3个manifolds的一种几何方式是将复杂的3个manifold切成沿2维表面的更简单的三维碎片的集合。例如，海加阿德（Heegaard）分裂是一种分解。使用这个想法，也可以通过对良好的手术进行手术来构建有用的新三杆。 PI计划使用Heegaard分裂和手术研究3个manifolds。该研究针对低维拓扑和结理论中的几个核心问题，该问题可能会影响科学研究的其他领域，例如DNA的拓扑结构。