Topics in low-dimensional topology

低维拓扑主题

• 批准号: 0705285
• 负责人: Tao Li
• 金额: $ 12.97万
• 依托单位: Boston College
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-07-01 至 2011-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0705285&HistoricalAwards=false
• 关键词: Topics; low; dimensional; topology

The PI plans to explore the interconnections between geometric, topological and combinatorial structures of 3-manifolds through efficient triangulations, branched surfaces and Heegaard splittings. The first part of the project is to construct geometric triangulations for certain 3-manifolds. The second part of the 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 the Cabling Conjecture and to give a new proof of the Property R theorem for knots. The third part of the project is to study various aspects of Heegaard splittings. There are many interesting connections between the research and several major questions in low-dimensional topology. Results from this project may pave the road toward the resolution of some difficult but important conjectures in low-dimensional topology. The PI plans to develop new tools and use techniques from his previous work, such as branched surfaces a nd laminations, 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. Three-manifolds have beautiful geometric and topological properties and these properties are encoded in certain combinatorial information. The PI will explore such combinatorial information through some geometric objects called efficient triangulation, branched surface and Heegaard splitting. 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 计划通过高效的三角剖分、分支曲面和 Heegaard 分裂来探索 3 流形的几何、拓扑和组合结构之间的互连。 该项目的第一部分是为某些 3 流形构建几何三角剖分。研究的第二部分是使用分支曲面和叠片来研究可约 3 流形中的结和链接的 Dehn 手术。 一个目标是证明布线猜想并给出结的属性 R 定理的新证明。 该项目的第三部分是研究 Heegaard 分裂的各个方面。 该研究与低维拓扑中的几个主要问题之间存在许多有趣的联系。 该项目的结果可能为解决低维拓扑中一些困难但重要的猜想铺平道路。 PI 计划开发新工具并使用他之前工作中的技术（例如分支曲面和层压）来实现这些目标。三流形是根据我们生活的 3 维空间建模的对象。甜甜圈和空间宇宙都是 3 流形的示例。 这些物体在物理和其他自然科学的许多背景下自然出现，并模拟许多有趣的现象。 三流形具有优美的几何和拓扑性质，并且这些性质被编码在某些组合信息中。 PI 将通过一些称为高效三角剖分、分支曲面和 Heegaard 分裂的几何对象来探索此类组合信息。 该研究针对低维拓扑和纽结理论中的几个核心问题，对其他科学研究领域（例如 DNA 的拓扑结构）具有潜在影响。