Heegaard Splitting and Topology of 3-Manifolds

三流形的 Heegaard 分裂和拓扑

• 批准号: 1906235
• 负责人: Tao Li
• 金额: $ 25.06万
• 依托单位: Boston College
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2019
• 资助国家: 美国
• 起止时间: 2019-07-01 至 2024-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1906235&HistoricalAwards=false
• 关键词: Heegaard; Splitting; Topology; Manifolds

Three-manifolds are objects modeled on the three-dimensional space that we live in. A donut and the spatial universe are both examples of three-manifolds. These objects arise naturally in many contexts in physical and other natural sciences, and can be used to model many interesting phenomena. The main goal of this project is to study the mathematical properties of three-manifolds. The PI plans to investigate some central questions in a branch of mathematics, known as low dimensional topology. These questions are concerned with how three-manifolds change under certain maps as well as operations called surgeries. The major tool that the PI uses is a topological structure called Heegaard splitting, which is a decomposition of a complicated three-manifold into simpler pieces along a two-dimensional surface. This research targets some of the fundamental questions in low-dimensional topology and knot theory. It also has a potential impact on other areas of scientific investigations, such as the topological structures of DNA. In this project, the PI will study the topology of three-manifolds. The project has three major parts. The first part is to explore a new approach to proving the Berge Conjecture. The Berge Conjecture can be divided into two halves: the first half is to prove the Berge Conjecture for knots with tunnel number one, and the second half is to show that if a nontrivial knot in the three-sphere admits a lens-space Dehn surgery, then the knot must have tunnel number one. The PI and his collaborators have carried out an in-depth study on the first half. The same approach may lead to a proof of the second half of the Berge Conjecture. The second part of the research is to study a long-standing conjecture concerning Heegaard genus and degree-one map. The PI will investigate this conjecture using special type of surgeries. The objective of the last part of the research is to study several fundamental questions in three-manifold topology concerning Heegaard splittings and curve complex. The PI plans to develop new tools and use techniques from his previous work to achieve these goals.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

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三个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，可用于模拟许多有趣的现象。该项目的主要目的是研究三个manifolds的数学特性。 PI计划在数学分支中调查一些中心问题，称为低维拓扑。这些问题涉及在某些地图和称为手术的操作下如何变化的三个manifolds。 PI使用的主要工具是拓扑结构，称为Heegaard拆分，这是复杂的三键式分解为沿二维表面较简单的碎片的分解。这项研究针对了低维拓扑和结理论中的一些基本问题。它还对其他科学研究的领域有潜在的影响，例如DNA的拓扑结构。在这个项目中，PI将研究三个manifolds的拓扑。该项目有三个主要部分。第一部分是探索一种证明Berge猜想的新方法。 BERGE的猜想可以分为两半：上半场是证明Berge的猜想，隧道第一，下半部分是表明，如果三个球形中的非平凡结会承认镜头空间Dehn手术，那么这个结必须是隧道第一。 PI和他的合作者在上半场进行了深入研究。同样的方法可能会导致Berge猜想的后半部分证明。该研究的第二部分是研究有关Heegaard属和学位图的长期猜想。 PI将使用特殊类型的手术研究这种猜想。这项研究的最后一部分的目的是研究有关Heegaard分裂和曲线复合体的三个manifold拓扑结构的几个基本问​​题。 PI计划开发新工具并使用他以前的工作以实现这些目标的技术。该奖项反映了NSF的法定任务，并使用基金会的知识分子优点和更广泛的影响评估标准，被认为值得通过评估来获得支持。