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.

三流形是以我们生活的三维空间为模型的物体。甜甜圈和空间宇宙都是三流形的例子。这些对象在物理和其他自然科学的许多情况下自然地出现，并且可以用来模拟许多有趣的现象。本课题的主要目的是研究三流形的数学性质。PI计划研究数学分支低维拓扑学中的一些核心问题。这些问题关注的是三流形在特定的地图下如何变化，以及被称为手术的操作。PI使用的主要工具是一种叫做Heegaard分裂的拓扑结构，它是沿着二维表面将复杂的三流形分解成更简单的部分。本研究针对低维拓扑学和结理论中的一些基本问题。它对其他科学研究领域也有潜在的影响，比如DNA的拓扑结构。在这个项目中，PI将研究三流形的拓扑结构。该项目有三个主要部分。第一部分是探索一种证明贝尔热猜想的新方法。Berge猜想可以分为两部分：前半部分是证明Berge猜想对于隧道数为1的结，后半部分是证明如果三球中的非平凡结允许透镜空间Dehn手术，则该结必须具有隧道数为1。PI和他的合作者对前半部分进行了深入的研究。同样的方法可以证明贝尔热猜想的后半部分。研究的第二部分是研究一个长期存在的关于Heegaard属和1度图的猜想。PI将使用特殊类型的手术来调查这一猜想。最后一部分的研究目的是研究三流形拓扑中有关Heegaard分裂和曲线复形的几个基本问题。PI计划开发新的工具并使用他以前工作中的技术来实现这些目标。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。