Surfaces in low-dimensional topology

低维拓扑中的表面

• 批准号: 0306506
• 负责人: Mark Brittenham
• 金额: --
• 依托单位: University of Nebraska-Lincoln
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-06-01 至 2008-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0306506&HistoricalAwards=false
• 关键词: Surfaces; low; dimensional; topology

The investigator proposes to study the topology of 3-dimensional manifolds,using 2-dimensional laminations and foliations as the basic tools. The maingoal is to show how these objects can be used to attack several long-standingconjectures in low-dimensional topology. In particular, he proposes to useconstructions of essential laminations, due to the investigator and others,to classify the exceptional Dehn surgeries on alternating knots. Heplans to use laminations to show that some large classes of 3-manifoldssatisfy the Virtual Haken Conjecture. He will also continue work aimed at providing a better understanding of the structure of sutured manifolddecompositions, a fundamental technique in building and using taut foliations,by developing examples of hyperbolic knots with large depth.We live in and move through a 3-dimensional universe, which we perceiveas extending infinitely in all directions. But recent observations hint at thepossibility that space may instead wrap around itself, so that we live in a finite universe. It is this very idea of a space wrapping around itself that lies at theheart of geometric topology, whose goal is to understand the global structure of objects which locally look like ordinary Euclidean space. The investigator plans to study several questions aimed at uncovering patterns in the global structure of 3-dimensional spaces, by using 2-dimensional surfaces to study the spaces in which they can be found. The main idea is to use surfaces as away of cutting the space into pieces; by studying the resulting pieces, we can gain insight into the large-scale structure of the 3-dimensional space.

研究者提出以二维层状和叶状为基本工具来研究3维流形的拓扑结构。主要目标是展示如何使用这些对象来攻击低维拓扑中的几个长期存在的猜想。特别是，由于研究者和其他人的建议，他建议使用基本叠层的结构，对交替结上的特殊 Dehn 手术进行分类。他计划使用叠片来证明一些大类的三流形满足虚拟哈肯猜想。他还将继续致力于通过开发大深度双曲结的例子，更好地理解缝合流形分解的结构，这是构建和使用拉紧叶状结构的基本技术。我们生活在一个 3 维宇宙中并在其中移动，我们认为它向各个方向无限延伸。但最近的观察表明，空间可能会自我环绕，因此我们生活在一个有限的宇宙中。几何拓扑的核心正是这种空间环绕自身的想法，其目标是理解局部看起来像普通欧几里得空间的对象的全局结构。研究人员计划研究几个旨在揭示 3 维空间全局结构模式的问题，通过使用 2 维表面来研究它们所在的空间。主要思想是利用表面来将空间切割成碎片；通过研究由此产生的碎片，我们可以深入了解 3 维空间的大规模结构。