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.

研究人员建议使用二维层压和叶子作为基本工具研究三维流形的拓扑。 Maingoal是展示如何使用这些对象来攻击低维拓扑中的几个长期观察。特别是，由于调查员和其他人，他建议对基本层压的用户构建，以对交替结的特殊手术进行分类。 Heplans使用层压表明，一些大型的3型ManifoldsSatsing对虚拟Haken的猜想感到满意。他还将继续工作，以更好地理解缝合的多种多样组合的结构，这是建立和使用拉紧叶子的基本技术，通过开发具有较大深度的双曲线结的示例，我们生活在三维的宇宙中，我们可以在所有方向上进行无限的范围。但是最近的观察暗示了空间可能会缠绕自身，因此我们生活在有限的宇宙中。正是这种空间缠绕自身的想法在于几何拓扑的心灵，其目标是了解当地看起来像普通欧几里得空间的物体的全球结构。研究人员计划通过使用二维表面研究可以找到它们的空间来研究几个旨在揭示三维空间全球结构模式的问题。主要的想法是将表面用作将空间切成碎片的范围。通过研究所得的碎片，我们可以深入了解3维空间的大规模结构。