Fibred 3-manifolds and beyond

纤维 3 歧管及以上

• 批准号: 0312442
• 负责人: Rachel Roberts
• 金额: --
• 依托单位: Washington University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-07-01 至 2011-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0312442&HistoricalAwards=false
• 关键词: Fibred; manifolds; beyond

DMS-0312442Rachel Roberts Professor Roberts will investigate several problems concerning the topology of 3-manifolds. In particular, she proposes investigating a series of questions suggested by her earlier work (joint with John Shareshian and Melanie Stein) demonstrating the existence of infinitely many hyperbolic manifolds containing no Reebless foliation. She also proposes a study of particular Reebless foliations and essential laminations as part of her proposed approach to proving the truth of the property P conjecture for knots, which says that no counterexample to the Poincar\'e Conjecture can be generated by Dehn surgery on knots. Three-manifolds are spaces formed by gluing together blocks of three-dimensional space according to certain prescribed rules. Globally, the spaces obtained are usually quite complex. These spaces arise naturally in many contexts in the physical and natural sciences and model many interesting phenomena. As an example of this, we note the recent joint research of topologists and cosmologists suggesting a perhaps surprising model for our spatial universe. As another example, the study of 3-manifolds is important in the study of the knotting and linking of strings in three-dimensional space, which in turn is important in the study of the structure of DNA. A primary goal of the study of 3-manifolds is to discover a useful description of all such spaces and to develop useful tools for working with them. In this research, Professor Roberts investigates questions related to this goal.

Roberts教授将研究有关3-流形拓扑的几个问题。特别是，她建议研究她早期的工作（与John Shareshian和Melanie Stein合作）提出的一系列问题，这些问题证明了无限多个不包含无叶理的双曲流形的存在性。她还提出了对特殊的无Reebless叶状和基本层状的研究，作为她提出的证明结的性质P猜想的真实性的方法的一部分，该猜想说，在结上的Dehn手术不能产生庞加莱猜想的反例。三流形是将三维空间的块按照一定的规则粘合在一起形成的空间。总的来说，得到的空间通常是相当复杂的。这些空间在物理和自然科学的许多环境中自然出现，并模拟了许多有趣的现象。作为一个例子，我们注意到最近拓扑学家和宇宙学家的联合研究为我们的空间宇宙提出了一个可能令人惊讶的模型。又如，对3流形的研究对于研究三维空间中弦的打结和连接是很重要的，而这对于研究DNA的结构也是很重要的。研究3流形的一个主要目标是发现所有这些空间的有用描述，并开发有用的工具来处理它们。在这项研究中，罗伯茨教授调查了与这一目标相关的问题。