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.

DMS-0312442Rachel Roberts Roberts 教授将研究有关 3 流形拓扑的几个问题。特别是，她建议调查她早期工作（与 John Shareshian 和 Melanie Stein 合作）提出的一系列问题，证明存在无限多个不包含 Reebless 叶状结构的双曲流形。她还提议对特定的 Reebless 叶状结构和基本层状结构进行研究，作为她提出的方法的一部分，以证明结的性质 P 猜想的真实性，该猜想表明，通过对结进行 Dehn 手术不能产生庞加莱猜想的反例。 三流形是由三维空间块按照一定的规定规则粘合在一起形成的空间。在全球范围内，获得的空间通常非常复杂。这些空间在物理和自然科学的许多背景下自然出现，并模拟了许多有趣的现象。作为一个例子，我们注意到拓扑学家和宇宙学家最近的联合研究提出了一个可能令人惊讶的空间宇宙模型。又如，3-流形的研究对于三维空间中弦的打结和连接的研究很重要，而这反过来对于DNA结构的研究也很重要。 三流形研究的主要目标是发现所有此类空间的有用描述，并开发用于处理它们的有用工具。在这项研究中，罗伯茨教授调查了与这一目标相关的问题。