Topology of Three Manifolds

三流形拓扑

• 批准号: 9971660
• 负责人: Marc Culler
• 金额: $ 15.39万
• 依托单位: University of Illinois at Chicago
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1999
• 资助国家: 美国
• 起止时间: 1999-06-01 至 2003-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9971660&HistoricalAwards=false
• 关键词: Topology; Three; Manifolds

Proposal: DMS-9971660 Principal Investigator: Marc Culler and Peter Shalen Abstract: Culler and Shalen are proposing to study the set of boundary slopes of a knot, both in the case of an arbitrary closed orientable 3-manifold and in the case of a manifold with cyclic fundamental group. This is a topic of intrinsic interest, which Culler and Shalen believe is also related to some of the most difficult unsolved problems in 3-manifold theory. Their program for applying results about essential surfaces in knot exteriors to 3-manifold theory have led Culler and Shalen to a number of questions about the character variety of a hyperbolic knot group, which they propose to work on. These ideas also lead to an approach to the very broad problem of giving a quantitative version of Thurston's Dehn surgery theorem; this can be seen as the problem underlying most of the existent work on Dehn surgery. The character variety has a distinguished 1-dimensional irreducible component X which is the one that contains the character of the representation that determines the hyperbolic structure on the knot. One question which will be addressed by Culler and Shalen is that of determining properties that distinguish surfaces associated to ideal points of the main component of the character variety from other essential surfaces in the knot exterior. Another question involves the relationship between the hyperbolic volume of the knot complement and properties of a certain factor of the A-polynomial, this factor being the defining equation of a plane curve closely related to X. There is a general notion of hyperbolic volume which applies to a representation whose character lies on X. A third question is that of understanding what properties of an essential surface associated to an ideal point of X guarantee that this generalized volume tends to 0 at the ideal point. Yet another relevant question is that of understanding how often it happens that the curve X is invariant under complex conjugation, and what it means, in terms of the topology of the 3-manfiold, for this to happen. In a somewhat different direction, Culler and Shalen have developed methods for relating the boundary slopes and the genera of essential surfaces in certain situations. They are proposing to develop extensions of this theory. Their techniques in this area may be relevant to the general question of whether every knot in a homotopy sphere has at least one nonzero integer boundary slope.A fundamental problem in many areas of mathematics is to classify all examples of a certain type of mathematical object. The objects of study in this proposal are 3-manifolds, which are mathematical models of 3-dimensional spaces. Since our universe is a 3-dimensional space, the classification of 3-manifolds is directly related to our understanding of nature itself. The classification problem for 3-manifolds is far from solved, but the work of many mathematicians over the last 20 years has at least produced a conjectural answer. The work supported by this problem forms part of the effort to verify the conjectured geometric classification of 3-manifolds.

提案：DMS-9971660 首席研究员：Marc Culler和Peter Shalen 摘要： Culler和Shalen提出研究纽结的边界斜率集，包括任意闭可定向3-流形和具有循环基本群的流形。 这是一个内在的兴趣的话题，Culler和Shalen认为这也与三维流形理论中一些最困难的未解决的问题有关。 他们将纽结外部的本质曲面的结果应用于3-流形理论的计划使Culler和Shalen提出了一些关于双曲纽结群的特征标簇的问题，他们提出了研究这些问题的想法，这些想法也导致了一个非常广泛的问题，即给出Thurston的Dehn手术定理的定量版本;这可以被看作是存在于Dehn手术的大部分工作中的问题。 字符品种有一个杰出的1维不可约的组成部分X这是一个包含的字符的表示，确定双曲结构的结。一个问题，这将是解决卡勒和沙伦是，确定的属性，区分表面相关联的理想点的主要组成部分的字符品种从其他基本表面的结外部。 另一个问题涉及纽结补的双曲体积与A-多项式的某个因子的性质之间的关系，该因子是与X密切相关的平面曲线的定义方程。 有一个双曲体积的一般概念，它适用于特征位于X上的表示。 第三个问题是理解与X的理想点相关联的本质曲面的什么性质保证这个广义体积在理想点趋于0。 另一个相关的问题是理解曲线X在复共轭下不变的频率，以及根据3-流形的拓扑，这意味着什么。 在一个稍微不同的方向上，Culler和Shalen已经开发了在某些情况下将边界斜率与本质曲面的属相关联的方法。 他们正在建议发展这一理论的扩展。 他们在这一领域的技术可能与同伦球面中的每个纽结是否至少有一个非零整数边界斜率的一般问题有关。在许多数学领域中的一个基本问题是对某种类型的数学对象的所有例子进行分类。 在这个建议中的研究对象是3-流形，这是三维空间的数学模型。 由于我们的宇宙是一个三维空间，三维流形的分类直接关系到我们对自然本身的理解。 三维流形的分类问题还远未解决，但许多数学家在过去20年的工作至少产生了一个理论上的答案。 这个问题所支持的工作形式的努力，以验证3-流形的几何分类。