Topology of Three Manifolds

摘要：Culler和Shalen提出研究任意闭合可定向3-流形和具有循环基群的流形的结的边界斜率集。这是一个具有内在兴趣的话题，Culler和Shalen认为这也与3流形理论中一些最难解决的问题有关。他们将结外基本曲面的结果应用于3流形理论的程序，使Culler和Shalen提出了许多关于双曲结群的特征变化的问题，他们打算对这些问题进行研究。这些想法也引出了一种方法来解决一个非常广泛的问题即给出瑟斯顿德恩手术定理的定量版本；这可以看作是Dehn手术中存在的问题。字符变体有一个独特的一维不可约分量X，它包含决定结上双曲结构的表示的字符。Culler和Shalen将解决的一个问题是，确定区分与角色多样性主要组成部分的理想点相关的表面与结外部其他基本表面的特性。另一个问题涉及到结补的双曲体积与a多项式的某个因子的性质之间的关系，这个因子是与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，适用于特征在X上的表示。第三个问题是理解与X的理想点相关的基本曲面的哪些性质保证该广义体积在理想点趋于0。另一个相关的问题是如何理解曲线X在复共轭下保持不变的频率，以及从3-曼场的拓扑结构来看，这种情况的发生意味着什么。在一个稍微不同的方向上，Culler和Shalen发展了在某些情况下将边界斜率和基本曲面的属联系起来的方法。他们正提议发展这一理论的延伸。他们在这方面的技术可能与同伦球中的每个结是否至少有一个非零整数边界斜率的一般问题有关。在许多数学领域的一个基本问题是对某一类数学对象的所有例子进行分类。本文的研究对象是三维流形，即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。3流形的分类问题还远未解决，但在过去20年里，许多数学家的工作至少产生了一个推测性的答案。这个问题支持的工作构成了验证3流形的猜想几何分类的努力的一部分。

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.