Topology of Three Manifolds

建议：DMS-9971660首席研究员：马克·卡勒和彼得·沙伦摘要：卡勒和沙伦建议研究纽结的边界斜率集，无论是在任意闭可定向3-流形的情况下，还是在具有循环基本群的流形的情况下。这是一个内在的有趣的话题，卡勒和沙伦认为这也与三维流形理论中一些最困难的悬而未决的问题有关。他们将关于纽结外的本质曲面的结果应用于3-流形理论的程序导致了卡勒和沙伦提出了一些关于双曲纽结群的特征变化的问题，他们建议对这些问题进行研究。这些想法还导致了一种非常广泛的问题的方法，即给出瑟斯顿德恩手术定理的量化版本；这可以被视为大多数现有德恩手术工作的基础问题。特征簇具有独特的1维不可约分量X，该分量X包含确定纽结上的双曲线结构的表示的特征。Culler和Shalen将要解决的一个问题是确定将与特征簇的主要分量的理想点相关联的曲面与纽结外部的其他基本曲面区分开来的性质。另一个问题涉及到结点补的双曲体积与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。存在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的理想点相关的本质曲面的什么性质保证这个广义体积在理想点上趋于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，以及就3-流形的拓扑而言，这意味着什么。在一个略有不同的方向上，卡勒和沙伦发展了一些方法，在某些情况下将边界坡度和基本曲面的属联系起来。他们正在提议发展这一理论的延伸。他们在这一领域的技巧可能与同伦球面中的每个结点是否至少有一个非零整数边界斜率的一般问题有关。许多数学领域的一个基本问题是对某一类型数学对象的所有示例进行分类。该方案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，对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远未解决，但许多数学家在过去20年的工作至少给出了一个猜想的答案。这个问题所支持的工作是验证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.