The Topology of Hyperbolic 3-Manifolds

Culler和Shalen正在继续研究双曲型3流形的拓扑结构。这项工作的背景是3流形的几何和拓扑理论的持续统一。这些结果可以用Gromov、Thurston和Jorgensen的一个定理来解释，即闭双曲3-流形的体积集是实数的良序集。本研究旨在了解体积小于给定阈值的3-流形的拓扑性质。或者，目标是确定与给定拓扑性质首次出现的序数相对应的体积。所使用的技术范围从非常经典的拓扑方法，如用于证明环定理的改进版本的塔结构，到最近的发展，包括证明马登驯良猜想和佩雷尔曼对利玛奇流下体积变化的估计。还涉及到与群论和组合拓扑的有趣联系。在这个研究项目中正在研究的空间，即三维流形，作为一个可能的物理宇宙的空间方面的数学模型。现代物理学和3流形的数学理论之间的新联系正在以迅速加速的速度被发现。数学理论传统上分为拓扑学和几何学，其中几何学关注的是可以测量的量，如长度、角度、面积或体积，而拓扑学关注的是即使几何特征被扭曲也能保持的全局属性。然而，这两个方面的主题是密切相关的，有许多例子的结果涉及到几何性质和拓扑性质。最近的数学成就，从Mostow刚性定理开始，一直到最近对Marden的驯服猜想和Thurston的几何化猜想的证明，都导致了3流形的几何和拓扑理论的统一。该基金支持的研究集中在双曲流形上，这类流形包括绝大多数具有齐次几何结构的3-流形，旨在从定量意义上理解拓扑复杂性如何依赖于流形的几何体积。

Culler and Shalen are continuing their study of the topological structure of hyperbolic 3-manifolds. The context for this work is the on-going unification of the geometric and topological theories of3-manifolds. The results can be viewed in terms of the theorem, which is a consequence of work of Gromov, Thurston and Jorgensen, that the set of volumes of closed hyperbolic 3-manifolds is a well-ordered set of real numbers. The proposed research aims to understand the topological properties of the 3-manifolds with volume less than a given threshold value. Alternatively, the goal is to determine the volume which corresponds to the ordinal at which a given topological property first appears. The techniques used range from very classical topological methods, such as a refined version of the tower construction used in proving the Loop Theorem, to the most recent developments, including the proof of the Marden Tameness conjecture and Perelman's estimates on the change of volume under Ricci flow. Interesting connections with group theory and combinatorial topology are also involved.The spaces which are being studied in this research project, namely 3-dimensional manifolds, serve as mathematical models of the spatial aspect of a possible physical universe. New connections between modern physics and the mathematical theory of 3-manifolds are being discovered at a rapidly accelerating pace. The mathematical theory has traditionally been divided into topology and geometry, where geometry focuses on quantities which can be measured, such as lengths, angles, areas or volumes, and topology focuses on global properties that are preserved even when the geometric features are distorted. However, these two aspects of the subject are closely related, andthere are many examples of results which relate geometric properties and topological properties. Recent mathematical achievements, beginning with the Mostow Rigidity theorem and continuing up to the recent proofs of Marden's Tameness Conjecture and Thurston's Geometrization Conjecture, are leading toward a unification of the geometrical and topological theories of 3-manifolds. The research supported by this grant concentrates on hyperbolic manifolds, a class which includes the vast majority of 3-manifolds with a homogeneous geometric structure, and aims to understand in a quantitative sense how topological complexity depends on the geometrical volume of themanifold.