The Topology of Hyperbolic 3-Manifolds

卡勒和沙伦继续研究双曲三维流形的拓扑结构。这项工作的背景是正在进行的统一的几何和拓扑理论的三维流形。这些结果可以根据Gromov，瑟斯顿和Jorgensen的工作的结果来看待，即闭双曲3-流形的体积集是实数的良序集。该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属性首次出现的序数相对应的体积。所使用的技术从非常经典的拓扑学方法，如用于证明Loop定理的塔结构的精化版本，到最新的发展，包括证明Marden Tenness猜想和Perelman对Ricci流下体积变化的估计。本研究还涉及到与群论和组合拓扑学的有趣联系。本研究项目中所研究的空间，即3维流形，作为可能的物理宇宙的空间方面的数学模型。现代物理学和三维流形数学理论之间的新联系正在以快速加速的速度被发现。传统上，数学理论分为拓扑学和几何学，其中几何学关注的是可测量的量，如长度、角度、面积或体积，而拓扑学关注的是即使几何特征扭曲时仍保持不变的全局属性。然而，这个主题的这两个方面是密切相关的，并且有许多与几何性质和拓扑性质相关的结果的例子。最近的数学成就，从Mostow刚性定理开始，一直到最近对Marden的温顺猜想和瑟斯顿的几何化猜想的证明，正导致三维流形的几何和拓扑理论的统一。这笔拨款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，目的是定量地了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.