Automorphisms of 3-manifolds

3-流形的自同构

• 批准号: 0102463
• 负责人: Darryl McCullough
• 金额: $ 6.27万
• 依托单位: University of Oklahoma Norman Campus
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-08-01 至 2003-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0102463&HistoricalAwards=false
• 关键词: Automorphisms; manifolds

AbstractAward: DMS-0102463Principal Investigator: Darryl McCulloughThe proposed work advances a number of research projects in thearea of low-dimensional topology. Their unifying theme is theautomorphisms of 3-dimensional manifolds, including homotopyequivalences, diffeomorphisms, and isometries. Specific projectsinclude: the Generalized Smale Conjecture for elliptic3-manifolds, the isomorphism problem for diffeomorphism groups ofelliptic 3-manifolds, generalization of the Abikoff-Maskitstructure theory for Kleinian groups using topological methods,investigation of free actions of finite groups on orientablehandlebodies using Nielsen equivalence classes of generatingsets, and fibration theorems for spaces of fiber-preservingdiffeomorphisms of manifolds having fiberings and singularfiberings.The primary mathematical constructs that will be investigated are3-manifolds, which are geometric objects locally modeled on the3-dimensional spatial structure of the physical universe, andgroups, which are algebraic systems with an operation akin to theaddition of ordinary numbers. Some of the ongoing work hasalready been applied in the theoretical physics of gravitation,but most of its applications are entirely within puremathematics. The guiding philosophy of most of the research is touse topological and geometric structure of 3-manifolds tounderstand groups of symmetries and other kinds ofautomorphisms. Groups of automorphisms of a mathematical objectoften exhibit their own interesting structure. A classic exampleof this is the finite-dimensional vector spaces. They are rathersimple objects, but their automorphism groups, the general lineargroups, have a rich structure and find wide-ranging uses inmathematics and physics. Within the proposed work, an example isthe 3-manifolds called handlebodies. These are among the simplest3-manifolds to describe topologically, but their groups ofsymmetries are subtle and varied. In fact, any finite group canbe a group of symmetries of some handlebody, and the number ofdistinct ways that a group can act as symmetries on a givenhandlebody can be quite large. A different use of the philosophyinvolves Kleinian groups, which are discrete groups of symmetriesof 3-dimensional hyperbolic space. Each Kleinian group produces aquotient 3-manifold, and one of the projects uses the topologicalstructure of these quotient 3-manifolds to give an algebraicclassification of Kleinian groups.

摘要奖：DMS-0102463主要研究人员：Darryl McCullow建议的工作推进了低维拓扑领域的一些研究项目。它们的统一主题是三维流形的自同构，包括同伦同构、微分同构和等距映射。具体项目包括：椭圆三维流形的广义斯梅尔猜想，椭圆三维流形的微分同构群的同构问题，用拓扑方法推广Klein群的Abikoff-Maskit结构理论，利用生成集的Nielsen等价类研究有限群在可定向手柄上的自由作用，以及具有纤维和奇异纤维的流形的保纤维微分同态空间的纤颤定理。将研究的主要数学结构是三维流形和群，它们是局部模拟在物理宇宙的三维空间结构上的几何对象，以及群，它们是具有类似于普通数相加的运算的大脑代数系统。一些正在进行的工作已经应用于引力的理论物理，但它的大多数应用完全在纯数学范围内。大多数研究的指导思想是利用三维流形的拓扑和几何结构来理解对称群和其他类型的自同构。数学对象的自同构群通常表现出它们自己的有趣结构。有限维向量空间就是一个典型的例子。它们是相对简单的物体，但它们的自同构群，即一般的线性群，具有丰富的结构，并在数学和物理中得到广泛的应用。在拟议的工作中，一个例子是称为手柄的3-流形。这些是最简单的拓扑学描述的3-流形之一，但它们的对称组是微妙的和不同的。事实上，任何有限群都可以是某个手柄上的对称性的群，而一个群可以作为给定手柄上的对称的不同方式的数量可能相当大。哲学的另一种用法涉及克莱因群，它是三维双曲空间的离散对称群。每个Klein群产生水的3-流形，其中一个项目利用这些商3-流形的拓扑结构给出了Klein群的代数分类。