Automorphisms of 3-manifolds

AbstractAward：DMS-0102463首席研究员：Darryl McCulloughThe拟议的工作推进了低维拓扑领域的一些研究项目。他们的统一主题是三维流形的自同构，包括同伦等价，同构和等距。具体项目包括：椭圆3-流形的广义Smale猜想，椭圆3-流形的同构群的同构问题，用拓扑方法推广Kleinian群的Abikoff-Maskit结构理论，用生成集的Nielsen等价类研究有限群在可定向体上的自由作用，和纤维空间的纤维化定理-具有纤维化和奇异纤维化的流形的超同构。主要的数学结构将被研究是3-流形，它们是在物理宇宙的三维空间结构上局部建模的几何对象，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学，但它的大部分应用完全在纯数学中。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。一个数学对象的自同构群常常表现出它们自己有趣的结构。一个经典的例子是有限维向量空间。它们是相当简单的对象，但它们的自同构群，即一般线性群，具有丰富的结构，在数学和物理学中有着广泛的用途。在所提出的工作中，一个例子是3-流形被称为单粒子体。这些是拓扑描述中最简单的三维流形，但它们的对称群是微妙而多样的。事实上，任何有限群都可以是某个对称体的对称群，并且一个群可以作为给定对称体上的对称的不同方式的数量可以相当大。该哲学的另一种用途涉及克莱因群，它们是3维双曲空间的离散对称群。每个Kleinian群都产生商3-流形，其中一个项目利用这些商3-流形的拓扑结构给出Kleinian群的代数分类。

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.