Mathematical Sciences: The Topology of Generalized Manifolds

数学科学：广义流形的拓扑

• 批准号: 9626624
• 负责人: Washington Mio
• 金额: $ 7.6万
• 依托单位: Florida State University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-08-15 至 1999-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9626624&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Topology; Generalized; Manifolds

9626624 Mio Topological n-manifolds are separable metric spaces that are locally homeomorphic to euclidean n-space. They occur ubiquitously in mathematics and its applications and are among the most important objects of study in all of mathematics. Their most observable topological properties are that of being finite dimensional, locally contractible, having the local homology of euclidean n-space, and allowing nicely embedded subspaces to be put in general position. A space is called a generalized n-manifold if it satisfies all of these properties, except possibly the last. If in addition, n is at least 5 and X allows general position, X is said to satisfy the disjoint disks property (DDP). The long standing conjecture that a generalized n-manifold X having the DDP must be a topological manifold was disproved recently by the investigators in joint work with S. C. Ferry and Shmuel Weinberger. Among the most significant questions that remain concerning generalized manifolds and that are the subject of this project is the question of whether generalized manifolds are topologically homogeneous. Other goals are to establish other structure theorems for DDP generalized manifolds that are known to hold for topological manifolds, such as the s-cobordism theorem, various splitting theorems, and regular neighborhood theorems. Splitting theorems, in particular, would be useful in attacking the conjecture that the pathology exhibited by generalized manifolds resides in dimension 4. It has been quite a shock for topologists to learn that n-dimensional manifolds are not characterized by a small collection of their most obvious properties, as had been conjectured. Since these spaces are the most heavily used in mathematics, it is highly desirable to understand this phenomenon better. The two co-investigators, who were among the four who dispatched the conjecture several years ago, are now intent on doing just this. As mentioned above, they will attempt to le arn whether, like ordinary topological manifolds, generalized manifolds exhibit the same topological structure in the vicinity of each of their points. The interest of these spaces arises from their potential as models for several phenomena observed in the study of dynamical systems, geometric group theory, and other areas, but lack of such homogeneity would seriously limit this potential. ***

9626624 MIO拓扑N-manifolds是可分离的度量空间，在欧几里得N空间上是本地同构的。 它们在数学及其应用中普遍存在，并且是所有数学中最重要的研究对象之一。 它们最可观察到的拓扑特性是具有有限的维度，本地合同，具有欧几里得N空间的局部同源性，并允许将嵌入式嵌入的子空间一般放置。 如果满足所有这些属性，则称为广义n-manifold，除了最后一个属性。 另外，n至少为5，x允许一般位置，则说x可以满足不连接磁盘属性（DDP）。 研究人员最近在与S. C. Ferry和Shmuel Weinberger的联合合作中驳斥了具有DDP的广义N-Manifold X必须是拓扑歧管的长期猜想。 关于广义流形的最重要的问题，并且是该项目的主题，这是一个问题，即广义流形在拓扑上是否是同质的。 其他目标是为DDP通用流形建立其他结构定理，该定理符合拓扑流形，例如S-cobordism定理，各种分裂定理和常规的邻里定理。 尤其是分裂定理对于攻击构想的猜想是有用的，即普遍流形的病理位于维度4中。拓扑师得知N维歧管的特征并没有像猜测那样具有最明显的特性，这是令人震惊的。 由于这些空间是数学中最广泛使用的空间，因此更好地理解这种现象是非常可取的。 几年前派遣了猜想的四名共同投资者，他们现在打算这样做。 如上所述，他们将试图尝试像普通的拓扑歧管一样，在每个观点附近表现出相同的拓扑结构。 这些空间的兴趣源于它们作为在动态系统，几何群体理论和其他领域的研究中观察到的几种现象模型的潜力，但是缺乏这种同质性将严重限制这一潜力。 ***