Mathematical Sciences: The Topology of Generalized Manifolds

9626624 Mio拓扑n流形是局部同纯欧氏n空间的可分离度量空间。它们在数学及其应用中无处不在，是所有数学中最重要的研究对象之一。它们最明显的拓扑性质是有限维的，局部可收缩的，具有欧氏n空间的局部同调，并允许嵌入的子空间被放置在一般位置。如果一个空间满足所有这些性质，那么它就被称为广义n流形，除了最后一个。另外，如果n至少为5，且X允许一般位置，则称X满足不相交磁盘属性（DDP）。具有DDP的广义n流形X一定是拓扑流形这一长期存在的猜想最近被研究人员与S. C. Ferry和Shmuel Weinberger联合证明是错误的。关于广义流形的最重要的问题之一是广义流形是否拓扑齐次，这也是本课题的主题。其他目标是为DDP广义流形建立其他结构定理，这些定理已知适用于拓扑流形，例如s协定理、各种分裂定理和正则邻域定理。特别是分裂定理，在驳斥广义流形所表现出的病态存在于4维的猜想时将是有用的。对于拓扑学家来说，了解到n维流形并不是像之前猜测的那样由它们最明显的性质的一小部分集合来表征的，这是相当震惊的。由于这些空间在数学中使用最多，因此非常希望更好地理解这一现象。这两名共同调查人员是几年前提出这一猜想的四人之一，他们现在正打算这样做。如上所述，他们将试图了解广义流形是否像普通拓扑流形一样，在其每个点的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动力系统、几何群论和其他领域研究中观察到的几种现象的模型的潜力，但缺乏这种同质性将严重限制这种潜力。＊＊＊

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. ***