Mathematical Sciences: The Topology of Generalized Manifolds

9626624 MIO拓扑n-流形是局部同胚于欧几里得n-空间的可分度量空间。它们普遍存在于数学及其应用中，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，局部可压缩的，具有欧几里得n-空间的局部同调，并允许很好地嵌入的子空间处于一般位置。一个空间称为广义n-流形，如果它满足所有这些性质，可能最后一个除外。此外，如果n至少为5且X允许一般位置，则称X满足不相交圆盘性质(DDP)。具有DDP的广义n-流形X一定是拓扑流形的猜想由来已久，但最近与S.C.Ferry和Shmuel Weinberger合作的研究人员证明了这一猜想是错误的。在关于广义流形的最重要的问题中，也是本项目的主题是广义流形是否拓扑齐次的问题。其他目标是建立DDP广义流形的其他结构定理，如S-余边定理、各种分裂定理和正则邻域定理。特别是，分裂定理将有助于破解广义流形表现出的病理驻留在4维的猜想。拓扑学家得知n维流形并不像已经猜测的那样具有最明显的性质的小集合，这对拓扑学家来说是相当令人震惊的。由于这些空间是数学中使用最多的空间，因此更好地理解这一现象是非常必要的。这两名联合调查人员是几年前做出这一猜测的四人之一，现在他们正致力于做这件事。如上所述，他们将试图研究广义流形是否像普通拓扑流形一样，在其每个点的附近显示相同的拓扑结构。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中观察到的几种现象的模型的潜力，但缺乏这种齐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. ***