Mathematical Sciences: The Topology of Generalized Manifolds

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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. ***