The Topology of Generalized Manifolds

DMS-0071693 Washington Mio和John BryantTopology n-流形是局部同胚于欧几里得n-空间的空间。流形的简单局部结构赋予了它们在数学中非常特殊的作用，因为在这些空间上建立模型的现象适用于大量的方法和工具。流形结构的研究几十年来一直是几何拓扑学中的一个重要课题。最近，与S.Ferry和S.Weinberger合作发现了称为广义流形的空间，它具有拓扑流形的大规模性质，但具有非常复杂的局部结构。之前人们猜测它们不存在，但它们可能存在的第一个理论证据出现在F.Quinn的工作中。这个项目建议继续研究广义流形的拓扑学。这项研究与流形的重要结构性质密切相关，如非球面流形的猜想刚性和Siebenmann周期现象。这个项目的长期目标是探索广义流形的局部结构，并通过证明这些奇异空间满足流形的大多数基本性质，如拓扑齐性、S-余边定理和α-逼近定理，在广义流形和拓扑流形之间建立更深层次的平行。更直接的研究目标包括控制改进问题、子流形和映射横截性问题、广义流形的法丛和嵌入问题。这些都是在研究核心问题时自然产生的因素，它们也代表了独立利益的重要后果。采用的主要方法是控制拓扑学，包括控制同伦理论、控制K-理论和应用于广义流形的面片表示以及最终应用于广义流形本身的控制运算。

DMS-0071693Washington Mio and John BryantTopological n-manifolds are spaces locally homeomorphic to Euclidean n-space. The simple local structure of manifolds lends them a very special role in mathematics since phenomena modeled on these spaces are amenable to a vast array of methods and tools. The study of the structure of manifolds has been an important theme in geometric topology for many decades. More recently, spaces called generalized manifolds, with the large-scale properties of topological manifolds but very intricate local structure, were discovered by the proposers in joint work with S. Ferry and S. Weinberger. Previously conjectured not to exist, the first theoretical evidence that they might exist appeared in the work of F. Quinn. This project proposes to continue the investigation of the topology of generalized manifolds. This study is intimately related to important structural properties of manifolds, such as the conjectural rigidity of aspherical manifolds and the Siebenmann periodicity phenomenon. Other indications of the relevance of these spaces are present in areas as diverse as Dynamical Systems, Geometric Group Theory and C-star-algebras.The long-term goal of this project is to probe the local structure of generalized manifolds and establish a deeper parallel between generalized manifolds and topological manifolds by showing that these exotic spaces satisfy most of the fundamental properties of manifolds, such as topological homogeneity, the s-cobordism theorem, and the alpha-approximation theorem. More immediate goals of the proposed investigation include the study of control improvement problems, submanifold and map transversality questions, normal bundle and embedding problems for generalized manifolds. These are elements that arise naturally in the study of the core problems, and they also represent important ramifications of independent interest. The main technique to be employed is controlled topology, including controlled homotopy theory, controlled K-theory, and controlled surgery applied to patch representations of generalized manifolds and, ultimately, to generalized manifolds themselves.