The Topology of Generalized Manifolds

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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.