Mathematical Sciences: Topology of 3-Manifolds

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9505253 Gabai The investigation focuses on several of the central open problems in three-dimensional manifold topology. E.g., is the homeomorphism type of a hyperbolic three-manifold determined by its homotopy type? Is a hyperbolic metric on a closed 3-manifold unique up to isotopy? Is every closed irreducible aspherical 3-manifold covered by 3-space? Do theorems about Haken manifolds or 3-manifolds with taut foliations generalize to manifolds with essential laminations? A three-dimensional manifold is a space M with the following property. Any sufficiently small observer in M believes that he is living in standard three-dimensional space. This assumes that the observer is nearsighted, i.e. his vision is restricted to small distances. Despite the fact that 3-manifolds have an extremely simple local structure, their global structures are in general quite complex. Though great efforts have been made in this century to understand the nature of 3-manifolds and much progress has been made these last 40 years, mathematicians still appear to be quite far from a complete understanding. This project addresses some of the most fundamental open problems in the subject. ***