Mathematical Sciences: Topology of 3-Manifolds

9505253加拜研究集中在三维流形拓扑中的几个中心公开问题上。例如，双曲三维流形的同胚型是否由它的同伦型决定？闭3-流形上的双曲度量在保序意义下是唯一的吗？每个封闭的不可约非球面3-流形都被3-空间覆盖吗？关于Haken流形或具有紧叶的3-流形的定理是否推广到具有本质层的流形？三维流形是具有以下性质的空间M。M中任何足够小的观察者都认为他生活在标准的三维空间中。这假设观察者是近视，即他的视野被限制在很小的距离内。尽管三维流形具有极其简单的局部结构，但它们的全局结构总体上是相当复杂的。尽管人们在本世纪做出了很大的努力来理解三维流形的本质，在过去的40年里也取得了很大的进展，但数学家们似乎仍然远未完全理解。这个项目解决了这个主题中一些最基本的公开问题。***

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. ***