Stratified Spaces and Controlled Topology

提案：DMS-9971367 PI：C。布鲁斯休斯摘要：这个项目应用控制拓扑学的技术来解决三个广泛领域的问题：分层空间，非正弯曲流形和复合体的末端。 一个重要的工具，以前开发的主要研究者是理论的流形近似纤维化。 它将被用来解决几个问题。 分层是将空间分解为多个部分。由于空间本身不需要是流形，几何方法不能应用，直到它是了解如何流形件配合在一起。 该项目最重要的方面之一是揭示了用于将多个部件连接在一起的几何结构。这种结构不仅可以进行分类和量化，而且还可以应用于研究分层空间的新几何工具的开发。在控制拓扑学的范畴内，层的邻域上的结构是一个丛理论，即流形分层近似纤维化。因此，控制拓扑成为车辆改造和应用经典流形技术来研究这些更复杂的流形奇点。 除了分层空间，非正曲率的流形和复形的端点都会导致受控拓扑和近似纤维化。非正曲流形的几何给出了其泛覆盖的有界性，然后可以将其转换为受控信息。复形的端点对无穷远附近近似纤维化的存在造成了障碍。拓扑学的最终目标是对流形进行分类，流形是局部同胚于欧几里德空间的空间。 自外科手术理论发展以来，控制拓扑已经成为实现这一目标的最有力的新技术。 更引人注目的是，受控拓扑学在研究分层空间或奇点流形时显示出了它的全部力量。 在受控拓扑学中，研究的对象配备了一个到度量空间的映射，称为控制空间。拓扑学的经典主题（例如，同伦等价，h-协边，和结束）进行了研究，在控制空间中测量的大小估计（以便经典拓扑恢复时，控制空间是一个点）。 在许多不同的数学领域的问题导致问题的流形与奇点。预计这里提出的研究将提供新的和有用的工具来处理这样的空间。

Proposal: DMS-9971367PI: C. Bruce HughesAbstract: This project applies techniques from controlled topology to solve problems in three broad areas: stratified spaces, nonpositively curved manifolds, and ends of complexes. An important tool developed previously by the principal investigator is the theory of manifold approximate fibrations. It will be used to attack several problems. A stratification presents a decomposition of the space into manifold pieces. Since the space need not be a manifold itself, geometric methods cannot be applied until it is understood how the manifold pieces fit together. One of the most important aspects of this project is the uncovering of the geometric structure used to join together the manifold pieces. This structure is not only subject to classification and quantification, but can also be applied for the development of new geometric tools for studying stratified spaces. The structure on the neighborhoods of the strata is a bundle theory, namely manifold stratified approximate fibrations, in the category of controlled topology. Thus, controlled topology becomes the vehicle for transforming and applying classical manifold techniques to study these more intricate manifolds with singularities. In addition to stratified spaces, both manifolds of nonpositive curvature and ends of complexes lead to controlled topology and approximate fibrations. The geometry of nonpositively curved manifolds give boundedness in their universal covers which can then be converted into controlled information. Ends of complexes lead to obstructions for the existence of approximate fibrations near infinity. This is then related to the topology of stratified spaces near the singularities.Topology's ultimate goal is to classify manifolds, the spaces which are locally homeomorphic to euclidean spaces. Controlled topology has become the most powerful new technique toward achieving this goal regarding high-dimensional manifolds since the development of surgery theory. Even more dramatically, controlled topology is showing its full force in the study of stratified spaces, or manifolds with singularities. In controlled topology, the objects of study come equipped with a map to a metric space, called the control space. The classical topics of topology (e.g., homotopy equivalences, h-cobordisms, and ends) are studied with sizing estimates measured in the control space (so that classical topology is recovered when the control space is a point). Problems in many different areas of mathematics lead to questions about manifolds with singularities. It is expected that the research proposed here will give new and useful tools for dealing with such spaces.