Stratified Spaces and Controlled Topology

分层空间和受控拓扑

• 批准号: 9971367
• 负责人: C. Bruce Hughes
• 金额: $ 7.28万
• 依托单位: Vanderbilt University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1999
• 资助国家: 美国
• 起止时间: 1999-07-15 至 2003-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9971367&HistoricalAwards=false
• 关键词: Stratified; Spaces; Controlled; Topology

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.

提案：DMS-9971367PI：C。Bruce Hughesabtract：该项目将来自受控拓扑的技术应用于三个广泛的领域：分层空间，非阳性弯曲的流形和复合物的末端。 主要研究者先前开发的一个重要工具是歧管近似振动的理论。 它将用于攻击几个问题。 一个分层将空间的分解成歧管。由于空间不必是一种歧管本身，因此在理解歧管碎片如何拟合在一起之前，才无法应用几何方法。 该项目最重要的方面之一是揭开用于将歧管碎片结合在一起的几何结构。这种结构不仅需要分类和定量，而且还可以用于开发用于研究分层空间的新几何工具。地层附近的结构是一个束理论，即受控拓扑类别的近似近似纤维，即歧管分层。因此，受控拓扑成为转换和应用经典的歧管技术来研究这些更复杂的奇异歧管的工具。 除了分层空间外，非阳性曲率的歧管和复合物的末端都导致了受控的拓扑结构和近似振动。非积极性弯曲的歧管的几何形状在其通用覆盖物中给出了界限，然后可以将其转换为受控信息。复合物的末端会导致障碍物存在无穷大附近的近似振动。然后，这与奇异性附近的分层空间的拓扑相关。topology的最终目标是对歧管进行分类，这些空间是局部同构对欧几里得空间的空间。 受控拓扑已成为自外科理论发展以来高维流形的这一目标的最强大的新技术。 更加戏剧化的是，受控拓扑表明其在分层空间或具有奇异性的歧管的研究中显示了其全部力量。 在受控拓扑结构中，研究对象配备了一个地图到指标空间，称为控制空间。研究了拓扑的经典主题（例如，同质等价，H-伴侣主义和末端）通过在控制空间中测得的尺寸估计值（因此，当控制空间是一个点时，恢复了经典拓扑）。 数学的许多不同领域的问题导致有关具有奇异性的流形的问题。预计这里提出的研究将为处理此类空间提供新的有用的工具。