Stratified Spaces and Controlled Topology

提议：DMS-9971367PI：C.Bruce Hughes摘要：这个项目应用控制拓扑学中的技术来解决三个广泛领域的问题：分层空间、非正曲线流形和复合体的末端。主要研究者以前开发的一个重要工具是流形近似纤颤理论。它将被用来解决几个问题。分层表示将空间分解成多个部分。由于空间本身不需要是流形，所以在了解流形部分如何组合在一起之前，几何方法是不能应用的。这个项目最重要的方面之一是揭开了用来连接多个部件的几何结构。这种结构不仅可以进行分类和量化，而且还可以用于研究分层空间的新几何工具的开发。在受控拓扑学范畴中，层邻域上的结构是一个丛理论，即流形分层近似纤维。因此，受控拓扑学成为变换和应用经典流形技术来研究这些更复杂的奇点流形的工具。除了分层空间外，非正曲率的流形和复形的两端都会导致受控拓扑和近似纤颤。非正曲线流形的几何给出了它们的泛覆盖的有界性，然后这些有界性可以转化为受控信息。络合物的末端导致近乎无穷远的近似纤颤的存在的障碍。然后，这与奇点附近的分层空间的拓扑有关。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氏空间的空间。自外科理论发展以来，可控拓扑已成为实现高维流形这一目标的最有力的新技术。更戏剧性的是，受控拓扑学在分层空间或奇点流形的研究中显示出了它的全部力量。在受控拓扑学中，研究对象配备了到度量空间的映射，称为控制空间。研究了拓扑学的经典主题(例如，同伦等价、h-余边线和端点)，并在控制空间中测量了大小估计(以便当控制空间为点时恢复经典拓扑)。数学中许多不同领域的问题导致了关于奇点流形的问题。预计这里提出的研究将为处理这类空间提供新的有用的工具。

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.