Mathematical Sciences: Manifolds, Stratified Spaces, and Controlled Topology

本课题主要研究控制拓扑学在流形和分层空间理论中的应用。特别重要的是一个发现分层空间几何结构的程序。问题是要了解奇异集的邻域。据推测，这样的邻里之间存在着“泪滴结构”。这一猜想的正解将使人们能够将许多以前仅为流形或特别严格的分层所知的基本结果推广到分层空间。其他的应用包括广义的Novikov猜想，作用在流形上的拓扑群，以及流形之间映射的改进。在这项研究中使用的中心工具是流形近似纤颤理论，以前由主要研究人员和他的合作者分类。该项目的一个方面是将这项工作扩展到分层环境。流形是拓扑学的主要研究对象。它们是局部欧几里得空间。最终目标是对流形及其对称性进行分类。自外科理论发展以来，可控拓扑学已成为高维流形研究中最有力的新技术。更戏剧性的是，受控拓扑学在奇点流形的研究中显示出了它的全部力量。这些分层空间包括多面体、复代数簇、流形上群作用的轨道空间和黎曼流形的某些极限。

This project is concerned with applications of controlled topology to the theory of manifolds and stratified spaces. Of particular significance is a program for discovering the geometric structure of stratified spaces. The problem is to understand the neighborhood of the singular set. It is conjectured that such a neighborhood has "teardrop structure." A positive solution to this conjecture would enable one to extend to stratified spaces many fundamental results previously known only for manifolds or for especially rigid stratifications. Other applications include generalized Novikov conjectures, groups acting topologically on manifolds, and the improvement of maps between manifolds. The central tool to be used in this research is the theory of manifold approximate fibrations, previously classified by the principal investigator and his collaborators. One aspect of this project is to extend that work to the stratified setting. Manifolds are the main object of study in topology. They are the spaces which are locally euclidean spaces. The ultimate goal is to classify manifolds and their symmetries. Controlled topology has become the most powerful new technique in the study of high- dimensional manifolds since the development of surgery theory. Even more dramatically, controlled topology is showing its full force in the study of manifolds with singularities. These stratified spaces include polyhedra, complex algebraic varieties, orbit spaces of group actions on manifolds, and certain limits of Riemannian manifolds.