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.
