Stratifications and Ends of Spaces

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Topology, as a branch of geometry, seeks to classify, characterize and explore those abstract spaces known as manifolds. Manifolds are locally like ordinary euclidean spaces(the line, the plane, etc.); however, they are allowed to have global twisting, curving and holes (e.g., circles, spheres, tori). They arise in many models of physical and biologicalphenomena. Manifolds with singularities, or stratified spaces, are even more ubiquitous as they appear as solution spaces of algebraic and differential equations and as limits, degenerations and compactifications of manifolds. Hughes has made a breakthrough in the understanding of topologically stratified spaces by establishing a theory of theneighborhoods of the singularities. This allows one to use geometric techniques almost as if the singularities were not present. The current project is largely concerned with exploiting this technique to further the understanding of manifolds with singularities. As an example, Hughes will investigate the extent of periodicity near the singular sets.The proposed research concerns the topology and geometry of manifolds, stratified spaces, trees and metric spaces. The investigations are in the areas of stratified spaces, trees, ultrametrics and noncommutative geometry and the fundamental theorem of algebraic K-theory. The main tools are controlled topology, surgery theory and C*-algebras of groupoids. Specific questions concern periodicity in the neighborhood of the singular set of a manifold stratified space, the classification of stratified h-cobordisms and stratified pseudoisotopies, and non-locally flat topological embeddings. In addition, techniques from noncommutative geometry will be applied to study the geometry of infinite trees and other non-compact spaces at infinity.