Algebraic and Geometric Topology

Indiana University, Bloomington IN

Investigators

Abstract

Davis and Orr investigate several issues concerning the classification of manifolds, as well as the classification of embeddings of manifolds in manifolds, i.e. knot theory. A variety of tools are used: surgery theory, localization and completion of groups, rings and spaces, index theory, von Neumann algebras, Whitney disks techniques, algebraic K- and L-theory, and controlled topology. Some problems of interest are the study of connected sums of manifolds and corresponding Nil and UNil phenomena, the concordance classification of knots, actions on a product of spheres, and the ribbon/slice problem. The goal, as usual in geometric topology, is to use a variety of algebraic, geometric, and analytic techniques to find and compute invariants for classification. Geometric topology is the study of manifolds. An n-dimensional manifold is a set of points locally modeled on n-dimensional Euclidean space. For instance, a 2-manifold is a surface and looks like a plane near each point. Many physical phenomenon are represented by manifolds, and as such, understanding the global structure of a manifold, and what possible manifolds exist, is fundamental to the sciences, as well as to mathematics. Additionally, one asks how manifolds can sit within manifolds, a subject known as knot theory. Knot theory has been used to model genetic structures and chemical bonds.