Embeddings, Intersections and Symmetries

Wayne State University, Detroit MI

Investigators

Abstract

This proposal involves various aspects of the homotopy theory of manifolds, all of which, in one way or another, have connections to the theory of embeddings. The proposal has four parts: 1) Embedding theory. The PI will use techniques arising from the homotopy theory of Poincare complexes to study embedding questions. 2) Group Actions. The study of equivariant manifold classification problems will be approached in a new way using homotopy theory. 3) String Topology. This is emerging field, created by Chas and Sullivan, studies algebraic structures associated with the free loop space of a manifold. The PI proposes to study homotopy theoretic aspects of string topology. 4) Thickenings. A "thickening" of a space is a manifold model for its homotopy type. The PI proposes to analyse the homotopy type of the moduli space of thickenings of a finite complex. A "manifold" is a space which is locally Euclidean. Manifolds arise naturally in physics, chemistry and biology as spaces of solutions of a suitably "nice" set of algebraic equations modeling the scientific object of study (space-time, atoms, dynamical systems, etc.) . Manifolds play a central role in mathematics. It is often the case that mathematical questions about manifolds can be formulated in terms of parametrized families of functions between spaces associated with manifolds. Homotopy theory is a subject designed to tackle questions about such families of functions. The PI proposes to study certain kinds of manifold questions which can be hopefully be analyzed from the homotopy theoretic toolbox. These involve problems related with intersections, embeddings and symmetries.