K-theory, Dynamics, and Intersection

Wayne State University, Detroit MI

Investigators

Abstract

The PI proposes two projects that interface algebraic and differential topology and K-theory. The first of these aims to strengthen the relationship between torsion invariants and the counting of closed orbits in dynamical systems. The goal here is to generalize Milnor's equation relating Reidemeister torsion to the Lefschetz zeta function. One aspect of the project will lead to a version of this equation which holds for families of dynamical systems that will relate a higher K-theory invariant to an invariant counting periodic orbits. The main tools will come from Waldhausen's work on algebraic K-theory and the functor TR, which is the topologists' version of the topological de Rham-Witt complex. The second project will study multi-relative version of intersection theory which will give insights beyond the metastable range. We will study obstructions to deforming a map from a manifold into another one off of a finite collection of pair-wise disjoint submanifolds, assuming that the map can be deformed off of the union of any proper sub-collection. The obstruction will live in a direct sum of bordism groups. In a certain range, the obstruction will capture the entire story. Applications of this theory will be given in embedding theory and also in the theory of link homotopy. The proposed research is largely about "manifolds" which are topological spaces that satisfy a certain homogeneity property. Locally speaking, all manifolds are alike in that at any point one sees a copy of Euclidean space. It is the global structure of manifolds that makes them interesting objects of study. Typically, algebraic topologists study manifolds by assigning certain algebraic quantities, called "invariants," to them, which measure global topological structure. Manifolds having different invariants can then be distinguished from one another. 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. K-theory is an algebraic theory which is a receptacle for invariants of manifolds. The PI will research certain kinds of manifold questions which can be analyzed using homotopy theory and K-theory.