Splitting homotopy equivalences: Applications, calculations, foundations

Vanderbilt University, Nashville TN

Investigators

Abstract

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). The goal of this project is to deepen and enhance our understanding of geometric topology, that is, the homeomorphism classification of closed manifolds (of dimension greater than three) within a given homotopy type. This project focuses on splitting homotopy equivalences along a two-sided codimension-one submanifold. The obstructions to this ideal situation are given by the Waldhausen Nil-groups and the Cappell UNil-groups. Hence, from an algebraic viewpoint, one theme of the project is the calculation and foundation of the splitting obstruction groups in L-theory. On the other hand, from a topological viewpoint, splitting is a special case of the coarser decomposition into big, non-simply connected pieces. The principal investigator proposes the use of these non-simply connected metablocks to study the Four-dimensional Surgery Conjecture. These metablocks allow for the more flexible notion of classifying space B_FΓ for families F of small subgroups of Γ. This approach would hybridize the techniques of three isolated communities that have sprung up in the past 25 years: controlled surgery, surgery on 4-manifolds with small fundamental group, and assembly maps for families. A manifold is a smooth shape, without any sharp edges or singularities. For example, a one-dimensional manifold is a curve, and a two-dimensional manifold is a surface. Three and four-dimensional manifolds occur in physics, such as general relativity. Higher-dimensional manifolds occur in string theory and also as configuration spaces for robot motion planning. The mathematical classification of manifolds is fundamental to understanding both the global structure of our universe and the hidden routes through which a closed physical system is connected to itself.