Homotopical Localizations and Complications of Spaces

Yale University, New Haven CT

Investigators

Abstract

Proposal: DMS-0071454 PI: Wojciech Chacholski This project is concerned with the study of the localization process and its dual, the colocalization process, in homotopy theory. One of the goals is to develop tools to analyze the colocalization functors analogous to those developed by Bousfield and Kan to study completion and localization. Another aspect of the project is to find geometric applications and interpretations of results on localization. This is related to certain geometric relations between spaces. The key idea is to use a fixed space as a basic building block and use operations like push-outs and wedges to construct more complicated spaces (in a similar way as spheres are used to build CW-complexes). For this purpose the P.I. has been working on understanding an invariant called complication. It roughly gives the minimal number of steps needed to build a new space out of a given one. There is a close relation between this invariant and the so called Bousfield lattice, the lattice of the localization functors on spaces. Understanding this relation has been one of the purposes of the P.I.'s work. Homotopy theory can be perceived as a study of the localization process. The idea is to ignore information which is of less interest to us. In this way homotopy theory works as a simplifying tool. For example, in order to simplify spaces, we can think about the unit interval as the point. In this way we ignore all the consequences caused by the difference between these two spaces. By doing so we identify for example the letter A and the letter O. This gives the standard homotopy theory. The P.I.'s research has been about understanding what happens if we treat other spaces as points.