Geometry and topology of Riemannian foliations via deformations

Abstract

We intend to extend the applications and techniques on regular Riemannian and Killing foliations, predominant in the works of Prof. Dr. Dirk Töben, and seek generalizations of those results for the case of singular Riemannian foliations. More specifically, we want to cover the following topics: elaboration of a survey on transverse geometry and topology of Riemannian foliations covering the developments in the field that are relevant to this project, invariance of the equivariant basic cohomology and other transverse invariants under deformations and applications to the localization of basic characteristic classes, generalizations of results on regular Riemannian foliations to singular ones using desingularization techniques. We also intend, through the development of the aforementioned topics, to obtain a methodology for approaching other related topics in this theme, such as the geometric finiteness of classifying spaces of Riemannian foliations and the construction of examples of such foliations via inversion of Molino's construction. (AU)