Holonomy transformations for singular foliations
Abstract
In order to understand the linearization problem around a leaf of a singular foliation, we extend the familiar holonomy map from the case of regular foliations to the case of singular foliations. To this aim we introduce the notion of holonomy transformation. Unlike the regular case, holonomy transformations can not be attached to classes of paths in the foliation, but rather to elements of the holonomy groupoid of the singular foliation. This assignment is injective. Holonomy transformations allow us to link the linearization problem with the compactness of the isotropy group of the holonomy groupoid, as well as with the linearization problem for proper Lie groupoids.
Cite
@article{arxiv.1205.6008,
title = {Holonomy transformations for singular foliations},
author = {Iakovos Androulidakis and Marco Zambon},
journal= {arXiv preprint arXiv:1205.6008},
year = {2014}
}
Comments
Final version, accepted for publication. The injectivity of the holonomy map, conjectured in version 1, is proven in full generality (Thm. 2.20). Further, we simplify the definition of holonomy transformation (Def. 2.4). 42 pages