Molino's description and foliated homogeneity

TL;DR

This paper refines Molino's topological description of equicontinuous foliated spaces by introducing a compact group action, characterizing foliated homogeneity, and providing examples where classical assumptions do not hold.

Contribution

It sharpens Molino's description by incorporating a compact group action and studies a smooth version, offering new characterizations of foliated homogeneity and minimality.

Findings

Characterization of compact minimal G-foliated spaces via group triviality

Introduction of a foliated action of a compact topological group

Examples where Molino's projection is not a principal bundle

Abstract

The topological Molino's description of equicontinuous foliated spaces, studied by the first author and Moreira Galicia, gives conditions to reduce their study to the particular case where the holonomy pseudogroup can be represented by a pseudogroup on some local group $G$ generated by some of its local left translations (a $G$-foliated space). That description is sharpened in this paper by introducing a foliated action of a compact topological group on the resulting $G$-foliated space, like in the case of Riemannian foliations. Moreover a $C^\infty$ version is also studied. The triviality of this compact group characterizes compact minimal $G$-foliated spaces, which are also characterized by their foliated homogeneity in the $C^\infty$ case. We also give an example where the projection of the Molino's description is not a principal bundle, and another example of positive topological codimension where the foliated homogeneity cannot be checked by only comparing pairs of leaves---in the case of zero topological codimension, weak solenoids with this property were given by Fokkink and Oversteegen, and later by Dyer, Hurder and Lukina.