Classifying matchbox manifolds (Q666702)

A matchbox manifold is a compact connected metrizable foliated space with totally disconnected transversals. Two homeomorphic matchbox manifolds have return-equivalent dynamics; thus invariants of return equivalence can be applied to distinguish nonhomeomorphic matchbox manifolds. \par In this work the authors study the converse implication, that is, when does return equivalence imply homeomorphisms? In particular, they show that, for the class of weak solenoidal matchbox manifolds, if the base manifolds satisfy a strong form of the Borel conjecture, then return equivalence for the dynamics of their foliations implies that the total spaces are homeomorphic. They also prove the following result: two equicontinuous $\mathbb{T}^n$-like matchbox manifolds of the same dimension are homeomorphic if and only if their corresponding restricted pseudogroups are return equivalent. However, these results cannot be extended to the ``adic surfaces'', a class of weak solenoids fibering over a closed surface of genus 2.