Classifying Matchbox Manifolds

Matchbox manifolds are foliated spaces whose transversal spaces are totally disconnected. In this work, we show that the local dynamics of a certain class of minimal matchbox manifolds classify their total space, up to homeomorphism. A key point is the use Alexandroff’s notion of a Y –like continuum, where Y is an aspherical closed manifold which satisfies the Borel Conjecture. In particular, we show that two equicontinuous Tn–like matchbox manifolds of the same dimension, are homeomorphic if and only if their corresponding restricted pseudogroups are return equivalent. With an additional geometric assumption, our results apply to Y -like weak solenoids where Y satisfies these conditions. At the same time, we show that these results cannot be extended to include classes of matchbox manifolds fibering over a closed surface of genus 2 manifold which we call “adic-surfaces”. These are 2–dimensional matchbox manifolds that have structure induced from classical 1-dimensional Vietoris solenoids. We also formulate conjectures about a generalized form of the Borel Conjecture for minimal matchbox manifolds.