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 ${\mathcal T}^n$--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.