  Matchbox manifolds are foliated spaces with totally disconnected
transversals. Two matchbox manifolds which are homeomorphic have return
equivalent dynamics, so that invariants of return equivalence can be applied to
distinguish non-homeomorphic matchbox manifolds. In this work we study the
problem of showing the converse implication: when does return equivalence imply
homeomorphism? For the class of weak solenoidal matchbox manifolds, we show
that if the base manifolds satisfy a strong form of the Borel Conjecture, then
return equivalence for the dynamics of their foliations implies the total
spaces are homeomorphic. In particular, we show that two equicontinuous
$\mT^n$--like matchbox manifolds of the same dimension are homeomorphic if and
only if their corresponding restricted pseudogroups are return equivalent. At
the same time, we show that these results cannot be extended to include the
"\emph{adic}-surfaces", which are a class of weak solenoids fibering over a
closed surface of genus 2.

Dynamical Systems;General Topology