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
nonhomeomorphic 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
–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 “adic surfaces”,
which are a class of weak solenoids fibering over a closed surface of genus
.
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