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Title: Classifying Matchbox Manifolds
Authors: Clark, Alex
Hurder, Steven
Lukina, Olga
First Published: 5-Mar-2019
Publisher: Mathematical Sciences Publishers (MSP) for Geometry and Topology Publications
Citation: Geometry & Topology, 2019, 23, pp. 1–27
Abstract: 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 Tn–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 2.
DOI Link: 10.2140/gt.2019.23.1
ISSN: 1465-3060
Version: Post-print
Status: Peer-reviewed
Type: Journal Article
Rights: Copyright © 2019, Mathematical Sciences Publishers (MSP) for Geometry and Topology Publications. Deposited with reference to the publisher’s open access archiving policy. (
Appears in Collections:Published Articles, Dept. of Mathematics

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