Geometry & Topology

Classifying matchbox manifolds

Alex Clark, Steven Hurder, and Olga Lukina

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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 T 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 “adic surfaces”, which are a class of weak solenoids fibering over a closed surface of genus 2 .

Article information

Geom. Topol., Volume 23, Number 1 (2019), 1-27.

Received: 7 November 2013
Revised: 22 October 2017
Accepted: 8 August 2018
First available in Project Euclid: 12 March 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 37B05: Transformations and group actions with special properties (minimality, distality, proximality, etc.) 37B45: Continua theory in dynamics 54C56: Shape theory [See also 55P55, 57N25] 54F15: Continua and generalizations 57R30: Foliations; geometric theory 58H05: Pseudogroups and differentiable groupoids [See also 22A22, 22E65]
Secondary: 20E18: Limits, profinite groups 57R65: Surgery and handlebodies

foliated spaces solenoids laminations Cantor pseudogroups


Clark, Alex; Hurder, Steven; Lukina, Olga. Classifying matchbox manifolds. Geom. Topol. 23 (2019), no. 1, 1--27. doi:10.2140/gt.2019.23.1.

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