Volume 23, issue 1 (2019)

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Classifying matchbox manifolds

Alex Clark, Steven Hurder and Olga Lukina

Geometry & Topology 23 (2019) 1–27

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.

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foliated spaces, solenoids, laminations, Cantor pseudogroups
Mathematical Subject Classification 2010
Primary: 37B05, 37B45, 54C56, 54F15, 57R30, 58H05
Secondary: 20E18, 57R65
Received: 7 November 2013
Revised: 22 October 2017
Accepted: 8 August 2018
Published: 5 March 2019
Proposed: Steve Ferry
Seconded: Leonid Polterovich, Yasha Eliashberg
Alex Clark
Department of Mathematics
University of Leicester
United Kingdom
Steven Hurder
Department of Mathematics
University of Illinois at Chicago
Chicago, IL
United States
Olga Lukina
Department of Mathematics
University of Illinois at Chicago
Chicago, IL
United States