## Geometry & Topology

### Classifying matchbox manifolds

#### 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 $T n$–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

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

Dates
Revised: 22 October 2017
Accepted: 8 August 2018
First available in Project Euclid: 12 March 2019

https://projecteuclid.org/euclid.gt/1552356077

Digital Object Identifier
doi:10.2140/gt.2019.23.1

Mathematical Reviews number (MathSciNet)
MR3921315

Zentralblatt MATH identifier
07034541

#### Citation

Clark, Alex; Hurder, Steven; Lukina, Olga. Classifying matchbox manifolds. Geom. Topol. 23 (2019), no. 1, 1--27. doi:10.2140/gt.2019.23.1. https://projecteuclid.org/euclid.gt/1552356077

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