Volume 23, issue 1 (2019)

Download this article
Download this article For screen
For printing
Recent Issues

Volume 28
Issue 7, 3001–3510
Issue 6, 2483–2999
Issue 5, 1995–2482
Issue 4, 1501–1993
Issue 3, 1005–1499
Issue 2, 497–1003
Issue 1, 1–496

Volume 27, 9 issues

Volume 26, 8 issues

Volume 25, 7 issues

Volume 24, 7 issues

Volume 23, 7 issues

Volume 22, 7 issues

Volume 21, 6 issues

Volume 20, 6 issues

Volume 19, 6 issues

Volume 18, 5 issues

Volume 17, 5 issues

Volume 16, 4 issues

Volume 15, 4 issues

Volume 14, 5 issues

Volume 13, 5 issues

Volume 12, 5 issues

Volume 11, 4 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 3 issues

Volume 7, 2 issues

Volume 6, 2 issues

Volume 5, 2 issues

Volume 4, 1 issue

Volume 3, 1 issue

Volume 2, 1 issue

Volume 1, 1 issue

The Journal
About the Journal
Editorial Board
Editorial Procedure
Subscriptions
 
Submission Guidelines
Submission Page
Policies for Authors
Ethics Statement
 
ISSN 1364-0380 (online)
ISSN 1465-3060 (print)
Author Index
To Appear
 
Other MSP Journals
Classifying matchbox manifolds

Alex Clark, Steven Hurder and Olga Lukina

Geometry & Topology 23 (2019) 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.

Keywords
foliated spaces, solenoids, laminations, Cantor pseudogroups
Mathematical Subject Classification 2010
Primary: 37B05, 37B45, 54C56, 54F15, 57R30, 58H05
Secondary: 20E18, 57R65
References
Publication
Received: 7 November 2013
Revised: 22 October 2017
Accepted: 8 August 2018
Published: 5 March 2019
Proposed: Steve Ferry
Seconded: Leonid Polterovich, Yasha Eliashberg
Authors
Alex Clark
Department of Mathematics
University of Leicester
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