Clark-Hurder-Lukina_GeomTop2018.pdf (394.36 kB)
Classifying Matchbox Manifolds
journal contribution
posted on 2019-08-21, 00:00 authored by Alex Clark, Steven Hurder, Olga LukinaMatchbox 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 non-homeomorphic 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.
Funding
2010 Mathematics Subject Classification. Primary 52C23, 57R05, 54F15, 37B45; Secondary 53C12, 57N55 . AC and OL supported in part by EPSRC grant EP/G006377/1.
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Copyright @ Mathematical Sciences PublishersCitation
Clark, A., Hurder, S., & Lukina, O. (2019). Classifying matchbox manifolds. Geometry & Topology. doi:10.2140/gt.2019.23.1Publisher
Mathematical Sciences PublishersLanguage
- en
issn
1465-3060Issue date
2013-11-01Usage metrics
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