Classifying matchbox manifolds

Clark, Alex¹ ; Hurder, Steven² ; Lukina, Olga²

¹ Department of Mathematics, University of Leicester, Leicester, United Kingdom
² Department of Mathematics, University of Illinois at Chicago, Chicago, IL, United States

Geometry & topology, Tome 23 (2019) no. 1, pp. 1-27.

Résumé

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 ${\mathbb{T}}^n$–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 $2$.

Détail

DOI : 10.2140/gt.2019.23.1

Classification : 37B05, 37B45, 54C56, 54F15, 57R30, 58H05, 20E18, 57R65
Keywords: foliated spaces, solenoids, laminations, Cantor pseudogroups

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