Low-dimensional solenoidal manifolds

Abstract

In this paper we survey n-dimensional solenoidal manifolds for n = 1, 2 and 3, and present new results about them. Solenoidal manifolds of dimension n are metric spaces locally modeled on the product of a Cantor set and an open n-dimensional disk. Therefore, they can be "laminated" (or "foliated") by n-dimensional leaves. By a theorem of A. Clark and S. Hurder, topologically homogeneous, compact solenoidal manifolds are McCord solenoids i.e., are obtained as the inverse limit of an increasing tower of finite, regular covers of a compact manifold with an infinite and residually finite fundamental group. In this case their structure is very rich since they are principal Cantor-group bundles over a compact manifold and they behave like "laminated" versions of compact manifolds, thus they share many of their properties. These objects codify the commensurability properties of manifolds.To Dennis Sullivan on occasion of his 80th birthday, for all the inspiration and mathematical ideas he has shared with me.

Lipschitz matchbox manifolds

“…The classification of solenoidal manifolds, up to homeomorphism, is studied by the authors in their works [12,26,27,28,29]. A solenoidal manifold can be considered as a generalized manifold, which is the point of view of the works of Sullivan [47,48,49] and Verjovsky [53,54], where they investigate their differential geometric properties.…”

“…Solenoidal manifolds are matchbox manifolds of dimension n in the terminology of [12]. The authors' works [12,13,26] and the survey by Verjovsky [54] discuss many examples of solenoidal manifolds.…”