Shape of matchbox manifolds

TL;DR

This paper introduces a method to analyze the shape of minimal matchbox manifolds without holonomy by constructing transverse Cantor foliations and approximating them with branched manifolds, generalizing classical techniques.

Contribution

It develops a new intrinsic approach to shape expansions of matchbox manifolds using transverse Cantor foliations without relying on embeddings.

Findings

Established a method for coding holonomy groups in foliated spaces.

Constructed transverse Cantor foliations using purely topological means.

Provided a framework for approximating matchbox manifolds with branched manifolds.

Abstract

In this work, we develop shape expansions of minimal matchbox manifolds without holonomy, in terms of branched manifolds formed from their leaves. Our approach is based on the method of coding the holonomy groups for the foliated spaces, to define leafwise regions which are transversely stable and are adapted to the foliation dynamics. Approximations are obtained by collapsing appropriately chosen neighborhoods onto these regions along a "transverse Cantor foliation". The existence of the "transverse Cantor foliation" allows us to generalize standard techniques known for Euclidean and fibered cases to arbitrary matchbox manifolds with Riemannian leaf geometry and without holonomy. The transverse Cantor foliations used here are constructed by purely intrinsic and topological means, as we do not assume that our matchbox manifolds are embedded into a smooth foliated manifold, or a smooth manifold.