A separable and metrizable
space X is a matchbox manifold if each point x of X has an open neighborhood
which is homeomorphic to Sx× ℝ for some zero-dimensional space Sx. Each arc
component of a matchbox manifold admits a parameterization by the reals ℝ in a
natural way. This is the main tool in defining the orientability of matchbox
manifolds. The orientable matchbox manifolds are precisely the phase spaces of
one-dimensional flows without rest points. We show in this paper that a compact
homogeneous matchbox manifold is orientable.
As an application a new proof is given of Hagopian’s theorem that a homogeneous
metrizable continuum whose only proper nondegenerate subcontinua are arcs must be
a solenoid. This is achieved by combining our work on matchbox manifolds with
Whitney’s theory of regular curves.
