On the Fundamental Groups of One-Dimensional Spaces. (with Jim Cannon)
Abstract: We show various properties of the fundamental groups of generic topological spaces and in particular one-dimensional spaces. The main result is that the fundamental group of a second countable, connected, locally path connected one-dimensional metric space is free if and only if it is countable, if and only if the space has a universal cover.
On the fundamental groups of planar sets. (with Jack Lamoreaux)
Abstract: In this paper we prove several results concerning the existence of universal covering spaces for separable metric spaces. To begin, we define several homotopy theoretic conditions which we then prove are equivalent to the existence of a universal covering space. We use these equivalencies to prove that every connected, locally path connected separable metric space whose fundamental group is a free group admits a universal covering space. As an application of these results, we prove the main result of this article which states that a connected, locally path connected subset of the Euclidean plane admits a universal covering space if and only if its fundamental group is free, if and only if its fundamental group is countable.