On the existence of universal covering spaces for metric spaces and subsets of the Euclidean plane

G. R. Conner, and J. W. Lamoreaux. "On the existence of universal covering spaces for metric spaces and subsets of the Euclidean plane." Fundamenta Mathematicae 187.2 (2005): 95-110. <http://eudml.org/doc/282973>.

@article{G2005,
abstract = {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 equivalences 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, 𝔼², admits a universal covering space if and only if its fundamental group is free, if and only if its fundamental group is countable.},
author = {G. R. Conner, J. W. Lamoreaux},
journal = {Fundamenta Mathematicae},
keywords = {planar; free group; fundamental group; covering space; metrizable},
language = {eng},
number = {2},
pages = {95-110},
title = {On the existence of universal covering spaces for metric spaces and subsets of the Euclidean plane},
url = {http://eudml.org/doc/282973},
volume = {187},
year = {2005},
}

TY - JOUR
AU - G. R. Conner
AU - J. W. Lamoreaux
TI - On the existence of universal covering spaces for metric spaces and subsets of the Euclidean plane
JO - Fundamenta Mathematicae
PY - 2005
VL - 187
IS - 2
SP - 95
EP - 110
AB - 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 equivalences 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, 𝔼², admits a universal covering space if and only if its fundamental group is free, if and only if its fundamental group is countable.
LA - eng
KW - planar; free group; fundamental group; covering space; metrizable
UR - http://eudml.org/doc/282973
ER -