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

Volume 187 / 2005

G. R. Conner, J. W. Lamoreaux Fundamenta Mathematicae 187 (2005), 95-110 MSC: Primary 20E05, 20F34, 55Q52, 55Q05, 57M05, 57M10, 57N05; Secondary 54C55, 54E35, 55M15. DOI: 10.4064/fm187-2-1

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, ${\mathbb E}^2$, admits a universal covering space if and only if its fundamental group is free, if and only if its fundamental group is countable.

Authors

  • G. R. ConnerMathematics Department
    Brigham Young University
    Provo, UT 84602, U.S.A.
    e-mail
  • J. W. LamoreauxMathematics Department
    Brigham Young University
    Provo, UT 84602

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