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

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Conner G., Lamoreaux J.

Fundamenta Mathematicae

2005

tom 187

nr 2

95-110

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.

The homotopy dimension of codiscrete subsets of the 2-sphere 𝕊²

100%

Cannon J., Conner G.

Fundamenta Mathematicae

2007

tom 197

nr 1

35-66

Andreas Zastrow conjectured, and Cannon-Conner-Zastrow proved, that filling one hole in the Sierpiński curve with a disk results in a planar Peano continuum that is not homotopy equivalent to a 1-dimensional set. Zastrow's example is the motivation for this paper, where we characterize those planar Peano continua that are homotopy equivalent to 1-dimensional sets. While many planar Peano continua are not homotopy equivalent to 1-dimensional compacta, we prove that each has fundamental group that embeds in the fundamental group of a 1-dimensional planar Peano continuum. We leave open the following question: Is a planar Peano continuum homotopically 1-dimensional if its fundamental group is isomorphic with the fundamental group of a 1-dimensional planar Peano continuum?

Ograniczanie wyników

2 Fundamenta Mathematicae

2 Conner G.

1 Cannon J.

1 Lamoreaux J.

1 2007

1 2005

Wyniki wyszukiwania

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

The homotopy dimension of codiscrete subsets of the 2-sphere 𝕊²