On the fundamental groups of one-dimensional spaces

We study here a number of questions raised by examining the fundamental groups of complicated one-dimensional spaces. The first half of the paper considers one-dimensional spaces as such. The second half proves related results for general spaces that are needed in the first half but have independent...

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Bibliographic Details
Published in:Topology and its applications 2006-08, Vol.153 (14), p.2648-2672
Main Authors: Cannon, J.W., Conner, G.R.
Format: Article
Language:English
Subjects:
2-set simple cover
Dendrite
Extended commutator subgroup
Fundamental group
Homology
Homotopically Hausdorff
Infinite multiplication
One-dimensional
Reduced path
Rich abelianization
Shape injective
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Summary:We study here a number of questions raised by examining the fundamental groups of complicated one-dimensional spaces. The first half of the paper considers one-dimensional spaces as such. The second half proves related results for general spaces that are needed in the first half but have independent interest. Among the results we prove are the theorem that the fundamental group of a separable, 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 and the theorem that the fundamental group of a compact, one-dimensional, connected metric space embeds in an inverse limit of finitely generated free groups and is shape injective.
ISSN:0166-8641
1879-3207
DOI:10.1016/j.topol.2005.10.008