Let X be a metrizable one-dimensional continuum. We describe the fundamental group of X as a subgroup of its Čech homotopy group. In particular, the elements of the Čech homotopy group are represented by sequences of words. Among these sequences the elements of the fundamental group are characterized by a simple stabilization condition. This description of the fundamental group is used to give a new algebro-combinatorial proof of a result due to Eda on continuity properties of homomorphisms from the fundamental group of the Hawaiian earring to that of X.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm223-2-2,
author = {Gerhard Dorfer and J\"org M. Thuswaldner and Reinhard Winkler},
title = {Fundamental groups of one-dimensional spaces},
journal = {Fundamenta Mathematicae},
volume = {220},
year = {2013},
pages = {137-169},
zbl = {1312.55013},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm223-2-2}
}
Gerhard Dorfer; Jörg M. Thuswaldner; Reinhard Winkler. Fundamental groups of one-dimensional spaces. Fundamenta Mathematicae, Tome 220 (2013) pp. 137-169. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm223-2-2/