Fundamental groups of one-dimensional spaces

Gerhard Dorfer ¹   ; Jörg M. Thuswaldner ²   ; Reinhard Winkler ³

¹ Institute of Discrete Mathematics and Geometry Vienna University of Technology Wiedner Hauptstr. 8-10/104 1040 Wien, Austria
² Chair of Mathematics and Statistics Department of Mathematics and Information Technology University of Leoben Franz-Josef-Straße 18 8700 Leoben, Austria
³ Institute of Discrete Mathematics and Geometry Vienna University of Technology Wiedner Hauptstr. 8-10/104 1040 Vienna, Austria

Fundamenta Mathematicae, Tome 223 (2013) no. 2, pp. 137-169

Résumé

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 $X$.

Détail

DOI : 10.4064/fm223-2-2

Keywords: metrizable one dimensional continuum describe fundamental group subgroup its ech homotopy group particular elements ech homotopy group represented sequences words among these sequences elements fundamental group characterized simple stabilization condition description fundamental group algebro combinatorial proof result due eda continuity properties homomorphisms fundamental group hawaiian earring