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June 2000 The Fundamental Groups of Certain One-Dimensional Spaces
Katsuya EDA
Tokyo J. Math. 23(1): 187-202 (June 2000). DOI: 10.3836/tjm/1255958814

Abstract

An infinitary version of edge path groups is introduced for applications to non-locally simply connected spaces (see Figure 1 in the text). (1) Edge path groups in this paper are subgroups of the free $\sigma$-product of copies of the integer group $\mathbf{Z}$, which is isomorphic to the fundamental groups of the Hawaiian earring of $I$-many circles for some index set $I$. (2) Let $Y$ be a subspace of the real line in the Euclidean plane $\mathbf{R}^2$ and $\mathcal{C}$ the set of all connected components of $Y$. Then, the fundamental group of $\mathbf{R}^2\backslash Y$ is isomorphic to a free product of infinitely many non-trivial groups, if and only if there exists an accumulation point of $\mathcal{C}$ in $Y\cup\{\infty\}\cup-\infty$.

Citation

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Katsuya EDA. "The Fundamental Groups of Certain One-Dimensional Spaces." Tokyo J. Math. 23 (1) 187 - 202, June 2000. https://doi.org/10.3836/tjm/1255958814

Information

Published: June 2000
First available in Project Euclid: 19 October 2009

zbMATH: 0960.55007
MathSciNet: MR1763511
Digital Object Identifier: 10.3836/tjm/1255958814

Rights: Copyright © 2000 Publication Committee for the Tokyo Journal of Mathematics

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Vol.23 • No. 1 • June 2000
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