The Fundamental Groups of Certain One-Dimensional Spaces

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 $Y\cup\{\infty\}\cup-\infty$.