### Abstract

Let ℍ be the so-called Hawaiian earring, i.e., ℍ = {(x,y): (x-1/n)^{2}+y^{2} = 1/n^{2}, 1 ≤ n ≤ ω} and o = (0,0). We prove: (1) If Y is a subspace of a line in the Euclidean plane ℝ^{2} and X its complement ℝ^{2}\Y with x ∈ X, then the fundamental group π_{1}(X, x) is isomorphic to a subgroup of π_{1}(ℍ, o). (2) Let Y be a subspace of a line in the Euclidean plane ℝ^{2}. Then, π_{1}(ℝ^{2}\Y, x) for x ∈ ℝ^{2}\Y is isomorphic to π_{1}(ℍ, o), if and only if there exists infinitely many connected components of Y which converge to a point outside of Y. (3) Every homomorphism from π_{1}(ℍ, o) to itself is conjugate to a homomorphism induced from a continuous map.

Original language | English |
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Pages (from-to) | 283-306 |

Number of pages | 24 |

Journal | Topology and its Applications |

Volume | 84 |

Issue number | 1-3 |

Publication status | Published - 1998 |

### Keywords

- σ-word
- Free σ-product
- Fundamental group
- Hawaiian earring
- Plane
- Spatial homomorphism
- Standard homomorphism

### ASJC Scopus subject areas

- Geometry and Topology

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## Cite this

Eda, K. (1998). Free σ-products and fundamental groups of subspaces of the plane.

*Topology and its Applications*,*84*(1-3), 283-306.