### Abstract

Let X be a one-dimensional metric space and ℍ be the Hawaiian earring.(1) Each homomorphism from π_{1}(ℍ) to π_{1}(X) is induced from a continuous map up to the base-point-change isomorphism on π_{1}(X).(2) Let X be a one-dimensional Peano continuum. Then X has the same homotopy type as that of ℍ if and only if π_{1}(X) is isomorphic to π_{1}(ℍ), if and only if X has a unique point at which X is not semi-locally simply connected. (3) Let X and Y be one-dimensional Peano continua which are not semi-locally simply connected at any point. Then, X and Y are homeomorphic if and only if π_{1}(X) and π_{1}(Y) are isomorphic. Moreover, each isomorphism from π_{1}(X) to π_{1}(Y) is induced by a homeomorphism from X to Y up to the base-point-change-isomorphism.

Original language | English |
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Pages (from-to) | 479-505 |

Number of pages | 27 |

Journal | Topology and its Applications |

Volume | 123 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2002 Sep 30 |

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### Keywords

- Fundamental group
- Hawaiian earring
- One-dimensional
- Spatial homomorphism

### ASJC Scopus subject areas

- Geometry and Topology

### Cite this

*Topology and its Applications*,

*123*(3), 479-505. https://doi.org/10.1016/S0166-8641(01)00214-0