### 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 |
---|---|

Pages (from-to) | 283-306 |

Number of pages | 24 |

Journal | Topology and its Applications |

Volume | 84 |

Issue number | 1-3 |

Publication status | Published - 1998 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Geometry and Topology

### Cite this

*Topology and its Applications*,

*84*(1-3), 283-306.

**Free σ-products and fundamental groups of subspaces of the plane.** / Eda, Katsuya.

Research output: Contribution to journal › Article

*Topology and its Applications*, vol. 84, no. 1-3, pp. 283-306.

}

TY - JOUR

T1 - Free σ-products and fundamental groups of subspaces of the plane

AU - Eda, Katsuya

PY - 1998

Y1 - 1998

N2 - Let ℍ be the so-called Hawaiian earring, i.e., ℍ = {(x,y): (x-1/n)2+y2 = 1/n2, 1 ≤ n ≤ ω} and o = (0,0). We prove: (1) If Y is a subspace of a line in the Euclidean plane ℝ2 and X its complement ℝ2\Y with x ∈ X, then the fundamental group π1(X, x) is isomorphic to a subgroup of π1(ℍ, o). (2) Let Y be a subspace of a line in the Euclidean plane ℝ2. Then, π1(ℝ2\Y, x) for x ∈ ℝ2\Y is isomorphic to π1(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 π1(ℍ, o) to itself is conjugate to a homomorphism induced from a continuous map.

AB - Let ℍ be the so-called Hawaiian earring, i.e., ℍ = {(x,y): (x-1/n)2+y2 = 1/n2, 1 ≤ n ≤ ω} and o = (0,0). We prove: (1) If Y is a subspace of a line in the Euclidean plane ℝ2 and X its complement ℝ2\Y with x ∈ X, then the fundamental group π1(X, x) is isomorphic to a subgroup of π1(ℍ, o). (2) Let Y be a subspace of a line in the Euclidean plane ℝ2. Then, π1(ℝ2\Y, x) for x ∈ ℝ2\Y is isomorphic to π1(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 π1(ℍ, o) to itself is conjugate to a homomorphism induced from a continuous map.

KW - σ-word

KW - Free σ-product

KW - Fundamental group

KW - Hawaiian earring

KW - Plane

KW - Spatial homomorphism

KW - Standard homomorphism

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UR - http://www.scopus.com/inward/citedby.url?scp=0000603416&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0000603416

VL - 84

SP - 283

EP - 306

JO - Topology and its Applications

JF - Topology and its Applications

SN - 0166-8641

IS - 1-3

ER -