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

Katsuya Eda

    Research output: Contribution to journalArticle

    21 Citations (Scopus)


    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.

    Original languageEnglish
    Pages (from-to)283-306
    Number of pages24
    JournalTopology and its Applications
    Issue number1-3
    Publication statusPublished - 1998



    • 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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