Trees, fundamental groups and homology groups

Katsuya Eda, Masasi Higasikawa

    Research output: Contribution to journalArticle

    1 Citation (Scopus)

    Abstract

    For a tree T of its height equal to or less than ω1, we construct a space XT by attaching a circle to each node and connecting each node to its successors by intervals. H is the Hawaiian earring and H1 T(X) denotes a canonical factor of the first integral singular homology group. The following equivalences hold for an ω1-tree T: (1) π1(Xω1) is embeddable into π1(XT), if and only if H1 T(X)ω1≃Π ω1 σZ is embeddable into H1 T(XT), if and only if T is not an Aronzajn tree. (2) π1(XT) is embeddable into ××ωZ≃π1(H) if and only if H1 T(XT) is embeddable into Zω≃H1 T(H) if and only if T is a special Aronzajn tree. (3) π1(XT) has a retract isomorphic to an uncountable free group, if and only if H1 T(XT) has a summand isomorphic to an uncountable free abelian group, if and only if T has an uncountable anti-chain.

    Original languageEnglish
    Pages (from-to)185-201
    Number of pages17
    JournalAnnals of Pure and Applied Logic
    Volume111
    Issue number3
    DOIs
    Publication statusPublished - 2001 Aug 30

    Keywords

    • σ-Word tree
    • 03E75
    • 20E05
    • 20F34
    • 54F50
    • 55N10
    • 55Q20
    • 55Q52
    • Aronzajn tree
    • Free σ-product
    • Fundamental group
    • Singular homology

    ASJC Scopus subject areas

    • Logic

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