Trees, fundamental groups and homology groups

Katsuya Eda; Masasi Higasikawa

doi:10.1016/S0168-0072(01)00025-2

Trees, fundamental groups and homology groups

Katsuya Eda^*, Masasi Higasikawa

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

2 Citations (Scopus)

Abstract

For a tree T of its height equal to or less than ω₁, we construct a space X_T 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(X_ω1) is embeddable into π₁(X_T), if and only if H₁ ^T(X)_ω1≃Π _ω1 ^σZ is embeddable into H₁ ^T(X_T), if and only if T is not an Aronzajn tree. (2) π₁(X_T) is embeddable into ××_ωZ≃π₁(H) if and only if H₁ ^T(X_T) is embeddable into Z^ω≃H₁ ^T(H) if and only if T is a special Aronzajn tree. (3) π₁(X_T) has a retract isomorphic to an uncountable free group, if and only if H₁ ^T(X_T) has a summand isomorphic to an uncountable free abelian group, if and only if T has an uncountable anti-chain.

Original language	English
Pages (from-to)	185-201
Number of pages	17
Journal	Annals of Pure and Applied Logic
Volume	111
Issue number	3
DOIs	https://doi.org/10.1016/S0168-0072(01)00025-2
Publication status	Published - 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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10.1016/S0168-0072(01)00025-2

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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.",

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AB - 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.

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Trees, fundamental groups and homology groups

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