### Abstract

For a tree T of its height equal to or less than ω_{1}, 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 H_{1}
^{T}(X) denotes a canonical factor of the first integral singular homology group. The following equivalences hold for an ω_{1}-tree T: (1) π_{1}(X_{ω1}) is embeddable into π_{1}(X_{T}), if and only if H_{1}
^{T}(X)_{ω1}≃Π _{ω1}
^{σ}Z is embeddable into H_{1}
^{T}(X_{T}), if and only if T is not an Aronzajn tree. (2) π_{1}(X_{T}) is embeddable into ××_{ω}Z≃π_{1}(H) if and only if H_{1}
^{T}(X_{T}) is embeddable into Z^{ω}≃H_{1}
^{T}(H) if and only if T is a special Aronzajn tree. (3) π_{1}(X_{T}) has a retract isomorphic to an uncountable free group, if and only if H_{1}
^{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 | |

Publication status | Published - 2001 Aug 30 |

### Fingerprint

### 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

### Cite this

*Annals of Pure and Applied Logic*,

*111*(3), 185-201. https://doi.org/10.1016/S0168-0072(01)00025-2

**Trees, fundamental groups and homology groups.** / Eda, Katsuya; Higasikawa, Masasi.

Research output: Contribution to journal › Article

*Annals of Pure and Applied Logic*, vol. 111, no. 3, pp. 185-201. https://doi.org/10.1016/S0168-0072(01)00025-2

}

TY - JOUR

T1 - Trees, fundamental groups and homology groups

AU - Eda, Katsuya

AU - Higasikawa, Masasi

PY - 2001/8/30

Y1 - 2001/8/30

N2 - 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.

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.

KW - σ-Word tree

KW - 03E75

KW - 20E05

KW - 20F34

KW - 54F50

KW - 55N10

KW - 55Q20

KW - 55Q52

KW - Aronzajn tree

KW - Free σ-product

KW - Fundamental group

KW - Singular homology

UR - http://www.scopus.com/inward/record.url?scp=0346487363&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0346487363&partnerID=8YFLogxK

U2 - 10.1016/S0168-0072(01)00025-2

DO - 10.1016/S0168-0072(01)00025-2

M3 - Article

AN - SCOPUS:0346487363

VL - 111

SP - 185

EP - 201

JO - Annals of Pure and Applied Logic

JF - Annals of Pure and Applied Logic

SN - 0168-0072

IS - 3

ER -