### 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 |

### 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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## Cite this

*Annals of Pure and Applied Logic*,

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