### Abstract

The intent of this article is to distinguish and study some ndimensional compacta (such as weak n-manifolds) with respect to embeddability into products of n curves. We show that if X is a locally connected weak n-manifold lying in a product of n curves, then rank H^{1}(X) ≥ n. If rankH^{1}(X) = n, then X is an n-torus. Moreover, if rank H^{1}(X) < 2n, then X can be presented as a product of an m-torus and a weak (n-m)-manifold, where m ≥ 2n - rank H^{1}(X). If rank H^{1}(X) < ∞, then X is a polyhedron. It follows that certain 2-dimensional compact contractible polyhedra are not embeddable in products of two curves. On the other hand, we show that any collapsible 2-dimensional polyhedron embeds in a product of two trees. We answer a question of Cauty proving that closed surfaces embeddable in a product of two curves embed in a product of two graphs. We construct a 2-dimensional polyhedron that embeds in a product of two curves but does not embed in a product of two graphs. This solves in the negative another problem of Cauty. We also construct a weak 2-manifold X lying in a product of two graphs such that H^{2}(X) = 0.

Original language | English |
---|---|

Pages (from-to) | 1509-1532 |

Number of pages | 24 |

Journal | Transactions of the American Mathematical Society |

Volume | 363 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2011 Mar |

Externally published | Yes |

### Fingerprint

### Keywords

- Embeddings
- Locally connected compacta
- Products of curves
- Weak manifolds

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Transactions of the American Mathematical Society*,

*363*(3), 1509-1532. https://doi.org/10.1090/S0002-9947-2010-05157-8

**Generalized manifolds in products of curves.** / Koyama, Akira; Krasinkiewicz, Józef; Spiez, Stanislaw.

Research output: Contribution to journal › Article

*Transactions of the American Mathematical Society*, vol. 363, no. 3, pp. 1509-1532. https://doi.org/10.1090/S0002-9947-2010-05157-8

}

TY - JOUR

T1 - Generalized manifolds in products of curves

AU - Koyama, Akira

AU - Krasinkiewicz, Józef

AU - Spiez, Stanislaw

PY - 2011/3

Y1 - 2011/3

N2 - The intent of this article is to distinguish and study some ndimensional compacta (such as weak n-manifolds) with respect to embeddability into products of n curves. We show that if X is a locally connected weak n-manifold lying in a product of n curves, then rank H1(X) ≥ n. If rankH1(X) = n, then X is an n-torus. Moreover, if rank H1(X) < 2n, then X can be presented as a product of an m-torus and a weak (n-m)-manifold, where m ≥ 2n - rank H1(X). If rank H1(X) < ∞, then X is a polyhedron. It follows that certain 2-dimensional compact contractible polyhedra are not embeddable in products of two curves. On the other hand, we show that any collapsible 2-dimensional polyhedron embeds in a product of two trees. We answer a question of Cauty proving that closed surfaces embeddable in a product of two curves embed in a product of two graphs. We construct a 2-dimensional polyhedron that embeds in a product of two curves but does not embed in a product of two graphs. This solves in the negative another problem of Cauty. We also construct a weak 2-manifold X lying in a product of two graphs such that H2(X) = 0.

AB - The intent of this article is to distinguish and study some ndimensional compacta (such as weak n-manifolds) with respect to embeddability into products of n curves. We show that if X is a locally connected weak n-manifold lying in a product of n curves, then rank H1(X) ≥ n. If rankH1(X) = n, then X is an n-torus. Moreover, if rank H1(X) < 2n, then X can be presented as a product of an m-torus and a weak (n-m)-manifold, where m ≥ 2n - rank H1(X). If rank H1(X) < ∞, then X is a polyhedron. It follows that certain 2-dimensional compact contractible polyhedra are not embeddable in products of two curves. On the other hand, we show that any collapsible 2-dimensional polyhedron embeds in a product of two trees. We answer a question of Cauty proving that closed surfaces embeddable in a product of two curves embed in a product of two graphs. We construct a 2-dimensional polyhedron that embeds in a product of two curves but does not embed in a product of two graphs. This solves in the negative another problem of Cauty. We also construct a weak 2-manifold X lying in a product of two graphs such that H2(X) = 0.

KW - Embeddings

KW - Locally connected compacta

KW - Products of curves

KW - Weak manifolds

UR - http://www.scopus.com/inward/record.url?scp=79951842551&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79951842551&partnerID=8YFLogxK

U2 - 10.1090/S0002-9947-2010-05157-8

DO - 10.1090/S0002-9947-2010-05157-8

M3 - Article

AN - SCOPUS:79951842551

VL - 363

SP - 1509

EP - 1532

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 3

ER -