### 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 |
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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 |

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### 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