### Abstract

In this article we mostly study algebraic properties of n-dimensional \cyclic" compacta lying either in products of n curves or in the nth symmetric product of a curve. The basic results have been obtained for compacta admitting essential maps into the n-sphere Sn. One of the main results asserts that if a compactum X admits such a mapping and X embeds in a product of n curves then there exists an algebraically essential map X → Tn into the n-torus; the same conclusion holds for X embeddable in the nth symmetric product of a curve. The existence of an algebraically essential mapping X Tn is equivalent to the existence of some elements a1; ⋯ an 2 H1(X) whose cup product a1 ⋯ an 2 Hn(X) is not zero, and implies that rankH1(X) n and catX > n. In particular, it follows that Sn, n ≥ 2, is not embeddable in the nth symmetric product of any curve. The case n = 2 answers in the negative a question of Illanes and Nadler.

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

Pages (from-to) | 611-641 |

Number of pages | 31 |

Journal | Houston Journal of Mathematics |

Volume | 38 |

Issue number | 2 |

Publication status | Published - 2012 |

Externally published | Yes |

### Fingerprint

### Keywords

- Cohomology groups
- Embeddings
- Products
- Symmetric products

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Houston Journal of Mathematics*,

*38*(2), 611-641.

**Embeddings into products and symmetric products - An algebraic approach.** / Koyama, Akira; Krasinkiewicz, Józef; Spiez, Stanis Law.

Research output: Contribution to journal › Article

*Houston Journal of Mathematics*, vol. 38, no. 2, pp. 611-641.

}

TY - JOUR

T1 - Embeddings into products and symmetric products - An algebraic approach

AU - Koyama, Akira

AU - Krasinkiewicz, Józef

AU - Spiez, Stanis Law

PY - 2012

Y1 - 2012

N2 - In this article we mostly study algebraic properties of n-dimensional \cyclic" compacta lying either in products of n curves or in the nth symmetric product of a curve. The basic results have been obtained for compacta admitting essential maps into the n-sphere Sn. One of the main results asserts that if a compactum X admits such a mapping and X embeds in a product of n curves then there exists an algebraically essential map X → Tn into the n-torus; the same conclusion holds for X embeddable in the nth symmetric product of a curve. The existence of an algebraically essential mapping X Tn is equivalent to the existence of some elements a1; ⋯ an 2 H1(X) whose cup product a1 ⋯ an 2 Hn(X) is not zero, and implies that rankH1(X) n and catX > n. In particular, it follows that Sn, n ≥ 2, is not embeddable in the nth symmetric product of any curve. The case n = 2 answers in the negative a question of Illanes and Nadler.

AB - In this article we mostly study algebraic properties of n-dimensional \cyclic" compacta lying either in products of n curves or in the nth symmetric product of a curve. The basic results have been obtained for compacta admitting essential maps into the n-sphere Sn. One of the main results asserts that if a compactum X admits such a mapping and X embeds in a product of n curves then there exists an algebraically essential map X → Tn into the n-torus; the same conclusion holds for X embeddable in the nth symmetric product of a curve. The existence of an algebraically essential mapping X Tn is equivalent to the existence of some elements a1; ⋯ an 2 H1(X) whose cup product a1 ⋯ an 2 Hn(X) is not zero, and implies that rankH1(X) n and catX > n. In particular, it follows that Sn, n ≥ 2, is not embeddable in the nth symmetric product of any curve. The case n = 2 answers in the negative a question of Illanes and Nadler.

KW - Cohomology groups

KW - Embeddings

KW - Products

KW - Symmetric products

UR - http://www.scopus.com/inward/record.url?scp=84893385242&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84893385242&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:84893385242

VL - 38

SP - 611

EP - 641

JO - Houston Journal of Mathematics

JF - Houston Journal of Mathematics

SN - 0362-1588

IS - 2

ER -