### Abstract

The Borsuk-Sieklucki theorem says that for every uncountable family {X_{α}}_{α∈A} of n-dimensional closed subsets of an n-dimensional ANR-compactum, there exist α ≠ β such that dim(X_{α} ∩ X_{β}) = n. In this paper we show a cohomological version of that theorem: THEOREM. Suppose a compactum X is clc_{ℤ}
^{n+1}, where n ≥ 1, and G is an Abelian group. Let {X_{α}}_{α∈J} be an uncountable family of closed subsets of X. If dim_{G}X = dim_{G}X_{α} = n for all α ∈ J, then dim_{G}(X_{α} ∩ X_{β}) = n for some α ≠ β. For G being a countable principal ideal domain the above result was proved by Choi and Kozlowski [C-K]. Independently, Dydak and Koyama [D-K] proved it for G being an arbitrary principal ideal domain and posed the question of validity of the Theorem for quasicyclic groups (see Problem 1 in [D-K]). As applications of the Theorem we investigate equality of cohomological dimension and strong cohomological dimension, and give a characterization of cohomological dimension in terms of a special base.

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

Pages (from-to) | 213-222 |

Number of pages | 10 |

Journal | Fundamenta Mathematicae |

Volume | 171 |

Issue number | 3 |

Publication status | Published - 2002 |

Externally published | Yes |

### Fingerprint

### Keywords

- ANR
- Cohomological dimension
- Cohomology locally n-connected compacta
- Descending chain condition

### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

*Fundamenta Mathematicae*,

*171*(3), 213-222.

**Borsuk-Sieklucki theorem in cohomological dimension theory.** / Boege, Margareta; Dydak, Jerzy; Jiménez, Rolando; Koyama, Akira; Shchepin, Evgeny V.

Research output: Contribution to journal › Article

*Fundamenta Mathematicae*, vol. 171, no. 3, pp. 213-222.

}

TY - JOUR

T1 - Borsuk-Sieklucki theorem in cohomological dimension theory

AU - Boege, Margareta

AU - Dydak, Jerzy

AU - Jiménez, Rolando

AU - Koyama, Akira

AU - Shchepin, Evgeny V.

PY - 2002

Y1 - 2002

N2 - The Borsuk-Sieklucki theorem says that for every uncountable family {Xα}α∈A of n-dimensional closed subsets of an n-dimensional ANR-compactum, there exist α ≠ β such that dim(Xα ∩ Xβ) = n. In this paper we show a cohomological version of that theorem: THEOREM. Suppose a compactum X is clcℤ n+1, where n ≥ 1, and G is an Abelian group. Let {Xα}α∈J be an uncountable family of closed subsets of X. If dimGX = dimGXα = n for all α ∈ J, then dimG(Xα ∩ Xβ) = n for some α ≠ β. For G being a countable principal ideal domain the above result was proved by Choi and Kozlowski [C-K]. Independently, Dydak and Koyama [D-K] proved it for G being an arbitrary principal ideal domain and posed the question of validity of the Theorem for quasicyclic groups (see Problem 1 in [D-K]). As applications of the Theorem we investigate equality of cohomological dimension and strong cohomological dimension, and give a characterization of cohomological dimension in terms of a special base.

AB - The Borsuk-Sieklucki theorem says that for every uncountable family {Xα}α∈A of n-dimensional closed subsets of an n-dimensional ANR-compactum, there exist α ≠ β such that dim(Xα ∩ Xβ) = n. In this paper we show a cohomological version of that theorem: THEOREM. Suppose a compactum X is clcℤ n+1, where n ≥ 1, and G is an Abelian group. Let {Xα}α∈J be an uncountable family of closed subsets of X. If dimGX = dimGXα = n for all α ∈ J, then dimG(Xα ∩ Xβ) = n for some α ≠ β. For G being a countable principal ideal domain the above result was proved by Choi and Kozlowski [C-K]. Independently, Dydak and Koyama [D-K] proved it for G being an arbitrary principal ideal domain and posed the question of validity of the Theorem for quasicyclic groups (see Problem 1 in [D-K]). As applications of the Theorem we investigate equality of cohomological dimension and strong cohomological dimension, and give a characterization of cohomological dimension in terms of a special base.

KW - ANR

KW - Cohomological dimension

KW - Cohomology locally n-connected compacta

KW - Descending chain condition

UR - http://www.scopus.com/inward/record.url?scp=0036037674&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0036037674&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0036037674

VL - 171

SP - 213

EP - 222

JO - Fundamenta Mathematicae

JF - Fundamenta Mathematicae

SN - 0016-2736

IS - 3

ER -