Abstract
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.
| 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 |
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Keywords
- ANR
- Cohomological dimension
- Cohomology locally n-connected compacta
- Descending chain condition
ASJC Scopus subject areas
- Algebra and Number Theory
Cite this
Borsuk-Sieklucki theorem in cohomological dimension theory. / Boege, Margareta; Dydak, Jerzy; Jiménez, Rolando; Koyama, Akira; Shchepin, Evgeny V.
In: Fundamenta Mathematicae, Vol. 171, No. 3, 2002, p. 213-222.Research output: Contribution to journal › Article
}
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 -