Borsuk-Sieklucki theorem in cohomological dimension theory

Margareta Boege, Jerzy Dydak, Rolando Jiménez, Akira Koyama, Evgeny V. Shchepin

Research output: Contribution to journalArticle

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 clc 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 languageEnglish
Pages (from-to)213-222
Number of pages10
JournalFundamenta Mathematicae
Volume171
Issue number3
Publication statusPublished - 2002
Externally publishedYes

Fingerprint

Dimension Theory
Cohomological Dimension
Principal ideal domain
Compactum
Uncountable
Theorem
n-dimensional
Closed
Subset
Abelian group
Countable
Equality
Arbitrary
Family

Keywords

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

ASJC Scopus subject areas

  • Algebra and Number Theory

Cite this

Boege, M., Dydak, J., Jiménez, R., Koyama, A., & Shchepin, E. V. (2002). Borsuk-Sieklucki theorem in cohomological dimension theory. Fundamenta Mathematicae, 171(3), 213-222.

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 journalArticle

Boege, M, Dydak, J, Jiménez, R, Koyama, A & Shchepin, EV 2002, 'Borsuk-Sieklucki theorem in cohomological dimension theory', Fundamenta Mathematicae, vol. 171, no. 3, pp. 213-222.
Boege M, Dydak J, Jiménez R, Koyama A, Shchepin EV. Borsuk-Sieklucki theorem in cohomological dimension theory. Fundamenta Mathematicae. 2002;171(3):213-222.
Boege, Margareta ; Dydak, Jerzy ; Jiménez, Rolando ; Koyama, Akira ; Shchepin, Evgeny V. / Borsuk-Sieklucki theorem in cohomological dimension theory. In: Fundamenta Mathematicae. 2002 ; Vol. 171, No. 3. pp. 213-222.
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