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 clcℤn+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 |
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Pages (from-to) | 213-222 |
Number of pages | 10 |
Journal | Fundamenta Mathematicae |
Volume | 171 |
Issue number | 3 |
DOIs | |
Publication status | Published - 2002 |
Externally published | Yes |
Keywords
- ANR
- Cohomological dimension
- Cohomology locally n-connected compacta
- Descending chain condition
ASJC Scopus subject areas
- Algebra and Number Theory