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Strong Cohomological Dimension

100%
Dydak J., Koyama A.
Bulletin of the Polish Academy of Sciences. Mathematics
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2008
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tom 56
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nr 2
183-189
EN
We characterize strong cohomological dimension of separable metric spaces in terms of extension of mappings. Using this characterization, we discuss the relation between strong cohomological dimension and (ordinal) cohomological dimension and give examples to clarify their gaps. We also show that $Ind_{G} X = dim_{G} X$ if X is a separable metric ANR and G is a countable Abelian group. Hence $dim_{ℤ} X = dim X$ for any separable metric ANR X.
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Borsuk-Sieklucki theorem in cohomological dimension theory

71%
Boege M., Dydak J., Jiménez R., Koyama A., Shchepin E.
Fundamenta Mathematicae
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2002
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tom 171
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nr 3
213-222
EN
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 $clc^{n+1}_{ℤ}$, 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.
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