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Borsuk-Sieklucki theorem in cohomological dimension theory

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Boege M., Dydak J., Jiménez R., Koyama A., Shchepin E.

Fundamenta Mathematicae

2002

tom 171

nr 3

213-222

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.

Ograniczanie wyników

1 Fundamenta Mathematicae

1 Boege M.

1 Dydak J.

1 Jiménez R.

1 Koyama A.

1 Shchepin E.

1 2002

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Borsuk-Sieklucki theorem in cohomological dimension theory