Borsuk-Sieklucki theorem in cohomological dimension theory

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

*この研究の対応する著者

研究成果: Article査読

抄録

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 clcn+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.

本文言語English
ページ(範囲)213-222
ページ数10
ジャーナルFundamenta Mathematicae
171
3
DOI
出版ステータスPublished - 2002
外部発表はい

ASJC Scopus subject areas

  • 代数と数論

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