Borsuk–Sieklucki theorem in cohomological dimension theory

Margareta Boege ¹   ; Jerzy Dydak ²   ; Rolando Jiménez ¹   ; Akira Koyama ³   ; Evgeny V. Shchepin ⁴

¹ Instituto de Matemáticas UNAM Av. Universidad S//N, Col. Lomas de Chamilpa 62210 Cuernavaca, Morelos, México
² Department of Mathematics University of Tennessee Knoxville, TN 37996, U.S.A.
³ Division of Mathematical Sciences Osaka Kyoiku University Kashiwara, Osaka 582-8582, Japan
⁴ Steklov Institute of Mathematics Gubkina 8 117966 Moscow GSP-1, Russia

Fundamenta Mathematicae, Tome 171 (2002) no. 3, pp. 213-222

Résumé

The Borsuk–Sieklucki theorem says that for every uncountable family $\{X_{\alpha}\}_{\alpha \in A}$ of $n$-dimensional closed subsets of an $n$-dimensional ANR-compactum, there exist $\alpha \ne \beta$ such that $\mathop{\rm dim} (X_{\alpha} \cap X_{\beta}) = n$. In this paper we show a cohomological version of that theorem:Theorem. Suppose a compactum $X$ is ${\rm clc}^{n+1}_{{\Bbb Z}}$, where $n\geq 1$, and $G$ is an Abelian group. Let $ \{X_{\alpha }\}_{\alpha \in J}$ be an uncountable family of closed subsets of $X$. If ${\rm dim} _GX={\rm dim} _GX_{\alpha }=n$ for all $ \alpha \in J$, then $\mathop{\rm dim} _G(X_{\alpha }\cap X_{\beta })=n$ for some $\alpha \neq \beta$.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.

Détail

DOI : 10.4064/fm171-3-2

Keywords: borsuk sieklucki theorem says every uncountable family alpha alpha n dimensional closed subsets n dimensional anr compactum there exist alpha beta mathop dim alpha cap beta paper cohomological version theorem theorem suppose compactum clc bbb where geq abelian group alpha alpha uncountable family closed subsets dim dim alpha alpha mathop dim alpha cap beta alpha neq beta being countable principal ideal domain above result proved choi kozlowski c k independently dydak koyama d k proved being arbitrary principal ideal domain posed question validity theorem quasicyclic groups see problem d k applications theorem investigate equality cohomological dimension strong cohomological dimension characterization cohomological dimension terms special base