# Wydawnictwa / Czasopisma IMPAN / Fundamenta Mathematicae / Wszystkie zeszyty

## Borsuk–Sieklucki theorem in cohomological dimension theory

### Tom 171 / 2002

Fundamenta Mathematicae 171 (2002), 213-222 MSC: 55M10, 54F45. DOI: 10.4064/fm171-3-2

#### Streszczenie

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.

#### Autorzy

• Margareta BoegeInstituto de Matemáticas
UNAM
Av. Universidad S//N, Col. Lomas de Chamilpa
62210 Cuernavaca, Morelos, México
e-mail
• Jerzy DydakDepartment of Mathematics
University of Tennessee
Knoxville, TN 37996, U.S.A.
e-mail
• Rolando JiménezInstituto de Matemáticas
UNAM
Av. Universidad S//N, Col. Lomas de Chamilpa
62210 Cuernavaca, Morelos, México
e-mail
• Akira KoyamaDivision of Mathematical Sciences
Osaka Kyoiku University
Kashiwara, Osaka 582-8582, Japan
e-mail
• Evgeny V. ShchepinSteklov Institute of Mathematics
Gubkina 8
117966 Moscow GSP-1, Russia
e-mail

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