Borsuk–Sieklucki theorem in cohomological dimension theory
Volume 171 / 2002
Abstract
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.
