Borsuk-Sieklucki theorem in cohomological dimension theory

Margareta Boege, et al. "Borsuk-Sieklucki theorem in cohomological dimension theory." Fundamenta Mathematicae 171.3 (2002): 213-222. <http://eudml.org/doc/282952>.

@article{MargaretaBoege2002,
abstract = {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.},
author = {Margareta Boege, Jerzy Dydak, Rolando Jiménez, Akira Koyama, Evgeny V. Shchepin},
journal = {Fundamenta Mathematicae},
keywords = {cohomological dimension; strong cohomological dimension; ANR-compactum; cohomology locally -connected compactum; Borsuk-Sieklucki theorem; Abelian group; descending chain condition},
language = {eng},
number = {3},
pages = {213-222},
title = {Borsuk-Sieklucki theorem in cohomological dimension theory},
url = {http://eudml.org/doc/282952},
volume = {171},
year = {2002},
}

TY - JOUR
AU - Margareta Boege
AU - Jerzy Dydak
AU - Rolando Jiménez
AU - Akira Koyama
AU - Evgeny V. Shchepin
TI - Borsuk-Sieklucki theorem in cohomological dimension theory
JO - Fundamenta Mathematicae
PY - 2002
VL - 171
IS - 3
SP - 213
EP - 222
AB - 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.
LA - eng
KW - cohomological dimension; strong cohomological dimension; ANR-compactum; cohomology locally -connected compactum; Borsuk-Sieklucki theorem; Abelian group; descending chain condition
UR - http://eudml.org/doc/282952
ER -