The Borsuk-Sieklucki theorem says that for every uncountable family of n-dimensional closed subsets of an n-dimensional ANR-compactum, there exist α ≠ β such that . In this paper we show a cohomological version of that theorem: Theorem. Suppose a compactum X is , where n ≥ 1, and G is an Abelian group. Let be an uncountable family of closed subsets of X. If for all α ∈ J, then 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm171-3-2, author = {Margareta Boege and Jerzy Dydak and Rolando Jim\'enez and Akira Koyama and Evgeny V. Shchepin}, title = {Borsuk-Sieklucki theorem in cohomological dimension theory}, journal = {Fundamenta Mathematicae}, volume = {173}, year = {2002}, pages = {213-222}, zbl = {1095.55001}, language = {en}, url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm171-3-2} }
Margareta Boege; Jerzy Dydak; Rolando Jiménez; Akira Koyama; Evgeny V. Shchepin. Borsuk-Sieklucki theorem in cohomological dimension theory. Fundamenta Mathematicae, Tome 173 (2002) pp. 213-222. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm171-3-2/