Borsuk-Sieklucki theorem cohomological dimension theory

The Borsuk-Sieklucki theorem says that for every uncountable family {X-alpha}alphais an element ofA of n-dimensional closed subsets of an n-dimensional ANR-compactum, there exist alphanot equalbeta such that dim(X-alpha boolean AND Xbeta) = n. In this paper we show a cohomological version of that theorem: THEOREM. Suppose a compactum X is clc(Z)(n+1), where n greater than or equal to 1, and G is an Abelian group. Let {X-alpha}(alpha=J) be an uncountable family of closed subsets of X. If dim(G)X = dim(G)X(alpha), = n for all alpha is an element of J, then dim(G) (X-alpha boolean AND X-beta) = n for some alpha not equal 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.