Borsuk-Sieklucki theorem cohomological dimension theory

被引:0
作者
Boege, M
Dydak, J
Jiménez, R
Koyama, A
Shchepin, EV
机构
[1] Univ Nacl Autonoma Mexico, Inst Matemat, Cuernavaca 62210, Morelos, Mexico
[2] Osaka Kyoiku Univ, Div Math Sci, Kashiwara, Osaka 5828582, Japan
[3] Univ Tennessee, Dept Math, Knoxville, TN 37996 USA
[4] VA Steklov Math Inst, Moscow 117966, Russia
关键词
cohomological dimension; cohomology locally n-connected compacta; ANR; descending chain condition;
D O I
10.4064/fm171-3-2
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
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.
引用
收藏
页码:213 / 222
页数:10
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