On approximation and embedding problems for cohomological dimension
type: Item
instance of: scholarly article
title: On approximation and embedding problems for cohomological dimension
zbMATH Open document ID: 0833.55002
DOI: 10.1016/0166-8641(94)90065-5
published in: Topology and its Applications
publication date: 1996-03-06T00:00:00Z
review text: The paper concerns questions of unstable intersections of compacta in Euclidean spaces, i.e. when the maps \(f: X\to \mathbb{R}^n\) and \(g: Y\to \mathbb{R}^n\) of compacta \(X\) and \(Y\) can be arbitrarily closely approximated by maps with disjoint images. The same authors have already published results of this type under different dimensional conditions on \(X\) and \(Y\). The main result on this line is the following theorem: Let \(X\) and \(Y\) be compacta such that \(\dim (X\times Y)<n\) and \((\text{codim } X)\cdot (\text{codim } Y)\geq n\), where \(\text{codim } X= n-\dim X\). Then every pair of maps \(f: X\to \mathbb{R}^n\) and \(g: Y\to \mathbb{R}^n\) can be approximated arbitrarily closely by maps \(f': X\to \mathbb{R}^n\) and \(g': Y\to \mathbb{R}^n\) such that \(f' (X)\cap g' (Y)= \emptyset\), provided \(\dim X\leq n-3\) and \(\dim Y\leq n-3\). This result improves the result obtained in their previous paper [Tsukuba J. Math. 17, 549-564 (1993; Zbl 0830.54017)], compare also [\textit{S. Spież} and \textit{H. Toruńczyk}, Topology Appl. 41, 193-204 (1991; Zbl 0770.54034)]. Beside this there are several statements on equivalences between notions defined in the paper and Section 2 brings a contemporary review of needed results of Bockstein [\textit{M. F. Bokshtejn}, Am. Math. Soc., Transl., II. Ser. 11, 173-385 (1959); translation from Tr. Mosk. Mat. Obshch. 5, 1-80 (1956; Zbl 0071.163)].
reviewed by: Ivan Ivanšić
Mathematics Subject Classification ID: 55M10
Mathematics Subject Classification ID: 57Q65
Mathematics Subject Classification ID: 54F45
Mathematics Subject Classification ID: 57Q55
Mathematics Subject Classification ID: 57N75
zbMATH DE Number: 500700
zbMATH Keywords: cohomological dimension
zbMATH Keywords: Bockstein algebra
zbMATH Keywords: dimension types of compacta
zbMATH Keywords: unstable intersections of compacta in Euclidean spaces
author: Alexander Dranishnikov
author: Dušan Repovš
author: Evgeny V. Shchepin
MaRDI profile type: Publication
full work available at URL: 0166-8641(94)90065-5
OpenAlex ID: W2095645921
cites work: Homological dimension theory
cites work: On Intersections of Compacta in Euclidean Space
cites work: On Intersection of Compacta in Euclidean Space II
cites work: On intersections of compacta of complementary dimensions in Euclidean space
cites work: On intersections of compacta in Euclidean space: The metastable case
cites work: On compacta that intersect unstably in Euclidean space
cites work: Q4087996
cites work: On a problem of Alexandroff concerning the dimension of product spaces. I, II
cites work: HOMOLOGICAL DIMENSION THEORY
cites work: Embeddings of $n$-dimensional locally compact metric spaces to $2n$-manifolds.
cites work: Moving compacta in \(\mathbb{R}{}^ m\) apart
cites work: THE EMBEDDING OF COMPACTA IN EUCLIDEAN SPACE
cites work: Finite-to-one mappings of topological spaces and cohomology manifolds
Wikidata QID: Q126807122
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description: scientific article; zbMATH DE number 500700