On approximation and embedding problems for cohomological dimension (Q1313937)

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)].