On linking of cycles in locally connected spaces (Q5936527)

Let \(G\) be an abelian group, \(M\) a topological space, \(X\) a compact subset of \(M\), and \(z\) a cycle in \(M\smallsetminus X\) with respect to \(G\). One says that \(z\) is linked with \(X\) in \(M\) if \(z\) is homologically trivial in \(M\) but is nontrivial in \(M\smallsetminus X\). In case \(M\) is a Menger manifold [\textit{M. Bestvina}, Characterizing \(k\)-dimensional universal Menger compacta, Mem. Am. Math. Soc. 71, No. 380 (1988; Zbl 0645.54029)], the authors obtain the following (see Theorem 1.2): NEWLINENEWLINENEWLINETheorem. Let \(M\) be an \(n\)-dimensional Menger manifold, \(G\) an abelian group, and \(C\) a compactum in \(M\) of dimension \(k<n\). Then \(C\) is not linked in \(M\) with any cycle over \(G\) of dimension less than \(n-k-1\). NEWLINENEWLINENEWLINENEWLINENEWLINENEWLINEThe second main consequence of this paper states that a compact subset of an ENR is a homology \(Z\)-set provided all its points are homology \(Z\)-sets.