Homological dimension and homogeneous ANR spaces (Q2400847)

The main object of the paper is to describe dimensional full-valuedness of metric compacta \(X\), using the equalities \(d_G X = \dim X\) fo some groups \(G\), where \(d_G X\) is the homological dimension of \(X\) defined by Alexandroff. Recall that the homological dimension \(d_G X\) of a metric compactum \(X\) is the maximum integer \(n\) with the property that there exist a closed subset \(F\) of \(X\) and a nontrivial element \(\gamma \in H_{n-1}(F; G)\) that is \(G\)-homologous to zero in \(X\), and that \(X\) is full-valued if \(\dim X\times Y = \dim X + \dim Y\) for all metric compacta \(Y\). As a consequence, the author proves that every two-dimensional \(lc^2\)-space is dimensionally full-valued. This improves a result of \textit{Y. Kodama} that every two-dimensional ANR is dimensionally full-valued [Fundam. Math. 44, 171--185 (1957; Zbl 0079.16801)]. The author also introduces the set \({\mathcal H}_{X, G}\), which is defined as the set of integers \(k\geq 1\) such that there exist a closed subset \(F\) of \(X\) and a nontrivial element \(\gamma \in H_{k-1}(F; G)\) that is \(G\)-homologous to zero in \(X\). Using this set, he obtains some properties of homogeneous ANR-spaces.