An inverse system approach to Menger manifolds
\textit{M. Bestvina} [Mem. Am. Math. Soc. 71, No. 380 (1988; Zbl 0645.54029)] characterized the Menger \(\mu^k\)-manifolds as \(k\)-dimensional \(LC^{k-1}\) compacta having the disjoint \(k\)-cell property. The purpose of the paper under review is to provide another approach to Bestvina's theory by replacing the method of triangulated \(\mu^k\)-manifolds of Bestvina by a method involving approximations by inverse sequences. The key role play properties of \(UV^n\) maps [Bestvina, Ferry, Chernavskij; compare the survey `Menger manifolds' by \textit{A. Chigogidze}, the author and \textit{E. D. Tymchatyn} in `Continua with the Houston problem book', Lect. Notes Pure Appl. Math. 170, 37-88 (1995)] which for the purposes of the paper under review can be treated as maps \(f:X\to Y\) between compacta such that for each \(y\) in \(Y\) and each neighbourhood \(U\) of \(y\) there exists a neighbourhood \(V\) of \(y\) contained in \(U\) such that the inclusion map \(f^{-1}(V)\subset f^{-1}(U)\) induces the zero homomorphisms between the homotopy groups in the dimensions from 0 to \(n\). The method of \(UV^n\) maps allows to translate the method of triangulated \(\mu^k\)-manifolds to the method of inverse sequences consisting of PL-manifolds and of \(UV^{k-1}\) maps as bonding maps. Although this description is equivalent to that of Bestvina, it presents a new viewpoint to the theory of Menger manifolds.
scientific article; zbMATH DE number 735686
scientific article