A characterization of compacta which admit acyclic \(UV^{n-1}\)-resolutions
R. D. Edwards' resolution theorem states in particular that every compact metric space \(X\) with cohomological dimension \(\dim_{\mathbb Z}X\leq n\) admits an acyclic resolution \(f:Y\to X\) with \(\dim Y\leq n\) [\textit{J. J. Walsh}, Shape theory and geometric topology, Proc. Conf., Dubrovnik 1981, Lect. Notes Math. 870, 105-118 (1981; Zbl 0474.55002)]. This can be generalized for all finitely generated abelian groups [\textit{A. N. Dranishnikov}, Math. USSR, Sb. 60, No. 2, 413-425 (1988); translation from Mat. Sb., Nov. Ser. 132(174), No. 3, 420-433 (1987; Zbl 0622.55002]. Also it was known that there is no generalization of that on infinitely generated groups. In the paper some technical definition of approximable cohomological dimension \(\text{a-}\dim_G\) is given which allows to prove a resolution theorem for all abelian groups. For finitely generated groups \(\text{a-}\dim_G=\dim_G\) for all spaces, generally \(\dim_G\leq \text{a-}\dim_G\).
scientific article; zbMATH DE number 1102802