A unified approach of characterizations and resolutions for cohomological dimension modulo \(p\)
The authors prove the following theorem: Let \(X\) be either compact Hausdorff or metrizable, with \(a\)-\(\dim_G X \leq n\). Then there exists a proper map \(f : Z \to X\) (\(Z\) either compact Hausdorff or metrizable resp.) \(\dim Z \leq n\) such that \(H^n (f^{-1} (x); G) = 0\) for all \(x\in X\). The new concept is that of approximable dimension with respect to an abelian group \(G\), \(a\)-\(\dim_G X\), constituting a new version of cohomology dimension of a space \(X\). Ordinary cohomology dimension and ordinary dimension of compact Hausdorff or metrizable spaces are related by the inequality \[ c \text{-dim}_G X \leq a \text{-dim}_G X \leq \dim X. \] Moreover for \(G = \mathbb{Z}\) or \(G = \mathbb{Z}_p\), \(p\) being a prime, and space \(X\) as above one has \[ a \text{-dim}_G X = c \text{-dim}_G X. \] As a result the main theorem of this paper comprises results of Edwards and J. J. Walsh, R. L. Rubin, A. N. Dranishnikov and S. Mardešić which were accomplished for different kinds of compact Hausdorff or metrizable spaces and the coefficient groups \(\mathbb{Z}\) resp. \(\mathbb{Z}_p\), \(p\) being a prime.
scientific article; zbMATH DE number 729213