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Realizing dimension functions via homology. (English) Zbl 0854.54034

Summary: The following theorem is the main result of the paper:
Theorem. Let \(G\) be an Abelian group and \(m> 0\). Let \({\mathcal G}\) be a countable family of countable Abelian groups and let \(D: {\mathcal G}\to \mathbb{Z}_+\) be a function. The following conditions are equivalent:
(1) For any CW complex \(P\) and any \(a\in H_m (P; G)- \{0\}\) there is a compactum \(X\) and a map \(\pi: X\to P\) such that \(\dim X= m\), \(\dim_H X\leq D(H)\) for each \(H\in {\mathcal G}\) and \(a\in \text{im} (\check H_m (X; G)\to \check H_m (P; G))\).
(2) \(\widetilde {H}_k (K (H, D(H)); G)= 0\) for all \(k< m\) and all \(H\in {\mathcal G}\).
As an application, we prove the existence of compacta realizing dimension functions, a result due to A. N. Dranishnikov [Sib. Math. J. 29, No. 1, 24-29 (1988); translation from Sib. Mat. Zh. 29, No. 1(167), 32-38 (1988; Zbl 0661.55002); ibid. 30, No. 1, 74-79 (1989); translation from ibid. 30, No. 1(173), 96-102 (1989; Zbl 0676.55004)].

MSC:

54F45 Dimension theory in general topology
55M10 Dimension theory in algebraic topology
55N99 Homology and cohomology theories in algebraic topology
55Q40 Homotopy groups of spheres
55P20 Eilenberg-Mac Lane spaces
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References:

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