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Universal Menger compacta and universal mappings. (English. Russian original) Zbl 0622.54026

Math. USSR, Sb. 57, 131-149 (1987); translation from Mat. Sb., Nov. Ser. 129(171), No. 1, 121-139 (1986).
For any natural n the author constructs a continuous map \(f_ n\) of a universal Menger n-dimensional compactum \(M_ n\) onto the Hilbert cube Q and a continuous map \(g_ n\) of \(M_ n\) onto itself which are universal within the class of continuous mappings of at most n-dimensional separable metric spaces, satisfying some softness conditions. Further it is proved that the preimage \(f_ n^{-1}(X)\) of any LC\({}^{n-1}\)- compactum \(X\subset Q\) is an n-dimensional Menger manifold and if X is an AR(n)-space then \(f_ n^{-1}(X)\) is homeomorphic to a Menger compactum \(M_ n\). From this result the author derives a lot of theorems, as e.g. characterizations of AR(n)-spaces, \(LC^{n-1}\)-spaces within the class of metrizable compacta, the stabilization theorem, the triangulation theorem and other results. The last part is devoted to universal objects and absolute extensors in the nonmetrizable case.
Reviewer: J.Chvalina

MSC:

54C55 Absolute neighborhood extensor, absolute extensor, absolute neighborhood retract (ANR), absolute retract spaces (general properties)
54C25 Embedding
54F35 Higher-dimensional local connectedness
57N15 Topology of the Euclidean \(n\)-space, \(n\)-manifolds (\(4 \leq n \leq \infty\)) (MSC2010)
54E45 Compact (locally compact) metric spaces
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