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Shape theory for G-pairs. (Russian) Zbl 0572.55010

This paper is an important contribution to shape theory and the study of G-spaces and G-pairs. The author continues his earlier works [ibid. 34, No.6 (210), 119-123 (1979; Zbl 0444.55006); Itogi Nauki Tekh. 19, 181-207 (1981; Zbl 0488.55008); Serdica 10, 223-228 (1984; Zbl 0556.54026)] concerning the shape theory of transformation groups.
The paper under review develops a construction of the equivariant shape theory for the categories of pairs corresponding to the following categories of G-spaces: the category of compact metric G-spaces (\({\mathcal M}{\mathcal C}_ G)\), the category of metric G-spaces (\({\mathcal M}_ G)\), the category of compact Hausdorff G-spaces (\({\mathcal C}_ G)\) and the category of p-paracompact G-spaces (\({\mathcal P}_ G)\), where G is a compact Hausdorff transformation group.
The paper is divided into two parts. The first part deals with the introduction for topological pairs and for G-pairs of the notions of AR, AE, ANR and ANE-object and with the proofs of some Kuratowski- Wojdyslawski type embedding theorems for the following categories of pairs: \({\mathcal M}^ 2\), \({\mathcal M}^ 2_ G\), \({\mathcal P}^ 2_ G\), \({\mathcal C}^ 2_ G.\)
In the second part, using the notion of associated inverse system, in the sense of Morita, the author defines the functor ass and he proves the existence of this functor for the homotopical categories of pairs H- \({\mathcal M}^ 2_ G\), H-\({\mathcal P}^ 2_ G\), H-\({\mathcal M}{\mathcal C}^ 2_ G\), H-\({\mathcal C}^ 2_ G\), establishing the density in these categories of the subcategories of ANE-pairs, for the categories \({\mathcal M}_ G\), \({\mathcal P}_ G\), \({\mathcal M}{\mathcal C}_ G\), \({\mathcal C}_ G\), respectively. Then the (homotopy) equivalence between two morphisms of associated systems of pairs is defined and the corresponding shape categories are introduced.
Concerning the proofs of the given results, the most technical difficulties appear especially in the case of the category \({\mathcal C}^ 2_ G\). In the introduction the author announced a third part entitled ”Equivariant homology, homotopy and cohomology of Alexandrov-Čech type” which is postponed for a more ample paper.
Reviewer: Ioan Pop (Iaşi)

MSC:

55P55 Shape theory
55P91 Equivariant homotopy theory in algebraic topology
54C56 Shape theory in general topology
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