Using non-cofinite resolutions in shape theory. Application to Cartesian products (Q2897912)

In [Strong shape and homology. Springer Monographs in Mathematics. Berlin: Springer (2000; Zbl 0939.55007)], the author defined the strong shape category of topological spaces \(SSh\) using the coherent homotopy category \(CH\) (whose objects are inverse systems of topological spaces indexed by cofinite directed sets) and showed there is a bijection between the set \(SSh(X,Y)\) of strong shape morphisms from \(X\) to \(Y\) and the set \(CH(X,Y)\) of the homotopy classes of coherent homotopy mappings from \(X\) to \(Y\) (here \(Y\) is a Hpol-resolution of the topological space \(Y\)). He extends the definition of this map from the case of cofinite Hpol-resolutions to the case of non-cofinite Hpol-resolutions and it is shown that such a bijection exits also in the case when \(Y\) is not cofinite. An application of this theory is the study of Cartesian products \(X \times P\), where \(X\) is a compact Hausdorff space and \(P\) is a polyhedron, using the standard non-cofinite Hpol-resolution of this product. The author reduces the question whether \(X \times P\) is a product of \(X\) and \(P\) in the category \(SSh\) to a question concerning homotopy classes of coherent homotopy mappings. Analogous results hold for the ordinary shape category of topological spaces and the pro-homotopy category of cofinite inverse systems of spaces.