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

The strong shape category of topological spaces SSh can be defined using the coherent homotopy category CH, whose objects are inverse systems consisting of topological spaces, indexed by cofinite directed sets. In particular, if X, Y are spaces and q : Y → Y is a cofinite HPol-resolution of Y , then there is a bijection between the set SSh(X, Y ) of strong shape morphisms F : X → Y and the set CH(X, Y ) of homotopy classes [f ] of coherent homotopy mappings f : X → Y . In the paper it is shown that such a bijection exists also in the case when Y is not cofinite. This fact makes it possible to study strong shape properties of the Cartesian product X × P of a compact Hausdorff space X and a polyhedron P using the standard resolution of X × P , which is a non-cofinite HPolresolution. As an application, one reduces the question whether X × P is a product of X and P in the category SSh to a question concerning homotopy classes of coherent homotopy mappings. Analogous results also hold for the ordinary shape category of topological spaces Sh and the pro-homotopy category of cofinite inverse systems of spaces. AMS subject classifications: 54C56, 54B10, 54B35