An existence theorem concerning strong shape of Cartesian products

The paper is devoted to the question is the Cartesian product X × P of a compact Hausdorff space X and a polyhedron P a product in the strong shape category SSh of topological spaces. The question consists of two parts. The existence part, which asks whether, for a topological space Z, for a strong shape morphism F : Z → X and a homotopy class of mappings [g] : Z → P , there exists a strong shape morphism H : Z → X×P , whose compositions with the canonical projections of X × P equal F and [g], respectively. The uniqueness part asks if H is unique. The main result of the paper asserts that H exists, whenever Z is either metrizable or has the homotopy type of a polyhedron. If X is a metric compactum, H exists for all topological spaces Z. The proofs use resolutions of spaces and coherent homotopies of inverse systems. It is known that, in the ordinary shape category Sh, H need not be unique, even in the case when Z is a metrizable space or a polyhedron. AMS subject classifications: 54C56, 54B10, 54B35