Phantom mappings and a shape-theoretic problem concerning products

In the paper one considers the Hawaiian earring (H,⁎)(H,⁎), the wedge (P,⁎)(P,⁎) of a sequence of 1-spheres and their Cartesian product (H×P,⁎)(H×P,⁎). One also considers the shape morphisms S[πH]S[πH], S[πP]S[πP], induced by the canonical projections πH:H×P→HπH:H×P→H, πP:H×P→PπP:H×P→P. The shape-theoretic problem asks if there exist a polyhedron Z   and a shape morphism H:Z→H×PH:Z→H×P, H≠S[⁎]H≠S[⁎], such that S[πH]H=S[⁎]S[πH]H=S[⁎] and S[πP]H=S[⁎]S[πP]H=S[⁎]. Here S[⁎]S[⁎] denotes the shape morphisms, induced by the constant mappings ⁎:Z→H×P⁎:Z→H×P, ⁎:Z→H⁎:Z→H, and ⁎:Z→P⁎:Z→P. Answering this problem affirmatively, would imply that the Cartesian product H×PH×P is not a product in the shape category of topological spaces. The main result of the paper establishes equivalence between the shape-theoretic problem and a problem involving phantom mappings.