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Phantom mappings and a shape-theoretic problem concerning products

In the paper one considers the Hawaiian earring (H, ∗), the wedge (P, ∗) of a sequence of 1-spheres and their Cartesian product (H×P, ∗). One also considers the shape morphisms S[πH], S[πP], induced by the canonical projections π_H : H×P → H, π_P : H×P → P. The shape-theoretic problem asks if there exist a polyhedron Z and a shape morphism H : Z → H×P, H not equals to S[∗], such that S[π_H]H = S[∗] and S[π_P]H = S[∗]. Here S[∗] denotes the shape morphisms, induced by the constant mappings ∗ : Z → H×P, ∗ : Z → H, and ∗ : Z → P. Answering this problem affirmatively, would imply that the Cartesian product H×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.

Hawaiian earring ; Pointed sum of 1-spheres ; Shape category ; Resolution ; Homotopy expansion ; Phantom mapping

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