Strong expansions of products and products in strong shape
Is the Cartesian product \(X \times Y\) of two spaces also the categorical product in the strong shape category SSh(Top)? The answer is in the affirmative whenever 1) \(X\) and \(Y\) are compact Hausdorff spaces or 2) whenever \(X\) is an FANR and \(Y\) a finitistic space. For ordinary shape it is not clear if there exists a product in the shape-category Sh(Top) at all. Due to his approach to strong shape the author investigates the following problems concerning strong expansions of spaces \(X\), \(Y\) in order to settle questions of this kind: Let \(\pmb p: X \rightarrow \pmb X \), resp. \(\pmb q: Y \rightarrow \pmb Y\) be strong expansions of spaces \(X\), \(Y\). A) Is \(\pmb p\times 1: X \times Y \rightarrow \pmb X \times \pmb Y\) a strong expansion? B. Is \(\pmb p \times \pmb q: X \times Y \rightarrow \pmb X \times \pmb Y\) a strong expansion? In general both questions have negative answers, however under certain conditions on the spaces (\(Y\) has the homotopy type of a polyhedron, resp. of a finite polyhedron) resp. conditions on the expansions of the space \(X\) (stationary movability) the author proves that the answer is positive. Reviewer's question: Is there any hope that in the strong shape category (not on the homotopy level) there does exist for arbitrary spaces, for which strong shape is defined, a product, which agrees with the Cartesian product \(X \times Y\)?
scientific article; zbMATH DE number 2093549