Strong expansions of products and products in strong shape
type: Item
instance of: scholarly article
title: Strong expansions of products and products in strong shape
zbMATH Open document ID: 1052.54018
author: Sibe Mardešić
published in: Topology and its Applications
publication date: 2004-08-20T00:00:00Z
review text: 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 \(X \times Y\)?
Mathematics Subject Classification ID: 54B35
Mathematics Subject Classification ID: 54C56
Mathematics Subject Classification ID: 55P55
zbMATH DE Number: 2093549
zbMATH Keywords: Inverse limit
zbMATH Keywords: Strong expansion
zbMATH Keywords: Homotopy expansion
zbMATH Keywords: Direct product
zbMATH Keywords: Shape
zbMATH Keywords: Strong shape
zbMATH Keywords: Finitistic space
reviewed by: Friedrich Wilhelm Bauer
MaRDI profile type: Publication
full work available at URL: j.topol.2003.08.014
OpenAlex ID: W2056483496
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DOI: 10.1016/J.TOPOL.2003.08.014
description: scientific article; zbMATH DE number 2093549