Movability in the sense of \(n\)-shape
All spaces are assumed to be compact metric ones. The paper deals with movable compact metric spaces, \(n\)-shape, \(n\)-movable spaces, and spaces \textit{movable in the sense of \(n\)-shape}. The latter is defined like ordinary movability, after replacing ordinary homotopy by \(n\)-homotopy: Two mappings \(f, g: X \longrightarrow Y\) are said to be \(n\)-homotopic (\(f\underset{n}{\simeq} g\)) whenever for any map \(h: B \longrightarrow X,\dim B \leq n\), the compositions \(fh \simeq gh\) are homotopic. The authors investigate the relations between \(n\)-movability and \textit{movable in the sense of \(n\)-shape}: Every compact metric ANR is \textit{movable in the sense of \(n\)-shape}. Every movable compactum is \textit{movable in the sense of \(n\)-shape}. Every \textit{movable in the sense of \(n\)-shape} compactum is \(n\)-movable. There exists a compactum which is \textit{movable in the sense of \(n\)-shape} but not movable. There exists a compactum which is \(1\)-movavble but not \textit{movable in the sense of \(1\)-shape}. Moreover the authors prove that for a product \( X = \prod^\infty_{i=1}\; X_i \), \(X\) is \textit{movable in the sense of \(n\)-shape} (resp. \(n\)-movable) if and only if all \(X_i\) are \textit{movable in the sense of \(n\)-shape} (resp. \(n\)-movable). The authors answer a question of A. Chigogidze concerning a characterization of hereditary \(n\)-shape equivalences.
scientific article; zbMATH DE number 2136453