On the Smale theorem in weak shape theory
type: Item
instance of: scholarly article
title: On the Smale theorem in weak shape theory
zbMATH Open document ID: 0621.54013
DOI: 10.1016/0166-8641(87)90027-7
published in: Topology and its Applications
publication date: 1987-01-01T00:00:00Z
review text: Weak shape, defined by \textit{K. Borsuk} [Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys. 18, 127-132 (1970; Zbl 0195.530)], coincides with compactly generated shape, defined by \textit{L. Rubin} and \textit{T. J. Sanders} [Gen. Topol. Appl. 4, 73-83 (1974; Zbl 0274.55012)], in locally compact metric spaces [\textit{T. J. Sanders}, Fundam. Math. 93, 37-40 (1976; Zbl 0339.55011)]. Compactly generated shape is defined using direct systems of compact spaces and shape morphisms. Therefore, its study reduces to the study of the category in-pro-C, where C is the homotopy category of polyhedra. Continuing the work of the second author [''The Hurewicz and Whitehead theorems with compact carriers'', Fundam. Math. (to appear)], the authors prove a Vietoris-Smale theorem for closed surjective maps \(F: X\to Y:\) If the fibers \(F^{-1}(y)\), \(y\in Y\), are approximately n-connected up to dimension n, then the induced morphism of the k-dimensional homotopy in-pro-groups is an isomorphism for \(k\leq n\) and an epimorphism for \(k=n+1\). This is more general than a similar result due to \textit{T. J. Sanders} [Fundam. Math. 118, 69-74 (1983; Zbl 0563.55008)] because F is not assumed to be proper. To obtain their result, the authors modify and generalize various results of \textit{J. Dydak} [Diss. Math. 156 (1979; Zbl 0405.55010)].
Mathematics Subject Classification ID: 54C56
Mathematics Subject Classification ID: 55P55
Mathematics Subject Classification ID: 55Q07
zbMATH DE Number: 4007219
zbMATH Keywords: pro-category
zbMATH Keywords: in-category
zbMATH Keywords: approximate connectedness
zbMATH Keywords: homotopy category of polyhedra
zbMATH Keywords: Vietoris-Smale theorem
zbMATH Keywords: k-dimensional homotopy in-pro- groups
author: Jack Segal
author: Stanisław Spież
MaRDI profile type: Publication
full work available at URL: 0166-8641(87)90027-7
OpenAlex ID: W2054957052
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description: scientific article; zbMATH DE number 4007219