Fiber shape theory
type: Item
instance of: scholarly article
title: Fiber shape theory
zbMATH Open document ID: 0604.54016
author: Tatsuhiko Yagasaki
published in: Tsukuba Journal of Mathematics
publication date: 1985-01-01T00:00:00Z
review text: The author developes a fiber shape theory for maps between metric spaces based on the Mardešić-Segal approach, usually called the inverse system approach. Instead of ANR's the author defines a fiber preserving version of ANR's calling them ''absolute neighborhood fiber retracts'' (ANFR) as follows: Let B be a fixed space. A map p:E\(\to B\) is an ANFR over B provided for any map \(q: X\to B\) and any fiber preserving closed embedding \(i: E\to X\) there exists a neighborhood U of i(E) in X and a map \(r: U\to E\) such that \(ri=id_ E\) and \(pr=q| U\). The construction of fiber shape category is based on inverse system approach [\textit{S. Mardešić} and \textit{J. Segal}, Shape Theory (1982; Zbl 0495.55001)]. For a given B the usual fiber homotopy category over B is considered containing a full subcategory which consists of those objects which are fiber homotopy dominated by some ANFR's over B. After showing that every map \(p: X\to B\) admits an expansion in the procategory of the mentioned full subcategory the shape fiber category is obtained. Further, properties of fiber shape category are investigated including the appropriate type of Chapman's complement theorem under the additional assumption of separability of spaces.
Mathematics Subject Classification ID: 54C56
Mathematics Subject Classification ID: 55P55
Mathematics Subject Classification ID: 54C55
Mathematics Subject Classification ID: 57N20
zbMATH DE Number: 3975843
zbMATH Keywords: movability
zbMATH Keywords: fiber shape theory
zbMATH Keywords: absolute neighborhood fiber retracts
zbMATH Keywords: ANFR
zbMATH Keywords: fiber shape category
zbMATH Keywords: inverse system approach
zbMATH Keywords: Chapman's complement theorem
reviewed by: Ivan Ivanšić
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.21099/tkbjm/1496160289
OpenAlex ID: W401212043
DOI: 10.21099/TKBJM/1496160289
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description: scientific article; zbMATH DE number 3975843