Fiber shape theory
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.
scientific article; zbMATH DE number 3975843
scientific article