Strong multihomotopy and Steenrod loop spaces
The authors give alternative models for the shape category and the strong shape category for compact metric spaces. The advantage of these models is that they do not rely on any external elements in their description. The basic concept used here is that of a fine multivalued map between the metric compacta \(X\) and \(Y\). Such a map is an upper semicontinuous multivalued map \(F : X \times R_+ \to Y\) such that for every \(\varepsilon > 0\) there is a \(t_0 \in R_+ = [0,\infty)\) such that for every \(t \geq t_0\), \(\text{diam} (F(x,t)) < \varepsilon\) for every \(x \in X\). The collection of fine multivalued maps, denoted in the paper by \(M(X,Y)\), is given a topology so that the arc components of the space correspond to the strong shape morphisms between \(X\) and \(Y\) and the components of \(M(X,Y)\) correspond to the shape morphisms between \(X\) and \(Y\). This makes possible a ready comparison between the strong shape morphisms and the shape morphisms. An application of this approach is given to the strong shape groups and the Steenrod loop spaces.
scientific article; zbMATH DE number 800608