Strong multihomotopy and Steenrod loop spaces
type: Item
instance of: scholarly article
title: Strong multihomotopy and Steenrod loop spaces
zbMATH Open document ID: 0842.55006
published in: Journal of the Mathematical Society of Japan
publication date: 1995-09-25T00:00:00Z
review text: 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.
reviewed by: J. Keesling
Mathematics Subject Classification ID: 55P55
Mathematics Subject Classification ID: 54C56
zbMATH DE Number: 800608
zbMATH Keywords: shape category
zbMATH Keywords: strong shape category
zbMATH Keywords: upper semicontinuous multivalued map
zbMATH Keywords: strong shape groups
zbMATH Keywords: Steenrod loop spaces
author: Antonio Giraldo
author: José Sanjurjo
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/04730475
OpenAlex ID: W2026094587
DOI: 10.2969/JMSJ/04730475
description: scientific article; zbMATH DE number 800608