Stabilization as a CW approximation
type: Item
instance of: scholarly article
title: Stabilization as a CW approximation
zbMATH Open document ID: 0940.55012
DOI: 10.1016/S0022-4049(99)00044-4
author: A. D. Elmendorf
published in: Journal of Pure and Applied Algebra
publication date: 2000-07-24T00:00:00Z
review text: Let \(S\) be the sphere spectrum and \({\mathcal M}_S\) denote the category of \(S\)-modules constructed by Elmendorf, Kriz, Mandell, and May. In this paper, it is shown that the map \(\Sigma^\infty:{\mathcal T}(X,Y)\to{\mathcal M}_S(\Sigma^\infty X,\Sigma^\infty Y)\) is a homeomorphism. Here \({\mathcal T}\) denotes the category of based compactly generated weak Hausdorff spaces. This implies that \(\Sigma^\infty\) is a full and faithful embedding of \({\mathcal T}\) in \({\mathcal M}_S\). In particular, the unstable homotopy class \([X,Y]\) is isomorphic to \(h{\mathcal M}_S(\Sigma^\infty X,\Sigma^\infty Y)\) and \(\Sigma^\infty X\) is not a stabilization of \(X\). This comes from the fact that all weak equivalences are not inverted. Suppose that \(X\) is homotopic to a CW complex. Then the homeomorphism \(\Sigma^\infty\) implies two theorems. One is that for a CW approximation \(\gamma:\Gamma \Sigma^\infty X\to\Sigma^\infty X\), the composite \({\mathcal T}(X, Y)@> \Sigma^\infty>>{\mathcal M}_S(\Sigma^\infty X,\Sigma^\infty Y)@> \gamma^*>>{\mathcal M}_S(\Gamma\Sigma^\infty X,\Sigma^\infty Y)\) induces the usual stabilization map \([X,Y]\to \{X,Y\}\). The other is that \(X\) is contractible, if \(\Sigma^\infty X\) is a cofibrant \(S\)-module. These are typical properties of the category \({\mathcal M}_S\) and seem to be ``well known to the experts''. The homeomorphism \(\Sigma^\infty\) is certified by showing that the top row in the following commutative diagram is an equalizer: \[ \begin{matrix} {\mathcal T}(X,Y) &\overset{\eta_*}\rightarrow &{\mathcal T}(X, \Omega^\infty\Sigma^\infty Y) & \overset{\Omega^\infty\widehat\xi_*}{\underset {\theta'} \rightrightarrows} &{\mathcal T}(X, \Omega^\infty F[{\mathcal L}(1), \Sigma^\infty Y))\\ \Sigma^\infty\downarrow && \downarrow\cong &&\downarrow \cong\\ {\mathcal M}_S(\Sigma^\infty X, \Sigma^\infty Y) &\overset\iota\rightarrow &{\mathcal S}{\mathcal U}(\Sigma^\infty X,\Sigma^\infty Y) & \overset{\widehat\xi_*} {\underset…
reviewed by: Katsumi Shimomura
Mathematics Subject Classification ID: 55P42
zbMATH DE Number: 1330823
MaRDI profile type: Publication
Wikidata QID: Q126352947
full work available at URL: s0022-4049(99)00044-4
OpenAlex ID: W2024275009
Recommended article: Stabilization Via Generalized Homogeneous Approximations
Recommended article: Q3975430
Recommended article: Q3770868
Recommended article: Stabilization by perturbation of ILL-conditioned cyclic reduction∗
Recommended article: Q4301356
Recommended article: Q3841735
Recommended article: Q4516223
Recommended article: Q4224247
Recommended article: On Stability with Respect to Linear Approximation
Recommended article: Stability under a nonlinear CFL condition
description: scientific article; zbMATH DE number 1330823