Cellular spaces, null spaces and homotopy localization
type: Item
instance of: scholarly article
title: Cellular spaces, null spaces and homotopy localization
zbMATH Open document ID: 0842.55001
author: Emmanuel D. Farjoun
published in: Lecture Notes in Mathematics
publication date: 1996-01-11T00:00:00Z
review text: The book gives a clear presentation of new methods in homotopy theory. It is well written and well organized. Each chapter starts with the statements of the results, followed by comments, examples, and explanations. The proofs are placed at the end of each chapter and are rather easy to follow. ``The book describes a certain framework for doing homotopy theory in which the function complexes play a central role. This approach emerged in the early 1990's but has roots in earlier work of Bousfield about localization and in the big advances made by Mahowald, Ravenel, Devinatz, Hopkins and Smith towards deeper understanding of the role of periodicity in stable homotopy theory.'' The key constructions are the homotopy idempotent localization functor \(L_f\) with respect to a map \(f\), and the augmented homotopy idempotent functor \(CW_A\) with respect to a space \(A\). The functor \(L_f\) assigns to a space \(X\) the ``smallest'' quotient \(X \to L_f X\), such that if \(\text{map}(f,Y)\) is a weak equivalence, then so is \(\text{map} (L_f X, Y) \to \text{map} (X,Y)\). For an appropriate \(f\), \(L_f\) is the localization functor with respect to a given homology theory. Therefore studying the functors \(L_f\) gives a general framework for looking at homological localizations. The functor \(CW_A\) assigns to a space \(X\) the biggest subobject \(CW_A X \to X\) such that \(\text{map}_* (A, CW_A X) \to \text{map}_*(A, X)\) is a weak equivalence. For example if \(A\) is the \(\mathbb{Z}/p\)-Moore space, the functor \(CW_A\) assigns to \(X\) the universal space \(CW_A X\) which has the same mod-\(p\) homotopy groups as \(X\). The author presents constructions of continuous versions of the functors \(L_f\) and \(CW_A\). An important issue of the book is the behavior of \(L_f\) and \(CW_A\) with respect to fibration sequences and mapping spaces, thus in particular with respect to loop spaces. The author proves that applying \(L_f\) and \(CW_A\) to a loop space produces again a …
Mathematics Subject Classification ID: 55-02
Mathematics Subject Classification ID: 55P60
zbMATH DE Number: 833160
zbMATH Keywords: localization
zbMATH Keywords: periodicity
zbMATH Keywords: stable homotopy theory
zbMATH Keywords: fibration sequences
zbMATH Keywords: mapping spaces
zbMATH Keywords: loop spaces
zbMATH Keywords: Morava \(K\)-theories
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/bfb0094429
OpenAlex ID: W612190972
DOI: 10.1007/BFB0094429
description: scientific article; zbMATH DE number 833160