Controlled simple homotopy theory and applications
type: Item
instance of: scholarly article
title: Controlled simple homotopy theory and applications
zbMATH Open document ID: 0548.57001
author: T. A. Chapman
published in: Lecture Notes in Mathematics
publication date: 1983-01-01T00:00:00Z
review text: The simple homotopy theory, invented by J. H. C. Whitehead, has found many applications. Today's reference on this subject is the excellent textbook by \textit{M. M. Cohen} [A course in simple-homotopy theory (1973; Zbl 0261.57009)]. This booklet brings us under the name of ''controlled'' simple homotopy theory'' a generalization of the ''classical'' simple homotopy theory which may now be attributed as ''uncontrolled''. The controlled setting comes from the notion of ''controlled homotopy'' in a space which means the following: Let p: \(A\to B\) be a map of a space A into a metric space B. Then a \(p^{-1}(\epsilon)\)-homotopy in A is a homotopy for which the p-image of the path of each point has the diameter \(<\epsilon\) in B. Now, let X and Y be polyhedra and p:\(Y\to B\) be a map (controlling map) into a metric space B (the parameter space). A map f: \(X\to Y\) is a \(p^{-1}(\epsilon)\)-equivalence provided there exists a map g: \(Y\to X\) for which there are homotopies \(fg\simeq id_ Y\), \(gf\simeq id_ X\) such that \(fg\simeq id_ Y\) is a \(p^{-1}(\epsilon)\)-homotopy in Y and that the composition of the homotopy \(gf\simeq id_ X\) with f gives a \(p^{-1}(\epsilon)\) homotopy in Y. This is the notion of ''controlled homotopy equivalence''. It turnes out that the following equivalent notion of the definition of simple homotopy equivalence is convenient for this generalization: f: \(X\to Y\) is a simple homotopy equivalence if there exist a compact polyhedron Z and CE-PL surjections r: \(Z\to X\) and s: \(Z\to Y\) such that fr is homotopic to s. Now, its controlled version reads as follows: A map f: \(X\to Y\) is \(p^{-1}(\epsilon)\)-simple provided there exist a compact polyhedron Z and CE-PL maps r:\(Z\to X\), s:\(Z\to Y\) such that fr and s are \(p^{- 1}(\epsilon)\)-homotopic in Y. These are two basic definitions of the controlled simple homotopy theory which is merely the obstruction theory that arises when one attempts to answer the analogue of the basic p…
Mathematics Subject Classification ID: 57-02
Mathematics Subject Classification ID: 57Q10
Mathematics Subject Classification ID: 57Q12
Mathematics Subject Classification ID: 57Q20
Mathematics Subject Classification ID: 57R80
zbMATH DE Number: 3874224
zbMATH Keywords: controlled homotopy
zbMATH Keywords: controlled homotopy equivalence
zbMATH Keywords: CE-PL maps
zbMATH Keywords: controlled Whitehead group
zbMATH Keywords: controlled Whitehead torsion
zbMATH Keywords: Controlled finiteness obstruction
zbMATH Keywords: controlled boundary theorem
zbMATH Keywords: controlled s- cobordism theorem
reviewed by: Ivan Ivanšić
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/bfb0066066
OpenAlex ID: W4210308830
DOI: 10.1007/BFB0066066
description: scientific article; zbMATH DE number 3874224