Boundedly controlled topology. Foundations of algebraic topology and simple homotopy theory
type: Item
instance of: scholarly article
title: Boundedly controlled topology. Foundations of algebraic topology and simple homotopy theory
zbMATH Open document ID: 0646.57001
author: Douglas R. Anderson
author: Hans Jørgen Munkholm
published in: Lecture Notes in Mathematics
publication date: 1993-06-05T00:00:00Z
review text: The theory of spaces parametrized over a space with control is the result of a new direction of research in algebraic and geometric topology. One aspect of this theory, the topology of \(\epsilon\)-control, has been initiated by \textit{F. Quinn} [Ann. Math., II. Ser. 110, 275-331 (1979; Zbl 0413.57012); Invent. Math. 68, 353-424 (1982; Zbl 0533.57008)] and \textit{T. A. Chapman} [Controlled simple homotopy theory and applications (Lect. Notes Math. 1009) (1983; Zbl 0548.57001)]. The second aspect, that of boundedly controlled topology, forms the subject of the present work. A boundedness control structure on a space Z is defined as a pair (P,C), where P is a class of subspaces of Z and \(C: P\to P\) is an order preserving function, satisfying certain inclusion, bounding, and covering properties. The pair (P,C) defines a category \({\mathcal P}\), whose abstraction leads to a general concept of ``category with endomorphism''. A boundedly controlled space (bc space) over a space Z with boundedness control structure is a pair (X,p), where \(p: X\to Z\) is a continuous map. This definition generates a category \({\mathcal T}op^ c/Z\) of bc spaces over Z; similarly, for CW complexes and finite CW complexes, the categories \({\mathcal C}{\mathcal W}^ c/Z\) and \({\mathcal C}{\mathcal W}^ c_ f/Z\) are obtained. In Chapter I the authors develop the category theoretic tools needed for the study of the category \({\mathcal T}op^ c/Z\). The focus is on certain categories of fractions \({\mathcal B}(\Sigma^{-1})\), where \({\mathcal B}\) is a category with endomorphism and \(\Sigma\) is a naturally defined family of morphisms in \({\mathcal B}\). An especially important case is that when \({\mathcal B}\) is a category of functors. The motivation for this work lies in the fact that most algebraic topological invariants of the spaces in \(Top^ c/Z\) are constructed in these categories of fractions. Chapter II is devoted to the algebraic topology of bc spaces. First the idea of a…
reviewed by: J.Weinstein
Mathematics Subject Classification ID: 57-02
Mathematics Subject Classification ID: 57Q10
Mathematics Subject Classification ID: 57R80
Mathematics Subject Classification ID: 55N35
Mathematics Subject Classification ID: 55Q70
Mathematics Subject Classification ID: 18F25
Mathematics Subject Classification ID: 55N15
Mathematics Subject Classification ID: 18B40
Mathematics Subject Classification ID: 18D99
Mathematics Subject Classification ID: 18A25
zbMATH DE Number: 193080
zbMATH Keywords: K-theory
zbMATH Keywords: boundedly controlled topology
zbMATH Keywords: boundedness control structure
zbMATH Keywords: category with endomorphism
zbMATH Keywords: boundedly controlled space
zbMATH Keywords: categories of fractions
zbMATH Keywords: fragmented space
zbMATH Keywords: bc CW complex
zbMATH Keywords: fundamental groupoid
zbMATH Keywords: homology invariants
zbMATH Keywords: homotopy groups
zbMATH Keywords: Hurewicz theorems
zbMATH Keywords: cellular approximation theorem
zbMATH Keywords: simple homotopy theory
zbMATH Keywords: Whitehead groups
zbMATH Keywords: algebraic Whitehead group
zbMATH Keywords: Stiefel-Whitney classes
zbMATH Keywords: Whitehead torsion
zbMATH Keywords: s- cobordism theorem
zbMATH Keywords: bc PL h-cobordism
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 193080