Delooping controlled pseudo-isotopies of Hilbert cube manifolds
type: Item
instance of: scholarly article
title: Delooping controlled pseudo-isotopies of Hilbert cube manifolds
zbMATH Open document ID: 0622.57012
DOI: 10.1016/0166-8641(87)90067-8
author: C. Bruce Hughes
published in: Topology and its Applications
publication date: 1987-01-01T00:00:00Z
review text: Let \(p: E\to B\) be a Hurewicz fibration, E a compact Hilbert cube manifold, and B a compact polyhedron (or a compact TOP manifold with a handle decomposition). The author defines the controlled Whitehead [respectively, pseudo-isotopy] ''space'' Wh(p) [respectively, \({\mathcal P}(p)]\). These ''spaces'' are semisimplicial complexes. It is proved that \(\Omega\) (Wh(p))\(\simeq {\mathcal P}(p)\), and that \(\pi_ 0(Wh(p))\) is the appropriate obstruction group for deciding whether a ''controlled'' homotopy equivalence \(f: M\to E\) between compact Hilbert cube manifolds is \(p^{-1}(\epsilon)\) homotopic to a homeomorphism for every \(\epsilon >0\). Here ''control'' means arbitrarily small \(\epsilon\) control in B.
Mathematics Subject Classification ID: 57N20
Mathematics Subject Classification ID: 57N37
Mathematics Subject Classification ID: 57Q10
zbMATH DE Number: 4009339
zbMATH Keywords: controlled homotopy equivalence
zbMATH Keywords: controlled simple homotopy theory
zbMATH Keywords: controlled Whitehead space
zbMATH Keywords: controlled pseudo-isotopy space
zbMATH Keywords: delooping
zbMATH Keywords: Hurewicz fibration
zbMATH Keywords: compact Hilbert cube manifold
zbMATH Keywords: manifold with a handle decomposition
reviewed by: Hans Jørgen Munkholm
MaRDI profile type: Publication
full work available at URL: 0166-8641(87)90067-8
OpenAlex ID: W1970118475
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description: scientific article; zbMATH DE number 4009339