Splitting manifold approximate fibrations
type: Item
instance of: scholarly article
title: Splitting manifold approximate fibrations
zbMATH Open document ID: 1013.57014
DOI: 10.1016/S0040-9383(01)00028-3
author: P. Kirby
author: John L. Bryant
published in: Topology
publication date: 2002-09-17T00:00:00Z
review text: Manifold approximate fibrations and their mapping cylinders play the role in the theory of topological manifolds that fiber bundles and tubular neighborhoods play in the theory of smooth manifolds. Manifold approximate fibrations are even more prominent in the study of generalized manifolds (ANR homology manifolds) and manifold (topologically) stratified spaces. This paper establishes important splitting properties of manifold approximate fibrations for generalized manifolds and manifold stratified spaces. However, the results are new and interesting for topological manifolds and their submanifolds. The results have already proved useful for transversality theorems for submanifolds of generalized manifolds [see \textit{J. Bryant}, Topology Appl. 94, 7-12 (1999; Zbl 0929.57012)]. For an example of the type of result proved here, let \(X\) be a closed generalized \(n\)-manifold, \(n\geq 5\), satisfying the disjoint disks property, and tamely embedded (codimension \(\geq 3\) and \(1\)-LCC) in a topological manifold \(M\). Suppose further that \(Y\) is a compact tame submanifold (topological or generalized) of \(X\). Then \(X\) has a mapping cylinder neighborhood in \(M\) such that the map \(p:W\to X\) giving the mapping cylinder is a manifold approximate fibration and \(p|_{p^{-1}(y)}:p^{-1}(Y)\to Y\) is also a manifold approximate fibration (that \(p\) is a manifold approximate fibration is trivial; the important new point is to be able to arrange for \(p\) to be a manifold approximate fibration over \(Y\)).
reviewed by: Bruce Hughes
Mathematics Subject Classification ID: 57N15
Mathematics Subject Classification ID: 57P99
Mathematics Subject Classification ID: 57N35
Mathematics Subject Classification ID: 57R99
Mathematics Subject Classification ID: 57N40
Mathematics Subject Classification ID: 57N80
Mathematics Subject Classification ID: 57N75
zbMATH DE Number: 1801891
zbMATH Keywords: generalized manifolds
zbMATH Keywords: embeddings
zbMATH Keywords: manifold approximate fibration
zbMATH Keywords: transversality
zbMATH Keywords: stratification
zbMATH Keywords: submanifold
zbMATH Keywords: mapping cylinder neighborhood
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1801891