An end theorem for stratified spaces
type: Item
instance of: scholarly article
title: An end theorem for stratified spaces
zbMATH Open document ID: 0930.57025
author: Frank Connolly
author: Bogdan Vajiac
published in: Inventiones Mathematicae
publication date: 2000-02-07T00:00:00Z
review text: A fundamental theorem of geometric topology characterizes those topological manifolds which are the interiors of compact manifolds with boundary. The general case of this theorem was proved by \textit{L. C. Siebenmann} (1965) in his Ph.D. Thesis and it says that a tame ended topological \(n\)-manifold \(X\) (\(n \geq 6\)) is the interior of a compact manifold with boundary if and only if the Siebenmann obstruction \(\sigma(X) \in \widetilde {K}_{0}({\mathbb Z}\pi)\) vanishes, where \(\pi\) is the fundamental group of the end of \(X\). \textit{F. Quinn} [Ann. Math., II. Ser. 110, 275-331 (1979; Zbl 0394.57022); Invent. Math. 68, 353-424 (1982; Zbl 0533.57008)] generalizes Siebenmann's result greatly using his controlled \(K\)-theory. For any locally compact pair \((X,A)\), where \(A\) is a closed and tame ANR in \(X\) and \(X-A\) is a topological \(n\)-manifold (\(n \geq 6\)), he defines a \(K\)-theoretic obstruction, which vanishes if and only if \(A\) has a mapping cylinder neighborhood in \(X\). In the paper under review, the authors show, using localized control methods of Quinn, that the Siebenmann theorem generalizes to the context of manifold stratified spaces in the sense of \textit{F. Quinn} [J. Am. Math. Soc. 1, No. 2, 441-499 (1988; Zbl 0655.57010)]. Roughly, a manifold stratified space is a locally compact Hausdorff space \(X\) together with a finite filtration by closed \(m\)-skeleta \(X^m\), \(X = X^n \supset X^{n-1} \supset \ldots \supset X^{-1} = \emptyset\), such that each stratum \(X_m = X^m - X^{m-1}\) is a separable \(m\)-manifold with neighborhoods in \(X_m \cup X_k\) (for \(k>m\)) which have the local homotopy properties of mapping cylinders of fibrations. \(X\) is tame ended if the one point compactification of \(X\) is again a manifold stratified space. The boundaries of all strata form the boundary \(\partial X\) of \(X\). For a manifold stratified space \(X\) with empty boundary, a completion of \(X\) is a compact stratified space \(\overlin…
Mathematics Subject Classification ID: 57N65
Mathematics Subject Classification ID: 57N80
Mathematics Subject Classification ID: 57N15
Mathematics Subject Classification ID: 57N45
Mathematics Subject Classification ID: 57N75
Mathematics Subject Classification ID: 54E20
Mathematics Subject Classification ID: 55N15
Mathematics Subject Classification ID: 55S35
Mathematics Subject Classification ID: 55M35
zbMATH DE Number: 1337382
zbMATH Keywords: filtered space
zbMATH Keywords: stratification
zbMATH Keywords: homotopically stratified set
zbMATH Keywords: manifold stratified space
zbMATH Keywords: stratum
zbMATH Keywords: homotopy link
zbMATH Keywords: space at infinity
zbMATH Keywords: tame subspace
zbMATH Keywords: tame ended space
zbMATH Keywords: completion
zbMATH Keywords: exhaustion
zbMATH Keywords: transversality
zbMATH Keywords: mapping cylinder neighborhood
zbMATH Keywords: Kirby-Siebenmann obstruction
zbMATH Keywords: Quinn obstruction
zbMATH Keywords: end obstruction
zbMATH Keywords: fundamental group at infinity
zbMATH Keywords: controlled topology
zbMATH Keywords: controlled \(K\)-theory
zbMATH Keywords: orbit space of a group action
reviewed by: Zbigniew Karno
MaRDI profile type: Publication
full work available at URL: s002220050294
OpenAlex ID: W2033354407
DOI: 10.1007/S002220050294
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description: scientific article; zbMATH DE number 1337382