\(\mathcal Z\)-compactifications of open manifolds
type: Item
instance of: scholarly article
title: \(\mathcal Z\)-compactifications of open manifolds
zbMATH Open document ID: 0951.57011
DOI: 10.1016/S0040-9383(98)00053-6
author: Fredric D. Ancel
author: Craig R. Guilbault
published in: Topology
publication date: 1999-08-16T00:00:00Z
review text: An open \(n\)-manifold \(M\) (\(n\geq 5\)) has a \(Z\)-compactification if there is a compact ANR \(\widehat M\) containing \(M\) as an open subset such that the \(Z\)-boundary \(\widehat M\smallsetminus M\) is a \(Z\)-set in \(\widehat M\). This latter condition means that there exist arbitrarily small deformations of \(\widehat M\) into \(M\). A simple example occurs when \(\widehat M\) is a compact manifold with manifold boundary \(\widehat M\smallsetminus M\), but, in general, neither \(\widehat M\) nor \(\widehat M\smallsetminus M\) need be a manifold. The quite interesting main results of this paper concern two open \(n\)-manifolds \(M\), \(N\) (\(n\geq 5\)) having \(Z\)-compactifications \(\widehat M\) and \(\widehat N\) with homeomorphic \(Z\)-boundaries. For such manifolds the authors prove the following three theorems. (1) The union \(\widehat M \cup_\partial \widehat N\) of \(\widehat M\) and \(\widehat N\) along their common \(Z\)-boundary is a manifold. The proof uses the Edwards-Quinn manifold recognition theorem. (2) If \(M\) and \(N\) are contractible, then \(\widehat M \cup_\partial \widehat N\) is a sphere. The proof of this result uses Newman's higher dimensional topological Poincaré conjecture. (3) If \(M\) and \(N\) are contractible, then \(M\) and \(N\) are homeomorphic. This result is established by using the fact that \(\widehat M\cup_\partial \widehat N\) is a sphere to construct a proper \(s\)-cobordism between \(M\) and \(N\) and then applying Siebenmann's proper \(s\)-cobordism theorem. The authors apply these results to the universal covers \(M\) and \(N\) of closed aspherical manifolds with isomorphic fundamental groups such that the group is word hyperbolic, CAT(0) or any other group leading to homeomorphic \(Z\)-boundaries. The conclusion is that \(M\) and \(N\) are homeomorphic. This elegant result is a weak version of the Borel conjecture which -- if true -- would say that the closed aspherical manifolds are themselves homeomorphic.
reviewed by: Bruce Hughes
Mathematics Subject Classification ID: 57N15
Mathematics Subject Classification ID: 54D35
Mathematics Subject Classification ID: 57N65
Mathematics Subject Classification ID: 54C55
zbMATH DE Number: 1317258
zbMATH Keywords: Z-compactification
zbMATH Keywords: aspherical manifold
zbMATH Keywords: Borel conjecture
zbMATH Keywords: CAT(0) group
zbMATH Keywords: word hyperbolic group
MaRDI profile type: Publication
full work available at URL: s0040-9383(98)00053-6
OpenAlex ID: W2072739610
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description: scientific article; zbMATH DE number 1317258