An obstruction to the resolution of homology manifolds
type: Item
instance of: scholarly article
title: An obstruction to the resolution of homology manifolds
zbMATH Open document ID: 0652.57011
author: Frank Quinn
published in: Michigan Mathematical Journal
publication date: 1987-01-01T00:00:00Z
review text: The author addresses an earlier result of his which asserted that, for \(n\geq 4\), every generalized n-manifold \((=ENR\) homology manifold) has a resolution (i.e., is the image of a cell-like mapping defined on a euclidean n-manifold). The focus is the point in the proof that asserted a certain integer obstruction vanished for ``dimension reasons''. It was pointed out by Sylvain Cappell that, in the generality needed, the obstruction may not vanish (an example is given that shows that the obstruction may not vanish in the situation that arises in the proof). The paper presents an analysis of the obstruction and reformulates the Resolution Theorem as follows: For every non-empty connected generalized n-manifold X, there is a ``local index'' \(i(X)\in (1+8{\mathbb{Z}})\) defined, which satisfies (a) if \(U\subset X\) is open, then \(i(U)=i(X)\), (b) \(i(\partial X)=i(X)\) if \(\partial X\neq \emptyset\), \(i(X\times Y)=i(X)i(Y)\), and (c) if dimX\(\geq 5\), or \(\dim X=4\) and \(\partial X\) is a manifold, then there is a resolution if and only if \(i(X)=1\). Whether there exists on ``exotic'' generalized manifold, that is, with index \(\neq 1\), remains unanswered.
Mathematics Subject Classification ID: 57P05
Mathematics Subject Classification ID: 57N15
zbMATH DE Number: 4064064
zbMATH Keywords: transversality
zbMATH Keywords: generalized n-manifold
zbMATH Keywords: ENR homology manifold
zbMATH Keywords: resolution
zbMATH Keywords: image of a cell-like mapping defined on a euclidean n- manifold
zbMATH Keywords: local index
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1029003559
OpenAlex ID: W2077443373
DOI: 10.1307/MMJ/1029003559
description: scientific article; zbMATH DE number 4064064