An obstruction to the resolution of homology manifolds
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.
image of a cell-like mapping defined on a euclidean n- manifold
scientific article; zbMATH DE number 4064064