\(\mathcal Z\)-compactifications of open manifolds
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.
scientific article; zbMATH DE number 1317258