Noncollarable ends of 4-manifolds: Some realization theorems (Q1325802)

In his thesis \textit{Siebenmann} has classified the collarable ends of noncompact \(n\)-manifolds, \(n \geq 6\), and \textit{F. Quinn} [J. Differ. Geom. 17, 503-521 (1982; Zbl 0533.57009)] extended these results to dimension 5 provided the fundamental group is a Freedman group. However, \textit{S. Kwasik} and \textit{R. Schultz} [Topology 27, No. 4, 443-457 (1988; Zbl 0664.57018)] gave examples of the form \(S^3 \times \mathbb{R}/G\) showing Siebenmann's theorem fails in dimension 4. In this paper the author shows that some of these exotic ends actually arise naturally as subsets of closed 4-manifolds. If \(E\) is a four- dimensional weak collar with \(\pi_1 (E) \simeq \mathbb{Z}_n\), and \(\partial E\) is \(\mathbb{Z}\)-homology equivalent to \(L(n,1)\), then there is a closed 4-manifold \(Y\) and a compactum \(\Sigma \subset Y\) such that \(\Sigma\) has the shape of \(S^2\) and \(\Sigma\) has a neighborhood \(N\) with \(N \smallsetminus \Sigma\) homeomorphic to \(E\). \(Y\) may be chosen to be \(S^2 \times S^2\) if \(n\) is even and \(\mathbb{C} \mathbb{P}^2 \# (- \mathbb{C} \mathbb{P}^2)\) if \(n\) is odd. Another class of Kwasik-Schultz counterexamples containing ends with fundamental group isomorphic to the Poincaré dodecahedral group is shown to be realized as complements of cell-like subsets of \(S^4\).