A non-\(\mathcal Z\)-compactifiable polyhedron whose product with the Hilbert cube is \(\mathcal Z\)-compactifiable (Q2717754)

A \(Z\)-compactification of a space \(X\) is a compact metric space \(\widetilde X\) containing \(X\) for which there is a homotopy \(h_t:X\to X\) (\(0\leq t\leq 1\)) such that \(h_0\) is the identity and \(h_t(\widetilde X)\subseteq X\) for all \(t > 0\). Such compactifications first appeared in Hilbert cube manifold theory [\textit{T. A. Chapman} and \textit{L. C. Siebenmann}, Acta Math. 137, No. 3-4, 171-208 (1976; Zbl 0361.57008)], but, more recently, they have been used in geometric group theory [\textit{M. Bestvina}, Mich. Math. J. 43, No. 1, 123-139 (1996; Zbl 0872.57005)]. The author gives an example of a locally compact polyhedron \(X\) that does not have a \textit{Z}-compactification, but the product of \(X\) and the Hilbert cube does have a \textit{Z}-compactification. This answers a question of Chapman and Siebenmann [op. cit.]. The example is \(2\)-dimensional and has a simple description: it is the infinite mapping telescope of a direct sequence \(S^1@>\theta>> S^1@>\theta>> S^1 @>\theta>> \cdots\) where \(\theta\) is a degree \(1\) map that wraps the circle around itself twice counterclockwise, then once back in clockwise dircetion. The proof that the example has the desired properties is quite complicated. The paper contains several open questions, one of which was recently answered by \textit{S. C. Ferry} [Mich. Math. J. 47, No. 2, 287-294 (2000; Zbl 0988.57013), see below] by proving that if \(X\) is a locally finite \(n\)-dimensional polyhedron and the product of \(X\) and the Hilbert cube has a \textit{Z}-compactification, then \(X\times I^{2n+5}\) also has a \textit{Z}-compactification.