Stabilizing fake Hilbert spaces (Q1093211)

Roughly, a space X is a fake Hilbert space if it is not homeomorphic to the Hilbert space \(\ell^ 2\) but it has many properties of \(\ell^ 2\). \textit{Ph. Bowers} [Pac. J. Math., to appear] showed that many fake Hilbert spaces become homeomorphic to the Hilbert space when crossed with a specific dendrite A. An instructive construction of A is given in the paper under review. By using the techniques of homotopy theory, it is shown that there exists a homologically correct fake Hilbert space whose product with A (or even with \(A^ n\), \(n=2,3,...)\) is not homeomorphic to the Hilbert space. On the other hand, it is shown that there exists a homologically correct fake Hilbert space, with the discrete approximation property for n-cells, whose product with A is homeomorphic to the Hilbert space. These results may be compared with the results of \textit{R. J. Daverman} and \textit{J. J. Walsh} [Am. J. Math. 103, 411-435 (1981; Zbl 0538.57006)] in the compact case.