Stabilizing fake Hilbert spaces
type: Item
instance of: scholarly article
title: Stabilizing fake Hilbert spaces
zbMATH Open document ID: 0628.57013
DOI: 10.1016/0166-8641(87)90050-2
author: Mladen Bestvina
published in: Topology and its Applications
publication date: 1987-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 57N20
Mathematics Subject Classification ID: 54F50
zbMATH DE Number: 4022172
zbMATH Keywords: Z-set
zbMATH Keywords: product with a dendrite
zbMATH Keywords: fake Hilbert space
zbMATH Keywords: discrete approximation property for n-cells
reviewed by: Sukhjit Singh
MaRDI profile type: Publication
full work available at URL: 0166-8641(87)90050-2
OpenAlex ID: W1997083412
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description: scientific article; zbMATH DE number 4022172