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Zbl 0684.55008
Thiemann, Hanns
The relation between homotopy limits and Bauer's shape singular complex. (English)
[J] Cah. Topologie Géom. Différ. Catég. 30, No. 2, 157-165 (1989). ISSN 1245-530X

Every compact metric space Y is the limit of a decreasing sequence $\underline{Y}$ of compact ANR's. The homotopy limit holim $\underline{Y}$ is the limit of the sequence $\tilde{\underline Y}$ obtained from $\underline{Y}$ by replacing the bonding mappings by homotopy equivalent fibrations. On the other hand, {\it F. W. Bauer} [Pac. J. Math. 64, 25-65 (1976; Zbl 0346.55014)] has associated with Y its (strong) shape singular complex $\bar S(Y)$ and a natural mapping ${\bar \omega}\sb Y: \vert \bar S(Y)\vert \to Y$ [see also {\it A. Koyama}, Tsukuba J. Math. 8, 261-295 (1984; Zbl 0564.55007)]. Using the mapping ${\bar \omega}{}\sb Y$ and properties of holim $\underline{Y}$, the author defines a natural mapping $q\sb Y: \vert \bar S(Y)\vert \to ho\lim \underline Y$ and proves that it is a weak homotopy equivalence. Moreover, Y is strong shape equivalent to a CW-complex if and only if $q\sb Y$ is a homotopy equivalence.
[S.Mardešić]

MSC 2000:: *55P55 Shape theory
55P10 Homotopy equivalences

Keywords: compact metric space; homotopy limit; (strong) shape singular complex; weak homotopy equivalence; strong shape equivalent

Citations: Zbl 0346.55014; Zbl 0564.55007

Cited in: Zbl 0989.55008

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