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Fine homotopy equivalence and injection. (English. Russian original) Zbl 0969.55002
Math. Notes 65, No. 6, 770-772 (1999); translation from Mat. Zametki 65, No. 6, 921-924 (1999).
From the text: The authors treat the problem of the coincidence between the classes of fine homotopy injections and equivalences of metric spaces, which was stated by F. D. Ancel in [Trans. Am. Math. Soc. 287, 1-40 (1985; Zbl 0507.54017)]. They prove:
Theorem 2. Let \(f: X\to Y\) be a map of metric spaces. Suppose that, for any cover \(\sigma\in \text{cov }Y\), there exists a map \(g: Y\to X\) such that (a) \(\text{Id}_Y\) is \(\sigma\)-close to the composition \(Y@> g>>X@> f>> Y\); (b) \(\text{Id}_X\) is \(f^{-1}(\sigma)\)-homotopic to the composition \(X@> f>> Y@> g>> X\); (c) \(f\simeq f\circ g\circ f[\text{rel }\sigma]\). Then the map \(f\) is a fine homotopy equivalence provided that \(X\) is complete with respect to some metric \(d\).
Corollary 1. Suppose that a map \(f: X\to Y\) of metric spaces is a fine homotopy injection and \(X\) is complete. Then \(f\) is a fine homotopy equivalence.
Corollary 2. Suppose that a map \(f: X\to Y\) of metric spaces is a hereditary shape equivalence and \(X\) is an ANE-space. Then \(f\) is a fine homotopy equivalence and \(Y\) is an ANE space.
MSC:
55P10 Homotopy equivalences in algebraic topology
54C99 Maps and general types of topological spaces defined by maps
55M10 Dimension theory in algebraic topology
54F45 Dimension theory in general topology
54E35 Metric spaces, metrizability
54C55 Absolute neighborhood extensor, absolute extensor, absolute neighborhood retract (ANR), absolute retract spaces (general properties)
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References:
[1] F. D. Ancel,Trans. Amer. Math. Soc.,287, 1–40 (1985). · doi:10.1090/S0002-9947-1985-0766204-X
[2] G. Kozlowski, ”Images of ANR’s”Trans. Amer. Math. Soc. (to appear).
[3] A. N. Dranishnikov,Uspekhi Mat. Nauk [Russian Math. Surveys],43, No. 5, 11–55 (1988).
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