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.

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) |

##### Keywords:

maps close with respect to a cover; maps homotopic with respect to a cover; metric spaces; fine homotopy injection; hereditary shape equivalence
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\textit{S. M. Ageev} and \textit{D. Repovš}, Math. Notes 65, No. 6, 770--772 (1999; Zbl 0969.55002); translation from Mat. Zametki 65, No. 6, 921--924 (1999)

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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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