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Compact open topology and CW homotopy type. (English) Zbl 1028.55010

Pour des CW-complexes \(X\) et \(Y\), soit \(Y^X\) l’espace des fonctions continues de \(X\) dans \(Y\) avec la topologie compacte-ouverte. L’auteur s’intéresse ici au problème de savoir quand \(Y^X\) a le type d’homotopie d’un CW-complexe. Le résultat principal de cet article est le suivant.
Soient \(X\) et \(Y\) des CW-complexes connexes et \(g: X\to Y\) une fonction continue. Supposons qu’il existe un sous-complexe fini \(T\) de \(X\) contenant le \(1\)-squelette de \(X\) et tel que tout sous-complexe \(L\) de \(X\) contenant \(T\) soit contenu dans un sous-complexe \(L'\) tel que \(L'/L\) soit fini et que \(H^i(L', T;(g|L')^*\pi_j(Y))= 0\) pour tout \(j\geq 2\) et tout \(i\leq j\) (cohomologie à coefficients locaux). Alors la composante connexe par arcs de \(Y^X\) contenant \(g\) est ouverte et fermée dans \(Y^X\) et a le type d’homotopie d’un CW-complexe.

MSC:

55P99 Homotopy theory
54C35 Function spaces in general topology
55P05 Homotopy extension properties, cofibrations in algebraic topology
54G20 Counterexamples in general topology
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