Dans cette note, on donne une version topologique des résultats obtenus par S. Kashiwabara (Tôhoku Math. J., 8 (1956), 13–28), (Tôhoku Math. J., 11 (1959), 327–350) et Ia. L. Sapiro (Izv. Bysh. Uceb. Zaved. Mat. no6, (1972), 78–85, Russian), (Izv. Bysh. Uceb. Zaved. Mat. no4, (1974), 104–113, Russian), à l’égard du théorème de décomposition de de Rham. On obtient ainsi la caractérisation d’une classe d’espaces topologiques ayant un espace de revêtement produit et on clarifie la structure géométrique de ces espaces. On caractérise aussi les morphismes de ces espaces et on donne quelques indications sur leur homotopie et homologie. Enfin, les résultats obtenus sont appliqués aux groupes topologiques et aux feuilletages différentiables. Dans ce dernier cas, on obtient une nouvelle manière pour traiter une classe de feuilletages étudiés par L. Conlon (Trans. Amer. Math. Soc., 194 (1974), 79–102) et une part d’un théorème de Cheeger-Gromoll-Lichnerowicz (J. of Diff. Geom., 6 (1971), 47–94).
In this note, a topological version of the results obtained, in connection with the de Rham reducibility theorem (Comment. Math. Helv., 26 ( 1952), 328–344), by S. Kashiwabara (Tôhoku Math. J., 8 (1956), 13–28), (Tôhoku Math. J., 11 (1959), 327–350) and Ia. L. Sapiro (Izv. Bysh. Uceb. Zaved. Mat. no6, (1972), 78–85, Russian), (Izv. Bysh. Uceb. Zaved. Mat. no4, (1974), 104–113, Russian) is given. Thus a characterization of a class of topological spaces covered by a product space is obtained and the geometric structure of these spaces is clarified. Also, the morphisms of such spaces are characterized and indications regarding the homotopy and homology of the space are given. Finally one applies the obtained results to topological groups and to differentiable foliations. In this last case an alternative treatment of a class of foliations studied by L. Conlon (Trans. Amer. Math. Soc., 194 (1974), 79–102) and a part of a Cheeger-Gromoll-Lichnerowicz theorem (J. of Diff. Geom., 6 (1971), 47–94) are obtained.
@article{AIF_1977__27_1_107_0,
author = {Vaisman, Izu},
title = {On some spaces which are covered by a product space},
journal = {Annales de l'Institut Fourier},
volume = {27},
year = {1977},
pages = {107-134},
doi = {10.5802/aif.644},
mrnumber = {55 \#11259},
zbl = {0336.55001},
language = {en},
url = {http://dml.mathdoc.fr/item/AIF_1977__27_1_107_0}
}
Vaisman, Izu. On some spaces which are covered by a product space. Annales de l'Institut Fourier, Tome 27 (1977) pp. 107-134. doi : 10.5802/aif.644. http://gdmltest.u-ga.fr/item/AIF_1977__27_1_107_0/
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