Non-unimodular transversely homogeneous foliations
Annales de l'Institut Fourier, Volume 71 (2021) no. 2, pp. 849-887.

We give sufficient conditions for the tautness of a transversely homogenous foliation defined on a compact manifold, by computing its base-like cohomology. As an application, we prove that if the foliation is non-unimodular then either the ambient manifold, the closure of the leaves or the total space of an associated principal bundle fiber over S 1 .

En calculant sa cohomologie basique, on donne des conditions suffisantes pour qu’un feuilletage transversalement homogène defini sur une variété compacte soit minimalisable. Comme application, on démontre que si le feuilletage est non unimodulaire alors soit la variété ambiante, soit l’adhérence des feuilles, soit un fibré principal associé au feuilletage, fibrent sur S 1 .

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Accepted:
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DOI: 10.5802/aif.3412
Classification: 57R30, 53C12
Keywords: Transversely homogeneous foliation, Lie foliation, base-like cohomology, unimodular foliation
Mot clés : Feuilletage transversalement homogène, feuilletage de Lie, cohomologie basique, feuilletage unimodulaire
Macías-Virgós, Enrique 1; Martín-Méndez, Pedro L. 1

1 Departamento de Matemáticas, Universidade de Santiago de Compostela (Spain)
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Macías-Virgós, Enrique; Martín-Méndez, Pedro L. Non-unimodular transversely homogeneous foliations. Annales de l'Institut Fourier, Volume 71 (2021) no. 2, pp. 849-887. doi : 10.5802/aif.3412. http://archive.numdam.org/articles/10.5802/aif.3412/

[1] Álvarez López, Jesús A.; Nozawa, Hiraku Secondary characteristic classes of transversely homogeneous foliations, CRM Preprint Series (2012) no. 1103, pp. 1-57

[2] Blumenthal, Robert A. Transversely Homogeneous Foliations, Ph. D. Thesis, Washington University in St. Louis (USA) (1978) | MR

[3] Blumenthal, Robert A. Transversely homogeneous foliations, Ann. Inst. Fourier, Volume 29 (1979) no. 4, pp. 143-158 | DOI | Numdam | MR | Zbl

[4] Blumenthal, Robert A. The base-like cohomology of a class of transversely homogeneous foliations, Bull. Sci. Math., Volume 104 (1980) no. 3, pp. 301-303 | MR | Zbl

[5] Carrière, Yves Flots riemanniens, Transversal structure of foliations (Toulouse, 1982) (Astérisque), Société Mathématique de France, 1984 no. 116, pp. 31-52 | Numdam | MR | Zbl

[6] Cheeger, Jeff; Ebin, David G. Comparison theorems in Riemannian geometry, North-Holland Mathematical Library, 9, North-Holland; Elsevier, 1975, viii+174 pages

[7] El Kacimi Alaoui, Aziz; Guasp, G.; Nicolau, Marcel On deformations of transversely homogeneous foliations, Topology, Volume 40 (2001) no. 6, pp. 1363-1393 | DOI | MR | Zbl

[8] El Kacimi Alaoui, Aziz; Nicolau, Marcel Structures géométriques invariantes et feuilletages de Lie, Indag. Math., New Ser., Volume 1 (1990) no. 3, pp. 323-333 | DOI | Zbl

[9] Faraut, Jacques Analysis on Lie groups. An introduction, Cambridge Studies in Advanced Mathematics, 110, Cambridge University Press, 2008, x+302 pages | DOI | MR

[10] Fedida, Edmond Feuilletages du plan, feuilletages de Lie (France), Ph. D. Thesis, Université Louis Pasteur (1973)

[11] Greub, Werner; Halperin, Stephen; Vanstone, Ray Connections, curvature, and cohomology. Vol. III: Cohomology of principal bundles and homogeneous spaces, Pure and Applied Mathematics, 47-III, Academic Press Inc., 1976, xxi+593 pages

[12] Hazewinkel, Mihil A duality theorem for cohomology of Lie algebras, Math. USSR, Sb., Volume 12 (1970), pp. 638-644 | DOI | Zbl

[13] Helgason, Sigurdur Differential geometry and symmetric spaces, Pure and Applied Mathematics, 12, Academic Press Inc., 1962, xiv+486 pages

[14] Hermann, Robert A sufficient condition that a mapping of Riemannian manifolds be a fibre bundle, Proc. Am. Math. Soc., Volume 11 (1960), pp. 236-242 | DOI | MR | Zbl

[15] Knapp, Anthony W. Lie groups, Lie algebras, and cohomology, Mathematical Notes, 34, Princeton University Press, 1988, xii+510 pages | MR

[16] Macias-Virgós, Enrique Homotopy groups in Lie foliations, Trans. Am. Math. Soc., Volume 344 (1994) no. 2, pp. 701-711 | DOI | MR | Zbl

[17] Macias-Virgós, Enrique; Martín-Méndez, Pedro Non-unimodular Lie foliations, C. R. Math. Acad. Sci. Paris, Volume 340 (2005) no. 5, pp. 359-362 | DOI | MR | Zbl

[18] Masa, Xosé Duality and minimality in Riemannian foliations, Comment. Math. Helv., Volume 67 (1992) no. 1, pp. 17-27 | DOI | MR | Zbl

[19] Molino, Pierre Riemannian foliations, Progress in Mathematics, 73, Birkhäuser, 1988, xii+339 pages (Translated from the French by Grant Cairns, With appendices by Cairns, Y. Carrière, É. Ghys, E. Salem and V. Sergiescu) | DOI

[20] Reinhart, Bruce L. Harmonic integrals on foliated manifolds, Am. J. Math., Volume 81 (1959), pp. 529-536 | DOI | MR | Zbl

[21] Royo Prieto, José I.; Saralegi-Aranguren, Martintxo; Wolak, Robert Cohomological tautness for Riemannian foliations, Russ. J. Math. Phys., Volume 16 (2009) no. 3, pp. 450-466 | DOI | MR | Zbl

[22] Scott, W. R. Group theory, Dover Publications, 1987, xiv+479 pages | MR

[23] Tischler, David On fibering certain foliated manifolds over S 1 , Topology, Volume 9 (1970), pp. 153-154 | DOI | MR | Zbl

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