Localization of basic characteristic classes
[Localisation de classes caractéristiques basiques]
Annales de l'Institut Fourier, Tome 64 (2014) no. 2, pp. 537-570.

Nous introduisons des classes et des nombres caractéristiques basiques d’un feuilletage riemannien. Si la variété riemannienne est complète, simplement connexe (ou plus généralement si le feuilletage est un feuilletage de Killing transversalement orientable) et si l’espace des ahérences des feuilles est compact, alors les nombres caractéristiques basiques sont déterminés par la dynamique infinitésimale du feuilletage en l’union des adhérences des feuilles fermées. En effet, ils peuvent être calculés avec un théorème de localisation de type Atiyah-Bott-Berline-Vergne pour la cohomologie équivariante basique.

We introduce basic characteristic classes and numbers as new invariants for Riemannian foliations. If the ambient Riemannian manifold M is complete, simply connected (or more generally if the foliation is a transversely orientable Killing foliation) and if the space of leaf closures is compact, then the basic characteristic numbers are determined by the infinitesimal dynamical behavior of the foliation at the union of its closed leaves. In fact, they can be computed with an Atiyah-Bott-Berline-Vergne-type localization theorem for equivariant basic cohomology.

DOI : 10.5802/aif.2857
Classification : 57R30, 53C12, 57R20
Keywords: Riemannian foliations, basic cohomology, equivariant cohomology, characteristic classes, localization
Mot clés : feuilletages riemanniens, cohomologie basique, cohomologie équivariante, classes caractéristiques, localisation
Töben, Dirk 1

1 Universidade Federal de São Carlos Departamento de Matemàtica Rod. Washington Luís, Km 235, C.P. 676 13565-905 São Carlos, SP (Brazil)
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Töben, Dirk. Localization of basic characteristic classes. Annales de l'Institut Fourier, Tome 64 (2014) no. 2, pp. 537-570. doi : 10.5802/aif.2857. http://www.numdam.org/articles/10.5802/aif.2857/

[1] Alekseev, A.; Meinrenken, E. Equivariant cohomology and the Maurer-Cartan equation, Duke Math. J., Volume 130 (2005) no. 3, pp. 479-521 | DOI | MR | Zbl

[2] Alexandrino, Marcos M.; Biliotti, Leonardo; Pedrosa, Renato H. L. Lectures on isometric actions, XV Escola de Geometria Diferencial. [XV School of Differential Geometry], Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 2008, pp. vi+224 | MR | Zbl

[3] Atiyah, M. F.; Bott, R. The moment map and equivariant cohomology, Topology, Volume 23 (1984) no. 1, pp. 1-28 | DOI | MR | Zbl

[4] Belfi, Victor; Park, Efton; Richardson, Ken A Hopf index theorem for foliations, Differential Geom. Appl., Volume 18 (2003) no. 3, pp. 319-341 | DOI | MR | Zbl

[5] Berline, Nicole; Vergne, Michèle Zéros d’un champ de vecteurs et classes caractéristiques équivariantes, Duke Math. J., Volume 50 (1983) no. 2, pp. 539-549 | DOI | MR | Zbl

[6] Bott, Raoul Vector fields and characteristic numbers, Michigan Math. J., Volume 14 (1967), pp. 231-244 | DOI | MR | Zbl

[7] Bott, Raoul; Tu, Loring W. Differential forms in algebraic topology, Graduate Texts in Mathematics, 82, Springer-Verlag, New York-Berlin, 1982, pp. xiv+331 | MR | Zbl

[8] do Carmo, Manfredo Perdigão Riemannian geometry, Mathematics: Theory & Applications, Birkhäuser Boston, Inc., Boston, MA, 1992, pp. xiv+300 (Translated from the second Portuguese edition by Francis Flaherty) | MR | Zbl

[9] El Kacimi-Alaoui, A.; Nicolau, M. On the topological invariance of the basic cohomology, Math. Ann., Volume 295 (1993) no. 4, pp. 627-634 | DOI | MR | Zbl

[10] Goertsches, O.; Töben, D. Equivariant basic cohomolog of Riemannian foliations (submitted, arXiv:1004.1043v1)

[11] Greub, Werner; Halperin, Stephen; Vanstone, Ray Connections, curvature, and cohomology. Vol. I: De Rham cohomology of manifolds and vector bundles, Academic Press, New York-London, 1972, pp. xix+443 (Pure and Applied Mathematics, Vol. 47) | MR | Zbl

[12] Guillemin, Victor W.; Sternberg, Shlomo Supersymmetry and equivariant de Rham theory, Mathematics Past and Present, Springer-Verlag, Berlin, 1999, pp. xxiv+228 | MR | Zbl

[13] Hurder, Steven; Töben, Dirk The equivariant LS-category of polar actions, Topology Appl., Volume 156 (2009) no. 3, pp. 500-514 | DOI | MR | Zbl

[14] Hurder, Steven; Töben, Dirk Transverse LS category for Riemannian foliations, Trans. Amer. Math. Soc., Volume 361 (2009) no. 11, pp. 5647-5680 | DOI | MR | Zbl

[15] Kamber, Franz W.; Tondeur, Philippe Foliated bundles and characteristic classes, Lecture Notes in Mathematics, Vol. 493, Springer-Verlag, Berlin-New York, 1975, pp. xiv+208 | MR | Zbl

[16] Lazarov, Connor; Pasternack, Joel Residues and characteristic classes for Riemannian foliations, J. Differential Geometry, Volume 11 (1976) no. 4, pp. 599-612 | MR | Zbl

[17] McCleary, John A user’s guide to spectral sequences, Cambridge Studies in Advanced Mathematics, 58, Cambridge University Press, Cambridge, 2001, pp. xvi+561 | MR | Zbl

[18] Molino, Pierre Riemannian foliations, Progress in Mathematics, 73, Birkhäuser Boston, Inc., Boston, MA, 1988, pp. xii+339 (Translated from the French by Grant Cairns, With appendices by Cairns, Y. Carrière, É. Ghys, E. Salem and V. Sergiescu) | MR | Zbl

[19] Mozgawa, Witold Feuilletages de Killing, Collect. Math., Volume 36 (1985) no. 3, pp. 285-290 | MR | Zbl

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

[21] Sakai, Takashi Riemannian geometry, Translations of Mathematical Monographs, 149, American Mathematical Society, Providence, RI, 1996, pp. xiv+358 (Translated from the 1992 Japanese original by the author) | MR | Zbl

[22] Sergiescu, Vlad Cohomologie basique et dualité des feuilletages riemanniens, Ann. Inst. Fourier (Grenoble), Volume 35 (1985) no. 3, pp. 137-158 | DOI | Numdam | MR | Zbl

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