Characteristic homomorphism for (F 1 ,F 2 )-foliated bundles over subfoliated manifolds
Carballés, José Manuel
Annales de l'Institut Fourier, Tome 34 (1984), p. 219-245 / Harvested from Numdam
Access to full text
Full (PDF)
Full (DJVU)

Dans ce travail on donne une construction des classes caractéristiques pour un sous-feuilletage (F 1 ,F 2 ), en suivant les méthodes de Kamber et Tondeur. Pour cela, on introduit la notion de fibré principal (F 1 ,F 2 )-feuilleté, et on définit un homomorphisme caractéristique qui lui est associé. On étudie la relation avec les homomorphismes caractéristiques des fibrés F i -feuilletés, i=1,2, et on calcule l’algèbre des classes caractéristiques en utilisant les résultats de Kamber et Tondeur sur la cohomologie de g-DG-algèbres. Finalement, on généralise les résultats de Goldman sur la restriction à une feuille d’un fibré feuilleté, et on définit l’homomorphisme d’holonomie d’une feuille d’un sous-feuilletage.

In this paper a construction of characteristic classes for a subfoliation (F 1 ,F 2 ) is given by using Kamber-Tondeur’s techniques. For this purpose, the notion of (F 1 ,F 2 )-foliated principal bundle, and the definition of its associated characteristic homomorphism, are introduced. The relation with the characteristic homomorphism of F i -foliated bundles, i=1,2, the results of Kamber-Tondeur on the cohomology of g-DG-algebras. Finally, Goldman’s results on the restriction of foliated bundles to the leaves of a foliation are generalized, and the holonomy homomorphism of a leaf of a subfoliation is defined.

@article{AIF_1984__34_3_219_0,
     author = {Carball\'es, Jos\'e Manuel},
     title = {Characteristic homomorphism for $(F\_1,F\_2)$-foliated bundles over subfoliated manifolds},
     journal = {Annales de l'Institut Fourier},
     volume = {34},
     year = {1984},
     pages = {219-245},
     doi = {10.5802/aif.984},
     mrnumber = {86c:57024},
     zbl = {0519.57022},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_1984__34_3_219_0}
}

Carballés, José Manuel. Characteristic homomorphism for $(F_1,F_2)$-foliated bundles over subfoliated manifolds. Annales de l'Institut Fourier, Tome 34 (1984) pp. 219-245. doi : 10.5802/aif.984. http://gdmltest.u-ga.fr/item/AIF_1984__34_3_219_0/

[1] G. Andrzejczak, Some characteristic invariants of foliated bundles, Institute of Mathematics, Polish Academy of Sciences, Preprint 182, Warszawa, 1979.

[2] R. Bott, Lectures on characteristic classes and foliations, Lecture Notes in Math., Vol. 279, Springer, Berlin, 1972. | MR 50 #14777 | Zbl 0241.57010

[3] L.A. Cordero and P.M. Gadea, Exotic characteristic classes and subfoliations, Ann. Inst. Fourier, Grenoble, 26-1 (1976), 225-237 ; errata, ibid. 27, fasc. 4 (1977). | Numdam | MR 53 #6584 | Zbl 0313.57010

[4] L.A. Cordero and X. Masa, Characteristic classes of subfoliations, Ann. Inst. Fourier, Grenoble, 31-2 (1981), 61-86. | Numdam | MR 83a:57033 | Zbl 0442.57009

[5] B.L. Feigin, Characteristic classes of flags of foliations, Funct. Anal. and its Appl., 9 (1975), 312-317. | MR 53 #6585 | Zbl 0328.57008

[6] R. Goldman, The holonomy ring of the leaves of foliated manifolds, J. Differential Geometry, 11 (1976), 411-449. | MR 56 #3852 | Zbl 0356.57016

[7] F.W. Kamber and Ph. Tondeur, Foliated bundles and characteristic classes, Lecture Notes in Math., Vol 493, Springer, Berlin, 1975. | MR 53 #6587 | Zbl 0308.57011

[8] X. Masa, Characteristic classes of subfoliations II, preprint.

[9] R. Moussu, Sur les classes exotiques des feuilletages, Lecture Notes in Math., Vol. 392, Springer, Berlin, 1974, 37-42. | MR 50 #14785 | Zbl 0292.57021

[10] B.L. Reinhart, Holonomy invariants for framed foliations, Lecture Notes in Math., Vol. 392, Springer, Berlin, 1974, 47-52. | MR 51 #1842 | Zbl 0291.57017