On the linear independence of certain cohomology classes in the classifying space for subfoliations

Domínguez, Demetrio

doi:10.1090/S0002-9947-1992-1024769-0

On the linear independence of certain cohomology classes in the classifying space for subfoliations
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Trans. Amer. Math. Soc. 329 (1992), 221-232 Request permission

Abstract:

The purpose of this paper is to establish the linear independence of certain cohomology classes in the Haefliger classifying space $B{\Gamma _{({q_1},{q_2})}}$ for sub-foliations of codimension $({q_{1,}}{q_2})$. The classes considered are of secondary type, not belonging to the subalgebra of $H(B{\Gamma _{({q_1},{q_2})}},R)$ generated by the union of the universal characteristic classes for foliations of codimension ${q_1}$ and ${q_2}$ respectively, and are elements of the kernel of the canonical homomorphism $H(B{\Gamma _{({q_1},{q_2})}},R) \to H(B{\Gamma _{{q_1}}} \times B{\Gamma _d},R)$ with $d = {q_2} - {q_1} > 0$.

References

Chal Benson and David B. Ellis, Characteristic classes of transversely homogeneous foliations, Trans. Amer. Math. Soc. 289 (1985), no. 2, 849–859. MR 784016, DOI 10.1090/S0002-9947-1985-0784016-8
Raoul Bott, Lectures on characteristic classes and foliations, Lectures on algebraic and differential topology (Second Latin American School in Math., Mexico City, 1971) Lecture Notes in Math., Vol. 279, Springer, Berlin, 1972, pp. 1–94. Notes by Lawrence Conlon, with two appendices by J. Stasheff. MR 0362335

Characteristic homomorphism for

-foliated bundles over subfoliated manifolds

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Luis A. Cordero, Sheaves and cohomologies associated to subfoliations, Results Math. 8 (1985), no. 1, 9–20. MR 812243, DOI 10.1007/BF03322655
L. A. Cordero and J. Masa, Characteristic class of subfoliations, Ann. Inst. Fourier (Grenoble) 31 (1981), no. 2, v, 61–86. MR 617241
Demetrio Domínguez, Sur les classes caractéristiques des sous-feuilletages, Publ. Res. Inst. Math. Sci. 23 (1987), no. 5, 813–840 (French, with English summary). MR 934673, DOI 10.2977/prims/1195176034
Demetrio Domínguez, Classes caractéristiques non triviales de sous-feuilletages localement homogènes, Kodai Math. J. 11 (1988), no. 2, 177–204 (French). MR 949127, DOI 10.2996/kmj/1138038872
Demetrio Domínguez, Classes caractéristiques non triviales des feuilles de sous-feuilletages localement homogènes, Geom. Dedicata 28 (1988), no. 2, 229–249 (French, with English summary). MR 971628, DOI 10.1007/BF00147454
B. L. Feĭgin, Characteristic classes of flags of foliations, Funkcional. Anal. i PriloŽen. 9 (1975), no. 4, 49–56 (Russian). MR 0402771
Dimitri Fuchs, Non-trivialité des classes caractéristiques de ${\mathfrak {g}}$-structures. Application aux classes caractéristiques de feuilletages, C. R. Acad. Sci. Paris Sér. A-B 284 (1977), no. 17, A1017–A1019. MR 433469
Dimitri Fuchs, Non-trivialité des classes caractéristiques de ${\mathfrak {g}}$-structures. Applications aux variations des classes caractéristiques de feuilletages, C. R. Acad. Sci. Paris Sér. A-B 284 (1977), no. 18, A1105–A1107. MR 433470
André Haefliger, Homotopy and integrability, Manifolds–Amsterdam 1970 (Proc. Nuffic Summer School), Lecture Notes in Mathematics, Vol. 197, Springer, Berlin, 1971, pp. 133–163. MR 0285027
André Haefliger, Sur les classes caractéristiques des feuilletages, Séminaire Bourbaki, 24ème année (1971/1972), Exp. No. 412, Lecture Notes in Math., Vol. 317, Springer, Berlin, 1973, pp. 239–260 (French). MR 0420655
James L. Heitsch, Deformations of secondary characteristic classes, Topology 12 (1973), 381–388. MR 321106, DOI 10.1016/0040-9383(73)90030-X
James L. Heitsch, Independent variation of secondary classes, Ann. of Math. (2) 108 (1978), no. 3, 421–460. MR 512428, DOI 10.2307/1971183
Franz W. Kamber and Philippe Tondeur, Foliated bundles and characteristic classes, Lecture Notes in Mathematics, Vol. 493, Springer-Verlag, Berlin-New York, 1975. MR 0402773
Franz W. Kamber and Philippe Tondeur, On the linear independence of certain cohomology classes of $B\Gamma _{q}$, Studies in algebraic topology, Adv. in Math. Suppl. Stud., vol. 5, Academic Press, New York-London, 1979, pp. 213–263. MR 527251
Robert Moussu, Sur les classes exotiques des feuilletages, Géométrie différentielle (Colloq., Univ. Santiago de Compostela, Santiago de Compostela, 1972) Lecture Notes in Math., Vol. 392, Springer, Berlin, 1974, pp. 37–42 (French). MR 0362343
Robert A. Wolak, Characteristic classes of almost-flag structures, Geom. Dedicata 24 (1987), no. 2, 207–220. MR 908974, DOI 10.1007/BF00150936