Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



The secondary characteristic classes of solvable foliations

Author: William T. Pelletier
Journal: Proc. Amer. Math. Soc. 88 (1983), 651-659
MSC: Primary 57R30; Secondary 53C12
MathSciNet review: 702294
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The secondary characteristic classes of a codimension $ q$ foliation of a manifold $ M$ are certain cohomology classes of $ M$ which are constructed from the curvature matrix of a torsion-free connection on the normal bundle of the foliation. We consider the foliation of a real or complex Lie group $ G$ by the left cosets of a closed connected subgroup $ H$ and any foliation obtained from it through dividing by a discrete subgroup $ \Gamma \subset G$. Such foliations are homogeneous. The Fuks-Pittie conjecture for homogeneous foliations is that the secondary classes are generated by $ \{ {h_1}{h_{{i_2}}} \cdots {h_{{i_\lambda }}}\tau _1^q\} $. We prove the Fuks-Pittie conjecture if $ H$ is reductive or solvable. We also prove an Addition Theorem for the classes which can be applied to reduce the general problem of calculating the secondary classes to the case in which $ G$ is semisimple.

References [Enhancements On Off] (What's this?)

  • [1] Daniel Baker, On a class of foliations and the evaluation of their characteristic classes, Comment. Math. Helv. 53 (1978), no. 3, 334–363. MR 0494144
  • [2] Raoul Bott, On the characteristic classes of groups of diffeomorphisms, Enseignment Math. (2) 23 (1977), no. 3-4, 209–220. MR 0488080
  • [3] D. B. Fuks, Finite-dimensional Lie algebras of formal vector fields and characteristic classes of homogeneous foliations, Izv. Akad. Nauk SSSR Ser. Mat. 40 (1976), no. 1, 57–64, 221 (Russian). MR 0413125
  • [4] Claude Godbillon, Cohomologies d’algèbres de Lie de champs de vecteurs formels, Séminaire Bourbaki, 25ème année (1972/1973), Exp. No. 421, Springer, Berlin, 1974, pp. 69–87. Lecture Notes in Math., Vol. 383 (French). MR 0418112
  • [5] Franz W. Kamber and Philippe Tondeur, Non-trivial characteristic invariants of homogeneous foliated bundles, Ann. Sci. École Norm. Sup. (4) 8 (1975), no. 4, 433–486. MR 0394700
  • [6] Harsh V. Pittie, The secondary characteristic classes of parabolic foliations, Comment. Math. Helv. 54 (1979), no. 4, 601–614. MR 552679, 10.1007/BF02566295

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 57R30, 53C12

Retrieve articles in all journals with MSC: 57R30, 53C12

Additional Information

Keywords: Foliation, homogeneous foliation, connection, curvature, Lie algebra, secondary characteristic class
Article copyright: © Copyright 1983 American Mathematical Society