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

DOI:
https://doi.org/10.1090/S0002-9939-1983-0702294-1

MathSciNet review:
702294

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The secondary characteristic classes of a codimension foliation of a manifold are certain cohomology classes of 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 by the left cosets of a closed connected subgroup and any foliation obtained from it through dividing by a discrete subgroup . Such foliations are *homogeneous*. The Fuks-Pittie conjecture for homogeneous foliations is that the secondary classes are generated by . We prove the Fuks-Pittie conjecture if 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 is semisimple.

**[1]**D. Baker,*On a class of foliations and evaluation of their characteristic classes*, Comment. Math. Helv.**53**(1978). MR**0494144 (58:13073)****[2]**R. Bott,*On the characteristic classes of groups of diffeomorphisms*, Enseign. Math. (2)**23**(1977). MR**0488080 (58:7651)****[3]**D. Fuks,*Finite dimensional Lie algebras of formal vector fields and secondary classes of foliations*, Uspehi Mat. Nauk**40**(1) (1976). MR**0413125 (54:1246)****[4]**C. Godbillon,*Cohomologies d'algèbres de Lie de champs de vecteurs formels*, Séminaire Bourbaki (1972-73), Exposé 421, Lecture Notes in Math., vol. 383, Springer-Verlag, New York, 1974. MR**0418112 (54:6156)****[5]**F. Kamber and P. Tondeur,*Non-trivial characteristic invariants of homogeneous foliated bundles*, Ann. Sci. École Norm. Sup. (4)**8**(1975), 433-486. MR**0394700 (52:15499)****[6]**H. Pittie,*The secondary characteristic classes of parabolic foliations*, Comment. Math. Helv.**54**(1979), 601-614. MR**552679 (82h:57022)**

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

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

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1983-0702294-1

Keywords:
Foliation,
homogeneous foliation,
connection,
curvature,
Lie algebra,
secondary characteristic class

Article copyright:
© Copyright 1983
American Mathematical Society