The secondary characteristic classes of solvable foliations

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.