Independent Variation of Secondary Classes

Abstract

teristic classes of a certain class of foliations. The examples we compute show that many of these secondary classes vary linearly independently. This generalizes a result due to Thurston [T] on the variation of one of these classes, the Godbillon-Vey class. It is also a partial analogue of the results of Bott [B2] and Lazarov-Pasternack [LP] on the independent variation of the secondary classes for holomorphic and Riemannian foliations respectively. We are also able to compute the Simons' characters of these foliations and we show that many of them vary linearly independently. The method we use is a generalization of the theory of residues of singular foliations due to Baum-Bott [BB]. We work with the natural foliation z on a flat vector bundle and a vector field X tangent to the fiber of the bundle which preserves the foliation. We assume the vector field has an isolated singularity along the zero section. This situation determines certain cohomology classes on the zero section, the residues of z- and X. We relate these residues to the secondary characteristic classes and Simons' characters of the foliation ' spanned by z and X off the zero section. Using this technique we compute examples which show that the residues of Z and X and thus the characteristic classes of Z vary linearly independently. In Section 2 we record some facts we will use later. In Section 3 we