Flat bundles and residues forfoliations

Abstract

In this note we compute the residues for a vector field preserving a foliation. We assume the vector field disappears only to first order along its singular set and that its singular set consists of closed separated leaves. Under these conditions, the residue is completely determined by the characteristic classes of the flat bundle v o, which is the normal bundle v of the foliation restricted to the singular set, and the action of the vector field on v o. For a foliation of even codimension q whose normal bundle has Euler class zero, we interpret the Pontrjagin ring of the normal bundle in dimension 2q as an obstruction to the existence of such a vector field. We show that if the Euler class of the normal bundle of a foliation is zero and it admits such a vector field, then many of its secondary characteristic classes must be also be zero. This is one of the few results relating the geometry of a foliation to its characteristic classes. We will make extensive use of the material in [6]. It will be very helpful to the reader to be familiar with this paper.