THE GODBILLON-VEY CYCLIC COCYCLE AND LONGITUDINAL DIRAC OPERATORS

The goal of this paper is to prove the index theorem for the pairing o f the Godbillon-Vey cyclic cocycle with the index class of the longitu dinal Dirac operator for a codimension one foliation. Let (X,F) be a f oliated S-1-bundle over an arbitrary spin manifold M. The Dirac operat or on M lifts to a longitudinal elliptic operator D, the longitudinal Dirac operator, on (X, F). The index class of D is an element of the K -0-group of the foliation C-algebra C*(X, F). A densely defined cycli c even-cocycle on C(X, F), the Godbillon-Vey cyclic cocycle, is const ructed. The main result gives a topological formula for the pairing of the Godbillon-Vey cyclic cocycle with the index class of D. The proof of the main theorem uses a new technique, the pairing with the graph projections.