The Godbillon-Vey cyclic cocycle and longitudinal Dirac operators

Hitoshi Moriyoshi and Toshikazu Natsume

Vol. 172 (1996), No. 2, 483–539

DOI: 10.2140/pjm.1996.172.483

Abstract

The goal of this paper is to prove the index theorem for the pairing of the Godbillon-Vey cyclic cocycle with the index class of the longitudinal Dirac operator for a codimension one foliation. Let (X,ℱ) be a foliated S¹-bundle over an arbitrary spin manifold M. The Dirac operator 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 K₀-group of the foliation C^∗-algebra C^∗(X,ℱ). A densely defined cyclic even-cocycle on C^∗(X,ℱ), the Godbillon-Vey cyclic cocycle, is constructed. 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.

Mathematical Subject Classification 2000

Primary: 58G12

Secondary: 46L80, 57R30

Milestones

Received: 16 February 1993

Revised: 22 March 1995

Published: 1 February 1996