ETA COCYCLES, RELATIVE PAIRINGS AND THE GODBILLON-VEY INDEX THEOREM

We prove a Godbillon-Vey index formula for longitudinal Dirac operators on a foliated bundle with boundary ; in particular, we define a Godbillon-Vey eta invariant on that is, a secondary invariant for longitudinal Dirac operators on type III foliations. Moreover, employing the Godbillon-Vey index as a pivotal example, we explain a new approach to higher index theory on geometric structures with boundary. This is heavily based on the interplay between the absolute and relative pairings of K-theory and cyclic cohomology for an exact sequence of Banach algebras, which in the present context takes the form with dense and holomorphically closed in and depending only on boundary data. Of particular importance is the definition of a relative cyclic cocycle for the pair ; is a cyclic cochain on defined through a regularization A la Melrose of the usual Godbillon-Vey cyclic cocycle tau (GV) ; sigma (GV) is a cyclic cocycle on , obtained through a suspension procedure involving tau (GV) and a specific 1-cyclic cocycle (Roe's 1-cocycle). We call sigma (GV) the eta cocycle associated to tau (GV) . The Atiyah-Patodi-Singer formula is obtained by defining a relative index class and establishing the equality . The Godbillon-Vey eta invariant eta (GV) is obtained through the eta cocycle sigma (GV) .