Relative pairings and the Atiyah-Patodi-Singer index formula for the Godbillon-Vey cocycle

We describe a Godbillon-Vey index formula for longitudinal Dirac operators on a foliated bundle (X, F) with boundary; in particular, we define a Godbillon-Vey eta invariant on (partial derivative X, F(partial derivative)), that is; a secondary invariant for longitudinal Dirac operators on type III foliations. Our theorem generalizes the classic Atiyah-Patodi-Singer index formula for (X, F). 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 pairing of K-theory and cyclic cohomology for an exact sequence of Banach algebras, which in the present context takes the form 0 -> J -> U -> B -> 0 with J dense and holomorphically closed in C*(X, F) and B depending only on boundary data.