Eta cocycles

Abstract

We announce 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 the boundary foliation, 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 $0\to J \to A \to B \to 0$ with J dense and holomorphically closed in the C^*-algebra of the foliation and B depending only on boundary data.