Eta cocycles, relative pairings and the Godbillon-Vey index theorem

Abstract

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 the boundary-foliation; this 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 $0\to J\to A\to B\to 0$, with J dense and holomorphically closed in the C^*-algebra of the foliation and B depending only on boundary data. Of particular importance is the definition of a relative cyclic cocycle $(τ_{GV}^r,σ_{GV})$ for the pair $A\to B$; $τ_{GV}^r$ is a cyclic cochain on A defined through a regularization, à la Melrose, of the usual Godbillon-Vey cyclic cocycle $τ_{GV}$; $σ_{GV}$ is a cyclic cocycle on B, obtained through a suspension procedure involving $τ_{GV}$ and a specific 1-cyclic cocycle (Roe's 1-cocycle). We call $σ_{GV}$ the eta cocycle associated to $τ_{GV}$. The Atiyah-Patodi-Singer formula is obtained by defining a relative index class $\Ind (D,D^\partial)\in K_* (A,B)$ and establishing the equality <\Ind (D),[τ_{GV}]>=<\Ind (D,D^\partial), [τ^r_{GV}, σ_{GV}]>$. The Godbillon-Vey eta invariant $η_{GV}$ is obtained through the eta cocycle $σ_{GV}$.

86 pages. This is the complete article corresponding to the announcement "Eta cocycles" by the same authors (arXiv:0907.0173)