A note on the higher Atiyah-Patodi-Singer index theorem on Galois coverings

Let $Gamma$ be a finitely generated discrete group satisfying the rapid decay condition. We give a new proof of the higher Atiyah-Patodi-Singer theorem on a Galois $Gamma$-coverings, thus providing an explicit formula for the higher index associated to a group cocycle $cin Z^k (Gamma;mathbb{C})$ which is of polynomial growth with respect to a word-metric. Our new proof employs relative K-theory and relative cyclic cohomology in an essential way.

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 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 $0to Jto Ato Bto 0$, 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 $(tau_{GV}^r,sigma_{GV})$ for the pair $Ato B$; $tau_{GV}^r$ is a cyclic cochain on A defined through a regularization, `a la Melrose, of the usual Godbillon-Vey cyclic cocycle $tau_{GV}$; $sigma_{GV}$ is a cyclic cocycle on B, obtained through a suspension procedure involving $tau_{GV}$ and a specific 1-cyclic cocycle (Roes 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 $Ind (D,D^partial)in K_* (A,B)$ and establishing the equality <Ind (D),[tau_{GV}]>=<Ind (D,D^partial), [tau^r_{GV}, sigma_{GV}]>$. The Godbillon-Vey eta invariant $eta_{GV}$ is obtained through the eta cocycle $sigma_{GV}$.

Eta cocycles

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