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 $0\to J\to A\to B\to 0$, 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 $(\tau_{GV}^r,\sigma_{GV})$ for the pair $A\to B$; $\tau_{GV}^r$ 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 (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 \Ind (D,D^\partial)\in K_* (A,B) and establishing the equality =<\Ind (D,D^\partial), [\tau^r_{GV}, \sigma_{GV}]>$. The Godbillon-Vey eta invariant$\eta_{GV}$is obtained through the eta cocycle$\sigma_{GV}$.Comment: 86 pages. This is the complete article corresponding to the announcement "Eta cocycles" by the same authors (arXiv:0907.0173

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 $c\in Z^k (\Gamma;\mathbb{C})$ 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

Synthesis, Crystal Structure, and Physical Properties of New Layered Oxychalcogenide La2O2Bi3AgS6

We have synthesized a new layered oxychalcogenide La2O2Bi3AgS6. From synchrotron X-ray diffraction and Rietveld refinement, the crystal structure of La2O2Bi3AgS6 was refined using a model of the P4/nmm space group with a = 4.0644(1) {\AA} and c = 19.412(1) {\AA}, which is similar to the related compound LaOBiPbS3, while the interlayer bonds (M2-S1 bonds) are apparently shorter in La2O2Bi3AgS6. The tunneling electron microscopy (TEM) image confirmed the lattice constant derived from Rietveld refinement (c ~ 20 {\AA}). The electrical resistivity and Seebeck coefficient suggested that the electronic states of La2O2Bi3AgS6 are more metallic than those of LaOBiS2 and LaOBiPbS3. The insertion of a rock-salt-type chalcogenide into the van der Waals gap of BiS2-based layered compounds, such as LaOBiS2, will be a useful strategy for designing new layered functional materials in the layered chalcogenide family.Comment: 11 pages, 1 table, 5 figure