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Register a new userEta cocycles, relative pairings and the Godbillon-Vey index theorem
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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 $sigma_{GV}$.
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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.
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.
Real index pairings of projections and unitaries on a separable Hilbert space with a real structure are defined when the projections and unitaries fulfill symmetry relations invoking the real structure, namely projections can be real, quaternionic, even or odd Lagrangian and unitaries can be real, quaternionic, symmetric or anti-symmetric. There are $64$ such real index pairings of real $K$-theory with real $K$-homology. For $16$ of them, the Noether index of the pairing vanishes, but there is a secondary $mathbb{Z}_2$-valued invariant. The first set of results provides index formulas expressing each of these $16$ $mathbb{Z}_2$-valued pairings as either an orientation flow or a half-spectral flow. The second and main set of results constructs the skew localizer for a pairing stemming from a Fredholm module and shows that the $mathbb{Z}_2$-invariant can be computed as the sign of its Pfaffian and in $8$ of the cases as the sign of the determinant of its off-diagonal entry. This is of relevance for the numerical computation of invariants of topological insulators.
The use of bundle gerbes and bundle gerbe modules is considered as a replacement for the usual theory of Clifford modules on manifolds that fail to be spin. It is shown that both sides of the Atiyah-Singer index formula for coupled Dirac operators can be given natural interpretations using this language and that the resulting formula is still an identity.
We show the Godbillon-Vey invariant arises as a `restricted Casimir invariant for three-dimensional ideal fluids associated to a foliation. We compare to a finite-dimensional system, the rattleback, where analogous phenomena occur.
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