2009/07/01 by Hitoshi Moriyoshi, Paolo Piazza, Moriyoshi, Hitoshi +1 · 1 citation
Mathematics · #19K56 #52J20 #58J28 #58J42 #Advanced Operator Algebra Research #Advanced Topics in Algebra #Differential Geometry (math.DG) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #K-Theory and Homology (math.KT) #math.DG #math.KT #msc:19K56 #msc:52J20 #msc:58J28 #msc:58J42
paper · pdf · doi:10.48550/arxiv.0907.0173
Abstract shortened. Sect. 5 modified. References added. Will appear with title "Relative pairings and the APS index formula for the Godbillon-Vey cocycle" in the Contemporary Mathematics volume "Non-commutative Geometry and Global Analysis. Proceedings of the conference in honor of Henri Moscovici". The corresponding complete paper, with proofs, has been posted on the arXiv on February 14 2011
openalex publication_date 2009/07/01 · arxiv created 2011/02/14 · arxiv updated 2011/02/15 · openalex created_date 2024/04/11 · openalex updated_date 2026/07/28
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→ J → A → B → 0 with J dense and holomorphically closed in the C^*-algebra of the foliation and B depending only on boundary data.