2011/02/14 by Hitoshi Moriyoshi, Paolo Piazza, Moriyoshi, Hitoshi +1
Mathematics · #19K56 #58J42 #Advanced Operator Algebra Research #Advanced Topics in Algebra #Differential Geometry (math.DG) #FOS: Mathematics #Holomorphic and Operator Theory #K-Theory and Homology (math.KT) #Primary: 58J20. Secondary: 58J22 #math.DG #math.KT #msc:19K56 #msc:58J20. #msc:58J22 #msc:58J42
paper · pdf · doi:10.48550/arxiv.1102.2876
86 pages. This is the complete article corresponding to the announcement "Eta cocycles" by the same authors (arXiv:0907.0173)
arxiv created 2011/02/14 · openalex publication_date 2011/02/14 · arxiv updated 2011/02/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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→ J→ A→ B→ 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 (τGVr,σGV) for the pair A→ B; τGVr is a cyclic cochain on A defined through a regularization, à la Melrose, of the usual Godbillon-Vey cyclic cocycle τGV; σGV is a cyclic cocycle on B, obtained through a suspension procedure involving τGV and a specific 1-cyclic cocycle (Roe's 1-cocycle). We call σGV the eta cocycle associated to τGV. The Atiyah-Patodi-Singer formula is obtained by defining a relative index class \Ind (D,D^∂)∈ K_* (A,B) and establishing the equality <\Ind (D),[τGV]>=<\Ind (D,D^∂), [τrGV, σGV]>. The Godbillon-Vey eta invariant ηGV is obtained through the eta cocycle σGV.