2003/08/19 by Éric Leichtnam, Eric Leichtnam, Paolo Piazza +2
Mathematics · #58J #Advanced Operator Algebra Research #Advanced Topics in Algebra #Differential Geometry (math.DG) #FOS: Mathematics #Holomorphic and Operator Theory #K-Theory and Homology (math.KT) #math.DG #math.KT #msc:58J
paper · pdf · doi:10.48550/arxiv.math/0308184
56 pages
arxiv created 2003/08/19 · openalex publication_date 2003/08/19 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let Γ be a discrete finitely generated group. Let M→ T be a Γ-equivariant fibration, with fibers diffeomorphic to a fixed even dimensional manifold with boundary Z. We assume that Γ→ M→ M/Γ is a Galois covering of a compact manifold with boundary. Let (D+ (θ))θ∈ T be a Γ-equivariant family of Dirac-type operators. Under the assumption that the boundary family is L2-invertible, we define an index class in the K-theory of the cross-product algebra, K0 (C0 (T)\rtimesr Γ). If, in addition, Γ is of polynomial growth, we define higher indeces by pairing the index class with suitable cyclic cocycles. Our main result is then a formula for these higher indeces: the structure of the formula is as in the seminal work of Atiyah, Patodi and Singer, with an interior geometric contribution and a boundary contribution in the form of a higher eta invariant associated to the boundary family. Under similar assumptions we extend our theorem to any G-proper manifold, with G an étale groupoid. We employ this generalization in order to establish a higher Atiyah-Patodi-Singer index formula on certain foliations with boundary. Fundamental to our work is a suitable generalization of Melrose b-pseudodifferential calculus as well as the superconnection proof of the index theorem on G-proper manifolds recently given by Gorokhovsky and Lott.