Etale Groupoids, eta invariants and index theory

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dc.creatorLeichtnam, Eric
dc.creatorPiazza, Paolo
dc.date2003-08-19
dc.date.accessioned2026-07-07T05:00:29Z
dc.date.available2026-07-07T05:00:29Z
dc.descriptionLet $Γ$ be a discrete finitely generated group. Let $\hat{M}\to T$ be a $Γ$-equivariant fibration, with fibers diffeomorphic to a fixed even dimensional manifold with boundary $Z$. We assume that $Γ\to \hat{M}\to \hat{M}/Γ$ is a Galois covering of a compact manifold with boundary. Let $(D^+ (θ))_{θ\in T}$ be a $Γ$-equivariant family of Dirac-type operators. Under the assumption that the boundary family is $L^2$-invertible, we define an index class in the K-theory of the cross-product algebra, $K_0 (C^0 (T)\rtimes_r Γ)$. 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.
dc.description56 pages
dc.identifierhttps://arxiv.org/abs/math/0308184
dc.identifierhttp://arxiv.org/abs/math/0308184
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/68346
dc.subjectDifferential Geometry
dc.subjectK-Theory and Homology
dc.subject58J
dc.titleEtale Groupoids, eta invariants and index theory
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