On the Algebraic Index for Riemannian Étale Groupoids

Abstract

Abstract. In this paper, we construct an explicit quasi-isomorphism to study the cyclic cohomology of a deformation quantization over a Riemannianétale groupoid. Such a quasi-isomorphism allows us to propose a general algebraic index problem for Riemanniań etale groupoids. We discuss solutions to that index problem when the groupoid is proper or defined by a constant Dirac structure on a 3-dimensional torus. Mathematics Subject Classification (2000). Primary 58J20; Secondary 53D55.Keywords. Riemannian foliation, deformation quantization, index, cyclic cohomology. IntroductionIn [17] and [18], the authors of this paper studied the algebraic index theory over orbifolds using noncommutative geometry and deformation quantization. In [17], we obtained an explicit topological formula for the Chern character of an elliptic operator on a compact Riemannian orbifold. With this paper we continue our study of algebraic index theory over singular spaces modeled by groupoids. More precisely, omitting the assumption of properness from [17] and [18] we study here the algebraic index theory of a singular space which can be obtained as the quotient of anétale groupoid equipped with an invariant Riemannian metric on the unit space. Such groupoids are called Riemannianétale groupoids and appear naturally in the study of Riemannian foliations.A Riemannian foliation [14] is a foliation (M, F) equipped with a bundle-like metric η. Such a bundle-like metric defines a holonomy invariant metric on the normal bundle of the foliation. Let X be a complete transversal to the foliation F which means that X is an immersed submanifold of M which intersects every leaf of F. The holonomy groupoid G associated to X is anétale groupoid and the

Equivariant Algebraic Index Theorem

“…In the case when the action of is free and proper, we recover the algebraic version of Connes-Moscovici higher index theorem. The case of proper actions has been considered in [22,23].…”