ABSTRACT. – In this paper, we develop twisted K-theory for stacks, where the twisted class is given by an S 1-gerbe over the stack. General properties, including the Mayer–Vietoris property, Bott periodicity, and the product structure K i α ⊗K j β → Ki+j α+β are derived. Our approach provides a uniform framework for studying various twisted K-theories including the usual twisted K-theory of topological spaces, twisted equivariant K-theory, and the twisted K-theory of orbifolds. We also present a Fredholm picture, and discuss the conditions under which twisted K-groups can be expressed by so-called “twisted vector bundles”. Our approach is to work on presentations of stacks, namely groupoids, and relies heavily on the machinery of K-theory (KK-theory) of C ∗-algebras. © 2004 Elsevier SAS RÉSUMÉ. – Dans cet article, nous développons la K-théorie tordue pour les champs différentiables, où la torsion s’effectue par une S 1-gerbe sur le champ en question. Nous en établissons les propriétés générales telles que les suites exactes de Mayer–Vietoris, la périodicité de Bott, et le produit K i α ⊗ K j β → Ki+j α+β. Notre approche fournit un cadre général permettant d’étudier diverses K-théories tordues, en particulier la K-théorie tordue usuelle des espaces topologiques, la K-théorie tordue équivariante, et la K-théorie tordue des orbifolds. Nous donnons également une définition équivalente utilisant des opérateurs de Fredholm, et nous discutons les conditions sous lesquelles les groupes de K-théorie tordue peuvent être réalisés à partir de “fibrés vectoriels tordus”. Notre approche consiste à travailler sur les réalisations concrètes des champs, à savoir les groupoïdes, et s’appuie de façon importante sur les techniques de K-théorie (KK-théorie) des C ∗-algèbres.

The purpose of this paper is to prove two theorems which concern the position of étale groupoids among general smooth (or ”Lie”) groupoids. Our motivation comes from the non-commutative geometry and algebraic topology concerning leaf spaces of foliations. Here, one is concerned with invariants of the holonomy groupoid of a foliation

We establish a close and previously unknown relation between quantales and groupoids, in terms of which the notion of étale groupoid is subsumed in a natural way by that of quantale. In particular, to each étale groupoid, either localic or topological, there is associated a unital involutive quantale. We obtain a bijective correspondence between localic étale groupoids and their quantales, which are given a rather simple characterization and are here called inverse quantal

Abstract. We give a superconnection proof of Connes ’ index theorem for proper cocompact actions of étale groupoids. This includes Connes ’ general foliation index theorem for foliations with Hausdorff holonomy groupoid. 1.

This paper is concerned with characteristic classes in the cohomology of leaf spaces of foliations. For a manifold M equipped with a foliation F it is well-known that the coarse (naive) leaf space M/F, obtained from M by identifying each leaf to a point,

Abstract. In this paper we construct an explicit geometric model for the group of gerbes over an orbifold X. We show how from its curvature we can obtain its characteristic class in H 3 (X) via Chern-Weil theory. For an arbitrary gerbe L, a twisting L Korb(X) of the orbifold K-theory of X is constructed, and shown to generalize previous twisting by Rosenberg [28], Witten [35], Atiyah-Segal [2] and Bowknegt et. al. [4] in the smooth case and by Adem-Ruan [1] for discrete torsion on an orbifold. Contents

Let G be a Lie groupoid over M such that the target-source map from G to M × M is proper. We show that, if O is an orbit of finite type (i.e. which admits a proper function with finitely many critical points), then the restriction G|U of G to some neighborhood U of O in M is isomorphic to a similar restriction of the action groupoid for the linear action of the transitive groupoid G|O on the normal bundle NO. The proof uses a deformation argument based on a cohomology vanishing theorem, along with a slice theorem which is derived from a new result on submersions with a fibre of finite type.

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Abstract. We show that Haefliger’s cohomology for étale groupoids, Moore’s cohomology for locally compact groups and the Brauer group of a locally compact groupoid are all particular cases of sheaf (or Čech) cohomology for topological simplicial spaces. 1.

Abstract. In this article, the cyclic homology theory of formal deformation quantizations of the convolution algebra associated to a proper étale Lie groupoid is studied. We compute the Hochschild cohomology of the convolution algebra and express it in terms of alternating multi-vector fields on the associated inertia groupoid. We introduce a noncommutative Poisson homology whose computation enables us to determine the Hochschild homology of formal deformations of the convolution algebra. Then it is shown that the cyclic (co)homology of such formal deformations can be described by an appropriate sheaf cohomology theory. This enables us to determine the corresponding cyclic homology groups in terms of orbifold cohomology of the underlying orbifold. Using the thus obtained description of cyclic cohomology of the deformed convolution algebra, we give a complete classification of all