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 structureOur 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

No abstract

For an orbifold X and α ∈ H 3 (X, Z), we introduce the twisted cohomology H * c (X, α) and prove that the non-commutative Chern character of Connes-Karoubi establishes an isomorphism between the twisted K-groups K * α (X) ⊗ C and the twisted cohomology H * c (X, α). This theorem, on the one hand, generalizes a classical result of Baum-Connes, Brylinski-Nistor, and others, that if X is an orbifold then the Chern character establishes an isomorphism between the K-groups of X tensored with C, and the compactlysupported cohomology of the inertia orbifold. On the other hand, it also generalizes a recent result of Adem-Ruan regarding the Chern character isomorphism of twisted orbifold K-theory when the orbifold is a global quotient by a finite group and the twist is a special torsion class, as well as Mathai-Stevenson's theorem regarding the Chern character isomorphism of twisted K-theory of a compact manifold.

The theory of principal G-bundles over a Lie groupoid is an important one unifying various types of principal G-bundles, including those over manifolds, those over orbifolds, as well as equivariant principal G-bundles. In this paper, we study differential geometry of these objects, including connections and holonomy maps. We also introduce a Chern-Weil map for these principal bundles and prove that the characteristic classes obtained coincide with the universal characteristic classes.As an application, we recover the equivariant Chern-Weil map of Bott-Tu. We also obtain an explicit chain map between the Weil model and the simplicial model of equivariant cohomology which reduces to the Bott-Shulman map S(g * )

We prove, under some mild conditions, that the equivariant twisted K