Twisted K-theory of differentiable stacks

Abstract

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

Sheaf theory for stacks in manifolds and twisted cohomology forS1–gerbes

“…One can build a two-category of topological groupoids by inverting Morita equivalence such that equivalence classes of U.1/-central extensions of groupoids representing X are indeed classified by H 3 .X I ‫/ޚ‬ (see, for example, Laurent-Gangoux, Tu and Xu [28]). …”

The Geometry of Unitary 2-Representations of Finite Groups and their 2-Characters

“…Similarly we define the geometric character of an equivariant gerbe as the space of sections over the fixed points of the gerbe (readers familiar with these ideas will recognize this as the push-forward of the transgression map in our context, as in [39]). Since the group acts on these fixed points by conjugation, the geometric character also produces an equivariant vector bundle over the group.…”

Mathematisches Institut Georg-August-Universität Göttingen Seminars Winter Term 2004/2005

“…, [TXL03], and [AS04] for a construction a twisted K-theory functor. We further assume that twisted K-theory admits a Mayer-Vietoris sequence and is a module over the untwisted K-theory.…”

“…Rather we assume that such a functor exists and has all necessary functorial properties. Actually we only need local quotient stacks, and the construction of the K-theory in this case was sketched in [FHT03] (see also [TXL03] and [AS04]). A verification of all functorial properties, in particular the construction of push-forward maps, is still a gap in the literature.…”