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Article Dans Une Revue Annales Scientifiques de l'École Normale Supérieure Année : 2004

Twisted K-theory of differentiable stacks

Résumé

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 $K^i_\alpha \otimes K^j_\beta \to K^{i+j}_{\alpha +\beta}$ 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 \emph{groupoids}, and relies heavily on the machinery of $K$-theory ($KK$-theory) of $C^*$-algebras.
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Dates et versions

hal-00104608 , version 1 (08-10-2006)

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Jean-Louis Tu, Ping Xu, Camille Laurent-Gengoux. Twisted K-theory of differentiable stacks. Annales Scientifiques de l'École Normale Supérieure, 2004, IV série 37, pp.841-910. ⟨hal-00104608⟩
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