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Twisted \(K\)-theory of differentiable stacks. (English) Zbl 1069.19006

This paper defines and develops the basic properties of a form of twisted \(K\)-theory sufficiently general to include as special cases the various forms of twisted \(K\)-theory which have occurred in the literature. These include twisted \(K\)-theory of spaces and of orbifolds and twisted equivariant \(K\)-theory. The results are stated in terms of twisted \(K\)-theory of differentiable stacks, with the twist provided by the action of an \(S^1\)-gerbe, however for the definition the gerbe and stack are represented by an \(S^1\)-central extension, \(\alpha\), of Lie groupoids \(S^1 \to \mathbb R \to \Gamma \rightrightarrows M\). Such an extension has associated to it a complex line bundle \(L = \mathbb R \times_{S^1} {\mathbb C}\) which may be considered as a Fell bundle of \(C^*\)-algebras over the Lie groupoid \(\Gamma \rightrightarrows M\). The twisted \(K\)-groups \(K^i_\alpha (\Gamma^\bullet)\) are defined to be the \(K\)-groups \(K^{-i}(C^*_r(\Gamma, \mathbb R))\) of the reduced \(C^*\)-algebra associated to \(L\).
The authors check that the groups defined in this way possess Bott periodicity and a Mayer-Vietoris sequence and that they specialize in the expected way if \(\Gamma\) is a manifold. They then show that these twisted \(K\)-groups may also be described as homotopy classes of invariant sections of Fredholm bundles associated to the groupoid (they refer to this result as the main theorem of the paper). The paper concludes with the construction of a product for this theory using the Kasparov product in \(KK\)-theory of \(C^*\)-algebras.

MSC:

19K35 Kasparov theory (\(KK\)-theory)
58J22 Exotic index theories on manifolds
19K56 Index theory
46L80 \(K\)-theory and operator algebras (including cyclic theory)
20L05 Groupoids (i.e. small categories in which all morphisms are isomorphisms)
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