Twisted K-theory of differentiable stacks

In this paper, we develop twisted K-theory for stacks, where the twisted class is given by an S¹-gerbe over the stack. General properties, including the Mayer-Vietoris property, Bott periodicity, and the product structure K_α ⁱ ⊗ K_β ^j → K_α+β ^i+j 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.