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Twisted K-theory of differentiable stacks

Jean-Louis Tu·Ping Xu·Camille Laurent-Gengoux
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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 structure $K^i_α\otimes K^j_β\to K^{i+j}_{α+β}$ 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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BibTeXRIS

Jean-Louis Tu, Ping Xu, Camille Laurent-Gengoux. 2004-09-13. Twisted K-theory of differentiable stacks. https://arxiv.org/abs/math/0306138

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