# Twisted K-theory

@article{Atiyah2004TwistedK, title={Twisted K-theory}, author={Michael Francis Atiyah and Graeme B. Segal}, journal={arXiv: K-Theory and Homology}, year={2004} }

Twisted complex K-theory can be defined for a space X equipped with a bundle of complex projective spaces, or, equivalently, with a bundle of C ∗ -algebras. Up to equivalence, the twisting corresponds to an element of H 3 (X; Z). We give a systematic account of the definition and basic properties of the twisted theory, emphasizing some points where it behaves differently from ordinary K-theory. (We omit, however, its relations to classical coho- mology, which we shall treat in a sequel.) We…

## 256 Citations

Models of twisted K -theory

- Mathematics
- 2008

This thesis concerns geometrical models in complete generality of twistings in complex K-theory, in particular of higher twistings. In the first three chapters we treat the non-equivariant situation.…

Twisted K theory of differentiable stacks

- Mathematics, Physics
- 2003

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…

The ring structure for equivariant twisted K-theory

- Mathematics, Physics
- 2006

Abstract We prove, under some mild conditions, that the equivariant twisted K-theory group of a crossed module admits a ring structure if the twisting 2-cocycle is 2-multiplicative. We also give an…

Twisted K-theory constructions in the case of a decomposable Dixmier-Douady class

- Mathematics
- 2012

Twisted K-theory on a manifold X, with twisting in the 3rd integral cohomology, is discussed in the case when X is a product of a circle T and a manifold M. The twist is assumed to be decomposable as…

Equivariant twisted Real K-theory of compact Lie groups

- Mathematics
- 2018

Abstract Let G be a compact, connected, and simply-connected Lie group viewed as a G -space via the conjugation action. The Freed–Hopkins–Teleman Theorem (FHT) asserts a canonical link between the…

Loop groups and twisted K-theory III

- Mathematics
- 2011

In this paper, we identify the Ad-equivariant twisted K-theory of a compact Lie group G with the “Verlinde group” of isomorphism classes of admissible representations of its loop groups. Our…

Twisted K-theory with coefficients in C*-algebras

- Mathematics
- 2011

We introduce a twisted version of $K$-theory with coefficients in a $C^*$-algebra $A$, where the twist is given by a new kind of gerbe, which we call Morita bundle gerbe. We use the description of…

Twisted equivariant K-theory of compact Lie group actions with maximal rank isotropy

- Mathematics, PhysicsJournal of Mathematical Physics
- 2018

We consider twisted equivariant K-theory for actions of a compact Lie group G on a space X where all the isotropy subgroups are connected and of maximal rank. We show that the associated rational…

General plan of the paper

- 2007

The motivation for this definition is to give in K-theory a satisfactory Thom isomorphism and Poincaré duality pairing which are analogous to the usual ones in cohomology with local coefficients.…

Twisted K-theory 3

- 2016

(0.1) Notation: Fix a separable Hilbert space H and denote by PU the projective unitary group of H , by Fred the space of Fredholm endomorphisms of H (both in the norm topology, unless otherwise…

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