In mathematics, twisted K-theory (also called K-theory with local coefficients) is a variation on K-theory, a mathematical theory from the 1950s that spans algebraic topology, abstract algebra and operator theory. More specifically, twisted K-theory with twist H is a particular variant of K-theory, in which the twist is given by an integral 3-dimensional cohomology class. It is special among the various twists that K-theory admits for two reasons. First, it admits a geometric formulation. This was provided in two steps; the first one was done in 1970 (Publ. Math. de l'IHÉS) by Peter Donovan and Max Karoubi; the second one in 1988 by Jonathan Rosenberg in Continuous-Trace Algebras from the Bundle Theoretic Point of View. In physics, it has been conjectured to classify D-branes, Ramond-Ramond field strengths and in some cases even spinors in type II string theory. For more information on twisted K-theory in string theory, see K-theory (physics). In the broader context of K-theory, in each subject it has numerous isomorphic formulations and, in many cases, isomorphisms relating definitions in various subjects have been proven. It also has numerous deformations, for example, in abstract algebra K-theory may be twisted by any integral cohomology class.
Definition To motivate Rosenberg's geometric formulation of twisted K-theory, start from the Atiyah–Jänich theorem, stating that
F r e d ( H ) , {\displaystyle Fred({\mathcal {H}}),}
the Fredholm operators on Hilbert space H {\displaystyle {\mathcal {H}}} , is a classifying space for ordinary, untwisted K-theory. This means that the K-theory of the space M {\displaystyle M} consists of the homotopy classes of maps
[ M → F r e d ( H ) ] {\displaystyle [M\rightarrow Fred({\mathcal {H}})]}
from M {\displaystyle M} to F r e d ( H ) . {\displaystyle Fred({\mathcal {H}}).}
A slightly more complicated way of saying the same thing is as follows. Consider the trivial bundle of F r e d ( H ) {\displaystyle Fred({\mathcal {H}})} over M {\displaystyle M} , that is, the Cartesian product of M {\displaystyle M} and F r e d ( H ) {\displaystyle Fred({\mathcal {H}})} . Then the K-theory of M {\displaystyle M} consists of the homotopy classes of sections of this bundle. We can make this yet more complicated by introducing a trivial
P U ( H ) {\displaystyle PU({\mathcal {H}})}
bundle P {\displaystyle P} over M {\displaystyle M} , where P U ( H ) {\displaystyle PU({\mathcal {H}})} is the group of projective unitary operators on the Hilbert space H {\displaystyle {\mathcal {H}}} . Then the group of maps
[ P → F r e d ( H ) ] P U ( H ) {\displaystyle [P\rightarrow Fred({\mathcal {H}})]_{PU({\mathcal {H}})}}
from P {\displaystyle P} to F r e d ( H ) {\displaystyle Fred({\mathcal {H}})} which are equivariant under an action of P U ( H ) {\displaystyle PU({\mathcal {H}})} is equivalent to the original groups of maps
[ M → F r e d ( H ) ] . {\displaystyle [M\rightarrow Fred({\mathcal {H}})].}
This more complicated construction of ordinary K-theory is naturally generalized to the twisted case. To see this, note that P U ( H ) {\displaystyle PU({\mathcal {H}})} bundles on M {\displaystyle M} are classified by elements H {\displaystyle H} of the third integral cohomology group of M {\displaystyle M} . This is a consequence of the fact that B P U ( H ) {\displaystyle BPU({\mathcal {H}})} topologically is a representative Eilenberg–MacLane space
K ( Z , 3 ) {\displaystyle K(\mathbf {Z} ,3)} . The generalization is then straightforward. Rosenberg has defined
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