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. The definition To motivate Rosenberg's geometric formulation of twisted K-theory, start from the Atiyah–Jänich theorem, stating that the Fredholm operators on Hilbert space , is a classifying space for ordinary, untwisted K-theory. This means that the K-theory of the space consists of the homotopy classes of maps from to A slightly more complicated way of saying the same thing is as follows. Consider the trivial bundle of over , that is, the Cartesian product of and . Then the K-theory of consists of the homotopy classes of sections of this bundle. We can make this yet more complicated by introducing a trivial bundle over , where is the group of projective unitary operators on the Hilbert space . Then the group of maps from to which are equivariant under an action of is equivalent to the original groups of maps This more complicated construction of ordinary K-theory is naturally generalized to the twisted case. To see this, note that bundles on are classified by elements of the third integral cohomology group of . This is a consequence of the fact that topologically is a representative Eilenberg–MacLane space . The generalization is then straightforward. Rosenberg has defined , the twisted K-theory of with twist given