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 M consists of the homotopy classes of maps
from M to
A slightly more complicated way of saying the same thing is as follows. Consider the trivial bundle of over M, that is, the Cartesian product of M and . Then the K-theory of M consists of the homotopy classes of sections of this bundle.
We can make this yet more complicated by introducing a trivial
bundle over M, 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 M are classified by elements H of the third integral cohomology group of M. This is a consequence of the fact that topologically is a representative Eilenberg-MacLane space
The generalization is then straightforward. Rosenberg has defined
- KH(M),
the twisted K-theory of M with twist given by the 3-class H, to be the space of homotopy classes of sections of the trivial bundle over M that are covariant with respect to a bundle fibered over M with 3-class H, that is
Equivalently, it is the space of homotopy classes of sections of the bundles associated to a bundle with class H.
Read more about this topic: Twisted K-theory
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