Twisted K-theory - The Definition

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

K_H(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.