How To Calculate It
Physicist typically want to calculate twisted K-theory using the Atiyah-Hirzebruch spectral sequence. The idea is that one begins with all of the even or all of the odd integral cohomology, depending on whether one wishes to calculate the twisted K0 or the twisted K1, and then one takes the cohomology with respect to a series of differential operators. The first operator, d3, for example, is the sum of the three-class H, which in string theory corresponds to the Neveu-Schwarz 3-form, and the third Steenrod square. No elementary form for the next operator, d5, has been found, although several conjectured forms exist. Higher operators do not contribute to the K-theory of a 10-manifold, which is the dimension of interest in critical superstring theory. Over the rationals Michael Atiyah and Graeme Segal have shown that all of the differentials reduce to Massey products of H.
After taking the cohomology with respect to the full series of differentials one obtains twisted K-theory as a set, but to obtain the full group structure one in general needs to solve an extension problem.
Read more about this topic: Twisted K-theory
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