Because the three-sphere is of dimension three, which is less than five, the third Steenrod square is trivial on its cohomology and so the first nontrivial differential is just . The later differentials increase the degree of a cohomology class by more than three and so are again trivial; thus the twisted -theory is just the cohomology of the operator which acts on a class by cupping it with the 3-class . Imagine that is the trivial class, zero. Then is also trivial. Thus its entire domain is its kernel, and nothing is in its image. Thus is the kernel of in the even cohomology, which is the full even cohomology, which consists of the integers. Similarly consists of the odd cohomology quotiented by the image of , in other words quotiented by the trivial group. This leaves the original odd cohomology, which is again the integers. In conclusion, and of the three-sphere with trivial twist are both isomorphic to the integers. As expected, this agrees with the untwisted -theory. Now consider the case in which is nontrivial. is defined to be an element of the third integral cohomology, which is isomorphic to the integers. Thus corresponds to a number, which we will call . now takes an element of and yields the element of . As is not equal to zero by assumption, the only element of the kernel of is the zero element, and so . The image of consists of all elements of the integers that are multiples of . Therefore, the odd cohomology, , quotiented by the image of , , is the cyclic group of order , . In conclusionIn string theory this result reproduces the classification of D-branes on the 3-sphere with units of -flux, which corresponds to the set of symmetric boundary conditions in the supersymmetric WZW model at level . There is an extension of this calculation to the group manifold of SU(3). In this case the Steenrod square term in , the operator , and the extension problem are nontrivial. See also K-theory (physics) Wess–Zumino–Witten model Bundle gerbe Notes References "Graded Brauer groups and K-theory with local coefficients", by Peter Donovan and Max Karoubi. Publ. Math. IHÉS Nr. 38, pp. 5–25 (1970). D-Brane Instantons and K-Theory Charges by Juan Maldacena, Gregory Moore and Nathan Seiberg Twisted K-theory and Cohomology by Michael Atiyah and Graeme Segal Twisted K-theory and the K-theory of