by the 3-class , to be the space of homotopy classes of sections of the trivial bundle over that are covariant with respect to a bundle fibered over with 3-class , that is Equivalently, it is the space of homotopy classes of sections of the bundles associated to a bundle with class . What is it? When is the trivial class, twisted K-theory is just untwisted K-theory, which is a ring. However, when is nontrivial this theory is no longer a ring. It has an addition, but it is no longer closed under multiplication. However, the direct sum of the twisted K-theories of with all possible twists is a ring. In particular, the product of an element of K-theory with twist with an element of K-theory with twist is an element of K-theory twisted by . This element can be constructed directly from the above definition by using adjoints of Fredholm operators and construct a specific 2 x 2 matrix out of them (see the reference 1, where a more natural and general Z/2-graded version is also presented). In particular twisted K-theory is a module over classical K-theory. 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 or the twisted , and then one takes the cohomology with respect to a series of differential operators. The first operator, , for example, is the sum of the three-class , which in string theory corresponds to the Neveu-Schwarz 3-form, and the third Steenrod square, soNo elementary form for the next operator, , has been found, although several conjectured forms exist. Higher operators do not contribute to the -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 . After taking the cohomology with respect to the full series of differentials one obtains twisted -theory as a set, but to obtain the full group structure one in general needs to solve an extension problem. Example: the three-sphere The three-sphere, , has trivial cohomology except for and which are both isomorphic to the integers. Thus the even and odd cohomologies are both isomorphic to the integers.