process

Twisted K-theory – old and new 141Grothendieck group. By the theory of characters, we see that R. zG/ l ˝Q is isomorphicto R.Z. zG// l ˝ Q, since the characters of such linear type representations of zG vanishoutside Z. zG/. On the other hand, if we denote by G 0 the image of Z. zG/ in G, theextension of abelian groupsQ=Z Z. zG/ G0splits (non canonically). This means that we can identify R. zG/ l ˝ Q with the representationring R.G 0 / ˝ Q as an R.G/ ˝ Q-module. This proves the last part of thetheorem.Theorem 6.11. Let us consider a graded algebra A D M 2n .C/ provided with a nonoriented action of an abelian group G (with respect to the grading). Then the gradedtwisted K-theory K A G .P / D GrK .A 0 /, with A 0 D G Ë A, is concentrated in a singledegree (0 or 1) and is a free Z-module. If we tensor this group with the complexnumbers and if we look at it as an R.G/ ˝ C D CŒG-module, it may be identifiedwith R.G 0 / ˝ Q for a suitable subgroup G 0 of G. In particular, the rank of K A G .P / divides the order of G.Proof. Let L be the orientation bundle of A (with respect to the action of G). If wechange A into A y˝ C.L/ and if we apply the Thom isomorphism theorem, we have tocompute K A G .L/ where G acts on A (resp. L) in an oriented way (resp. non orientedway).Let us apply Theorem 6.10 in this situation: since G is abelian, the function f ofthe theorem must be equal to 0 on the elements of zG which are not in Z. zG/. Therefore,the relevant group K A G .L/ is reduced to K G 0.L/ (after tensoring with C and whereG 0 is the image of Z.G/ in G). We now consider two cases:1. the action of G 0 on L is oriented, in which case we only find R.G 0 / (with ashift of dimension). Therefore, the dimension of KG A .P / ˝ C is the order of G0which divides the order of G.2. the action of G 0 on L is not oriented. We then find a direct sum of copies of C,each one corresponding to an element of G 0 such that .g 0 / D 1. The dimensionof KG A.L/ ˝ C is therefore half the order of G0 , hence divides jGj=2.Remark 6.12. For an oriented action of G on M 2n .C/, Theorem 3.5 enables us tosolve the analogous problem in ungraded twisted K-theory, which is done in 6.14.On the other hand, as we already mentioned, the two last theorems are related toresults of P. Hu and I. Kriz [26].In [31] we also perform analogous types of computations when A is a Cliffordalgebra and G is any finite group, again using the Thom isomorphism. For instance,if G is the symmetric group S n acting on the Clifford algebra of R n via the canonicalrepresentation of S n on R n , there is a nice relation between K-theory and the pentagonalidentity of Euler (cf. [31], p. 532).