process

Twisted K-theory – old and new 137Theorem 6.4. Let G be a finite group. Then the previous homomorphismis bijective.W 3 W Br.G/ ! H 3 .GI Z/Proof. First of all, we remark that H 3 .GI Z/ Š H 2 .GI Q=Z/ is the direct limit ofthe groups H 2 .GI m / through the maps H 2 .GI m / ! H 2 .GI p / when m dividesp. This stabilization process corresponds on the level of algebras to the tensor productA 7! A ˝ End.V /, where V is a G-vector space of dimension p=m. Therefore, themap W 3 is injective.The proof of the surjectivity is a little bit more delicate (see also 5.9). We can sayfirst that it is a particular case of a much more general result proved by A. Fröhlich andC. T. C. Wall [24] about the equivariant Brauer group of an arbitrary field k: there is asplit exact sequence (with their notations)0 Br.k/ BM.k; G/ H 2 .GI U.k// 0where Br.k/ is the usual Brauer group of k, U.k/ is the group of invertible elements ink and BM.k; G/ is a group built out of central simple algebras over k with a G-action.Since Br.C/ D 0 and H 2 .GI U.k// D H 2 .GI Q=Z/, the theorem is an immediateconsequence.Here is an elementary proof suggested by the referee (for k D C). If we start witha central extension1 n zG G 1we consider the finite dimensional vector spaceH Dff 2 L 2 . zG/ such that f ./ D 1 f./with .; / 2 n zGg:Then zG acts by left translation on H in such a way that this action is of linear typeas described in 6.2. Therefore, we have a projective representation of G on H andEnd.H / is a matrix algebra where G acts.Remark 6.5. Let us consider an arbitrary central extension zG of G by n Š Z=nassociated to a cohomology class c 2 H 2 .GI Z=n/. We are interested in the set ofelements g of G such that the conjugacy class hgi splits into n conjugacy classes in zG 1 .This set only depends on the image of c in H 3 .GI Z/ via the Bockstein homomorphismH 2 .GI Z=n/ ! H 3 .GI Z/ (cf. 6.6). In other words, two central extensions of G byZ=n with the same associated image by the Bockstein homomorphism have the sameset of n-split conjugacy classes.