process

138 M. KaroubiIn order to show this fact, let us consider the following diagram: n nmzG 1 zG G G,and an element g 1 of zG 1 . The conjugacy class of g D .g 1 / splits into n conjugacyclasses in zG 1 if and only if there is a trace function f on zG 1 with values in C such thatf.g 1 c/ D f.g 1 /c when c 2 n . Such a trace function extends obviously to zG, whichyields to the result, since the direct limit of the groups H 2 .GI Z=nm/ is preciselyH 2 .GI Q=Z/ Š H 3 .GI Z/.This remark may be generalized as follows according to a suggestion of J.-P. Serre:let zG 1 and zG be two group extensions (not necessary central) of G by abelian groupsC 1 and C of orders m 1 and m respectively, such that the following diagram commutes(with ˛ injective):˛C 1CzG 1 zG G G.The previous argument shows that if an element g of G splits into m conjugacy classesin zG, it splits into m 1 conjugacy classes in zG 1 : take trace functions f on zG with valuesin C such that f.gc/ D f.g/c (we write multiplicatively the abelian group C ). Theconverse is true if the extension of G by C is central.Let us now assume that X is not reduced to a point. We can use the Baum–Connes–Kuhn–Słomińska Chern character [11], [33], [41] which is defined on K .X/ (for anyfinite group ), with values in the direct sum L hi H even .X g / C./ . In this formula,hi runs through all the conjugacy classes of G, C./ being the centralizer of (thecohomology is taken with complex coefficients). One of the main features of this“Chern character”K .X/ L hi H even .X / C./is the isomorphism it induces between K .X/˝Z C and the cohomology with complexcoefficients on the right-hand side. If E is a -vector bundle, the map is definedexplicitly by Formula 1.13, p. 170 in [11].Let us now take for the group zG previously considered and let us analyze theformula in this case. We shall view the right-hand side not just as a function on the