process

Twisted K-theory – old and new 145Another way to proceed is to use a variation of the method developed in [19], p. 21,for any A. We consider the following diagram (with X connected):F.X;S 1 /F.X;S 1 /u nU 0 .A y˝n/A n Aut 0 .A y˝n /vU 0 .A y˝ A/ y˝n Aut 0 .A y˝ A/ y˝n .In this diagram A n is the alternating group (for n 3), n is the cartesian product ofA n and U 0 .A y˝n / over Aut 0 .A y˝n / and the horizontal maps between the U ’s and theAut’s are essentially given by g 7! g ˝ g. Note that these maps induce a map fromF.X;S 1 / to itself which is f.z/7! f.z 2 /.By the general theory developed in 7.2, we know that there exists a canonicalmap from A n to U 0 .A y˝ A/ y˝n such that its composite with the projection fromU 0 .A y˝A/ y˝n to Aut 0 .A y˝A/ y˝n is the composite of the last horizontal maps. Let usnow define the subgroup C n of n which elements x are defined by '..x// D v.u.x//.It is clear that C n is a double cover of A n which is either the product of A n by Z=2 orthe Schur group C n [40] (since H 2 .A n I Z=2/ is isomorphic to Z=2 when n>3).Using the methods of §6, we have therefore the following composite maps:K A .X/ KA y˝nA n.X/ KA y˝n.C n / .X/ KA y˝n.X/ ˝ R.C n /,where the notation K A y˝n.C n / .X/ means that the group C n acts trivially on A y˝n . Followingagain Atiyah [2], we see then that any group homomorphism R.C n / ! Z gives rise toan operation K A .X/ ! K A y˝n.X/ Therefore, in the graded case, the Schur group C nreplaces the symmetric group S n , which we have used in the ungraded case.In order to define more computable operations, we may replace the alternatinggroup A n by the cyclic group Z=n (when n is odd) as it was already done in [19].The reduction of the central extension C n becomes trivial and we have a commutativediagramU 0 .A y˝n / Z=n Aut.A y˝n /.The Adams operation ‰ n is then given by the same formula as in 7.4 and [19]‰ n W K A .X/ ! K A˝n .X/ ˝ nZ