process

118 M. Karoubi[35], M. F. Atiyah and G. Segal [8], this restriction is in fact not necessary. In orderto avoid it, one may use for instance the Atiyah–Jänich theorem [2], [27] about therepresentability of K-theory by the space of Fredholm operators (already quoted in[19] for the cup-product which cannot be defined otherwise).In the present paper, we would like to make a synthesis between different viewpointson the subject: [19], [38], [35] and [8] (partially of course) 3 . We hope to have been“pedagogical” in some sense to the non experts.However, this paper is not just historical. It presents the theory with another pointof view and contains some new results. We extend the Thom isomorphism to this moregeneral setting (see also [16]), which is important in order to relate the “ungraded”and “graded” twisted K-theories. We compute many interesting equivariant twistedK-groups, complementing the basic papers [35], [8] and [9]. For this purpose, we usethe “Chern character” for finite group actions, as defined by Baum, Connes, Kuhn andSłomińska [11], [33], [41], together with our generalized Thom isomorphism. Theselast computations are related to some previous ones [31] and to the work of manyauthors. Finally, we introduce new cohomology operations which are complementaryto those defined in [19] and [9].We don’t pretend to be exhaustive in a subject which has already many ramifications.In an appendix to this paper we try to give a short historical survey and a list of interestingcontributions of many authors related to the results quoted here.General plan of the paperLet us first recall the point of view developed in [19], in order to describe the backgroundmaterial. We consider a locally trivial bundle of Z=2-graded central simple complexalgebras A, i.e. modelled on M 2n .C/ or M n .C/ M n .C/, with the obvious gradings 4 .Then A has a well-defined class ˛ in the group GBr.X/ (as introduced above). Onthe other hand, one may consider the category of “A-bundles” , whose objects arevector bundles provided with an A-module structure (fibrewise). We call this categoryE A .X/; the graded objects of this category are vector bundles which are modules overA y˝ C 0;1 , where C 0;1 is the Clifford algebra C C D CŒx=.x 2 1/. The groupK˛.X/ is now defined as the “Grothendieck group” of the forgetful functorE A y˝C 0;1 .X/ ! E A .X/:We refer the reader to [28], p. 191, for this definition which generalizes the usualGrothendieck group of a category. For our purpose, we make it quite explicit at the endof §1, using the concept of “grading”.Despite its algebraic simplicity, this definition of K˛.X/ is not quite satisfactoryfor various reasons. For instance, it is not clear how to define in a simple way a3 As it was pointed out to me by J. Rosenberg, one should also add the following reference, in the spirit of[18]: Ellen Maycock Parker, The Brauer group of graded continuous trace C*-algebras, Trans. Amer. Math.Soc. 308 (1988), Nr. 1, 115–132.4 up to graded Morita equivalence.