process

Twisted K-theory – old and new 147On the other hand, the theorem of Atiyah and Jänich [2], [27] showed that Fredholmoperators play a crucial role in K-theory. It was discovered independently in [10] and[29], [32] that Clifford modules may also be used as a variation for the spectrum of(real and complex) K-theory. In particular, the cup-product from K n .X/ K p .Y / toK nCp .X Y/may be reinterpreted in a much simpler way with this variation. Whatappeared as a convenience for usual K-theory in these previous references became anecessity for the definition of the product in the twisted case, as it was shown in [19].The next step was taken by J. Rosenberg [38], some twenty years later, in redefiningGBr.X/ for any class in H 3 .XI Z/ (not just torsion classes), thanks to the new roleplayed by C*-algebras in K-theory. The need for such a generalization was not clearin the 60’s (although all the tools were there) since we were just interested in usualPoincaré pairing. In passing, we should mention that one can define more generaltwistings in any cohomology theory and therefore in K-theory (see for instance [8],p. 18). However, the geometric ones are the most interesting for the purpose of ourpaper.We already mentioned the paper of E. Witten [43] in the introduction which madeuse of this more general twisting of J. Rosenberg, together with [35], [8], [9]. Butwe should also quote many other papers: [13] for the relations with the theory ofgerbes and D-branes in Physics; [1] for the relation with orbifolds; [36] who used theChern character defined byA. Connes and M. Karoubi in order to prove an isomorphismbetween twisted K-theory and a computable “twisted cohomology” (at least rationally);[22], [21] about the relation between loop groups and twisted K-theory; [23] (again) forthe relation with equivariant cohomology and many other works which are mentionedinside the paper and in the references. In particular, M. F. Atiyah and M. Hopkins [7]introduced another type of K-theory, denoted by them K˙X/, linked to deep physicalproblems, which is a particular case of twisted equivariant K-theory. This definitionof K˙.X/ was already given in [28], §3.3, in terms of Clifford bundles with a groupaction (see 6.16 and also [31] for the details).We invite the interested reader to use the classical research tools on the Web formany other references in mathematics and physics.References[1] A. Adem and Y. Ruan, Twisted Orbifold K-theory, Commun. Math. Phys. 237 (2003),533–556.[2] M. F. Atiyah, K-theory, Notes by D. W. Anderson, 2nd ed., Adv. Book Classics, Addison-Wesley Publishing Company, Redwood City, CA, 1989.[3] M. F. Atiyah, Bott periodicity and the index of elliptic operators, Quart. J. Math. OxfordSer. 19 (1968), 113–140.[4] M. F. Atiyah and F. Hirzebruch, Riemann-Roch theorems for differentiable manifolds, Bull.Amer. Math. Soc. 65 (1959), 276–281.[5] M. F. Atiyah and F. Hirzebruch, Vector bundles and homogeneous spaces, in Proc. Sympos.Pure Math. 3, Amer. Math. Soc., Providence, R.I., 1961, 7–38.