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Twisted K-theory – old and newby Max KaroubiSome history and motivation about this paperThe subject “K-theory with local coefficients”, now called “twisted K-theory”, wasintroduced by P. Donovan and the author in [19] almost forty years ago. 1 It associatesto a compact space X and a “local coefficient system”˛ 2 GBr.X/ D Z=2 H 1 .XI Z=2/ Tors.H 3 .XI Z//an abelian group K˛.X/ which generalizes the usual Grothendieck–Atiyah–HirzebruchK-theory of X when we restrict ˛ being in Z=2 (cf. [5]). This “graded Brauer group”GBr.X/ has the following group structure: if ˛ D ."; w 1 ;W 3 / and ˛0 D ." 0 ;w1 0 ;W0 3 /are two elements, one defines the sum ˛C˛0 as ."C" 0 ;w 1 Cw1 0 ;W 3CW3 0Cˇ.w1w1 0 //,where ˇ W H 2 .XI Z=2/ ! H 3 .XI Z/ is the Bockstein homomorphism. With thisdefinition, one has a generalized cup-product 2 :K˛.X/ K˛0.X/ ! K˛C˛0.X/:The motivation for this definition is to give in K-theory a satisfactory Thom isomorphismand Poincaré duality pairing which are analogous to the usual ones in cohomologywith local coefficients. More precisely, as proved in [28], if V is a real vector bundleon a compact space X with a positive metric, the K-theory of the Thom space of V isisomorphic to a certain “algebraic” group K C.V / .X/ associated to the Clifford bundleC.V /, viewed as a bundle of Z=2 graded algebras. A careful analysis of this groupshows that it depends only on the class of C.V / in GBr.X/, the three invariants beingrespectively the rank of V mod. 2, w 1 .V / and ˇ.w 2 .V // D W 3 .V /, where w 1 .V /,w 2 .V / are the first two Stiefel–Whitney classes of V . In particular, if V is a c spinorialbundle of even rank, one recovers a well-known theorem of Atiyah and Hirzebruch.On the other hand, if X is a compact manifold, it is well known that such a Thomisomorphism theorem induces a pairing between K-groupsK˛.X/ K˛0.X/ ! Zif ˛ C ˛0 is the class of C.V / in GBr.X/, where V is the tangent bundle of X.The necessity to revisit these ideas comes from a new interest in the subject becauseof its relations with Physics [43], as shown by the number of recent publications.However, for these applications, the first definition recalled above is not completesince the coefficient system is restricted to the torsion elements of H 3 .XI Z/. Asit was pointed out by J. Rosenberg [38] and later on by C. Laurent, J.-L. Tu, P. Xu1 See the appendix for a short history of the subject.2 Strictly speaking, this product is defined up to non canonical isomorphism ; see 2.1 for more details.