Twisted K-theory – old and new

Abstract

Abstract. Twisted K-theory has its origins in the author's PhD thesis [27] and in a paper with P. Donovan [19]. The objective of this paper is to revisit the subject in the light of new developments inspired by Mathematical Physics. See for instance E. Witten [42], J. Rosenberg [37], C. Laurent-Gentoux, J.-L. Tu, P. Xu [41] and M.F. Atiyah, G. Segal [8], among many authors. We also prove some new results in the subject: a Thom isomorphism, explicit computations in the equivariant case and new cohomology operations. Mathematics Subject Classification (2000). 55N15, 19K99, 19L47, 46L80Keywords. K-theory, Brauer group, Fredholm operators, Poincaré duality, Clifford algebras, Adams operations. Some history and motivation about this paperThe subject "K-theory with local coefficients", now called "twisted K-theory", was introduced by P. Donovan and the author in [19] almost forty years ago.1 It associates to a compact space X and a "local coefficient system"an abelian group K α (X) which generalizes the usual Grothendieck-Atiyah-Hirzebruch K-theory of X when we restrict α being in Z/2 (cf. [5]). This "graded Brauer group" GBr(X) has the following group structure: if α = (ε, w 1 , W 3 ) andis the Bockstein homomorphism. With this definition, one has a generalized cup-product 2 .The motivation for this definition is to give in K-theory a satisfactory Thom isomorphism and Poincaré duality pairing which are analogous to the usual ones in cohomology with local coefficients. More precisely, as proved in [27], if V is a real 1 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. 2Max KAROUBI vector bundle on a compact space X with a positive metric, the K-theory of the Thom space of V is isomorphic to a certain "algebraic" group K C(V ) (X) associated to the Clifford bundle C(V ), viewed as a bundle of Z/2−graded algebras. A careful analysis of this group shows that it depends only on the class of C(V ) in GBr(X), the three invariants being respectively the rank of V mod. 2, w 1 (V ) and β(w 2 (V )) = 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 spinorial bundle 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 Thom isomorphism theorem induces a pairing between K-groups, where V is the tangent bundle of X.The necessity to revisit these ideas comes from a new interest in the subject because of its relations with Physics [42], as shown by the number of recent publications. However, for these applications, the first definition recalled above is not complete since the coefficient system is restricted to the torsion elements of H 3 (X; Z). 3 . We hope have been "pedagogical" in some sense to the non experts.However, this paper is not just historical. It presents the theory with another point of view and contains some new results. We extend the Thom isomorphism to ...

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