1 Twisted K - Theory Old and New

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 [K1] (a precursor to twisted K-theory), if V is a real 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 group KC(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, w1(V) and β(w2(V)) = W3(V), where w1(V), w2(V) are the first two Stiefel-Whitney classes of V. In particular, if V is a cspinorial bundle of even rank, one recovers a well-known theorem of Atiyah and Hirzebruch [AH]. On the other hand, if X is a compact connected manifold, it is well-known that such a Thom isomorphism theorem induces a pairing between K-groups