process

Twisted K-theory – old and new 131We shall now provide two different proofs of the theorem.The first one, more elementary in spirit, consists in using a Mayer–Vietoris argumentwhich we can apply here since the two sides of the formula above behave as cohomologytheories 21 with respect to the base X. Therefore, we may assume that A and V aretrivial: this is a special case of the theorem stated in [30], pp. 211/212.The second one is more subtle and may be generalized to the equivariant case. Letus first describe the Thom isomorphism for complex V . This is a slight modificationof Atiyah’s argument using the elliptic Dolbeault complex [3]. More precisely, weconsider the composite map W K A .X/ ! K A y˝C.V / .X/ t ! K A .V /:The first map is the cup-product with the algebraic “Thom class” which is ƒV providedwith the classical Clifford graded module structure. This map is an isomorphismfrom well-known algebraic considerations (Morita equivalence). Therefore t is anisomorphism if and only if is an isomorphism. On the other hand, is just thecup-product with the topological Thom class T V which belongs to the usual topologicalK-theory K.V / of Atiyah and Hirzebruch [5], [6]. In order to prove that is anisomorphism, we may now use the exact sequence in 3.3 to reduce ourselves to theungraded twisted case. In other words, it is enough to show that the cup-product withT V induces an isomorphism‰ W K .A/ .X/ ! K .A/ .V /:In order to prove this last point, Atiyah defines a reverse map 22‰ 0 W K .A/ .V / ! K .A/ .X/:He shows that ‰ 0 ‰ D Id and, by an ingenious argument, deduces that ‰‰ 0 D Id aswell.Now, as soon as Theorem 4.2 is proved for complex V , the general case followsfrom a trick already used in [30], p. 241: we consider the following three Thom homomorphismswhich behave “transitively”:K A y˝C.V /y˝C.V /y˝C.V / .X/ KA y˝C.V /y˝C.V / .V / KA y˝C.V / .V ˚ V/ K A .V ˚ V ˚ V/.We know that the composites of two consecutive arrows are isomorphisms since V ˚Vcarries a complex structure. It follows that the first arrow is an isomorphism, which isessentially the theorem stated (using Morita equivalence again).21 Strictly speaking, one has to “derive” the two members of the formula, which can be done easily sincethey are Grothendieck groups of graded Banach categories.22 More precisely, one has to define an index map parametrized by a Banach bundle, which is also classical[20].