process

36 R. Meyer5.3 The Dirac-dual-Dirac method and geometry. Let us compare the approachesin §5.1 and §5.2! The bad thing about the C -algebraic approach is that it applies tofewer theories. The good thing about it is that Kasparov theory is so flexible that anycanonical map between K-theory groups has a fair chance to come from a morphismin KK G which we can construct explicitly.For some groups, the Dirac morphism in KK G .P; C/ is a KK-equivalence:Theorem 73 (Higson–Kasparov [26]). Let the group G be amenable or, more generally,a-T-menable. Then the Dirac morphism for G is a KK G -equivalence, so that G satisfiesthe Baum–Connes conjecture with coefficients.The class of groups for which the Dirac morphism has a one-sided inverse is evenlarger. This is the point of the Dirac-dual-Dirac method. The following definitionin [38] is based on a simplification of this method:Definition 74. A dual Dirac morphism for G is an element 2 KK G .C;P/ with ı D D id P .If such a dual Dirac morphism exists, then it provides a section for the assemblymap F .P ˝ A/ ! F .A/ for any functor F W KK G ! C and any A 22 KK G , so thatthe assembly map is a split monomorphism. Currently, we know no group without adual Dirac morphism. It is shown in [16], [18], [19] that the existence of a dual Diracmorphism is a geometric property of G because it is related to the invertibility of anotherassembly map that only depends on the coarse geometry of G (in the torsion-free case).Instead of going into this construction, we briefly indicate another point of view thatalso shows that the existence of a dual Dirac morphism is a geometric issue. Let P bean abstract dual for some space X (like E.G; F /). The duality isomorphisms in Definition71 are determined by two pieces of data: a Dirac morphism D 2 KK G .P; C/ anda local dual Dirac morphism ‚ 2 RKK G .XI C;P/. The notation is motivated by thespecial case of a Spin c -manifold X with P D C 0 .X/, where D is the K-homology classdefined by the Dirac operator and is defined by a local construction involving pointwiseClifford multiplications. If X D E.G; F /, then it turns out that 2 KK G .C;P/is adual Dirac morphism if and only if the canonical map KK G .C;P/! RKK G .XI C;P/maps 7! ‚. Thus the issue is to globalise the local construction of ‚. This is possibleif we know, say, that X has non-positive curvature. This is essentially how Kasparovproves the Novikov conjecture for fundamental groups of non-positively curved smoothmanifolds in [31].References[1] M. F. Atiyah, Global theory of elliptic operators, in Proc. Internat. Conf. on FunctionalAnalysis and Related Topics (Tokyo, 1969), University of Tokyo Press, Tokyo 1970, 21–30.[2] William Arveson, An invitation to C -algebras, Grad. Texts in Math. 39, Springer-VerlagNew York 1976.[3] Saad Baaj, Georges Skandalis, C -algèbres de Hopf et théorie de Kasparov équivariante,K-Theory 2 (1989), 683–721.