process

34 R. Meyerfor all A 22 KK G ; this can be proved like the similar adjointness statements in §4.1,using the embedding A ! Res H G IndG H.A/ as functions supported on H G andthe correspondence IndH G ResH G .A/ Š C 0.G=H; A/ ! K.`2G=H / ˝ A A. It isimportant here that H G is an open subgroup. By the way, if H G is a cocompactsubgroup (which means finite index in the discrete case), then Res H Gis the left-adjointof IndH G instead.Definition 68. We let CIbe the subcategory of all objects of KK G of the form Ind G H .A/for A 22 KK H and H 2 F . Let hCIi be the smallest class in KK G that contains CIand is closed under KK G -equivalence, countable direct sums, suspensions, and exacttriangles.Equivalently, hCIi is the localising subcategory generated by CI. This is oursubstitute for the category of .G; F /-CW-complexes.Definition 69. Let CC be the class of all objects of KK G with Res H G.A/ Š 0 for allH 2 F .Theorem 70. If P 2hCIi, N 2 CC, then KK G .P; N / D 0. Furthermore, for anyA 22 KK G there is an exact triangle P ! A ! N ! †P with P 2hCIi, N 2 CC.Definitions 68–69 and Theorem 70 are taken from [38]. The map F .P / ! F .A/for a functor F W KK G ! C is analogous to the assembly map in Definition 67 anddeserves to be called the Baum–Connes assembly map for F .We can use the tensor product in KK G to simplify the proof of Theorem 70: oncewe have a triangle P C ! C ! N C ! †P C with P C 2hCIi, N C 2 CC, thenA ˝ P C ! A ˝ C ! A ˝ N C ! †A ˝ P Cis an exact triangle with similar properties for A. It makes no difference whether weuse ˝min or ˝max here. The map P C ! C in KK G .P C ; C/ is analogous to the mapE.G; F / ! ?. It is also called a Dirac morphism for G because the K-homologyclasses of Dirac operators on smooth spin manifolds provided the first important examples[31].The two assembly map constructions with spaces and C -algebras are not justanalogous but provide the same Baum–Connes assembly map. To see this, we mustunderstand the passage from the homotopy category of spaces to KK. Usually, wemap spaces to operator algebras using the commutative C -algebra C 0 .X/. But thisconstruction is only functorial for proper continuous maps, and the functoriality iscontravariant. The assembly map for, say, G D Z is related to the non-proper mapp W R ! ?, which does not induce a map C ! C 0 .R/; even if it did, this map wouldstill go in the wrong direction. The wrong-way functoriality in KK provides an elementp Š 2 KK 1 .C 0 .R/; C/ instead, which is the desired Dirac morphism up to a shift in thegrading. This construction only applies to manifolds with a Spin c -structure, but it canbe generalised as follows.On the level of Kasparov theory, we can define another functor from suitable spacesto KK that is a covariant functor for all continuous maps. The definition uses a notion