process

Categorical aspects of bivariant K-theory 23The functor KK G S W S ! KKG S is the universal split-exact C -stable functor; inparticular, KK G .S/ is an additive category. In addition, it has the following propertiesand is, therefore, universal among functors on S with some of these extra properties:• it is homotopy invariant;• it is exact for G-equivariantly cp-split extensions;• it satisfies Bott periodicity, that is, in KK G there are natural isomorphismsSus 2 .A/ Š A for all A 22 KK G .Corollary 51. Let F W S ! C be split-exact and C -stable. Then F factors uniquelythrough KK G S , is homotopy invariant, and satisfies Bott periodicity. A KKG -equivalenceA ! B in S induces an isomorphism F .A/ ! F.B/.We will view the universal property of Theorem 50 as a definition of KK G and thusof the groups KK G 0 .A; B/. We also letKK G n .A; B/ WD KKG A; Sus n .B/ Isince the Bott periodicity isomorphism identifies KK G 2 Š KKG 0 , this yields a Z=2-gradedtheory.Now we describe KK G 0 .A; B/ more concretely. Recall A K WD A ˝ K.L 2 G/.Proposition 52. Let A and B be two G-C -algebras. There is a natural bijectionbetween the morphism sets KK G 0 .A; B/ in KKG and the set Œq.A K /; B K ˝ K.`2N/ ofhomotopy classes of G-equivariant -homomorphisms from q.A K / to B K ˝ K.`2N/.Proof. The canonical functor G-C sep ! KK G is C -stable and split-exact, andtherefore homotopy invariant by Theorem 46 (this is already asserted in Theorem 50).Proposition 44 yields that it is additive for coproducts. Split-exactness for the splitextension q.A/ Q.A/ A shows that id A 0W Q.A/ ! A restricts to a KK G -equivalence q.A/ A. Similarly, C -stability yields KK G -equivalences A A K andB B K ˝ K.`2N/. Hence homotopy classes of -homomorphisms from q.A K / toB K˝K.`2N/ yield classes in KK G 0 .A; B/. Using the concrete description of Kasparovcycles, which we have not discussed, it is checked in [36] that this map yields a bijectionas asserted.Another equivalent description isKK G 0 .A; B/ Š Œq.A K/ ˝ K.`2N/;q.B K / ˝ K.`2N/Iin this approach, the Kasparov product becomes simply the composition of morphisms.Proposition 52 suggests that q.A K / and B K ˝K.`2N/ may be the cofibrant and fibrantreplacement of A and B in some model category related to KK G . But it is not clearwhether this is the case. The model category structure constructed in [28] is certainlyquite different.