process

Categorical aspects of bivariant K-theory 27Definition 55. A triangle A ! B ! C ! †A in KK G is called exact if it isisomorphic as a triangle to the mapping cone triangleSus.B/ ! Cone.f / ! A f ! Bfor some G-equivariant -homomorphism f .Alternatively, we can use G-equivariantly cp-split extensions in G-C sep. Anysuch extension I E Q determines a class in KK G 1 .Q; I / Š KKG 0 .Sus.Q/; I /,so that we get a triangle Sus.Q/ ! I ! E ! Q in KK G . Such triangles are calledextension triangles. A triangle in KK G is exact if and only if it is isomorphic to theextension triangle of a G-equivariantly cp-split extension [38].Theorem 56. With the suspension automorphism and exact triangles defined above,KK G is a triangulated category. So is KK G .S/ if S G-C sep is closed undersuspensions, G-equivariantly cp-split extensions, and Morita–Rieffel equivalence as inTheorem 50.Proof. That KK G is triangulated is proved in detail in [38]. We do not discuss thetriangulated category axioms here. Most of them amount to properties of mappingcone triangles that can be checked by copying the corresponding arguments for thestable homotopy category (and reverting arrows). These axioms hold for KK G .S/because they hold for KK G . The only axiom that requires more care is the existenceaxiom for exact triangles; it requires any morphism to be part of an exact triangle. Wecan prove this as in [38] using the concrete description of KK G 0 .A; B/ in Proposition 52.For some applications like the generalisation to KK G .S/, it is better to use extensiontriangles instead. Any f 2 KK G 0 .A; B/ Š KKG 1 .Sus.A/; B/ can be represented bya G-equivariantly cp-split extension K.H B / E Sus.A/, where H B is a fullG-equivariant Hilbert B-module, so that K.H B / is G-equivariantly Morita–Rieffelequivalent to B. The extension triangle of this extension contains f and belongs toKK G .S/ by our assumptions on S.Since model category structures related to C -algebras are rather hard to get (compare[28]), triangulated categories seem to provide the most promising formal setupfor extending results from classical spaces to C -algebras. An earlier attempt can befound in [54]. Triangulated categories clarify the basic bookkeeping with long exactsequences. Mayer–Vietoris exact sequences and inductive limits are discussed fromthis point of view in [38]. More importantly, this framework sheds light on more advancedconstructions like the Baum–Connes assembly map. We will briefly discussthis below.4.3 The Universal Coefficient Theorem. There is a very close relationship betweenK-theory and Kasparov theory. We have already seen that K .A/ Š KK .C;A/ isa special case of KK. Furthermore, KK inherits deep properties of K-theory such asBott periodicity. Thus we may hope to express KK .A; B/ using only the K-theory of