process

20 R. MeyerF.B/ F.E 0 / F .A/ is a split extension in C. Since both F.f 0 / and F.fC 0 /are sections for it, we get a map F.fC 0 / F.f 0 /W F .A/ ! F.B/. Thus a quasihomomorphisminduces a map F .A/ ! F.B/if F is split-exact. The formal propertiesof this construction are summarised in [11].GivenaC -algebra A, there is a universal quasi-homomorphism out of A. LetQ.A/ WD A t A be the coproduct of two copies of A and let A W Q.A/ ! A be thefolding homomorphism that restricts to id A on both factors. Let q.A/ be its kernel.The two canonical embeddings A ! A t A are sections for the folding homomorphism.Hence we get a quasi-homomorphism A Q.A/ F q.A/. The universalproperty of the free product shows that any quasi-homomorphism yields a G-equivariant -homomorphism q.A/ ! B.Theorem 46. Suppose S is closed under split extensions and tensor products withC.Œ0; 1/ and K.`2N/. IfF W S ! C is C -stable and split-exact, then F is homotopyinvariant.This is a deep result of Nigel Higson [24]; a simple proof can be found in [11].Besides basic properties of quasi-homomorphisms, it uses that inner endomorphismsact identically on C -stable functors (Example 40).Actually, the literature only contains Theorem 46 for functors on C alg. But theproof in [11] works for functors on categories S as above.3.4.3 Exact functorsDefinition 47. We call F exact if F.K/ ! F.E/ ! F.Q/ is exact (at F.E/) forany extension K E Q in S. More generally, given a class E of extensionsin S like, say, the class of equivariantly cp-split extensions, we define exactness forextensions in E.It is easy to see that exact functors are additive.Most functors we are interested in satisfy homotopy invariance and Bott periodicity,and these two properties prevent a non-zero functor from being exact in the strongersense of being left or right exact. This explains why our notion of exactness is muchweaker than usual in homological algebra.It is reasonable to require that a functor be part of a homology theory, that is,a sequence of functors .F n / n2Z together with natural long exact sequences for allextensions [54]. We do not require this additional information because it tends to behard to get a priori but often comes for free a posteriori:Proposition 48. Suppose that F is homotopy invariant and exact (or exact for equivariantlycp-split extensions). Then F has long exact sequences of the form!F Sus.K/ ! F Sus.E/ ! F Sus.Q/ ! F.K/ ! F.E/ ! F.Q/for any (equivariantly cp-split) extension K E Q. In particular, F is splitexact.