process

66 A. Bartels, S. Echterhoff, and W. LückThis will be done by constructing a map of spectramap H .H=‹; X/ C ^Or.H / E H ! map G .G=‹; ˛X/ C ^Or.G/ E G :We follow the constructions in [13, Section 1]. The homomorphism ˛ induces a covariantfunctor Or.˛/W Or.H / ! Or.G/ by sending H=L to ˛.H=L/ D G=˛.L/.Given a contravariant functor Y W Or.H / ! Spaces, we can assign to it its inductionwith Or.˛/ which is a contravariant functor ˛Y W Or.G/ ! Spaces. Given a contravariantfunctor Z W Or.G/ ! Spaces, we can assign to it its restriction which isthe contravariant functor ˛Z WD Z ı Or.˛/W Or.G/ ! Spaces. Induction ˛ and˛ form an adjoint pair. Given an H -CW-complex X, there is a natural identification˛.map H .H=‹; X// D map G .G=‹; ˛X/. Using [13, Lemma 1.9] we get for anH -CW-complex X a natural map of spectramap H .H=‹; X/ C ^Or.H / ˛E G ! map G .G=‹; ˛X/ C ^Or.G/ E G :Given an H -set S, we obtain a functor of groupoids t H .S/ ! t G .˛S/sending s 2 Sto .1; s/ 2 G ˛ S and a morphism in t H .S/ given by a group element h to the onein t G .˛S/ given by ˛.h/. This yields a natural transformation of covariant functorsOr.H / ! Groupoids # from t H ! t G ı Or.˛/. Composing with the functor Egives a natural transformation of covariant functors Or.H / ! Spectra from E H to˛E G . It induces a map of spectramap H .H=‹; X/ C ^Or.H / E H ! map H .H=‹; X/ C ^Or.H / ˛E G :Its composition with the maps of spectra constructed beforehand yields the desired mapof spectra map H .H=‹; X/ C ˝Or.H / E H ! map G .G=‹; ˛X/ C ˝Or.G/ E G .We omit the straightforward proof that the axioms of an induction structure aresatisfied. This finishes the proof of Theorem 6.1.The statement about the natural transformation TW E ! F is obvious.8 Some K-theory spectra associated to groupoidsThe last step in completing the proof of Theorem 6.1 is to prove the following Theorem8.1 because then we can apply it in combination with Lemma 7.1. (Actuallywe only need the version of Theorem 8.1, where G is given by a group .) LetGroupoids finker # G be the subcategory of Groupoids # G which has the same objectsand for which a morphism from F 0 W G 0 ! G to F 1 W G 1 ! G given by a functorF W G 0 ! G 1 satisfying F 1 ı F D F 0 has the property that for every object x 2 G 0the group homomorphism aut G0 .x/ ! aut G1 .F .x// induced by F has a finite kernel.Denote by Rings, -Rings, and C -C Algebras the categories of rings, rings withinvolution and C -algebras.Theorem 8.1. Let G be a fixed groupoid. Let R W G ! Rings, R W G !-Rings, orAW G ! C Algebras respectively be a covariant functor. Then there exist covariant