process

Inheritance of isomorphism conjectures under colimits 65Then we can associate to it an equivariant homology theory H ‹.;I E/ (with valuesin Z-modules) over such that for every group .G; / over and subgroup H Gwe have a natural identificationH H n .fgI E/ D H G n .G=H; E/ D n.E.H //:If TW E ! F is a natural transformation of such functors Groupoids # !Spectra, then it induces a transformation of equivariant homology theories over H ‹ . I T/W H ‹ .; I E/ ! H ‹ .; I F/such that for every group .G; / over and subgroup H G the homomorphismHn H .fgI T/W H n H .fgI E/ ! H n H .fgI F/ agrees under the identification above with n .T.H //W n .E.H // ! n .F.H //.Proof. We begin with explaining how we can associate to a group .G; / over aG-homology theory H G . I E/ with the property that for every subgroup H G wehave an identificationHn G .G=H; E/ D n.E.H //:We just follow the construction in [13, Section 4]. Let Or.G/ be the orbit category of G,i.e., objects are homogenous spaces G=H and morphisms are G-maps. Given a G-set S,the associated transport groupoid t G .S/ has S as set of objects and the set of morphismsfrom s 0 2 S to s 1 2 S consists of the subset fg 2 G j gs 1 D s 2 g of G. Compositionis given by the group multiplication. A G-map of sets induces a functor betweenthe associated transport groupoids in the obvious way. In particular the projectionG=H ! G=G induces a functor of groupoids pr S W t G .S/ ! t G .G=G/ D G. Thust G .S/ becomes an object in Groupoids # by the composite ı pr S . We obtaina covariant functor t G W Or.G/ ! Groupoids # . Its composition with the givenfunctor E yields a covariant functorNow defineE G WD E ı t G W Or.G/ ! Spectra:H G .X; AI E/ WD H G . I EG /;where H G . I EG / is the G-homology theory which is associated to E G W Or.G/ !Spectra and defined in [13, Section 4 and 7]. Namely, if X is a G-CW-complex, we canassign to it a contravariant functor map G .G=‹; X/W Or.G/ ! Spaces sending G=Hto map G .G=H; X/ D X H and put H G n .XI EG / WD n .map G .G=‹; X/ C ^Or.G/ E G /for the spectrum map G .G=‹; X/ C ^Or.G/ E G (which is denoted in [13] bymap G .G=‹; X/ C ˝Or.G/ E G ).Next we have to explain the induction structure. Consider a group homomorphism˛ W .H; / ! .G; / of groups over and an H -CW-complex X. We have to constructa homomorphismH H n .XI E/ ! H G n .˛XI E/: