process

Inheritance of isomorphism conjectures under colimits 51• Compatibility with the boundary homomorphisms. @ G n ı ind˛ D ind˛ ı@ H n .• Functoriality. Let ˇ W .G; / ! .K; / be another morphism of groups over .Then we have for n 2 Zindˇı˛ D H K n .f 1/ ı indˇ ı ind˛ W H H n .X; A/ ! H K n ..ˇ ı ˛/ .X; A//;where f 1 W ˇ˛.X; A/ Š ! .ˇ ı ˛/ .X; A/, .k;g;x/7! .kˇ.g/; x/ is the naturalK-homeomorphism.• Compatibility with conjugation. Let .G; / be a group over and let g 2 G be anelement with .g/ D 1. Then the conjugation homomorphisms c.g/W G ! Gdefines a morphism c.g/W .G; / ! .G; / of groups over . Let f 2 W .X; A/ !c.g/ .X; A/ be the G-homeomorphism which sends x to .1; g 1 x/ in G c.g/.X; A/.Then for every n 2 Z and every G-CW-pair .X; A/ the homomorphismagrees with H G n .f 2/.ind c.g/ W H G n .X; A/ ! H G n .c.g/ .X; A//• Bijectivity. If ˛ W .H; / ! .G; / is a morphism of groups over such thatthe underlying group homomorphism ˛ W H ! G is an inclusion of groups, thenind˛ W Hn H .fg/ ! H n G.˛fg/ D Hn G .G=H / is bijective for all n 2 Z.Definition 2.3 reduces to the one of an equivariant homology in [28, Section 1] ifone puts Df1g.Lemma 2.5. Let ˛ W .H; / ! .G; / be a morphism of groups over . Let .X; A/ bean H -CW-pair such that ker.˛/ acts freely on X A. Thenis bijective for all n 2 Z.ind˛ W H H n .X; A/ ! H G n .˛.X; A//Proof. Let F be the set of all subgroups of H whose intersection with ker.˛/ is trivial.Obviously, this is a family, i.e., closed under conjugation and taking subgroups. A H -CW-pair .X; A/ is called a F -H -CW-pair if the isotropy group of any point in X Abelongs to F .AH -CW-pair .X; A/ is a F -H -CW-pair if and only if ker.˛/ acts freelyon X A.The n-skeleton of ˛.X; A/ is ˛ applied to the n-skeleton of .X; A/. Let .X; A/ bean H -CW-pair and let f W A ! B be a cellular H -map of H -CW-complexes. Equip.X [ f B;B/ with the induced structure of a H -CW-pair. Then there is an obviousnatural isomorphism of G-CW-pairs˛.X [ f B;B/ Š ! .˛X [˛f ˛B;˛B/: