process

52 A. Bartels, S. Echterhoff, and W. LückNow we proceed as in the proof of Lemma 2.2 but now considering the transformationsind˛ W H H n .X; A/ ! H G n .˛.X; A//only for F -H -CW-pairs .X; A/. Thus we can reduce the claim to the special case.X; A/ D H=L for some subgroup L H with L \ ker.˛/ Df1g. This specialcase follows from the following commutative diagram whose vertical arrows are bijectiveby the axioms and whose upper horizontal arrow is bijective since ˛ induces anisomorphism ˛j L W L ! ˛.L/:H L n .fg/ind˛jL W L!˛.L/Hn˛.L/.fg/H H nind H Lind˛.H=L/ ind G˛.L/HGn .˛H=L/ D Hn G.G=˛.L//.3 Equivariant homology theories and colimitsFix a group and an equivariant homology theory H ‹ with values in ƒ-modules over .Let X be a G-CW-complex. Let ˛ W H ! G be a group homomorphism. Denoteby ˛X the H -CW-complex obtained from X by restriction with ˛. We have alreadyintroduced the induction ˛Y of an H -CW-complex Y . The functors ˛ and ˛ areadjoint to one another. In particular the adjoint of the identity on ˛X is a naturalG-mapf.X;˛/W ˛˛X ! X: (3.1)It sends an element in G ˛ ˛X given by .g; x/ to gx.Consider a map ˛ W .H; / ! .G; / of groups over . Define the ƒ-mapa n D a n .X; ˛/W Hn H ind˛.˛X/ ! Hn G.˛˛X/ H n G.f .X;˛// ! Hn G.X/:If ˇ W .G; / ! .K; / is another morphism of groups over , then by the axioms ofan induction structure the composite Hn H .˛ˇX/ a n.ˇ X;˛/! Hn G.ˇX/ a n.X;ˇ/!Hn K.X/ agrees with a n.X; ˇ ı ˛/W Hn H .˛ˇX/ D Hn H ..ˇ ı ˛/ X/ ! Hn G .X/ fora K-CW-complex X.Consider a directed system of groups fG i j i 2 I g with G D colim i2I G i andstructure maps i W G i ! G for i 2 I and i;j W G i ! G j for i;j 2 I;i j .Weobtain for every G-CW-complex X a system of ƒ-modules fH G i .i X/ j i 2 I gwith structure maps a n .j X; i;j/W H G i .i X/ ! H G j .j X/. We get a map ofƒ-modulestn G .X; A/ WD colim i2I a n .X; i /W colim i2I H G in . i .X; A// ! H n G .X; (3.2) A/:The next definition is an extension of [4, Definition 3.1].