process

Inheritance of isomorphism conjectures under colimits 55is contained in the image of the map (3.5).Choose an index j with j i and g 2 im. j /. Then the structure map for i j isa map H G in .i G=H / ! H G jn .j G=H / which sends the summand corresponding toG i .gH / 2 G i n.i G=H / to the summand corresponding to G j .1H / 2 G j n.j G=H /which is by definition the image ofH G jn .k j /W H G jn .G j = j1 .H // ! H G jn . j G=H /:Obviously the image of composite of the last map with the structure mapH G jn . j G=H / ! colim i2I H G in . i G=H /is contained in the image of the map (3.5). Hence the map (3.5) is surjective. Thisfinishes the proof of Lemma 3.4.4 Isomorphism conjectures and colimitsA family F of subgroups of G is a collection of subgroups of G which is closed underconjugation and taking subgroups. Let E F .G/ be the classifying space associated toF . It is uniquely characterized up to G-homotopy by the properties that it is a G-CWcomplexand that E F .G/ H is contractible if H 2 F and is empty if H … F . For moreinformation about these spaces E F .G/ we refer for instance to the survey article [29].Given a group homomorphism W K ! G and a family F of subgroups of G, definethe family F of subgroups of K by F D fH K j .H/ 2 F g: (4.1)If is an inclusion of subgroups, we also write F j K instead of F .Definition 4.2 (Isomorphism conjecture for H ‹ ). Fix a group and an equivarianthomology theory H ‹ with values in ƒ-modules over .A group .G; / over together with a family of subgroups F of G satisfies theisomorphism conjecture (for H ‹) if the projection pr W E F .G/ !fg to the one-pointspacefg induces an isomorphismH G n .pr/W H G n .E F .G// Š ! H G n .fg/for all n 2 Z.From now on fix a group and an equivariant homology theory H ‹ over .Theorem 4.3 (Transitivity principle). Let .G; / be a group over . Let F G befamilies of subgroups of G. Assume that for every element H 2 G the group .H; j H /over satisfies the isomorphism conjecture for F j H .Then the up to G-homotopy unique map E F .G/ ! E G .G/ induces an isomorphismHn G.E F .G// ! Hn G.E G .G// for all n 2 Z. In particular, .G; / satisfies theisomorphism conjecture for G if and only if .G; / satisfies the isomorphism conjecturefor F .