process

Inheritance of isomorphism conjectures under colimits 57Proof. (i) This follows from Theorem 4.4 (i) since C.G i / D C.G/j Gi holds for i 2 I .(ii) If G is the colimit of the directed system fG i j i 2 I g, then the subgroup K Gis the colimit of the directed system fi1 .K/ j i 2 I g. Hence we can assume G D Kwithout loss of generality.Since C is closed under quotients by assumption, we have C.G i / i C.G/ forevery i 2 I . Hence we can consider for any i 2 I the compositionH G in.E C.G i /.G i // ! H G in.E i C.G/.G i // ! H G in .fg/:Because of Theorem 4.4 (ii) it suffices to show that the second map is bijective. Byassumption the composition of the two maps is bijective. Hence it remains to show thatthe first map is bijective. By Theorem 4.3 this follows from the assumption that theisomorphism conjecture holds for every subgroup H G i and in particular for anyH 2 i C.G/ for C.G i/j H D C.H /.5 Fibered isomorphism conjectures and colimitsIn this section we also deal with the fibered version of the isomorphism conjectures.(This is not directly needed for the purpose of this paper and the reader may skip thissection.) This is a stronger version of the Farrell–Jones conjecture. The Fibered Farrell–Jones conjecture does imply the Farrell–Jones conjecture and has better inheritanceproperties than the Farrell–Jones conjecture.We generalize (and shorten the proof of) the result of Farrell–Linnell [16, Theorem7.1] to a more general setting about equivariant homology theories as developedin Bartels–Lück [3].Definition 5.1 (Fibered 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 the fibered isomorphism conjecture(for H ‹ ) if for each group homomorphism W K ! G the group .K; ı / over satisfies the isomorphism conjecture with respect to the family F .Theorem 5.2. Let .G; / be a group over . Let F be a family of subgroups of G. LetfG i j i 2 I g be a directed system of groups with G D colim i2I G i and structure mapsi W G i ! G. Suppose that H ‹ is strongly continuous and for every i 2 I the fiberedisomorphism conjecture holds for .G i ; ı i / andi F .Then the fibered isomorphism conjecture holds for .G; / and F .Proof. Let W K ! G be a group homomorphism. Consider the pullback of groupsK i i G iS iiK G.