process

Inheritance of isomorphism conjectures under colimits 59(i) Let G be the directed union G D S i2I G i of subgroups G i . Suppose that H ‹ is continuous and that the fibered isomorphism conjecture is true for .G i ;j Gi /and C.G i / for all i 2 I .Then the fibered isomorphism conjecture is true for .G; / and C.G/.(ii) Let fG i j i 2 I g be a directed system of groups with G D colim i2I G i andstructure maps i W G i ! G. Suppose that H ‹ is strongly continuous and thatthe fibered isomorphism conjecture is true for .G i ; ı i / and C.G i // for alli 2 I .Then the fibered isomorphism conjecture is true for .G; / and C.G/.Proof. (i) The proof is analogous to the one in [4, Proposition 3.4], where the case Df1g is considered.(ii) Because C is closed under taking quotients we conclude C.G i / i C.G/.Now the claim follows from Theorem 5.2 and Lemma 5.3.Corollary 5.7. (i) Suppose that H ‹ is continuous. Then the ( fibered) isomorphismconjecture for .G; / and C.G/ is true for all groups .G; / over if and only if it istrue for all such groups where G is a finitely generated group.(ii) Suppose that H ‹ is strongly continuous. Then the fibered isomorphism conjecturefor .G; / and C.G/ is true for all groups .G; / over if and only if it is true forall such groups where G is finitely presented.Proof. Let .G; / be a group over where G is finitely generated. Choose a finitelygenerated free group F together with an epimorphism W F ! G. Let K be the kernelof . Consider the directed system of finitely generated subgroups fK i j i 2 I g of K.Let SK i be the smallest normal subgroup of K containing K i . Explicitly SK i is givenby elements which can be written as finite products of elements of the shape fk i f 1for f 2 F and k 2 K i . We obtain a directed system of groups fF=SK i j i 2 I g,where for i j the structure map i;j W F=SK i ! F= SK j is the canonical projection.If i W F=SK i ! F=K D G is the canonical projection, then the collection of mapsŠf i j i 2 I g induces an isomorphism colim i2I F=SK i ! G. By construction for eachi 2 I the group F=SK i is finitely presented and the fibered isomorphism conjectureholds for .F= SK i ; ı i / and C.F= SK i / by assumption. Theorem 5.6 (ii) implies thatthe fibered Farrell–Jones conjecture for .G; / and C.G/ is true.6 Some equivariant homology theoriesIn this section we will describe the relevant homology theories over a group andshow that they are (strongly) continuous. (We have defined the notion of an equivarianthomology theory over a group in Definition 2.3.)