process

Inheritance of isomorphism conjectures under colimits 53Definition 3.3 ((Strongly) continuous equivariant homology theory). An equivarianthomology theory H ‹ over is called continuous if for every group .G; / over andevery directed system of subgroups fG i j i 2 I g of G with G D S i2I G i the ƒ-map(see (3.2))tn G .fg/W colim i2I H G in .fg/ ! H n G .fg/is an isomorphism for every n 2 Z.An equivariant homology theory H ‹ over is called strongly continuous if forevery group .G; / over and every directed system of groups fG i j i 2 I g withG D colim i2I G i and structure maps i W G i ! G for i 2 I the ƒ-mapis an isomorphism for every n 2 Z.tn G .fg/W colim i2I H G in .fg/ ! H n G .fg/Here and in the sequel we view G i as a group over by ı i W G ii W G i ! G as a morphism of groups over .! andLemma 3.4. Let .G; / be a group over . Consider a directed system of groupsfG i j i 2 I g with G D colim i2I G i and structure maps i W G i ! G for i 2 I . Let.X; A/ be a G-CW-pair. Suppose that H ‹ is strongly continuous.Then the ƒ-homomorphism (see (3.2))is bijective for every n 2 Z.t G n .X; A/W colim i2I H G in . i .X; A// Š! HGn .X; A/Proof. The functor sending a directed systems of ƒ-modules to its colimit is an exactfunctor and compatible with direct sums over arbitrary index maps. If .X; A/ is a pairof G-CW-complexes, then .i X; i A/ is a pair of G i-CW-complexes. Hence thecollection of maps ftn G .X; A/ j n 2 Zg is a natural transformation of G-homologytheories of pairs of G-CW-complexes which satisfy the disjoint union axiom. Hence inorder to show that tn G .X; A/ is bijective for all n 2 Z and all pairs of G-CW-complexes.X; A/, it suffices by Lemma 2.2 to prove this in the special case .X; A/ D .G=H; ;/.For i 2 I let k i W G i =i 1 .H / !i .G=H / be the G i-map sending g 1 i i.H /to i .g i /H . Consider a directed system of ƒ-modules fH G in .G i =i1 .H // j i 2 I gwhose structure maps for i;j 2 I;i j are given by the compositeH G in .G i= i1 .H // ind i;j! H G jn .G j i;j G i = i 1 .H //H G jn .f i;j /! H G jn .G j = 1j .H //for the G j -map f i;j W G j i;j G i = 1i.H / ! G j = 1j.H / sending .g j ;g i 1i.H //