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50 A. Bartels, S. Echterhoff, and W. Lück• Disjoint union axiom. Let fX i j i 2 I g be a family of G-CW-complexes. Denoteby j i W X i ! `i2I X i the canonical inclusion. Then the mapMHn G .j i/W M Hn G .X i/ Š a ! HnG X ii2I i2Ii2Iis bijective.Let H G and KG be G-homology theories. A natural transformation T W H G !K G of G-homology theories is a sequence of natural transformations T n W H G n ! KG nof functors from the category of G-CW-pairs to the category of ƒ-modules which arecompatible with the boundary homomorphisms.Lemma 2.2. Let T W HG ! KG be a natural transformation of G-homology theories.Suppose that T n .G=H / is bijective for every homogeneous space G=H and n 2 Z.Then T n .X; A/ is bijective for every G-CW-pair .X; A/ and n 2 Z.Proof. The disjoint union axiom implies that both G-homology theories are compatiblewith colimits over directed systems indexed by the natural numbers (such as the systemgiven by the skeletal filtration X 0 X 1 X 2 [ n0 X n D X). The argumentfor this claim is analogous to the one in [40, 7.53]. Hence it suffices to prove thebijectivity for finite-dimensional pairs. Using the axioms of a G-homology theory, thefive lemma and induction over the dimension one reduces the proof to the special case.X; A/ D .G=H; ;/.Next we present a slight variation of the notion of an equivariant homology theoryintroduced in [28, Section 1]. We have to treat this variation since we later want to studycoefficients over a fixed group which we will then pullback via group homomorphismswith as target. Namely, fix a group . A group .G; / over is a group G togetherwith a group homomorphism W G ! . A map ˛ W .G 1 ; 1 / ! .G 2 ; 2 / of groupsover is a group homomorphisms ˛ W G 1 ! G 2 satisfying 2 ı ˛ D 1 .Let ˛ W H ! G be a group homomorphism. Given an H -space X, define theinduction of X with ˛ to be the G-space denoted by ˛X which is the quotient of G Xby the right H -action .g; x/ h WD .g˛.h/; h 1 x/ for h 2 H and .g; x/ 2 G X.If ˛ W H ! G is an inclusion, we also write ind G H instead of ˛. If .X; A/ is anH -CW-pair, then ˛.X; A/ is a G-CW-pair.Definition 2.3 (Equivariant homology theory over a group ). An equivariant homologytheory H ‹ with values in ƒ-modules over a group assigns to every group .G; /over a G-homology theory HG with values in ƒ-modules and comes with the followingso called induction structure: given a homomorphism ˛ W .H; / ! .G; / ofgroups over and an H -CW-pair .X; A/, there are for each n 2 Z natural homomorphismssatisfyingind˛ W H H n .X; A/ ! H G n .˛.X; A// (2.4)