Assembly Maps for Group Extensions in K-theory and L-theory

In this paper we show that the Farrell-Jones isomorphism conjectures are inherited in group extensions for assembly maps in algebraic K-theory and L-theory. Introduction Under what assumptions are the Farrell-Jones isomorphism conjectures inherited by group extensions or subgroups? We will formulate a version of the standard conjectures (see Farrell-Jones [9]) with twisted coefficients in an additive category, and then study these questions via the continuously controlled assembly maps of [10, §7]. A formulation using the Davis-Lück assembly maps [8] has already been given by Bartels and Reich [3], and applied there to show inheritance by subgroups. Recall that the Farrell-Jones conjecture in algebraic K-theory asserts that certain “assembly” maps H n (EVCG;KR) → Kn(RG) are isomorphisms, for a given ring R, and all n ∈ Z. Here the space EVCG is the universal G-CW-complex for G-actions with virtually cyclic isotropy, and the left-hand side denotes equivariant homology with coefficients in the non-connective K-theory spectrum for the ring R. Theorem A. Let N → G π −→ K be a group extension, where N ⊳ G is a normal subgroup, and K is the quotient group. Let A an additive category with G-action. Suppose that (i) The group K satisfies the Farrell-Jones conjecture in algebraic K-theory, with twisted coefficients in any additive category with K-action. (ii) Every subgroup of G containing N as a subgroup, with virtually cyclic quotient, satisfies the Farrell-Jones conjecture in algebraic K-theory, with twisted coefficients in A. Then the group G satisfies the Farrell-Jones conjecture in algebraic K-theory, with twisted coefficients in A. This is a special case of a more general result (see Theorem 4.7). The same statement holds for algebraic L-theory as well, where the coefficient categories are additive categories with involution. The corrresponding result for the Baum-Connes conjecture was obtained by Oyono-Oyono [11], and our proof follows the outline given there. One of the Date: Sept. 4, 2007. Partially supported by NSERC grant A4000 and NSF grant DMS 9104026. The authors also wish to thank the SFB 478, Universität Münster, for hospitality and support. 1 2 IAN HAMBLETON, ERIK K. PEDERSEN, AND DAVID ROSENTHAL main points is that the most effective methods known for proving the standard FarrellJones conjectures (for particular groups G) also work for the twisted coefficient versions (compare [1], [2], [5], [6], [14], [15], and [16]). The following result implies the fibered isomorphism conjecture of [9]. Theorem B. Suppose that φ : H → G is a group homomorphism. Then the FarrellJones conjecture holds for G, with twisted coefficients in any G-category, if and only if the assembly map for H relative to the family generated by the subgroups φ(V ), V ⊂ G virtually cyclic, is an isomorphism with twisted coefficients in any H-category. The corresponding result for the Davis-Lück assembly maps was obtained by BartelsReich [3], who also pointed out a number of applications (see also Example 4.8, Example 4.9 and Corollary 4.10 below). One can check as in [10] that those assembly maps are equivalent to the continuously controlled assembly maps used in this paper. 1. Assembly via Controlled Categories The controlled categories of Pedersen [12], Carlsson-Pedersen [5], [7] are our main tool for identifying various different assembly maps. We will recall the definition of these categories, and then the usual assembly maps are obtained by applying functors H : G-CW -Complexes → Spectra as described in [10]. We will extend the earlier definitions in order to allow an additive category as coefficients, instead of just working with modules over a ring R. A formulation for assembly maps with coefficients in the setting of [8] has already been given in [3]. Following the method of [10], one can check that the two different descriptions give the same assembly maps. LetG be any discrete group, and letX be a G-CW complex (we will use a left G-action). Subspaces of the form G·D ⊂ X, withD compact in X, are called G-compact subspaces of X. More generally, a subspace whose closure has this form is called relatively G-compact. A resolution of X is a pair (X, p), where X is a free G-CW complex and p : X → X is a continuous G-equivariant map, such that for every G-compact set G ·D ⊂ X there exists a G-compact set G ·D ⊂ X such that p(G ·D) = G ·D. The notion of resolution comes from [12], and was developed further in [1, §3]. The original example was X = G × X, with the diagonal G-action and first factor projection. Let A be an additive category with involution, and suppose that A has a right Gaction compatible with the involution. This is a collection of covariant functors {g : A → A, ∀g ∈ G}, such that (g ◦ h) = h ◦ g and e = id. We require that the functors g commute with the involution ∗ : A → A (an involution is a contravariant functor with square the identity). Definition 1.1. Let (Z,X) be a G-CW pair, where X is a closed G-invariant subspace. Let Y = Z − X, and fix a resolution p : Z → Z, whose restriction to Y is denoted Y . The category D(Z,X;A) has objects A = (Ay) consisting of a collection of objects of A, indexed by y ∈ Y , and morphisms φ : A → B consisting of collections φ = (φy) of morphisms φy : Ay → Bz in A, indexed by y, z ∈ Y , satisfying: ASSEMBLY FOR GROUP EXTENSIONS 3 (i) the support {y ∈ Y |Ay 6= 0} is locally finite in Y , and relatively G-compact in Z. (ii) for each morphism φ : A → B, and for each y ∈ Y , the set {z |φy 6= 0 or φ y z 6= 0} is finite. (iii) the morphisms φ : A → B are continuously controlled atX ⊂ Z. For every x ∈ X, and for every Gx-invariant neighbourhood U of x in Z, there is a Gx-invariant neighbourhood V of x in Z so that φy = 0 and φ y z = 0 whenever p(y) ∈ (Y − U) and p(z) ∈ (V ∩ U ∩ Y ). If X = ∅, we use the shorter notation D(Z;A) := D(Z, ∅;A), and in this case the continous control condition (iii) on morphisms is vacuous. If S is a discrete left G-set, we denote by Dl(S × Z, S × X;A) the subcategory where the morphisms are S-levelpreserving: φ (s,z) (s,y) = 0 if s 6= s ′ ∈ S, for any y, z ∈ Y . The category D(Z,X;A) is an additive category with involution, where the dual of A is given by (A)y = A ∗ y for all y ∈ Y . It depends functorially on the pair (Z,X) of G-CW complexes. The actions of G on A and Z induce a right G-action on D(Z,X;A). For g ∈ G, we set (gA)y = g Agy and (gφ) z y = g (φ gy). The fixed subcategory will be denoted D(Z,X;A). If G = {e} is the trivial group, we use the notation D(Z,X;A). We have not included the resolution (Z, p) in the notation, because two different resolutions give G-equivalent categories (see [1, Prop. 3.5]). We can compare these fixed subcategories to the equivariant category BG(Z,X;R) defined in [10, §7]. Lemma 1.2. There is an equivalence of categories BG(Z,X;R) ≃ D (Z,X;A), when A is the category of finitely-generated free R-modules. Proof. We define a functor F : D(Z,X;A) → BG(Z,X;R) by sending an object A to the free R-module F (A)y = ⊕g∈GyA(g,y), for all y ∈ Y , with the obvious reference map to Y . Similarly, for a morphism φ : A → B, we define F (φ)y = (φ g,z g,y )g,g′∈G, for all y, z ∈ Y . The verification that this definition makes sense will be left to the reader. Conversely, we can define a functor F ′ : BG(Z,X;R) → D (Z,X;A) on objects by decomposing an object A = (Ay) of BG(Z,X;R) as Ay = ⊕g∈Gy (Ay)g, since Ay is a finitely-generated free RGy-module. Now we let F (A)(g,y) = (Ay)g, for all y ∈ Y , g ∈ G, and on morphisms by letting F (φ) ,z g,y = φ gz gy . Again the verifications will be left to the reader (technically we should work with a category equivalent to BG(Z,X;R), in which the objects are based: each A = R[T ], where T is a free G-set, and T is equipped with a reference map to X × [0, 1]). For applications to assembly maps, we will letX be a G-CW complex and Z = X×[0, 1] so that Y = X × [0, 1). The category just defined will be denoted D(X × [0, 1);A) := D(X × [0, 1], X × 1;A) . Let D(X × [0, 1);A)∅ denote the full subcategory of D (X × [0, 1);A) with objects A such that the intersection with the closure supp(A) = {(x, t) ∈ X × [0, 1) |A(x,t) 6= 0} ∩ (X × 1) 4 IAN HAMBLETON, ERIK K. PEDERSEN, AND DAVID ROSENTHAL is the empty set. Example 1.3. If A is the additive category of finitely generated free R-modules, then D(X× [0, 1);A)∅ is equivalent to the category of finitely generated free RG-modules, for any G-CW complex X. The quotient category will be denoted D(X × [0, 1);A), and we remark that this is a germ category (see [10, §7], [13], [5]). The objects are the same as in D(X × [0, 1);A) but morphisms are identified if they agree close to X = X × 1 (i.e. on the complement of a neighbourhood of X × 0). Here is a useful remark. Lemma 1.4 ([10]). The forgetful functor D l (S ×X × [0, 1);A) >0 → D(S ×X × [0, 1);A) is an equivalence of categories. Proof. In the germ category, every morphism has a representative which is level-preserving with respect to projection on S. The category D(X×[0, 1);A) is an additive category with involution, and we obtain a functor G-CW -Complexes → AddCat. The results of [4, 1.28, 4.2] now show that the functors F λ : G-CW -Complexes → Spectra defined by (1.5) F λ G(X;A) := { K (D(X × [0, 1);A)) L (D(X × [0, 1);A)) , where λ = K−∞ or λ = L−∞ respectively, are G-homotopy invariant and G-excisive. We can now extend the definition of the assembly maps to allow coefficients in any additive category with G-action. Definition 1.6. We define the continuously controlled assembly map with coefficients in A to be the map F λ G(X;A) → F λ G(•;A). From the methods of [10], the continuously controlled assembly map with coefficients is homotopy equivalent to the assembly map with coefficients constructed in [3]. The most important example to consider is when X = EVCG, in which case the Farrell-Jones conjecture with coefficients asserts that this assembly map is an equivalence. Given a discrete group G, a family of subgroups F of G, and coefficients A, we will refer to F λ G(EFG;A) → F λ G(•;A) as the (G,F ,A)-assembly map. By applying K−∞ or L−∞ to the sequence of additive categories (with involution): D(X × [0, 1);A)∅ → D (X × [0, 1);A) → D(X × [0, 1);A) we obtain a fibration of spectra [5]. As in [10], we have the following description for the assembly map. ASSEMBLY FOR GROUP EXTENSIONS 5 Theorem 1.7 ([10, §7]). The continuously controlled assembly map F λ G(X;A) → F λ G(•;A) is homotopy equivalent to the connecting map λ(D(X × [0, 1);A)) → Ωλ(D(X × [0, 1);A)∅ for λ = K−∞ or λ = L−∞. See [10, §2] for the definition of homotopy equivalent functors from G-CW -Complexes → Spectra, and [8, 5.1] for the result that any functor E : Or(G) → Spectra out of the orbit category of G may be extended uniquely (up to homotopy) to a functor E% : G-CW -Complexes → Spectra which is G-homotopy invariant and G-excisive. This will be our method for comparing functors. The orbit category Or(G) is the category with objects G/K, for K any subgroup of G, and the morphisms are G-maps. 2. Change of Coefficients We will need some ‘change of coefficient’ properties for the categories defined in the last section. The first three properties are essentially just translations of [3, Proposition 2.8] into our language. The corresponding versions for additive categories with involution are needed to apply these change of coefficient functors to L-theory. Definition 2.1. Let K and G be groups, A an additive category with commuting right K and G-actions, and S a K-G biset. Then, the category D(S;A) has a right G-action via (g · A)y = g Ayg−1 and (g · φ) z y = g ∗φ −1 yg−1, for all y, z ∈ S. We will mostly use the level-preserving subcatetory D l (S;A). Our first result is used in the arguments below. If T is a G-set, and S is a transitive K-G biset, let K × G act on S × T by the formula (k, g) · (s, t) := (ksg, gt) for all (k, g) ∈ K ×G and all (s, t) ∈ S × T . Lemma 2.2. Let T be a left G-set, and S be a transitive K-G biset. Then there is an additive functor F : D l (S × T × [0, 1);A) → D G l (T × [0, 1);D K l (S;A)) which induces an equivalence of categories D l (S × T ;A) ≃ D G l (T ;D K l (S;A)) . Proof. We will take the standard resolutions S = K × S, with elements denoted (k, s), for k ∈ K and s ∈ S, and T = G× T × [0, 1], with elements denoted (g, t), for g ∈ G and t ∈ T × [0, 1]. Therefore S × T = K ×G× S × T × [0, 1] is a resolution for S × T × [0, 1]. We define the functor F : D l (S × T × [0, 1);A) → D G l (T × [0, 1);D K l (S;A)) 6 IAN HAMBLETON, ERIK K. PEDERSEN, AND DAVID ROSENTHAL on objects by setting B = F (A)(g,t) in D K l (S;A) as the object B = (B(k,s)) with B(k,s) = A(k,g,s,t) in A. We use a similar formula for morphisms: