Assembly Maps for Group Extensions in K-theory and L-theory with Twisted Coefficients

In this paper we show that the Farrell-Jones isomorphism conjectures are inherited in group extensions for assembly maps in K-theory and L-theory with twisted coefficients. 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 [10]) with twisted coefficients in an additive category, and then study these questions via the continuously controlled assembly maps of [11, §7]. A formulation using the Davis-Lück assembly maps [9] has already been given by Bartels and Reich [4], 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 be 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 Date: March 31, 2008. 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 obtained by Oyono-Oyono [12], and our proof follows the outline given there. One of the 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], [3], [6], [7], [15], [16], and [17]). An immediate corollary to Theorem A is the following. Corollary (Corollary 4.10). The Farrell-Jones conjecture with twisted coefficients is true for G1×G2 if and only if it is true for G1, G2, and every product V1×V2, where V1 ≤ G1 and V2 ≤ G2 are virtually cyclic subgroups. The fibered isomorphism conjecture of Farrell and Jones [10] for a group G and a ring R asserts that for every group homomorphism, φ : H → G, the assembly map for H relative to the family generated by the subgroups φ(V ), V ⊂ G virtually cyclic, is an isomorphism. This conjecture implies the Farrell-Jones conjecture and has better inheritance properties. For example, the fibered version of our Theorem A is also true (see, for example, [2, Section 2.3]). The following result shows that the Farrell-Jones conjecture with twisted coefficients implies the Fibered Farrell-Jones conjecture. Theorem B. Suppose that φ : H → G is a group homomorphism. Then the Farrell-Jones conjecture for G, with twisted coefficients in any G-category, implies that 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 [4], who also pointed out a number of applications of the assembly map with twisted coefficients, including the study the Kand L-theory of twisted group rings (see also Example 4.8 and Example 4.9 below). One can check as in [11] 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 [13], Carlsson-Pedersen [6], [8] 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 [11]. 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 [9] has already been given in [4]. Following the method of [11], 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 ASSEMBLY FOR GROUP EXTENSIONS 3 a G-compact set G ·D ⊂ X such that p(G ·D) = G ·D. The notion of resolution comes from [13], 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: (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 [11, §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 4 IAN HAMBLETON, ERIK K. PEDERSEN, AND DAVID ROSENTHAL 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) 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 [11, §7], [14], [6]). 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 ([11]). Let S be a discrete left G-set. 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 [5, 1.28, 4.2] now show that the functors F λ : G-CW -Complexes → Spectra defined by (1.5) F λ G(X;A) := {