Fibrations with non commutative fibers

Fibrations with non commutative fibers Siegfried Echteroff, Ryszard Nest, Hervé Oyono-Oyono To cite this version: Siegfried Echteroff, Ryszard Nest, Hervé Oyono-Oyono. Fibrations with non commutative fibers. Journal of Noncommutative Geometry, 2009, 3, pp.377-417. ฀hal-00483033฀ HAL Id: hal-00483033 https://hal.science/hal-00483033 Submitted on 12 May 2010 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. January 4, 2009 FIBRATIONS WITH NONCOMMUTATIVE FIBERS SIEGFRIED ECHTERHOFF, RYSZARD NEST, AND HERVE OYONO-OYONO Abstract. We study an analogue of fibrations of topological spaces with the homotopy lifting property in the setting of C ∗ -algebra bundles. We then derive an analogue of the Leray-Serre spectral sequence to compute the K-theory of the fibration in terms of the cohomology of the base and the K-theory of the fibres. We present many examples which show that fibrations with noncommutative fibres appear in abundance in nature. 0. Introduction In recent years the study of the topological properties of C*-algebra bundles plays a more and more prominent rôle in the field of Operator algebras. The main reason for this is two-fold: on one side there are many important examples of C*-algebras which do come with a canonical bundle structure. On the other side, the study of C*-algebra bundles over a locally compact Hausdorff base space X is the natural next step in classification theory, after the far reaching results which have been obtained in the classification of simple C*-algebras. To fix notation, by a C*-algebra bundle A(X) over X we shall simply mean a C0 (X)-algebra in the sense of Kasparov (see [17]): it is a C*-algebra A together with a non-degenerate ∗-homomorphism Φ : C0 (X) → ZM(A), where ZM(A) denotes the center of the multiplier algebra M(A) of A. For such C0 (X)-algebra A, the fibre over x ∈ X is then Ax = A/Ix , where Ix = {Φ(f ) · a; a ∈ A and f ∈ C0 (X) such that f (x) = 0}, and the canonical quotient map qx : A → Ax is called the evaluation map at x. We shall often write A(X) to indicate the given C0 (X)-structure of A. We shall recall the basic constructions and properties of C0 (X)-algebras in the preliminary section below. We refer to [8] for further notations concerning C0 (X)-algebras. 2000 Mathematics Subject Classification. Primary 19K35, 46L55, 46L80, 46L85 ; Secondary 14DXX, 46L25, 58B34, 81R60, 81T30. Key words and phrases. Non-commutative Fibrations, Non-commutative Principal Torus Bundles, K-theory, Leray-Serre spectral sequence. This work was partially supported by the Deutsche Forschungsgemeinschaft (SFB 478) and the European training network EU-NCG (MRTN-CT-2006-031962). 1 2 ECHTERHOFF, NEST, AND OYONO-OYONO The main problem when studying bundles from the topological point of view is to provide good topological invariants which help to understand the local and global structure of the bundles. A good example is given by the class of separable continuous-trace C*-algebras, which are, up to Morita equivalence, just the section algebras of locally trivial bundles over X with fibres the compact operators K ∼ = K(l2 (N)). Using the standard classification of fibre bundles, these algebras (or rather the underlying bundle structure) are classified up to Morita equivalence by a corresponding Dixmier-Douady class in Ȟ 3 (X, Z). Another interesting class of examples are the non-commutative principle torus bundles, which have been studied by the authors in [8]. A basic example of a non-commutative principal 2-torus bundle is given by the C*-algebra C ∗ (H) of the discrete rank 3 Heisenberg group H, which has a canonical structure of a C*-algebra bundle over the circle T where the fibre Az over z ∈ T is the non-commutative 2-torus Aθ if z = e2πiθ . This shows in particular, that such bundles are in general far away from being section algebras of locally trivial C*-algebra bundles (but see [8, §2] for a classification based on classical methods). The main purpose of [8] was the study of the K-theoretic properties of the principle non-commutative Tn -bundles after forgetting the Tn -actions. Using Kasparov’s RKK(X; ·, ·)-theory as the version of K-theory which is probably most adapted to the study C*-algebra bundles, we show in [8, Corollary 3.4] that the non-commutative Tn -bundles are always locally RKK-trivial, which means that for each x ∈ X there exists a neighbourhood U of x such that the restriction A(U) of A(X) to U is RKK(U; ·, ·)-equivalent to C0 (U × Tn ). As usual, the global picture is much more difficult. Using the local RKKtriviality we show in [8] that to each non-commutative torus bundle A(X) we may associate a corresponding bundle of K-theory groups which comes equipped with a canonical action of the fundamental group π1 (X) of the base X. Using this associated group bundle allows us to obtain at least a partial classification result up to RKK-equivalence (see [8, Theorem 7.5]). In this paper we want to extend the studies of [8] from a more general perspective. Indeed, we are interested in C*-algebra bundles which are noncommutative analogues of classical fibrations in topology which satisfy certain weak versions of the homotopy lifting property. Indeed, the important point implied by the homotopy lifting property in classical topology is that for any fibration q : Y → X with this property, the space Y looks, in a topological sense, locally like a product space U × F . The phrase “in a topological sense” means that any homotopy invariant (co-)homology theory cannot differentiate between p−1 (U) and U × F . FIBRATIONS WITH NONCOMMUTATIVE FIBERS 3 Since it seems to be impossible to rephrase the homotopy lifting property in the non-commutative setting, we shall give a definition of this property in dependence of a given (co-)homology theory on the category of C*-algebras. For example, a (section algebra of a) C*-algebra bundle A(X) over X is called a K-fibration if for any positive integer p, for any p-simplex ∆p and for any continuous map f : ∆p → X the pull-back f ∗ A(∆p ) of A(X) via f is Ktheoretically trivial in the sense that the evaluation homomorphism qv : f ∗ A(∆p ) → Af (v) ∼ Ki (Af (v) ) for all induces an isomorphism of K-theory groups Ki (f ∗ A(∆p )) = p v ∈ ∆ . In a similar way we can define KK-fibrations, RKK-fibrations or h-fibrations, when (hn )n∈Z (resp (hn )n∈Z ) is any given (co-)homology theory on a suitable category of C*-algebras. In case of K-theory, the strongest notion will be that of an RKK-fibration (which implies that such bundles are automatically KK- and K-fibrations) and we shall indicate that there exist many natural examples of such fibrations. For instance, the principle noncommutative torus bundles of [8] are always RKK-fibrations. The main result of this paper will be the proof of a non-commutative analogue of the Leray-Serre spectral sequence for general h-fibrations. Indeed, if (hn )n∈Z (resp. (hn )n∈Z ) is any given (co-)homology theory on a suitable category of C*-algebras, and if A(X) is an h-fibration over the geometric realisation of a simplicial complex X, we show that we can associate to A the group bundle H = {hq (Ax ) : x ∈ X} which carries a canonical action of π1 (X). The Leray Serre spectral sequence for A(X) then converges to h(A(X)) and has (co-)homology groups H p (X, Hq ) as E2 -terms. Thus, at least in principle we can use the spectral sequence for computation of the K-theory groups of any K-fibration A(X). In particular this applies to the principal non-commutative Tn -bundles as studied in [8]. The spectral sequence also serves as an obstruction for RKK-equivalence of two bundles A(X) and B(X)—any such equivalence induces an isomorphism between the respective spectral sequences. It is certainly an interesting question to what extend the converse might hold, at least in case where A(X) and B(X) are RKK-fibrations (or locally RKK-trivial). In a final section we apply the spectral sequence to the study of the noncommutative torus bundles of [8] and show that it gives the missing tool for deciding which noncommutative torus bundles are globally RKK-trivial. We further give an explicit computation of the spectral sequences in the case of non-commutative 2-torus bundles over T2 . The results show that there are noncommutative principle torus bundles with isomorphic spectral sequences for which we do not know at this point whether they are RKK-equivalent. 4 ECHTERHOFF, NEST, AND OYONO-OYONO 1. Some preliminaries 1.1. Homology theories on C ∗ -algebras. Let Call denote the category of all C ∗ -algebras with ∗-homomorphisms as morphisms. By a good subcategory of Call we shall understand any subcategory C of Call with C ∈ C and which is closed under taking ideals, quotients, extensions and suspension in the sense that if A ∈ Ob(C), then SA := C0 (R, A) ∈ Ob(C). Moreover, for simplicity, we shall assume that MorC (A, B) = MorCall (A, B) for all A, B ∈ C and that if A ∼ = B in Call and A ∈ C, then B ∈ C. In many cases considered below, the above assumption on the morphisms could probably be weakened to the assumptions given in [1, 21.1], but we don’t want to bother with this extra generality. Standard examples of good subcategories of Call are given by the category Csep of separable C ∗ -algebras, the category Cnuc of nuclear C ∗ -algebras or the category Ccom of commutative C ∗ -algebras. Following [1, 21.1] we now give the following Definition 1.1. A homology theory on a good subcategory C of Call is a sequence {hn }n∈Z of covariant functors hn from C to the category Ab of abelian groups satisfying the following axioms: (H) If f0 , f1 : A → B are homotopic, then f0,∗ = f1,∗ : hn (A) → hn (B) for all n ∈ Z. q i (LX) If 0 → J → A → B → 0 is a short exact sequence in C, then for each n ∈ Z there are connecting maps ∂n : hn (B) → hn−1 (J), natural with respect to morphisms of short exact sequences, making exact the following long sequence ∂n+1 i q∗ ∂ i ∗ n ∗ · · · −→ hn (J) −→ hn (A) −→ hn (B) −→ hn−1 (J) −→ ··· Similarly, we define a cohomology theory on a good subcategory as a sequence {hn }n∈Z of contravariant functors hn : C → Ab which satisfy the obvious reversed axioms (e.g. see [1, 21.1]). Remark 1.2. (1) It follows from these axioms that all hn : C → Ab are additive in the sense that ! ! hn (A1 A2 ) = hn (A1 ) hn (A2 ) ∗ and " that hn (A) # = {0} if A is a contractible C -algebra. Since CA := C (−∞, ∞], A is contractible, it follows from (LX) applied to the short exact sequence 0 → SA → CA → A → 0 that hn+1 (A) = hn (SA) (resp. hn−1 (A) = hn (SA)) for all A ∈ C. (2) A covariant (resp. contravariant) functor F : Csep → Ab is called stable if ip : A → A ⊗ K; ip (a) = a ⊗ p FIBRATIONS WITH NONCOMMUTATIVE FIBERS ∼ = 5 ∼ = induces an isomorphism ip,∗ : F (A) → F (A⊗K) (resp. i∗p : F (A⊗K) → F (A)) for every one-dimensional projection p ∈ K. It is shown in [1, Corollary 22.3.1] (the result is originally due to Cuntz [5]) that every stable (co-)homology theory {hn } (resp. {hn }) on Csep satisfies Bott-periodicity hn+2 (A) = hn (S 2 A) ∼ = hn (A)). Hence, every stable = hn (A) (resp. hn+2 (A) ∼ (co-)homology theory on Csep is Z/2Z-graded and the long exact sequence (LX) then becomes a cyclic six-term exact sequence. (3) A homology theory {hn } (resp.{hn }) is called σ-additive (resp. σ$ % multiplicative) if hn (A) = i∈I hn (Ai ) (resp. hn (A) = i∈I hn (Ai )) whenever A ∈ C is a countable direct sum of objects Ai ∈ C, i ∈ I, and similar for cohomology theories. (4) The main example of a homology theory on Call (or any good subcategory C of Call ) is given by K-theory, and K-homology serves as the main example for a cohomology theory on Call . Note that K-theory is σ-additive and K-homology is σ-multiplicative. Assume now that C is a good subcategory of Call and suppose that A(X) ∈ C is a C0 (X)-algebra. In what follows we write ∆p =< v0 , . . . , vp > for the standard p-simplex with vertices v0 , . . . , vp . It follows from the properties of a good subcategory of Call #that if A ∈ C and f : ∆p → X is any continuous map, " then f ∗ A = C(∆p ) ⊗ A /If , with If is a suitable ideal in C(∆p ) ⊗ A, is again an object in C. In particular, all fibers Ax for x ∈ X are in C. The following definition is motivated by the notations and results presented in [12, Chapter I]: Definition 1.3. Suppose that C is a good subcategory of Call and that {hn } is a homology theory on C (resp. {hn } is a cohomology theory on C). Suppose further that A = A(X) is a C0 (X)-algebra in C. Then (i) A(X) is called an h-fibration if for all continuous maps f : ∆p → X and for every point v ∈ ∆p , the quotient map qv : f ∗ A → Af (v) induces an isomorphism qv,∗ : hn (f ∗ A) → hn (Av ) (resp. qv∗ : hn (Af (v) ) → hn (f ∗ A)). (ii) If C = Csep , then A(X) is called a KK-fibration, if for all continuous maps f : ∆p → X and for every element v ∈ ∆p the quotient map qv : f ∗ A → Af (v) is a KK-equivalence. (iii) If C = Csep , then A(X) is called an RKK-fibration, if f ∗ A is RKK(∆p ; ·, ·)-equivalent to C(∆p , Af (v) ) for any continuous map f : ∆p → X and for any element v of ∆p . Remark 1.4. (1) Any RKK-fibration is a KK-fibration. This follows from the fact that if x ∈ RKK(∆p ; C(∆p , Af (v) ), f ∗ A) is an RKK-equivalence, then 6 ECHTERHOFF, NEST, AND OYONO-OYONO we get the following commutative diagram in KK: x C(∆p , Af (v) ) −−∼−→ f ∗ A =   qv  qv '∼ = ' Af (v) ∼ = −−−→ Af (v) , x(v) where all arrows except of the right vertical one are known to be isomorphisms in KK. But then all arrows are KK-equivalences. We shall formulate below a partial converse of this easy observation, which follows from a result of Dâdârlat. (2) It is a direct consequence of [1, Corollary 22.3.1] that if A(X) is a KKfibration, then A(X) is an h-fibration for any stable (co-)homology theory {hn } (resp.{hn }) on Csep . (3) Every locally trivial C*-algebra bundle A(X) is an RKK-fibration. This follows from the fact that a pull-back of a locally trivial C*-algebra bundle is again locally trivial, and that any locally trivial bundle over a contractible space is trivial (e.g., see [16]). (4) All non-commutative principal n-tori as considered in [8] are RKKfibrations. This follows from [8, Proposition 3.1]. (5) Being an h-fibration (resp. KK-fibration, resp. RKK-fibration) is preserved by taking pull-backs inside C. This follows from the fact that if A is a C0 (X)-algebra in C and g : Y → X is any continuous map such that g ∗ (A) ∈ C, and if f : ∆p → Y is any continuous map, then f ∗ (g ∗ (A)) = (g ◦ f )∗ (A), and hence evaluation at any vertex induces isomorphisms in h-theory. (6) Being a KK-fibration is preserved under taking maximal tensor products with arbitrary separable C ∗ -algebras and by minimal tensor products with separable exact C ∗ -algebras. This follows from the fact that taking maximal or minimal tensor products of a KK-equivalence x ∈ KK0 (C, D) with a fixed C ∗ -algebra B gives a KK-equivalence x ⊗ B ∈ KK0 (C ⊗(max) B, D ⊗(max) B). Similar statements hold for RKK-fibrations. In what follows next we want to show that in many situations being a KKfibration is actually equivalent to being an RKK-fibration. Recall that a C*algebra bundle (i.e., a C0 (X)-algebra) A(X) is called a continuous C*-algebra bundle if for all a ∈ A the map x *→ +ax + is a continuous function on X. We need the following deep theorem of Dâdârlat (see [6, Theorem 1.1]). Theorem 1.5. Let X be a compact metrizable finite dimensional space and let A(X) and B(X) be separable nuclear continuous C*-algebra bundles over X. Suppose further that σ ∈ RKK(X; A(X), B(X)) is such that FIBRATIONS WITH NONCOMMUTATIVE FIBERS 7 σ(x) ∈ KK(Ax , Bx ) is invertible for all x ∈ X. Then σ is invertible in RKK(X; A(X), B(X)). As a direct corollary we get the partial converse to the observation made in item (2) of Remark 1.4: Corollary 1.6. Suppose that A(X) is a separable nuclear continuous C*algebra bundle over some locally compact space X. Then A(X) is a KKfibration, if and only if it is an RKK-fibration. Proof. Since every RKK-fibration is a KK-fibration by item (1) of Remark 1.4 we only have to show the converse. Write ∆ := ∆p and let f : ∆ → X be any continuous map. Since A is a KK-fibration, there exists the inverse qv−1 ∈ KK(Af (v) , f ∗A) of the evaluation map qv . Consider the image of qv−1 under the composition of maps σ∆,C(∆) KK(Af (v) , f ∗A) −−−−→ RKK(∆; C(∆) ⊗ Af (v) , C(∆) ⊗ f ∗ A) µ∗ −−−→ RKK(∆; C(∆, Af (v) ), f ∗ A), where µ : C(∆) ⊗ f ∗ A → f ∗ A; µ(g ⊗ a) = g · a is the multiplication homomorphism. If we evaluate this class at a point w ∈ ∆, we obtain the class (qv−1 ) ⊗ qw ∈ KK(Af (v) , Af (w) ), which is invertible since A is a KK-fibration. Hence the result follows from Dâdârlat’s theorem. ! Another interesting problem is the relation between locally RKK-triviality, which was discussed in [8] in connection with non-commutative torus bundles and the RKK-fibrations considered here. Let us recall that a C*-algebra bundle A(X) is called locally RKK-trivial, if for every x ∈ X there exists a neighbourhood V of x such that the restriction A(V ) of A to V is RKK(V ; ·, ·)equivalent to C0 (V, Ax ). We have seen in [8] that all principal non-commutative torus bundles are locally RKK-trivial. The proof of the following proposition is then straightforward. Proposition 1.7. Suppose that X is locally euclidean, i.e. every x ∈ X has a neighbourhood U which is homeomorphic to an open ball in some Rn . Then, if A(X) is an RKK-fibration it follows that A(X) is locally RKK-trivial. A bit surprisingly, the converse of the above proposition seems to be much more complicated. We shall obtain it later as a corollary of another remarkable theorem of Dâdârlat (see [6, Theorem 2.5]), which states that every separable and nuclear continuous C*-algebra bundle over some compact metrizable space X is RKK(X; ·, ·)-equivalent to a continuous bundle of simple Kirchberg algebras, i.e., each fibre is a separable nuclear purely infinite C*-algebra. As a direct consequence we get 8 ECHTERHOFF, NEST, AND OYONO-OYONO Proposition 1.8. Suppose that A(X) is a separable nuclear continuous C*algebra bundle over the compact metrizable finite dimensional space X such that A(X) is locally RKK-trivial. Then A(X) is RKK-equivalent to a locally trivial bundle of stable Kirchberg algebras. Proof. By Dâdârlat’s theorem, we may assume that A(X) is a C*-algebra bundle of simple Kirchberg algebras, and by stabilizing this bundle, we may assume that all fibers are stable. If A(X) is locally RKK-trivial, we can find for each x ∈ X a compact neighbourhood Vx such that A(Vx ) ∼RKK C(Vx , Ax ). It is then a consequence of [6, Theorem 2.7] that this equivalence is actually realized by an isomorphism A(Vx ) ∼ = C(Vx , Ax ) of C*-algebra bundles over Vx . ! As a corollary we get Corollary 1.9. If A(X) is a separable nuclear continuous field of C*-algebras over a locally compact space X. If A(X) is locally RKK-trivial, then A(X) is an RKK-fibration. Proof. If f : ∆p → A(X) is any continuous map, the pull-back f ∗ A(∆p ) satisfies all requirements of the above proposition. Since ∆p is contractible, every locally trivial bundle over ∆p is trivial. Thus it follows from the proposition that f ∗ A(∆p ) is RKK-equivalent to a trivial bundle. ! 2. Examples In this section we want to show that K-fibrations and KK-fibrations do appear quite often in nature. We already mentioned above that all locally trivial C*-algebra bundles are RKK-fibrations. Since being an RKK-fibration is stable under C0 (X)-linear Morita equivalence, this implies also that all continuous-trace C*-algebras with spectrum X are RKK-fibrations. Although these classes of C*-algebra bundles are certainly interesting, it would probably not give enough motivation for a general study of fibrations as we do in this paper. A class of interesting algebras which are, in general, far away from being locally trivial bundles of C*-algebras are the non-commutative principal torus bundles as studied in [8], and we already pointed out that all of them are RKKfibrations. Recall that the principal non-commutative torus bundles are, by definition, crossed products of the form C0 (X, K)!Zn , where Zn acts fibre-wise on the trivial bundle C0 (X, K). We shall now see that, with the help of the Baum-Connes conjecture, one can construct many other examples of RKK-, KK-, or K-fibrations via a similar crossed product construction. FIBRATIONS WITH NONCOMMUTATIVE FIBERS 9 Suppose that A is a C*-algebra bundle and α : G → Aut(A) is any C0 (X)linear action of the locally compact group G on A, i.e., we have αs (f · a) = f · αs (a) for all s ∈ G, f ∈ C0 (X), and a ∈ A. (We simply write f · a for Φ(f )a if Φ : C0 (X) → ZM(A) is the C0 (X)structure map of the bundle). Then α induces actions αx : G → Aut(Ax ) on the fibres Ax via αsx (a+Ix ) = αs (a)+Ix . The full and reduced crossed products A !(r) G have canonical structures as C0 (X)-algebras via the composition of the given C0 (X)-structure Φ : C0 (X) → ZM(A) of A with the canonical embedding M(A) → M(A !(r) G). For the full crossed product A ! G, the fibre over x ∈ X is then given by the full crossed product Ax ! G, which follows from the exactness of full crossed with respect to short exact sequences of G-algebras. For the reduced crossed products the situation can be more complicated. However, if G is exact in the sense of Kirchberg and Wassermann (which is true for a large class of groups—see [18]), then the fibre of A !r G over x ∈ X is Ax !r G. Note also that if f : Y → X is any continuous map, and if α : G → Aut(A) is a C0 (X)-linear action of G on A, then we get a C0 (Y )-linear pull-back action f ∗ (α) : G → Aut(f ∗ A) given on elementary tensors g ⊗ a ∈ f ∗ A = C0 (Y ) ⊗C0 (X) A by the formula f ∗ (α)s (g ⊗ a) := g ⊗ αs (a). It is then easily checked (e.g. see [10]), that f ∗ A ! G ∼ = f ∗ (A ! G) as C0 (Y )algebras and f ∗ A !r G ∼ = f ∗ (A !r G) if G is exact. In what follows next, we want to give some conditions which imply that the C0 (X)-algebras A ! G and A !r G are either K∗ -fibrations, KK-fibrations, or even RKK-fibrations. As the basic tool for this we shall use the Baum-Connes conjecture for G. Recall that for any G-algebra A, the topological K-theory of G with coefficient A is defined as K∗top (G; A) = lim KKG ∗ (C0 (Z), A), Z where Z runs through the G-compact subspaces of a universal proper G-space E(G). In [2], Baum, Connes and Higson constructed an assembly map µA : K∗top (G; A) → K∗ (A !r G) and they conjectured that this map should always be an isomorphism of groups. Although this conjecture turned out to be false in general (e.g., see [15]), the conjecture has been shown to be true for very large classes of groups including all amenable and, more general, a-T -menable groups (see [14]). In what follows, if A is a fixed G-algebra, we shall say that G satisfies BC for A if the map is an isomorphism for this special G-algebra A. 10 ECHTERHOFF, NEST, AND OYONO-OYONO A-T -menable groups satisfy in fact a stronger version of the Baum-Connes conjecture, which can be stated as follows. Recall that a G-algebra D is said to be a proper G-algebra, if D is a C0 (Z)-algebra for some proper G-space Z in such a way that the structure map Φ : C0 (Z) → ZM(D) is G-equivariant. A group G is said to have a γ-element if there exists an element γG ∈ KKG (C, C) and a proper G-algebra D such that γG can be written as a Kasparov product β ⊗D δ for some β ∈ KKG (C, D) and δ ∈ KKG (D, C), and such that the K restriction resG K (γ) = 1 ∈ KK (C, C) for all compact subgroups K of G. If G has a γ-element as above, then, by work of Kasparov and Tu [17, 21] (extended in [4, Theorem 1.11] to the weaker notion of a γ-element used here) the Baum-Connes assembly map is known to be split injective with image µA (K∗top (G; A)) = γG · K∗ (A !r G). We say that G satisfies the strong BaumConnes conjecture if γG = 1G in KKG 0 (C, C). By the results of Higson and Kasparov in [14], every a-T -menable group satisfies the strong Baum-Connes conjecture. It is clear from the above discussion that every group G which satisfies the strong Baum-Connes conjecture satisfies BC for all G-algebras A. Proposition 2.1. Suppose that A and B are G-algebras and that q ∈ (r) KKG (A, B). Let jG (q) ∈ KK0 (A !(r) G, B !(r) G) denote the descent of q for the full (resp. reduced) crossed products. For every compact subgroup K of G let ϕK : K∗ (A ! K) → K∗ (B ! K); ϕK (x) = x ⊗ jK (resG K ([q])). Then the following are true: (i) If G satisfies BC for A and B and if ϕK is an isomorphism for every compact subgroup K of G, then · ⊗ jGr (q) : K∗ (A !r G) → K∗ (B !r G) is an isomorphism. (ii) If G satisfies the strong Baum-Connes conjecture and if ϕK is an isomorphism for every compact subgroup K of G, then (r) · ⊗ jG (q) : K∗ (A !(r) G) → K∗ (B !(r) G) is an isomorphism for the full and reduced crossed products. (iii) If G satisfies the strong Baum-Connes conjecture and if jK (resG K (q)) is a KK-equivalence between A ! K and B ! K for all compact subgroups (r) K of G, then jG (q) is a KK-equivalence between A!(r) G and B !(r) G, for the full and reduced crossed products. Proof. Since G satisfies BC for A and B item (i) follows if we can show that taking Kasparov product with q induces an isomorphism from K∗top (G; A) to K∗top (G; B). But since all ϕK are isomorphisms, this follows from [7, Proposition 1.6]. The proof of (ii) is a consequence of (i) and the fact that the strong BaumConnes conjecture implies the Baum-Connes conjecture for all G-algebras and FIBRATIONS WITH NONCOMMUTATIVE FIBERS 11 it implies also that G is K-amenable which implies that the regular representation L : A ! G → A !r G induces an isomorphism in K-theory [22]. Finally, the proof of (iii) follows from the second part of [20, Proposition 8.5], since under the assumption of the strong Baum-Connes conjecture, the derived crossed products A !L G and B !L G of [20, Proposition 8.5] are KKequivalent to the full and reduced crossed products A !(r) G and B !(r) G, respectively. ! The above proposition now implies: Proposition 2.2. Suppose that A = A(X) is a separable C*-algebra bundle over X and let α : G → Aut(A) be a C0 (X)-linear action of the second countable locally compact group G on A(X). Assume that for each compact subgroup K of G the C0 (X)-algebra A(X) ! K is a K∗ -fibration. Then (i) If G is exact and satisfies BC for f ∗ A for all continuous f : ∆p → X, p = 0, 1, 2, .. (in particular, if G satisfies BC for all G-algebras B), then the reduced crossed product A(X) !r G is a K∗ -fibration. (ii) If G satisfies the strong Baum-Connes conjecture, then the full crossed product A(X) ! G is a K∗ -fibration. If, in addition, G is exact, the same is true for the reduced crossed product A(X) !r G. (iii) If G satisfies the strong Baum-Connes conjecture and if A(X) ! K is a KK-fibration for every compact subgroup K ⊆ G, then A(X) ! G is a KK-fibration. If, in addition, G is exact, then A(X) !r G is a KK-fibration, too. Proof. If G is exact, then A!r G is a C0 (X)-algebra with fibres Ax !r G and we have f ∗ A!r G ∼ = f ∗ (A!r G) for all continuous f : ∆p → X. By the assumption on the compact subgroups of G we see that the quotient map qv : f ∗ A → Af (v) induces an isomorphism ∼ = K∗ (f ∗ A ! K) −→ K∗ (Af (v) ! K) for all compact subgroups K of G. Item (i) then follows from part (i) of Proposition 2.1. Similarly, (ii) and (iii) follow from parts (ii) and (iii) of Proposition 2.1 together with the fact that the C0 (X)-algebra A(X) ! G has fibres Ax ! G. If G is exact, the same argument works for A(X) !r G. ! Remark 2.3. (1) If G has no compact subgroups (e.g., G = Rn , G = Zn or G = Fn , the free group with n generators), then the requirement that A(X) ! K being K∗ -fibration (resp. KK-fibration) in the above Proposition reduces to the requirement that A(X) is a K∗ -fibration (resp. KK-fibration). Thus, if any of the groups G = Rn , Zn , Fn acts fibrewise on a K∗ -fibration (resp. KK-fibration) A(X), then A(X) !(r) G is also a K∗ -fibration (resp. 12 ECHTERHOFF, NEST, AND OYONO-OYONO KK-fibration), since all of these groups are exact and satisfy the strong BaumConnes conjecture. Of course, there are many other examples of such groups. (2) It follows from [3, Proposition 3.1] that if G is exact and has a γ-element in the sense of Kasparov [17], and if A(X) is a continuous C*-algebra bundle over X, then G satisfies BC for f ∗ A for all f : ∆p → X if (and only if) G satisfies BC for Ax for every fibre Ax of A. (The only if direction follows from taking the constant map f : ∆p → X; f (v) = x and using the fact that f ∗ A = C(∆p , Ax ) is KKG -equivalent to Ax ). If we specialize to continuous-trace algebras A with base X, we can improve the results. For notation, we let K = K(l2 (N)) denote the compact operators on the infinite dimensional separable Hilbert space. Recall that if A(X) is any separable continuous-trace algebra with spectrum X, then A(X) ⊗ K is a locally trivial C*-algebra bundle with fibre K. Using this we get: Corollary 2.4. Suppose that G is a second countable locally compact group acting fibre-wise on a separable continuous-trace C ∗ -algebra A(X) with spectrum X. Then (i) If G satisfies the strong Baum-Connes conjecture (e.g., if G is a-T menable), then A(X) ! G is a KK-fibration. If, in addition, G is exact, the same holds for A !r G. (ii) If G is exact and satisfies BC for C(∆p , K) for all fibre-wise actions on C(∆p , K), p ≥ 0, then A !r G is a K∗ -fibration. (iii) If G is exact and has a γ-element, and G satisfies BC for K, for all actions of G on K, then A !r G is a K∗ -fibration. Notice that by the results of [3] condition (iii) is satisfied for all almost connected groups and for all linear algebraic groups over Qp . Proof of Corollary 2.4. The corollary will follow from Proposition 2.2 and Remark 2.3 if we can show that (f ∗ A ! K) ⊗ K is a trivial C(∆p )-algebra for all continuous maps f : ∆p → X, since this will imply that A ! K is a KKfibration. For this we first note that (f ∗ A ! K) ⊗ K ∼ = (f ∗ A ⊗ K) ! K, where K acts trivially on K. Using this we may simply assume that f ∗ A = C(∆p , K). But it follows then from [7, Proposition 1.5] that any fibre-wise action of a compact group K on C(∆p , K) = C(∆p ) ⊗ K is exterior equivalent to a diagonal action id ⊗αv , with αv the action on the fibre K = C(∆p , K)v . Thus C(∆p , K) ! K ! is isomorphic to C(∆p , K !αv K) as bundles over ∆p . So far we only considered K∗ - or KK-fibrations, but we promised at the beginning of this section that we will provide also examples of RKK-fibrations. Indeed, combining the above results with Corollary 1.6 gives FIBRATIONS WITH NONCOMMUTATIVE FIBERS 13 Corollary 2.5. Suppose that A(X) is a separable nuclear and locally trivial C*-algebra bundle and let G be a second countable amenable group acting fibrewise on A(X). Then the following are true (i) If G has no compact subgroups then A(X) ! G is an RKK-fibration. (ii) If A(X) is a continuous trace algebra with spectrum X, then A(X) ! G is an RKK-fibration. Proof. Since G is amenable, it satisfies the strong Baum-Connes conjecture by [14]. Moreover, a crossed product of a continuous C*-algebra bundle by a fibre-wise group action of an amenable group G is again a continuous C*algebra bundle by [23]. Since nuclearity is also preserved under taking crossed products by amenable groups, it follows that A(X) ! G is a nuclear separable and continuous C*-algebra bundle. Thus it follows from Corollary 1.6 that A(X) ! G is an RKK-fibration if and only if it is a KK-fibration. Hence the result follows from Remark 2.3 and Corollary 2.4. ! Of course, as an example of the above corollary we get a new proof of the fact that the non-commutative principal torus bundles of [8] are RKK-fibrations, since, by definition, they are crossed products of the form C0 (X, K) ! Zn by C0 (X)-linear actions of Zn on C0 (X, K). 3. The group bundle corresponding to an h-fibration Suppose that X is a locally compact space. By an (abelian) group bundle G := {Gx : x ∈ X} we understand a functor from the homotopy groupoid of X to the category of (abelian) groups. It is given by a family of groups Gx , x ∈ X, together with group isomorphisms cγ : Gx → Gy for each continuous path γ : [0, 1] → X which starts at x and ends at y, such that the following additional requirements are satisfied: (i) If γ and γ % are homotopic paths from x to y, then cγ = cγ " . (ii) If γ1 : [0, 1] → X and γ2 : [0, 1] → X are paths from x to y and from y to z, respectively, then cγ1 ◦γ2 = cγ1 ◦ cγ2 , where γ1 ◦ γ2 : [0, 1] → X is the usual composition of paths. It follows from the above requirements, that if X is path connected, then all groups Gx are isomorphic and that we get a canonical action of the fundamental group π1 (X) on each fibre Gx . A morphism between two group bundles G = {Gx : x ∈ X} and G % = {G%x : x ∈ X} is a family of group homomorphisms φx : Gx → G%x which commutes with the maps cγ . The trivial group bundle is the bundle with every Gx equal to a fixed group G and all maps cγ being the identity. We then write 14 ECHTERHOFF, NEST, AND OYONO-OYONO X × G for this bundle. If X is path connected, then a given group bundle G on X can be trivialized if and only if the action of π1 (X) on the fibres Gx are trivial. In that case every path γ from base points x to y induces the same morphism cx,y : Gx → Gy and if we choose a fixed base point x0 , the family of maps {cx,x0 : x ∈ X} is a group bundle isomorphism between the trivial group bundle X × Gx0 and the given bundle G = {Gx : x ∈ X}. It is now easy to check that every h∗ -fibration A(X) gives rise to a group bundle H∗ := {h∗ (Ax ) : x ∈ X}: Proposition 3.1. Suppose that A(X) is an h∗ -fibration. For any path γ : [0, 1] → X with starting point x and endpoint y let cγ : h∗ (Ax ) → h∗ (Ay ) denote the composition (3.1) ǫ−1 1,∗ ǫ0,∗ = = h∗ (Ay ) −−∼−→ h∗ (γ ∗ A) −−∼−→ h∗ (Ax ) Then H∗ (A) := {h∗ (Ax ) : x ∈ X} together with the above defined maps cγ is a group bundle over X. A similar result holds for a cohomology theory h∗ if A(X) is an h∗ -fibration (with arrows in (3.1) reversed). Proof. It is clear that constant paths induce the identity maps and that cγ◦γ " = cγ ◦ cγ " , where γ ◦ γ % denotes composition of paths. Moreover, if Γ : [0, 1] × [0, 1] → X is a homotopy between the paths γ0 and γ1 with equal starting and endpoints, then cγ0 and cγ1 both coincide with the composition ǫ(0,0),∗ ◦ ǫ−1 (1,1),∗ , ∗ where ǫ(0,0) and ǫ(1,1) denote evaluations of Γ A at the respective corners of [0, 1]2 . Hence we see that cγ only depends on the homotopy class of γ. ! Definition 3.2. Suppose that h is a (co)homology theory on a good category C of C ∗ -algebras and let A(X) be an h-fibration. Then H∗ := {h∗ (Ax ) : x ∈ X} (resp. H∗ (A) := {h∗ (Ax ) : x ∈ X} if h is a cohomolgy theory) together with the maps cγ : h∗ (Ay ) → h∗ (Ax ) is called the h∗ -group bundle associated to A(X). Remark 3.3. If A(X) is a KK-fibration, then it is in particular a K∗ - and a K ∗ -fibration, where K∗ and K ∗ denote ordinary K-theory and K-homology. We shall denote the resulting group bundles by K∗ (A) and K∗ (A), respectively. 4. The Leray-Serre spectral sequence In this section we want to proof an analogue of the classical Leray-Serre spectral sequence for topological Serre-fibrations. From the last remark of the previous section we know that if A(X) is a h-fibration for a (co-)homology theory h, then we get the group bundle H∗ (A) over X. It is well-known in topology that one can use such bundles as coefficients for singular or simplicial (co-)homology on X. It is our aim to show that every h-fibration over a finite FIBRATIONS WITH NONCOMMUTATIVE FIBERS 15 dimensional simplicial complex X admits a spectral sequence with E2 -terms isomorphic to the (co-)homology of X with coefficient in H∗ (resp. H∗ ). Assume that X is a locally compact CW-complex and that A is any C0 (X)algebra. For p ≥ 0 let Xp denote the p-skeleton of X and we set Xp = ∅ for a negative integer p < 0. We will always assume that X is finite dimensional so that there exists a smallest integer d (the dimension of X) such that Xp = X for all p ≥ d. For all p we write Ap := A|Xp and Ap,p−1 := A|Xp \Xp−1 , where we use A|∅ := {0}. We then obtain short exact sequences 0 → Ap,p−1 → Ap → Ap−1 → 0. If h∗ is any homology theory on a good subcategory C of Call such that all algebras Ap and Ap,p−1 are in C, naturality of the long exact sequences (4.1) ∂n+1 q∗ i ∂ i ∗ n ∗ · · · −→ hn (J) −→ hn (A) −→ hn (B) −→ hn−1 (J) −→ ··· gives the following commutative diagram: ? ? ? ? q∗ y q∗ y ∂ ι ∂ ι ι ∂ ι ∗ ∗ −→ −→ hq (Ap+2 ) −−−−−→ hq−1 (Ap+3,p+2 ) −−−− −−−−−→ hq+1 (Ap+1 ) −−−−−→ hq (Ap+2,p+1 ) −−−− ? ? ? ? q∗ y q∗ y −−−−−→ hq+1 (Ap ) ? ? q∗ y −−−−−→ ∂ hq (Ap+1,p ) ∗ ∗ −→ −−−− −→ hq (Ap+1 ) −−−−−→ hq−1 (Ap+2,p+1 ) −−−− ? ? q∗ y ∂ hq (Ap,p−1 ) ∗ −−−− −→ −−−−−→ hq+1 (Ap−1 ) −−−−−→ ? ? q∗ y ι hq (Ap ) ? ? q∗ y ∂ −−−−−→ hq−1 (Ap+1,p ) ι ∗ −−−− −→ Here the vertical arrows are induced by the quotient maps q : Ap → Ap−1 , the maps ι∗ : hq (Ap,p−1) → hq (Ap ) are induced by the inclusions ι : Ap,p−1 → Ap and the maps ∂ : hq+1 (Ap−1 ) → hq (Ap,p−1) denote the boundary maps in the long exact sequence (4.1). Hence, the upper staircase of this diagram forms the sequence (4.1). Now writing H p,q := hq (Ap ), E1p,q := hq (Ap,p−1), $ $ p,q H := p,q H p,q and E := E1 we obtain an exact couple H ""! q∗ !! !! ι∗ !! E !! H " " "" "" ∂ " "## from which we obtain by the general procedure (which, for example, is exq p,q plained in [19]) a spectral sequence {Erp,q , dr } with E∞ -terms E∞ = Fpq /Fp+1 " # with Fpq := ker hq (A) → hq (Ap ) . Since Fpq = hq (A) for p < 0 and Fpq = {0} for p ≥ d, the dimension of X, it follows that the spectral sequence converges to hq (A). This means that we obtain a filtration q q {0} = Fdq ⊆ Fd−1 ⊆ · · · ⊆ F−1 = hq (A) 16 ECHTERHOFF, NEST, AND OYONO-OYONO of subgroups Fpq of hq (A) such that the sub-quotients can be computed (at least in principle) by our spectral sequence. Similarly, if we start with a cohomology theory h∗ on C, we consider the diagram ? ? ? ? q∗ y q∗ y ←−−−−− hq−1 (Ap ) ? ? q∗ y ι∗ hq (Ap,p−1 ) ←−−−−− ∂ hq (Ap+1,p ) ←−−−−− hq (Ap+1 ) ←−−−−− hq+1 (Ap+2,p+1 ) ←−−−−− ? ? q∗ y ←−−−−− ∂ hq (Ap ) ? ? q∗ y ∂ ι∗ ∂ ←−−−−− hq−1 (Ap−1 ) ←−−−−− ? ? q∗ y ←−−−−− hq+1 (Ap+1,p ) ←−−−−− ι∗ ∂ ι∗ ι∗ ∂ ι∗ ←−−−−− hq−1 (Ap+1 ) ←−−−−− hq (Ap+2,p+3 ) ←−−−−− hq (Ap+2 ) ←−−−−− hq+1 (Ap+31,p+2 ) ←−−−−− ? ? ? ? q∗ y q∗ y r ∞ which provides a" spectral sequence {Ep,q , dr } with E ∞ -terms Ep,q := Fqp /Fqp−1 # where Fqp := im hq (Ap ) → hq (A) . Again, since X is finite dimensional, the spectral sequence converges to hq (A). Hence, at this stage we arrive at Proposition 4.1. Suppose that X is a finite dimensional CW-complex and let A(X) be a C*-algebra bundle over X. Suppose that C is a good subcategory of Call so that Ap := A(Xp ) ∈ C for every p-skeleton of X. Then, if h∗ is a homology theory (resp. h∗ is a cohomology theory) on X there exists a spectral r sequence {Erp,q , dr } (resp. {Ep,q , dr }) which converges to h∗ (A) (resp. h∗ (A)) as described above. Remark 4.2. Let us denote by {Uip : i ∈ Ip } the open p-cells of X. We then have ! A(Uip ). Ap,p−1 = i∈Ip If X is a finite simplicial complex this sum is finite and it follows from additivity of h∗ (resp. h∗ ) that ! ! 1 hq (A(Uip )) ). hq (A(Uip )) (resp. Ep,q = E1p,q = i∈Ip i∈Ip Of course, if h∗ (resp. h∗ ) is σ-additive or σ-multiplicative we get similar infinite direct sum or product decompositions in case where X is a σ-finite (i.e., X has countably many cells). In any case we shall assume that X is locally finite. The d1 -differential is then determined by the maps (p,i),(p+1,j) d1,q : hq (A(Uip )) → hq−1 (A(Ujp+1 )) given by the composition Iqp,i hq (A(Uip )) −−−→ hq (Ap,p−1) ∂ i ∗ −−− → p+1,j Qq−1 hq (Ap ) −−−→ hq−1 (Ap+1,p ) −−−−→ hq−1 (A(Ujp+1 )) FIBRATIONS WITH NONCOMMUTATIVE FIBERS 17 where I p,i : A(Uip ) → Ap,p−1 denotes the inclusion and Qp,i : Ap,p−1 → A(Uip ) denotes the quotient map. Similarly, for a cohomology theory h∗ we get maps p q q+1 d1,q (A(Ujp−1 )) (p,i),(p−1,j) : h (A(Ui )) → h which are given by the compositions Qqp,i hq (A(Uip )) −−−→ i∗ hq (Ap,p−1) q+1 −−−→ h ∂ −−−→ hq+1 (Ap−1 ) q+1 Ip−1,j (Ap−1,p−2) −−−→ hq+1 (A(Ujp−1 )). 2 ) is with similar meanings for Qp,i and Ip,i . It follows then that E2p,q (resp. Ep,q the cohomology (resp. homology) of the complex build out of the above given data. (p,i),(p+1,j) We now want to study the groups h∗ (A(Uip )) and the maps d1,q more closely in case where A(X) is an h∗ -fibration and X is a (finite) simplicial complex. In particular, we want to give a better computation of the E2 -terms. We shall restrict to the case of a homology theory on a good category C of C*-algebras throughout, noting that similar arguments work for a cohomology theory as well. We start with introducing some notation: As before, we let ∆p :=< v0 , . . . , vp > denote the oriented closed n-simplex with vertices v0 , . . . , vp , we let ∆ ◦ p denote its interior and we let ∂∆p denote its boundary. If 0 ≤ k ≤ p we shall consider ∆k =< v0 , . . . , vk > as a subset of ∆p and for i ∈ {0, . . . , n} we write ∆p−1 :=< v0 , . . . , vi−1 , vi+1 , . . . vp > for the oriented ith face of ∆p . If X is i our given simplicial complex we write {∆p (j) : j ∈ Ip } for the set of closed p-simplexes in X, and we let σ p (j) : ∆p → ∆p (j) ⊆ X denote an explicit affine homeomorphism between the standard simplex ∆p and ∆p (j). 2 1 To study the differential d : Ep,q → Ep+1,q−1 in the above remark, we first p need to study the simple case where X = ∆ itself. Recall that if A = A(X) is an h∗ -fibration with X simply connected, then for each x, y ∈ X there are unique isomorphisms Φx,y : h∗ (Ax ) → h∗ (Ay ), which, for any chosen path γ : [0, 1] → X with γ(0) = y, γ(1) = x, satisfy the equations Φx,y ◦ ev1,∗ = ev0,∗ : h∗ (γ ∗ A[0, 1]) → h∗ (Ay ). Lemma 4.3. Suppose that A = A(∆P ) is an h∗ -fibration (resp. h∗ -fibration) with p ≥ 1. Then for every x ∈ ∆p evaluation at x induces an isomorphism evx,∗ : h∗ (A(∆p )) → h∗ (Ax ). 18 ECHTERHOFF, NEST, AND OYONO-OYONO Moreover, if y is any other point in ∆p , then evy,∗ = Φx,y ◦ evx,∗ (and similar statements for h∗ -fibrations). Proof. The first statement holds by definition of an h∗ -fibration. So we only have to check that evy,∗ = Φx,y ◦evx,∗ for any pair x, y ∈ ∆p . Let γ : [0, 1] → ∆p denote any path connecting y = γ(0) with x = γ(1). Since γ is a proper map, [8, Lemma 1.3] provides a ∗-homomorphism Φγ : A(∆p ) → γ ∗ A([0, 1]) and it is clear from the construction of Φγ that evx = ev1 ◦Φγ and evy = ev0 ◦Φγ . The result now follows from the definition of Φx,y . ! Lemma 4.4. Let 1 ≤ p and suppose that the C(∆p )-algebra A is an h∗ fibration. Let W ⊆ ∆p be any set which is obtained from ∆p by removing a union of k faces of dimension p − 1 from ∆p with 1 ≤ k ≤ p. Then h∗ (A(W )) = 0. Proof. The proof is by induction on p and k. If p = 1 then k = 1 and W is homeomorphic to [0, 1). Since evaluation A([0, 1]) → A1 induces an isomorphism of h∗ -groups, it follows from the long exact sequence corresponding to 0 → A([0, 1)) → A([0, 1]) → A1 → 0 that h∗ (A(W )) = 0. Suppose now that p > 1. If ∆p−1 is any closed face of ∆p , then i it follows from the properties of h∗ -fibrations that the quotient map qi,∗ qi : A(∆p ) → A(∆p−1 ) induces an isomorphism hn (A(∆p )) −→ hn (A(∆p−1 )), i i p−1 since composition with evaluation at any vertex v of ∆i induces isomorp−1 p ∼ ∼ , the phisms hn (A(∆ )) = hn (A(∆i )) = hn (Av ). Hence, if W = ∆p " ∆p−1 i long exact sequence of h∗ -groups corresponding to the short exact sequence 0 → A(W ) → A(∆p ) → A(∆p−1 ) → 0 implies that h∗ (A(W )) = 0. i Suppose now that k > 1 and let F := ∆p " W . Then we can write F as a union F % ∪∆p−1 , where F % is a union of k−1 faces. By the induction assumption i we know that h∗ (A(W % )) = 0 for W % := ∆p " F % . Moreover, W % " W is equal to W %% := ∆p−1 \ F %% , where F %% is some union of l p − 2-dimensional faces with i 1 ≤ l ≤ p − 1. Hence, by induction assumption, we have h∗ (A(W %% )) = 0 and then all terms in the exact sequence hq (A(W % )) → hq (A(W )) → hq (A(W %% )) must be zero. ! Definition and Remark 4.5. For the p-simplex ∆p =< v0 , . . . , vp > let Wi := ∆ ◦p ∪ ∆ ◦ p−1 . If the C(∆p )-algebra A is an h∗ -fibration, then it foli lows from the above lemma that h∗ (A(Wi )) = 0, which then implies that the boundary map ∂i : hq (A(∆ ◦ p−1 )) → hq−1 (A(∆ ◦ p )) corresponding to the short i exact sequence 0 → A(∆ ◦ p ) → A(Wi ) → A(∆ ◦ p−1 ) → 0 is an isomorphism for i all q. In particular, there is a chain of isomorphisms evv ,∗ ∂ ∂ ∂p 0 1 2 hq (A(∆p )) −→ hq (Av0 ) −→ hq−1 (A(∆ ◦ 1 )) −→ · · · −→ hq−p (A(∆ ◦ p )), FIBRATIONS WITH NONCOMMUTATIVE FIBERS 19 which we shall call the canonical oriented isomorphism ∼ = ◦ p )). Φpq : hq (A(∆p )) −→ hq−p (A(∆ It is important for us to understand how the canonical isomorphisms depends on the orientation of the simplex ∆p . We start this investigation with two basic observations. The first considers the case ∆1 = [0, 1]: Lemma 4.6. Suppose that A([0, 1]) is a C*-algebra bundle which is an h∗ fibration. Let ∂0 : hq (A0 ) → hq−1 (A(0, 1)) and ∂1 : hq (A1 ) → hq−1 (A(0, 1)) denote the isomorphisms given by the connecting maps in the long exact sequences related to evaluation of A([0, 1)) and A((0, 1]) at 0 and 1, respectively. Then ∂0 = − ∂1 ◦ Φ0,1 . Proof. Consider the long exact sequence · · · → hq (A(0, 1)) → hq (A[0, 1]) → hq (A0 corresponding to ! ∂ A1 ) → hq−1 (A(0, 1)) → · · · 0 → A((0, 1)) → A([0, 1]) → A0 ! A1 → 0. The connecting map in this sequence equals ∂0 + ∂1 , which follows from naturality of the long exact sequence together with the diagram 0 −−−→ A((0, 1)) −−−→ A([0, 1)) −−−→     =' ι' A0 −−−→ 0  ι ' $ 0 −−−→ A((0, 1)) −−−→ A([0, 1]) −−−→ A0 A1 −−−→ 0 ( ( ( ι   = ι  0 −−−→ A((0, 1)) −−−→ A((0, 1]) −−−→ A1 −−−→ 0. Exactness then gives 0 = ∂0 ◦ ev0,∗ +∂1 ◦ ev1,∗ where evi,∗ : hq (A([0, 1])) → hq (Ai ), i = 0, 1, denotes the evaluation isomorphism. Thus, we get ∂0 ◦ ev0,∗ = − ∂1 ◦ ev1,∗ and composing both sides with ev−1 ! 0,∗ on the right gives the lemma. In the next lemma we compare the isomorphisms ∼ hq (A(∆ ◦ p−1 ◦ p )) p−1 )) = hq−1 (A(∆ and ∼ ◦ p )) hq (A(∆ ◦ p−1 p )) = hq−1 (A(∆ as defined in 4.5 for the p-th and the (p − 1)-st face of ∆p : 20 ECHTERHOFF, NEST, AND OYONO-OYONO Lemma 4.7. Suppose that p > 1. Then the compositions ∼ ◦ p )) ◦ p−1 hr (A(∆ ◦ p−2 )) ∼ = hr−1 (A(∆ p )) = hr−2 (A(∆ and ∼ ◦ p )) ◦ p−1 hr (A(∆ ◦ p−2 )) ∼ = hr−1 (A(∆ p−1 )) = hr−2 (A(∆ differ by the factor −1. Proof. Let ◦ p, W =∆ ◦ p−2 ∪ ∆ ◦ p−1 ∪∆ ◦ p−1 p−1 ∪ ∆ p W0 = ∆ ◦ p−2 ∪ ∆ ◦ p−1 p , W1 = ∆ ◦ p−2 ∪ ∆ ◦ p−1 p−1 , W0,1 = ∆ ◦ p−1 ∪∆ ◦ p−1 ◦ p−2 . p p−1 ∪ ∆ Then it follows from Lemma 4.4 that all groups h∗ (A(W )) = h∗ (A(W1 )) = h∗ (A(W0 )) = h∗ (∆ ◦p ∪ ∆ ◦ p−1 ◦p ∪ ∆ ◦ p−1 p−1 ) p ) = h∗ (∆ vanish. Since h∗ (A(W )) = 0, the boundary map in the long exact sequence for 0 → A(∆ ◦ p ) → A(W ) → A(W0,1 ) → 0 induces an isomorphism ∂0,1 : hl (A(W0,1 )) → hl−1 (A(∆ ◦ p )). Naturality of the long exact sequences and the diagram ) −−−→ 0 ◦ p−1 0 −−−→ A(∆ ◦ p ) −−−→ A(∆ ◦p ∪ ∆ ◦ p−1 p ) −−−→ A(∆  p   ι  =' ι' ' 0 −−−→ A(∆ ◦ p ) −−−→ (  = −−−→ A(W0,1 ) −−−→ 0 ( ι  A(W ) (  ι 0 −−−→ A(∆ ◦ p ) −−−→ A(∆ ◦p ∪ ∆ ◦ p−1 ◦ p−1 p−1 ) −−−→ A(∆ p−1 ) −−−→ 0 together with the Five-Lemma shows that the inclusions of ∆ ◦ p−1 and ∆ ◦ p−1 p−1 p into W0,1 induce isomorphisms ∼ = ιp : hl (A(∆ ◦ p−1 p )) → hl (A(W0,1 )) and ∼ = ιp−1 : hl (A(∆ ◦ p−1 p−1 )) → hl (A(W0,1 )) such that (4.2) ∂0,1 ◦ ιp = ∂p and ∂0,1 ◦ ιp−1 = ∂p−1 , FIBRATIONS WITH NONCOMMUTATIVE FIBERS 21 where, as before, ∂i : hl (A(∆ ◦ pi )) → hl−1 (A(∆ ◦ p )) denotes the isomorphism for the ith face of ∆p . We now look at the diagram 0 −−−→ (4.3) A(∆ ◦ p−1 ) (p  q −−−→ A(W0 ) −−−→ A(∆ ◦ p−2 ) −−−→ 0 ( (  = q  A(∆ ◦ p−1 p−1 ) −−−→ A(W1 ) −−−→ A(∆ ◦ p−2 ) −−−→ 0 0 −−−→ A(∆ ◦ p−1 ∪∆ ◦ p−1) −−−→ A(W0,1 ) −−−→ A(∆ ◦ p−2 ) −−−→ 0 p  p−1   =   q' q' ' 0 −−−→ It implies that the composition of the boundary map hr (A(∆ ◦ p−2 )) → hr−1 (A(∆ ◦ p−1 ∪∆ ◦ p−1 p p−1 )) followed by the projections onto hr−1 (A(∆ ◦ p−1 ◦ p−1 p−1 )), respecp )) and hr−1 (A(∆ tively, coincide with the boundary maps ◦ p−1 ∂pp−2 : hr (A(∆ ◦ p−2 )) ∼ = hr−1 (A(∆ p )) and p−2 ∂p−1 : hr (A(∆ ◦ p−2 )) ∼ ◦ p−1 = hr−1 (A(∆ p−1 )), respectively. On the other hand, it is clear that the isomorphisms ∼ ιp : hr−1 (A(∆ ◦ p−1 p )) = hr−1 (A(W0,1 )) and ∼ ιp−1 : hr−1 (A(∆ ◦ p−1 p−1 )) = hr−1 (A(W0,1 )) factorise via the inclusions of hr−1 (A(∆ ◦ p−1 ◦ p−1 p−1 )) as direct sump )) and hr−1 (A(∆ p−1 p−1 mands of hr−1 (A(∆ ◦p ∪ ∆ ◦ p−1)), and that these inclusions invert the projections on the summands which are induced from the quotient maps in the first vertical row of Diagram (4.3). This implies exactness of the sequence hr (A(∆ ◦ p−2 ∂pp−2 L p−2 ∂p−1 )) −−−−−−−−→ hr−1 (A(∆ ◦ p−1 p )) $ ιp +ιp−1 hr−1 (A(∆ ◦ p−1 −−−−→ hr−1 (A(W0,1 )). p−1 )) − Combining this with Equation (4.2) gives p−2 p−2 , ) = ∂p ◦ ∂pp−2 + ∂p−1 ◦ ∂p−1 0 = ∂0,1 ◦ (ip ◦ ∂pp−2 + ip−1 ◦ ∂p−1 which finally finishes the proof. ! We now want to consider an arbitrary permutation ϕ : {0, . . . , p} → {0, . . . , p}. We shall denote by the same letter the unique affine isomorphism ∼ = ∆p −→ ∆p which is induced by applying ϕ on the vertices. Let Ψϕ : A(∆p ) → ϕ∗ A(∆p ) be the isomorphism of [8, Lemma 1.3]. It clearly restricts to an isomorphism, also denoted Ψϕ , between the ideals A(∆ ◦ p ) and ϕ∗ A(∆ ◦ p ). We then get 22 ECHTERHOFF, NEST, AND OYONO-OYONO Proposition 4.8. Let sign(ϕ) denote the sign of the permutation ϕ : {0, . . . , p} → {0, . . . , p}. Then the following diagram commutes Ψϕ,∗ hq (A(∆p )) −−−→   Φqp ' (4.4) Ψϕ,∗ hq (ϕ∗ A(∆p ))  sign(ϕ)Φq p ' hq+p (A(∆ ◦ p )) −−−→ hq+p (ϕ∗ A(∆ ◦ p )). Proof. Since every permutation is a product of transpositions which interchange two neighbours in {0, . . . , p}, we may assume without loss of generality that ϕ interchanges l with l + 1. If we identify A(∆p ) with ϕ∗ A(∆p ) via Ψϕ , and if we write ∆lϕ for the ldimensional face < ϕ(v0 ), . . . , ϕ(vl ) >⊆ ∆p , the above diagram restricts to showing that the isomorphism given by the composition evvϕ(0) ,∗ ◦ 1ϕ )) hq (A(∆p )) −→ hq (Aϕ(v0 ) ) −→ hq+1 (A(∆ −→ hq+2 (A(∆ ◦ 2ϕ )) −→ · · · −→ hq+p (A(∆ ◦ p )) differs from the canonical isomorphism Φpq : hq (A(∆p )) → hq+p (A(∆ ◦ p )) by sign(ϕ). For this we first remark, that by Lemma 4.3 we have the equality Φv0 ,ϕ(v0 ) ◦ evv0 ,∗ = evϕ(v0 ),∗ : hq (A(∆p )) → hq (Aϕ(v0 ) ), which then implies that we have to prove that the isomorphisms ◦ p )) ◦ 2 )) · · · ∼ ◦ 1 )) ∼ Θ : hq (Av0 ) ∼ = hq−p (A(∆ = hq−2 (A(∆ = hq−1 (A(∆ and ◦ p )) ◦ 2ϕ )) · · · ∼ ◦ 1ϕ )) ∼ Θϕ : hq (Aϕ(v0 ) ) ∼ = hq−p (A(∆ = hq−2 (A(∆ = hq−1 (A(∆ are related via Θ = sign(ϕ)Θϕ ◦ Φv0 ,ϕ(v0 ) . If ϕ permutes 0 with 1, then we have ∆jϕ = ∆j for all j ≥ 1, so the above equation reduces to the case p = 1. This case is taken care for by Lemma 4.6 above. If ϕ permutes l with l + 1 for some l > 0, then we have ∆jϕ = ∆j for all j ≥ l + 1, so we may assume without loss of generality that l = p − 1. We then also have ∆p−2 = ∆p−2 and all we have to show is that the compositions ϕ ∼ ◦ p )) ◦ p−1 hr (A(∆ ◦ p−2 )) ∼ = hr−1 (A(∆ p )) = hr−2 (∆ and ∼ hr (A(∆ ◦ p−2 )) ∼ ◦ p−1 ◦ p )) = hr−1 (A(∆ p−1 )) = hr−2 (∆ differ by the factor −1. But this is shown in Lemma 4.7. This completes the proof. ! FIBRATIONS WITH NONCOMMUTATIVE FIBERS 23 In order to state the following important corollary, let us note that if A(∆p ) is a face of ∆p , then the quotient map A(∆p ) → is an h∗ -fibration and if ∆p−1 l p−1 A(∆l ) induces an isomorphism (4.5) resl : h∗ (A(∆p )) → h∗ (A(∆p−1 )) l which follows from the simple fact that evaluation at any common vertex induces an isomorphism in h∗ -theory for both algebras. With this notation we now get Corollary 4.9. Suppose that A(∆p ) is an h∗ -fibration and let ∂l : hq−p+1 (A(∆ ◦ p−1 )) → hq−p (A(∆ ◦ p )) l denote the isomorphism of Definition 4.5. Let Φqp : hq (A(∆p )) → hq−p (A(∆ ◦ p )) denote the canonical isomorphism and let Φqp−1 : hq (A(∆p−1 )) → hq−p+1 (A(∆ ◦ p−1 )) l l be the canonical isomorphism for A(∆p−1 ), with respect to the orientation of l p ∆p−1 inherited from ∆ . Then l ∂l ◦ Φqp−1 ◦ resl = (−1)p−l Φqp . Proof. Consider the permutation ϕ : {0, . . . , p} → {0, . . . , p} defined by • ϕ(k) = k for k = 0, . . . , l − 1; • ϕ(k) = k + 1 for k = l, . . . , p − 1; • ϕ(p) = l. Then sign(ϕ) = (−1)p−l . If we identify ϕ∗ A with A via Φϕ , as done in the proof of Proposition 4.8, then the isomorphism Φqp for the fibration ϕ∗ A(∆p ) just becomes ∂l ◦ Φqp−1 . By Proposition 4.8 this differs from the isomorphism Φqp for A(∆p ) by the factor sign(ϕ) = (−1)p−l . ! The Leray-Serre spectral theorem. We are now going back to the situation of Proposition 4.1 in the special case where X is a finite dimensional simplicial complex. In what follows, we write Cp for the set of oriented closed p-simplexes in X. To be more precise, we consider any element σ ∈ Cp as a given affine realization σ : ∆p → ∆pσ ⊆ X of the closed p-cell ∆pσ of X, which then induces an orientation on ∆pσ . If σ ∈ Cp , we write σl : ∆p−1 → ∆p−1 l σ,l ⊆ X for the l-th face of σ. Then there exists a unique element τ ∈ Cp−1 such that τ (∆p−1 ) = p−1 σl (∆p−1 ) and then a unique affine transformation ϕlσ,τ : ∆p−1 → ∆p−1 ∼ =∆l l such that (4.6) τ = σl ◦ ϕlσ,τ . These give precisely the gluing data for our simplicial complex X. 24 ECHTERHOFF, NEST, AND OYONO-OYONO As outlined in Remark 4.2, under suitable finiteness conditions explained there, the E1 terms are then given by ! hq (A(∆ ◦ pσ )) E1p,q = σ∈Cp and the differential d : E1p−1,q+1 → E1p,q applied to the direct summand p−1,q+1 hq+1 (A(∆ ◦ p−1 projects to the direct summand hq (A(∆ ◦ pσ )) of E1p,q τ )) of E1 via the chain of maps p−1 Iq+1 (τ ) (4.7) hq+1 (A(∆ ◦ p−1 τ )) −→ hq+1 (Ap−1,p−2 ) ι Qpq (σ) ∂ ∗ −→ hq+1 (Ap−1 ) −→ hq (Ap,p−1) −→ hq (A(∆ ◦ pσ )), $ where as before, Ap = A(Xp ) , Ap,p−1 = A(Xp \ Xp−1 ) ∼ ◦ pσ ), = σ∈Cp A(∆ I p−1 (τ ) : A(∆ ◦ p−1 τ ) → Ap−1,p−2 is the inclusion map corresponding to the open p cell ∆ ◦ σ and Qp (σ) : Ap,p−1 → A(∆ ◦ pσ ) is the quotient map. We now apply the inverses of the canonical isomorphisms Φpp+q : hp+q (A(∆pσ )) → hq (A(∆ ◦ pσ )) (see Definition 4.5) to each simplex σ ∈ Cp (and similarly for τ ∈ Cp−1 ) which gives us isomorphisms ! ! hp+q (A(∆p−1 hp+q (A(∆pσ )) and E1p−1,q+1 ∼ E1p,q ∼ = = τ )). τ ∈Cp−1 σ∈Cp In this picture, the differential is described on the summands via (4.8) Φp−1 p+q p−1 Iq+1 (τ ) ι ∗ dστ : hp+q (A(∆p−1 ◦ p−1 τ )) −→ hq+1 (A(∆ τ )) −→ hq+1 (Ap−1,p−2 ) −→ hq+1 (Ap−1 ) ∂ Qpq (σ) (Φpq+p )−1 −→ hq (Ap,p−1) −→ hq (A(∆ ◦ pσ )) −→ hp+q (A(∆pσ )). We then show: Proposition 4.10. Suppose that A(X) is an h∗ -fibration over the finite dimensional simplicial complex X. Then the map dστ in (4.8) is zero, if ∆p−1 is τ p not a face of ∆σ , and we have dστ = (−1)p−l sign(ϕlσ,τ )(resστ )−1 , ∼ = p−1 σ p if ∆p−1 = ∆p−1 τ σ,l , where we denote by resτ : h∗ (A(∆σ )) −→ h∗ (A(∆τ )) the isomorphism induced by the quotient map A(∆pσ ) → A(∆pτ ). For the proof of the proposition, we need the following lemma: FIBRATIONS WITH NONCOMMUTATIVE FIBERS 25 Lemma 4.11. Suppose that σ ∈ Cp and τ ∈ Cp−1 are as above. Suppose further that Z is any closed simplicial sub-complex of Xp such that ∆p−1 and τ p σ ∆σ are contained in Z. Then the differential dτ of (4.8) coincides with the same map, if we replace X by Z. Proof. Apply the quotient map q : A(X) → A(Z) to all ingredients of the composition of maps in (4.7) and use naturality of the long exact sequences in h∗ -theory. Since the quotient map induces the identity in the first and in the last place of that chain, and since, by naturality, all other maps are linked by commutative diagrams (note that all maps in the chain are maps taken from appropriate long exact sequences linked by factorizations of the quotient map A(X) → A(Z)), the result follows. ! Proof of Proposition 4.10. Suppose first that ∆p−1 is not a face of ∆pσ . To see τ that dστ is then the zero map, we actually show that the chain of maps of (4.7) relative to the sub-complex Z = ∆p−1 ∪ ∆pσ is the zero map. We then have τ $ A(Zp−1 ) = A(∆p−1 A(∆pσ \ ∆ ◦ pσ ), and the chain of maps in (4.7) becomes τ ) the composition ! ι∗ ∂ p−1 hq+1 (A(∆ ◦ p−1 )) −→ h (A(∆ )) hq+1 (A(∆pσ \ ∆ ◦ pσ )) −→ hq (A(∆ ◦ pσ )). q+1 τ τ But the first map takes its image in the first summand of the middle term, which lies in the kernel of the second map. So we can now restrict to the case where ∆pτ coincides with the l-th face of ∆pσ . By Lemma 4.11, we may also assume without loss of generality that X = ∆pσ . Using once again naturality of long exact sequences in h∗ -theory, and applying this to the inclusion of the ideal A(∆ ◦ p−1 ◦ pσ ) into A(∆pσ \ Xp−2) σ,l ∪ ∆ (which is the restriction to the complement of the p − 2-skeleton of ∆pσ ) shows that in this situation the chain of maps in (4.7) reduces to the boundary map ∂l in the long exact sequence ∂ l → hq+1 (A(∆ ◦ p−1 ◦ pσ )) → hq+1 (A(∆ ◦ p−1 ◦ pσ )) → . σ,l )) → hq (A(∆ σ,l ∪ ∆ Moreover, it follows from Proposition 4.8, that replacing the orientation of l ∆p−1 = ∆p−1 τ σ,l given by τ to that given by σl results to the factor sign(ϕσ,τ ) p−1 in the canonical oriented isomorphism Φp−1 ◦ p−1 p+q : hp+q (A(∆τ )) → hq (A(∆ τ )). l σ Taking this into account, the map dτ becomes sign(ϕσ,τ )-times the composition Φp−1 p+q ∂ (Φpp+q )−1 l hp+q (A(∆p−1 ◦ p−1 ◦ pσ )) −→ hp+q (A(∆pσ )). σ,l )) −→ hq+1 (A(∆ σ,l )) −→ hq (A(∆ But it follows from Corollary 4.9 that this composition coincides with (−1)l (resστ )−1 (which plays the role of resl in that corollary). Hence we arrived at the equation dστ = sign(ϕlσ,τ )(−1)p−l (resστ )−1 , which finishes the proof. ! 26 ECHTERHOFF, NEST, AND OYONO-OYONO We now recall the definition of the simplicial cohomology of a finite dimensionaland locally finite simplicial complex X with coefficients in a group bundle ) G = {Gx : x ∈ X}. We refer to Section 3 for the definition of a group bundle and for the notion of the group bundle H∗ associated to an h∗ -fibration. If Y is any simply connected subset of X, a section g ∈ Γ(Y, G) is said to be constant, if g becomes constant in any trivialization of the bundle over Y , which is equivalent to saying that g(y) = Φx,y g(x) for all x, y ∈ Y . We denote by Γconst (Y, G) the group of constant sections of Y . It is clear that, if Y is simply connected and Z ⊆ Y is any (simply connected) subset, then the restriction map resYZ : Γconst (Y, G) → Γconst (Z, G) is an isomorphism. In particular, Γconst (Y, G) is isomorphic to Gx for every x∈Y. We now define the p-cochains for simplicial cohomology on X with coefficient G as C p (X; G) := {Cp ∋ σ *→ f (σ) ∈ Γconst (∆pσ ; G)}, i.e., as the set of all maps which assign a p-simplex σ ∈ Cp to a constant section on ∆pσ . Moreover, we define Cfpin (X; G) as the subgroup of C p (X; G) consisting of all finitely supported functions. The boundary map ∂ : C p−1 (X; G) → C p (X; G) is defined by p * (−1)l (resσσl )−1 (f (σl )), ∂f (σ) = l=0 l where we define f (σl ) = sign(ϕσ,τ )f (τ ), where τ ∈ Cp−1 is the unique element l with ∆p−1 = ∆p−1 τ σ,l , and ϕσ,τ is defined as in (4.6). It restricts to boundary map p on Cf in (X; G). We define H p (X; G) as the p-th cohomology of the chain complex (C p (X; G), ∂) and Hfpin (X; G) as the p-th cohomology of (Cfpin (X; G), ∂). Of course, both cohomology groups coincide on finite complexes. The computations in Proposition 4.10 now immediately give Theorem 4.12 (Leray-Serre spectral sequence for h∗ -fibrations). Let h∗ be a homology theory on a good category of C ∗ -algebras and suppose that X is a finite dimensional σ-finite and locally finite simplicial complex. Suppose that A = A(X) is an h∗ -fibration with associated group bundle H∗ . If h∗ is σ-additive or if X is finite, the E2 -term in the spectral sequence of Proposition 4.1 is given by E2p,q = Hfpin (X; Hp+q ). FIBRATIONS WITH NONCOMMUTATIVE FIBERS 27 If h∗ is σ-multiplicative, then E2p,q = H p (X; Hp+q ). Remark 4.13. (1) As mentioned earlier, the main example we have in mind for the above theorem is the case where h∗ is the K-theory functor. But the result applies also to other interesting functors like the functor KK(B, ·) for a fixed C*-algebra B. In general, these functors are only finitely additive, so we should restrict to finite simplicial complexes in these situations. (2) We should note that the cohomology groups H p (X; G) we defined above coincide with the usual singular cohomology with local coefficients as defined in many standard text books (e.g., see [11]), while the groups Hfpin (X; G) are known as the simplicial cohomology with local coefficients with finite supports. Although we don’t want to go through all the details for the proof of the Leray-Serre spectral sequence for cohomology theories on C*-algebras (the steps are similar as for homology theory with all arrows reversed) we want at least give a proper statement of the result. As mentioned earlier, K-homology serves as a main example of such theory, but other examples are given by the functors KK(·, B) for a fixed C*-algebra B. Recall that the simplicial homology Hp (X; G) with coefficient in a group bundle G is defined as the homology of the chain complex (Cp (X; G), d), where * aσ σ : aσ ∈ Γconst (∆pσ ; G)}, Cp (X; G) := { σ∈Cp with boundary map d : Cp (X; G) → Cp−1 (X; G) given by d(aσ σ) = p * (−1)l sign(ϕlσ,τ ) resσσi (aσ )τl l=0 where for each 0 ≤ l ≤ p, τl is the unique element in Cp−1 with image ∆p−1 σ,l and sign(ϕlσ,τ ) is as before. Again, if we restrict to finite sums, we obtain the theory Hpf in (X; G) with finite supports. The Leray-Serre theorem then reads as follows Theorem 4.14 (Leray-Serre spectral sequence for h∗ -fibrations). Let h∗ be a cohomology theory on a good category of C ∗ -algebras and let X be as in Theorem 4.12. Suppose that A = A(X) is an h∗ -fibration with associated group bundle H∗ . If h∗ is σ-additive or if X is finite, the E 2 -term in the spectral sequence of Proposition 4.1 is given by 2 Ep,q = Hpf in (X; Hp+q ). If h∗ is σ-multiplicative, then 2 Ep,q = Hp (X; Hp+q ). 28 ECHTERHOFF, NEST, AND OYONO-OYONO We want to close this section with a discussion how the spectral sequences considered here give new invariants for RKK-equivalence of C*-algebra bundles. The following lemma might be well known to the experts, but since we rely heavily on it, we give the argument here. For notation, we let KK denote the category whose objects are separable C*-algebras and the morphisms between two objects A and B are the elements in KK(A, B). Recall that for any pair of bundles A(X) and B(X) and any continuousinclusion map f : Y ֒→X there exists a canonical pull-back map f ∗ : RKK(X; A(X), B(X)) → RKK(Y ; f ∗ A(Y ), f ∗ B(Y )). In particular, if Y ⊆ X, then we obtain restrictions x *→ resX Y (x) from RKK(X, A(X), B(X)) → RKK(Y ; A(Y ), B(Y )) given via the inclusion of Y into X (see [17, Proposition 2.2]). Recall also that a short exact extension q 0 → J → A → A/J → 0 of C*-algebras is semi-split if there exists a completely positive section s : A/J → A. Note that this is always true if A/J is nuclear (which follows if A is nuclear). Lemma 4.15. Suppose that A(X) and B(X) are C*-algebra bundles over X and suppose that U ⊂ X is open. Let x ∈ RKK(X; A, B) and let iA : A(U) → A(X) and qA : A(X) → A(X \U) denote the inclusion and quotient maps (and similarly for B(X)). Then the diagram i q i qB A A → A(X) −−− → A(X \ U) A(U) −−−    resX (x)   x (x) resX ' X\U ' ' U B B(U) −−− → B(X) −−−→ B(X \ U) commutes in the category KK. Moreover, if both extensions in the above diagram are semi-split (we do not require that the c.p. sections are C0 (X)-linear), then the diagram ∂ A SA(X \ U) −−− → A(U)    resX (x) resX (x) ' ' U X\U ∂ B SB(X \ U) −−− → B(U) also commutes in KK, where SA and SB denote the suspensions of A and B, respectively. Proof. Let (E(X), T ) be a Kasparov cycle representing x. We may assume that A(X) acts nondegenerately on E(X). If f : Y → X is a continuous map, then f ∗ (x) is represented by the cycle (C0 (Y ) ⊗C0 (X) E(X), 1 ⊗ T ) ∼ = (E(X) ⊗C0 (X) C0 (Y ), T ⊗ 1) FIBRATIONS WITH NONCOMMUTATIVE FIBERS 29 depending on whether we want tensor C0 (Y ) from the left or from the right. The first square of the first diagram then follows from an obvious isomorphism of KK(A(U), B(X))-cycles (A(U) ⊗A(X) E(X), 1 ⊗ T ) ∼ = (C0 (U) ⊗C (X) E(X), 1 ⊗ T ). 0 The left cycle represents [iA ]⊗A(X) x and the right cycle represents resX U (x)⊗B(U ) [iB ]. Similarly, the second square of the first diagram follows from the observation that both products [qA ] ⊗A(X\U ) resX X\U (x) and x ⊗B(X) [qB ] are represented by the module (E(X) ⊗C0 (X) C0 (X \ U), T ⊗ 1) with the canonical module actions. So let us now assume that both extensions are semi-split. Then the boundary map ∂A in the second diagram is given by Kasparov product with an element [∂A ] ∈ KK1 (A(X \ U), A(U)) constructed as follows: let " # " # Z = (0, 1] × (X \ U) ∪ {1} × U ⊆ (0, 1] × X. Let p : Z → X denote the canonical projection and write A(Z) for the pullback p∗ A(Z). Note that, as an algebra, A(Z) is just the mapping cone CqA of the homomoprhism qA . Let eA : A(U) → A(Z) be the inclusion map given by identifying U with the open set {1} × U ⊆ Z. It is shown in [1, Theorem 19.5.5] that" eA is a KK-equivalence. Let uA denote its inverse and let # jA : SA(X \ U) = A (0, 1) × (X \ U) → A(Z) denote the inclusion. Then it is shown in [1, Theorem 19.5.7] that [∂A ] = [jA ] ⊗A(Z) uA . The same construction applies to B(X). Let p∗ (x) be the pull-back of x in RKK(Z; A(Z), B(Z)). Then the commutativity of the second diagram follows from the commutativity of " # jA e A (0, 1) × (X \ U) −−− → A(Z) ←−A−− A(U)    resZ (p∗ (x))   resZ (p∗ (x))' p∗ (x)' ' U (0,1)×(X\U ) " # jB e B (0, 1) × (X \ U) −−− → B(Z) ←−B−− B(U) which is a consequence of the commutativity of the first diagram in the lemma. ! We say that a (co-)homology theory on Csep is KK-representable if there exists a C*-algebra B such that the (co-)homology theory is given by A *→ KK∗ (B, A) (resp. A *→ KK(A, B)). Of course K-theory and K-homology are important examples, but also K-theory with coefficients in Z/nZ is an example of such theory. Note that every KK-fibration is automatically an h∗ -fibration (resp. h∗ -fibration) if h∗ (resp h∗ ) is KK-representable. Corollary 4.16. Let h∗ be a KK-representable homology theory on Csep . Assume that A(X) and B(X) are h∗ -fibrations. Then any class x ∈ RKK(X; A, B) induces a morphism between the associated group bundles H∗ (A) and H∗ (B). If X is a CW-complex and A(X) and B(X) are nuclear, 30 ECHTERHOFF, NEST, AND OYONO-OYONO then X induces a morphism between the associated exact couples for the Leray-Serre spectral sequence. In particular, if x is a RKK-equivalence, then it induces an isomorphism between the Leray-Serre spectral sequences for h∗ (A) and h∗ (B). A similar statement holds for KK-representable cohomology theories. Remark 4.17. Note that in the case of K-theory one can omit the nuclearity assumption on A(X) and B(X) in the above lemma. The reason is that for U ⊆ X open, and Z as in the proof of Lemma 4.15, we always have e∗ an isomorphism K∗ (A(U)) ∼ = K∗ (A(Z)) such that the boundary map ∂A : K∗+1 (A(X \ U)) → K∗ (A(U)) is given via the composition ∂A = (e∗ )−1 ◦ jA,∗ , with eA and jA as in the proof of the lemma. Thus the same argument as in the lemma shows that for each such U the transformation given by Kasparov product with the appropriate restrictions of x gives a transformation between the K-theory long exact sequences for A and B corresponding to U. This is all we need to obtain a well-defined morphism between the exact couples. 5. Applications to non-commutative torus bundles Recall from [8] that a non-commutative principal Tn -bundle (or NCP Tn bundle for short) is defined as a C*-algebra bundle A(X) equipped with a fibrewise action of Tn such that A(X) ! Tn is isomorphic to C0 (X, K). By Takesaki-Takai duality, every such bundle can be realized up to stabilization by a crossed product C0 (X, K)!Zn for some fiberwise action of Zn on C0 (X, K). Using results from [9, 10] we showed in [8, §2] that the Tn -equivariant stable isomorphism classes of NCP Tn -bundles over a given space X can be classified by the pairs ([Y ], f ), where [Y ] denotes the isomorphism class of a classical n(n−1) q principal Tn -bundle Y → X and f : X → T 2 is a continuous map. The NCP Tn -bundle corresponding to the pair ([Y ], f ) is then given by " # " #Tn Y ∗ f ∗ (C ∗ (Hn )) = C0 (Y ) ⊗C0 (X) f ∗ (C ∗ (Hn )) . Let’s recall the ingredients of this construction: C ∗ (Hn ) is the group C*-algebra of the group Hn = 4g1 , . . . , gn , fij : 1 ≤ i < j ≤ n5 with relations gi gj = fij gj gi and fij central for all 1 ≤ i < j ≤ n. This group n(n−1) has center Zn = 4fij : 1 ≤ i < j ≤ n5 ∼ = Z 2 and, therefore, C ∗ (Hn ) is a n(n−1) +n via the inclusion continuous C*-algebra bundle over T 2 ∼ =Z Φ : C(T n(n−1) 2 )∼ = C ∗ (Zn ) → Z(C ∗ (Hn )). # Moreover, if we equip C ∗ (Hn ) with the dual action of Tn ∼ =H n /Zn it becomes n(n−1) n an NCP T -bundle over T 2 . We shall denote by U1 , . . . , Un the unitaries FIBRATIONS WITH NONCOMMUTATIVE FIBERS 31 of C ∗ (Hn ) corresponding to g1 , . . . , gn , respectively, and by Wi,j the unitaries n(n−1) corresponding to fi,j for 1 ≤ i < j ≤ n. If f : X → T 2 is a continuous map, then the pull-back f ∗ (C ∗ (Hn )) becomes an NCP Tn -bundle over X. By taking the C0 (X)-balanced tensor product of f ∗ (C ∗ (Hn )) with C0 (Y ) (with C0 (X)-action on C0 (Y ) induced by q : Y → X in the obvious way), and then taking the algebra of fixed points with respect to the diagonal action (with action by inverse# automorphism on one factor) provides the NCP " the n ∗ ∗ T -bundle Y ∗ f (C (Hn )) . The Tn -action is induced by the given Tn -action on Y . In [8] we studied the topological nature of the C*-algebra bundles A(X) after “forgetting” the underlying Tn -actions. In particular, we were interested in the question under what conditions two such bundles are K-theoretically equivalent fibrations, i.e., when are two such bundles RKK-equivalent. We arrived at the following result: Theorem 5.1 ([8, Theorem 7.2]). Let A(X) be a NCP Tn -bundle over the path n(n−1) connected space X and let f : X → T 2 be the continuous map associated to A(X) as above. Then the following are equivalent: (i) f is homotopic to a constant map. (ii) The K-theory bundle K∗ (A(X)) is trivial. (iii) A(X) is RKK-equivalent to C0 (Y ) for some (commutative) principal Tn -bundle q : Y → X. In this section we will use the Leray-Serre spectral sequence to obtain the following triviality result. Theorem 5.2. Let A(X) be the NCP Tn -bundle corresponding to the pair ([Y ], f ) as explained above such that X is a finite dimensional locally finite and σ-finite simplicial complex. Then the following are equivalent: (i) A(X) is RKK-equivalent to C0 (X × Tn ). (ii) The K-theory bundle K∗ (A) is trivial and all d2 -differentials in the Leray-Serre spectral theorem for A(X) vanish. (iii) f is homotopic to a constant map and Y ∼ = X × Tn as Tn -bundles. The proof depends on explicit calculations of the d2 maps in the spectral sequence of a commutative principal Tn -bundle q : Y → X, and then transporting this result to the spectral sequence for the K∗ -fibration A(X). In what follows we shall always denote by Yx the fibre of a given principal Tn -bundle over a point x ∈ X and we write {Erp,q (Y ), dr } for the spectral sequence corresponding to a fixed triangulation of the base X. We always assume that X is finite dimensional and σ-finite. The following proposition is certainly well-known to the experts, but since we didn’t find an appropriate reference we give a proof. 32 ECHTERHOFF, NEST, AND OYONO-OYONO q Proposition 5.3. Let Y → X be a principal T-bundle. Then the (0, 1)-degree component of the differential 0 1 2 0 d0,1 2 : H (X, K (Yx )) → H (X, K (Yx )) q on E2∗,∗ (Y ) vanishes if and only if Y → X is trivial. The proof of this proposition will require some preliminary work. Let X−1 = ∅ ⊂ X0 ⊂ · · · ⊂ Xn = X be the skeleton decomposition of X and let us set Yk = q −1 (Xk ). For σ a simplex of X, let us denote by Vσ the closure of the ∗-neighboorhood of σ. We may assume that • for all simplices σ there exist continuous maps Ψσ : q −1 (Vσ ) → T; • for all pairs of simplices σ and σ % which are faces of a common simplex, there exists a continuous map hσ,σ" : Vσ ∩ Vσ" → R such that (i) q −1 (Vσ ) → Vσ × T; y *→ (q(y), Ψσ (y)) is a T-equivariant homeomorphism. (ii) Ψσ" = Ψσ · e2iπhσ,σ" on Vσ ∩ Vσ" for all pairs of simplices σ and σ % which are faces of a common simplex. We will denote by V the corresponding atlas. Notice that this atlas provides an idenfication K 1 (Yx ) ∼ = K 1 (T) induced by Yx → T; y *→ Ψσ (y) for any x in the simplex σ. Although the identification Yx ∼ = T depends on σ, it follows 1 1 ∼ from (ii) that the induced map K (Yx ) = K (T) does not. Let X0 = {x0 , x1 , x2 , x3 , . . .} be the set of vertices of X. If xi and xj are connected by an edge, we will denote by ei,j the oriented edge starting at xi and ending at xj . For t ∈ [0, 1], we define xi,j (t) = (1 − t)xi + txj in ei,j . Let UV : Y0 → T be the continuous map such that UV and Ψxi coincide on q −1 (xi ). We extend UV to a continuous map WV : Y1 → T in the following way: If xi and xj are connected by the oriented edge ei,j , we define WV on q −1 (ei,j ) by WV (z) = e2iπthxi ,xj (q(z)) Ψxi (z) for q(z) = xi,j (t). We have E10,1 (Y ) = K 1 (Y0 \ Y−1 ) = K 1 (Y0 ) E11,2 (Y ) = K 0 (Y1 \ Y0 ) and d0,1 : K 1 (Y0 ) → K 0 (Y1 \ Y0 ) is the boundary map ∂ associated the the 1 pair (Y0 , Y1 ). Since UV is the restriction of WV to Y0 , the class [UV ] of UV in K 1 (Y0 ) satisfies ∂[UV ] = 0 and thus [UV ] defines a class ωV in E20,1 (Y ) ∼ = H 0 (X, K 1 (Yx )) ∼ = Z which is thereby a generator. Lemma 5.4. With the notations above and up to the canonical identification 1,0 K 0 (Yx ) ∼ = H 2 (X, Z) is the = Z (which sends [1] to 1) d2 ωV ∈ H 2 (X, K 0 (Yx )) ∼ first Chern class of Y . FIBRATIONS WITH NONCOMMUTATIVE FIBERS 33 Proof. We extend WV : Y 1 → T to a continuous map φV : Y2 → C given on Yσ = q −1 (σ) ∼ = σ × T for a 2-simplex σ in X with boundary ∂σ and center xσ by φV (tx + (1 − t)xσ , z) = tWV (x, z) for t in [0, 1], x in ∂σ and z ∈ T. Then , φV −(1 − φV φ̄V )1/2 VV := (1 − φV φ̄V )1/2 φ̄V / . W 0 is a lift in U2 (C(Y2 )) for 0V W V and thus ∂[WV ] = [PV ] − [( 10 00 )], where , , 1 0 φV φ̄V φV (1 − φV φ̄V )1/2 ∗ PV = VV · · VV = 0 0 φ̄V (1 − φV φ̄V )1/2 1 − φV φ̄V is a projector in M2 (C(Y2 )). Then, up to Bott periodicity, d1,0 2 ω is the class in H 2 (X, K 0 (Yx )) = E22,0 (Y ) of the simplicial 2-cocycle c, with value on a 2-simplex σ oriented by its boundary ∂σ o o c(σ) = i∗ ◦ ∂[WV ] ∈ K 0 (q −1 (σ)) ∼ = K 0 (σ ×T), σ o o where σ= σ \ ∂σ and iσ is the inclusion map iσ :σ֒→ Y2 \ Y1 . Finally, we get o c(σ) = [PV |σ] − [( 10 00 )] ∈ K 0 (q −1 (σ)), where PV |σ ∈ M2 (C(q −1 (σ)) is the restriction of PV to q −1 (σ). Let us denote by Φσ the inverse of the trivialisation map q −1 (σ) → σ × T; y *→ (q(y), Ψσ (y)). We then get isomorphisms o o (5.1) K 0 (q −1 (σ)) ∼ = Z, = K 0 (σ) ∼ where o o • the first isomorphism is induced by σ→ q −1 (σ); x *→ Φσ (x, 1) (if we o o o o identify q −1 (σ) with σ ×T via Ψσ , this simply becomes σ→σ ×T; x *→ (x, 1)); o • the second map is the Bott periodicity for the interior σ ∼ = R2 of the oriented simplex σ. Let us define for a continuous map f : σ → C with |f | ≤ 1 and |f | = 1 on ∂σ the projector , |f |2 f (1 − |f |2 )1/2 . Pf = ¯ f (1 − |f |2 )1/2 1 − |f |2 Then [Pf ] − [( 10 00 )] is the image of [f |∂σ ] ∈ K 1 (∂σ) under the boundary map in K-theory associated to the exact sequence o 0 → C0 (σ) → C(σ) → C(∂σ) → 0. In particular [Pf ] − [( 10 00 )] only depends on the winding number of f |∂σ on ∂σ and this winding number is precisely the image of [Pf ] − [( 10 00 )] under the second isomorphism of equation 5.1. Consequently, if we set fV,σ : σ → C; x *→ Φσ (x, 1), 34 ECHTERHOFF, NEST, AND OYONO-OYONO then the image of c(σ) under the chain of isomorphism of equation 5.1 is the winding number of the restriction of fV,σ to the oriented boundary ∂σ. If σ has vertices xi , xj and xk connected by oriented edges ei,j , ej,k and ek,i , then since hσ,xi + hxi ,xj − hσ,xj and hσ,xj + hxj ,xk − hσ,xk are integers, we have fV,σ = e2iπhσ,xi · gV,σ where gV,σ (xi,j (t)) = exp(2iπthxi ,xj (xi,j (t))) gV,σ (xj,k (t)) = exp(2iπ(hxi ,xj (xj,k (t)) + thxj ,xk (xj,k (t)))) gV,σ (xk,i (t)) = exp(2iπ(hxi ,xj (xk,i (t)) + hxj ,xk (xk,i (t)) + thxk ,xi (xk,i (t)))). But e2iπhσ,xi has winding number 0, and hxi ,xj and hxj ,xk can be pushed forward homotopically to the edge ek,i , and thus the restriction of gV,σ to ∂σ is a unitary homotopic to xi,j (t) *→ 1 xj,k (t) *→ 1 xk,i (t) *→ exp(2iπt(hxi ,xj (xk,i (t)) + hxj ,xk (xk,i (t)) + hxk ,xi (xk,i (t)))). Since hxi ,xj + hxj ,xk + hxk ,xi is an integer mσ , the restriction to ∂σ of gV,σ and hence of fV,σ has winding number mσ . Up to the composition of the two isomorphisms of equation 5.1, we finally get that c(σ) = mσ , which is precisely the cocycle defining the first Chern class of Y . ! Proof of Proposition 5.3. Since ωV is a generator for E20,1 (Y ) ∼ = Z, we see from 0,1 0,1 2,0 Lemma 5.4 that d2 : E2 (Y ) → E2 (Y ) is vanishing if and only if the first Chern class of Y vanishes, i.e if and only if Y is trivial. ! Let us now generalise this result to Tn -principal bundles. For this, we define Tni = {(z1 , . . . , zn ) ∈ T such that zi = 1} and we let Yi = Y /Tni denote the quotient space for the action of Tni on a principal Tn -bundle Y → X. Then Yi is a T-principal bundle with action induced by the inclusion αi : T ֒→ Tn of the i-th factor and with base space X. Moreover Y is isomorphic as a Tn -bundle to α1∗ Y1 ∗ · · · αn∗ Yn , where αi∗ Yi is the Tn -bundle induced from Yi by αi . In consequence, Y is completly determined (up to isomorphism of Tn -bundle) by the first Chern classes of the principal T-bundle Yi and Y is a trivial Tn -bundle if and only if all the Yi are trivial. q Proposition 5.5. Let Y → X be a principal Tn -bundle. Then the (0, 1)-degre component of the differential 0 1 2 0 d0,1 2 : H (X, K (Yx )) → H (X, K (Yx )) q on E2∗,∗ (Y ) vanishes if and only if Y → X is trivial. FIBRATIONS WITH NONCOMMUTATIVE FIBERS 35 Proof. Since K ∗ (Y ) is equipped with an algebra structure, d2 is a map of differential algebra. The unital algebra K ∗ (Yx ) being generated by the image of K ∗ (Yi,x ) under the morphism induced by the projection map Y → Yi , the map d0,1 2 is completly determined by the image of elements coming from K ∗ (Yi,x ). The projection map Y → Yi provides a morphism of spectral sequences (Erp,q (Yi ), dr ) −→ (Erp,q (Y ), dr ). In particular, we get a commutative diagram d0,1 H 0 (X, K 1 (Yx,i)) −−2−→ H 2 (X, K 0 (Yx,i))     ' ' d0,1 H 0 (X, K 1 (Yx )) −−2−→ H 2 (X, K 0 (Yx )), where the vertical arrows are induced by the projection of the fiber πi : Yx → Yi,x . Since the inclusion K 0 (Yi,x ) → K 0 (Yx ) is injective (since it sends [1] to [1]) the right vertical map is injective, too. Thus it follows from Proposition 5.3 that the range of the left vertical map of the diagram lies in the kernel of 0 1 2 0 ! d0,1 2 : H (X, K (Yx )) → H (X, K (Yx )) if and only if Yi is trivial. Remark 5.6. ∗,∗ ∗,∗ (i) If Y ∼ = X × Tn is the trivial Tn -bundle over X, then d2 : E2 → E2 vanishes completely. To see this, we can use the Künneth formula in K-theory to show that it is enough to prove the result for the LeraySerre spectral sequence associed to C0 (X), i.e., the Hirzebruch spectral sequence for the K-theory of X. Then E2p,q (X) = H p (X, Z) if p − q is even and E2p,q (X) = 0 if p − q is odd. Since d2 maps E2p,q (X) to E2p+2,q+1(X), it follows that either the source or the target of this map must be zero. Thus d2 = 0. As a direct consequence of this observation q and of Proposition 5.5 we now see that a principal Tn -bundle Y → X is trivial if and only if all d2 -differentials in the associated spectral sequence vanish. q (ii) More generally, if Y → X is a principal Tn -bundle, then K ∗ (Y ) $ p,q is endowed with a K ∗ (X)-module structure. Then E2 (Y ) = $ p $ p−q p p−q H (X, K (Yx )) has a H (X, K ({∗}))-module structure $ 0 provided by the cup product. Thereby, H (X, K p−q (Yx )) being a $ p generating set and since the differential d2 is H (X, K p−q ({∗}))linear, then d2 is completly determined by the Chern classes ci . Proof of Theorem 5.2. Using the fact that RKK-equivalence induces an equivalence of spectral sequences, the result is now a direct consequence of Theorem 5.1 and the above remark. ! A natural question is then: let A(X) be a NCP-Tn -bundle with classifying data ([Y ], f ). Can we recover from the exact sequence {Er∗,∗ (A)), dr } derived 36 ECHTERHOFF, NEST, AND OYONO-OYONO from the K∗ -fibration A(X) any information concerning Y ? As we shall see below this is not always the case. Using the above notations, we let qi : Y → Yi = Y /Tni denote the quotient map. There are canonical C0 (X)-linear ∗-homomorphisms Λi : C(Yi ) → A(X) = Y ∗ f ∗ (C"∗ (H))(X) given as follows: First we define a ∗" # Tn homomorphism Λ̃i : C0 (Yi ) → C0 (Y ) ⊗ C(T)) by Λ̃i (φ) (y, t) = φ(tqi (y)), where T is acted upon by Tn using the projection on the i-th component. Next we identify C(Ti ) with C ∗ (Ui ) via functional calculus to obtain from this a well defined C0 (X)-linear ∗-homomorphism (also called Λ̃i ) from C0 (Yi ) " #Tn to C0 (Y ) ⊗ C ∗ (Ui ) . Since Ui commutes with all Wkj , the C ∗ -algebra C ∗ 4Ui , Wkj ; 1 ≤ k < j ≤ n5 is a C(Tn(n−1)/2 )-subalgebra of C ∗ (Hn ) isomorphic to C(Tn(n−1)/2 ) ⊗ C ∗"(Ui ) and hence, as C0 (X)-algebras,# we can n n identify (C0 (Y ) ⊗ C ∗ (Ui ))T ∼ = C0 (Y ) ⊗C0 (X) C(Tn(n−1)/2 ) ⊗ C ∗ (Ui ) )T with n " # T a subalgebra of Y ∗ f ∗ (C ∗ (Hn )) = C0 (Y ) ⊗C0 (X) f ∗ C ∗ (Hn ) . The map Λi is then given by the composition of Λ̃i with this inclusion. The morphism Λi : C0 (Yi ) → A(X) induces a morphism p,q p,q {Λp,q i,r : Er (Yi ) → Er (A)} of spectral sequences. At the E2 -term, the morphism p p−q Λp,q (T)) → H p (X, Kp−q (A)) i,2 : H (X, K is induced by the morphism of group bundles (Λi,x,∗)x∈X : X × K ∗ (T) → K∗ (A). In particular, if ci ∈ H 2 (X, Z) is the Chern class of Yi then using the notations of Lemma 5.4 we get (5.2) 0,1 2,0 d0,1 2 (Λi,2 ([ων ]) = Λi,2 (ci ). According to [11], if X is path connected with base point x, then the cohomology group H ∗ (X, K∗ (A)) can be described in the following way: fix a simplicial decomposition of X and lift it to a π1 (X)-invariant simplicial de0 Let S∗ (X) 0 be the simplicial complex obtained from this composition of X. 0 Then S∗ (X) 0 is endowed with an action of simplicial decomposition of X. Γ = π1 (X) by automorphisms and H ∗ (X, K∗ (A)) is then the cohomology of the 0 K∗ (Ax )) of Γ-equivariant homomorphisms from S∗ (X) 0 complex HomΓ (S∗ (X), to K∗ (Ax ). In particular, in degree zero we get H 0 (X, K∗ (A)) = InvΓ K∗ (Ax ), where for an abelian group N equipped with an action of Γ, InvΓ N stands for the set of Γ-invariant elements of N. Since we will need it later on, we can also define at this point the coinvariant for N to be CoinvΓ N = N/4x − γx; x ∈ N and γ ∈ Γ5. FIBRATIONS WITH NONCOMMUTATIVE FIBERS 37 Using this, and noticing that the classes {[U1,x ], . . . , [Un,x ]} of K1 (Ax ) are invariant, we get that Λ0,1 i,2 ([ων ]) = [Ui,x ] and thus according to equation 5.2 0,1 we finally obtain that d2 ([Ui,x ]) = Λ2,0 i,2 (ci ). Thus we can find the first Chern classes of the Yi in our spectral sequence iff Λ2,0 i,2 (ci ) does not vanish. However, 2,0 as we shall see below, the map Λi,2 is not injective in general. The end of the section is devoted to the study of the spectral sequences of NCP T2 -bundles with base T2 . In case where the underlying function f : T2 → T is homotopic to a constant, we get a complete description by Remark 5.6. If f is not homotopic to a constant, then the only part of the differential d2 which does not vanish automatically is 0 2 1 2 2 2 d0,1 2 : H (T , K (T )) → H (T , K0 (A)). since we shall see below that H 0 (T2 , K0 (A)) is given by the invariants in K0 (T2 ) under the action of Z2 ∼ = π(T2 ), and hence is generated by the class [1] of the unit. Since this class trivially extends to a class in K0 (A(X)), it must vanish under any differential in the spectral sequence. To proceed let us first remark that if F : [0, 1]×X → Tn(n−1)/2 is a homotopy between f0 : X → Tn(n−1)/2# and f1 : X → Tn(n−1)/2 and if q : Y → X is any " Tn -bundle, then Y × [0, 1] ∗ F ∗ (C ∗ (Hn )) is a homotopy of NCP -Tn -bundles and thus according to [8, Proposition 3.2], Y ∗ f0∗ (C ∗ (Hn )) and Y ∗ f1∗ (C ∗ (Hn )) are RKK-equivalent. q For X = T2 , the classifying data are ([Y → T2 ], f ), where f : T2 → T is a continuous function. According to the previous remark, we can replace f by a homotopic function and thus we can assume that there exist integers k and l such that f (z1 , z2 ) = z1k z2l for every (z1 , z2 ) in T2 . Let us compute H ∗ (T2 , K0 (A)). We have π1 (T2 ) ∼ = Z2 with action of the generators (1, 0) and 2 0 2 (0, 1) of Z on K (T ) in the base ([1], β) given by the matrices ( 10 k1 ) and ( 10 1l ), respectively (see [8, Proposition 5.2]). Let us fix a simplicial decomposition of T2 . Then since R2 → T2 is the classifying covering for Z2 , the simplicial complex S(R2 ) is a free resolution for Z[Z2 ] and hence for any abelian group M equipped with an action of Z2 , the cohomology of the complex HomZ2 (S(R2 ), M) is naturally isomorphic to H ∗ (Z2 , M). Recall from [13] that for an abelian group M equipped with an action of Zn , the cohomology group H ∗ (Zn , M) can be computed recursively in the following way: • For n = 0 we have that H 0 (Zn , M) ∼ = M and H k (Zn , M) = {0} for k ≥ 1. • Let us consider the action of Zn−1 on M using the n − 1 last factors of Zn . Then the action of the first factor of Zn induces an action of Z on 38 ECHTERHOFF, NEST, AND OYONO-OYONO H k (Zn−1 , M) and there is an natural exact sequence 0 −→ CoinvZ H k (Zn−1 , M) −→ H k (Zn , M) −→ InvZ H k−1(Zn−1 , M)) −→ 0. From this, it is straightforward to check that H n (Zn , M) is naturally isomorphic to CoinvZn M. In the case M = Z equipped with the trivial action of Zn , the corresponding identification H n (Zn , Z) ∼ = Z is given = CoinvZn Z ∼ n by pairing with the fundamental class of Hn (Z , Z). Under the identification H∗ (Zn , Z) ∼ = H∗ (Tn , Z), this class can be viewed as the fundamental class [Tn ] of Hn (Tn , Z). Combining all this, we are now in the position to describe the d2 map of the spectral sequence derived from a NCP T2 -bundle A(T2 ) with classifying q data ([Y → T2 ], f ). Let k be the greatest common divisor of the winding numbers of the two components of f . We can assume that k 7= 0, otherwise f is homotopic to a constant map and thus A(T2 ) is RKK-equivalent to C(Y ). Then • H 2 (T2 , K0 (A)) ∼ = Z/kZ = CoinvZ2 K 0 (T2 ) ∼ ! Z, where the image in CoinvZ2 K0 (T2 ) of the class [1] ∈ K0 (T2 ) is a generator for Z/kZ, and where the image of the Bott element β ∈ K0 (T2 ) is a generator for Z. • Up to this identification, d0,1 2 has range in Z/kZ and 2 d0,1 2 ([Ui,z ]) = 4ci , [T ]5 mod k, for z = f (1, 1), i = 1, 2, and where ci is the Chern class of the T-bundle Y /T2i → X. Remark 5.7. (i) In particular, for the function f (z1 , z2 ) = z1 , the d2 map vanishes for any principal T2 -bundle Y → T2 . 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Tu. La conjecture de Novikov pour les feuilletages hyperboliques, K-theory, 16, 129-184 (1999). [22] J.-L. Tu. La conjecture Baum-Connes pour les feuilletages moyennable, K-theory, 17, 215-264 (1999). [23] D.P. Williams. The structure of crossed products by smooth actions. J. Austral. Math. Soc. Ser. A 47 (1989), no. 2, 226–235. S. Echterhoff: Westfälische Wilhelms-Universität Münster, Mathematisches Institut, Einsteinstr. 62 D-48149 Münster, Germany E-mail address: echters@uni-muenster.de R. Nest: Department of Mathematics, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen, Denmark E-mail address: rnest@math.ku.dk Université Blaise Pascal de Clermont-Ferrand, Laboratoire de Mathématiques, Plateau des Cézeaux, 63177 Aubière Cedex, France E-mail address: oyono@math.cnrs.fr