Cohomology of Stacks

Transcription

1 Cohomology of Stacks K. Behren Department of Mathematics, University of British Columbia, Vancouver, Canaa Lectures given at the School an Conference on Intersection Theory an Mouli Trieste, 9-27 September 2002 LNS

2 Abstract We construct the e Rham cohomology of ifferentiable stacks via a ouble complex associate to any Lie groupoi presenting the stack. This is a straightforwar generalization of the Čech-e Rham complex of a ifferentiable manifol. We explain the relationship to the Cartan moel of equivariant cohomology. To get a theory of Poincaré uality, we construct the companion theory of cohomology with compact supports. We explain how cohomology acts on cohomology with compact supports, an how to integrate compact support cohomology classes. We specialize to ifferentiable stacks of Deligne-Mumfor type (these inclue orbifols), where we prove that one can calculate cohomology, as well as compact support cohomology, via the complex of global ifferential forms. Finally, for proper ifferentiable stacks of Deligne- Mumfor type we prove Poincaré uality between e Rham cohomology an itself. We go on to efine the cohomology class of a close substack an intersection numbers. We o a few sample calculations from the theory of mouli stacks. In the secon part of these notes we construct the singular homology an cohomology of a topological stack. Among the results we prove is that singular homology equals equivariant homology in the case of a quotient stack. We also prove that the Q-value singular cohomology of a Deligne-Mumfor stack is equal to that of its coarse mouli space. We conclue with a iscussion of Chern classes an a few examples.

3 Contents Introuction 253 De Rham cohomology 254 The simplicial nerve of a Lie groupoi Čech cohomology The e Rham complex Equivariant Cohomology Multiplicative structure Cohomology with compact supports Moule structure The integral, Poincaré uality Deligne-Mumfor stacks The class of a substack, intersection numbers Example: the stack of elliptic curves The Lefschetz trace formula Singular homology 279 The singular chain complex of a topological groupoi What o cycles look like? Examples Invariance uner Morita equivalence Equivariant homology Cohomology Relation to e Rham cohomology Relation to the cohomology of the coarse mouli space Chern classes References 294

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5 Cohomology of Stacks 253 Introuction In these lectures we give a short introuction to the cohomology of stacks. We first focus on the e Rham theory for ifferentiable stacks. Then we go on to singular homology of topological stacks. All the technical tools we use are explaine in [2]. Our constructions are often straightforwar generalizations of constructions in [ibi.] When efining the cohomology groups H k (X, R) for a manifol X with values in the real numbers R, there are two approaches: (i) resolve the coefficient sheaf R using, for example, the e Rham complex R Ω 0 Ω 1 Ω 2... an efine H k (X, R) as the cohomology groups of the complex of global sections Γ(X, Ω ). (ii) resolve the manifol X, using a goo cover {U i } (a cover where all intersections are iffeomorphic to R n ), an efine H k (X, R) as the the Čech cohomology groups of the constant sheaf R with respect to the covering. Approach (i) is justifie because for every q, the higher cohomology groups of Ω q on any manifol vanish, i.e., because the sheaves Ω q are acyclic over manifols. Approach (ii) is justifie because for every p, the manifol U p = i 0,...,i p U i0... U ip has no higher cohomology groups (with values in R). In other wors U p is acyclic, for every p. It can be useful to combine the two approaches into one by consiering the Čech-e Rham complex, a ouble complex mae up from all Ωq (U p ). (See [2], Chapter II.) For stacks things are less simple. Approach (i) breaks own, because the Ω q are not acyclic over stacks(see Exercise 6). Approach (ii) breaks own, because of the lack of goo covers. But we can always resolve a stack using a simplicial manifol an all Ω q are acyclic for manifols. Therefore the combination of the two approaches using the Čech-e Rham complex works well for stacks. We will explain this approach in etail.

6 254 K. Behren Remark It is possible to efine the cohomology of stacks by resolving the coefficient sheaf alone. For this we woul have to use injective (or other) resolutions in the big site of the stack. For etails, see [1]. To efine the cohomology of a stack by resolving the stack alone, we woul have to use hypercovers. Neither of these two approaches will be iscusse in these notes, except for the brief Remark 11. De Rham cohomology Differentiable stacks are stacks over the category of ifferentiable manifols. They are the stacks associate to Lie groupois. A groupoi X 1 X 0, is a Lie groupoi if both X 0 an X 1 are ifferentiable (i.e., C ) manifols, all structure maps are ifferentiable an source an target map are (ifferentiable) submersions. Two Lie groupois X 1 X 0 an Y 1 Y 0 give rise to essentially the same stack, if an only if they are Morita equivalent, which means that there is a thir Lie groupoi Z 1 Z 0, together with Morita morphisms Z X an Z Y. A morphism of Lie groupois f : X Y is a Morita morphism if f 0 : X 0 Y 0 is a surjective submersion an the iagram (s,t) X 1 X 0 X 0 f 1 f 0 f 0 (1) (s,t) Y 1 Y 0 Y 0 is cartesian, i.e., a pullback iagram of ifferentiable manifols. We say that a Morita morphism f : X Y amits a section if X 0 Y 0 amits a section. Recall that a functor is an equivalence of categories if it is fully faithful an essentially surjective. Diagram (1) being cartesian translates into f being fully faithful. The requirement on f 0 is much stronger than essential surjectivity (an there are certain weakenings of the notion of Morita morphism taking this into account). For abstract categories, any equivalence has an inverse. For a Morita morphism of Lie groupois, this is not the case, unless it amits a section. Any section s : X 0 Y 0 of a Morita morphism f : X Y inuces uniquely a groupoi morphism s : Y X with the properties

7 Cohomology of Stacks 255 f s = i Y, s f = i X, which means that there exits a 2-isomorphism θ : s f i X. (Recall that a 2-isomorphism between two morphisms of Lie groupois, θ : f g, where f, g : X Y are morphisms, f X θ Y g is a ifferentiable map θ : X 0 Y 1 satisfying the formal properties of a natural transformation between functors.) Another construction from the theory of groupois we will use is restriction. Let X 1 X 0 be a Lie groupoi. Let Y 0 X 0 be a submersion. Define Y 1 by the cartesian iagram Y 1 Y 0 Y 0 X 1 X 0 X 0 (Note that we nee the assumption that Y 0 X 0 is a submersion, in orer that Y 1 will be a manifol an the two maps Y 1 Y 0 will be submersions.) One checks that Y is again a Lie groupoi. It is calle the restriction of X via Y 0 X 0. Note also that if Y 0 X 0 is surjective, then the natural morphism Y X is a Morita morphism. The simplicial nerve of a Lie groupoi Let X 1 X 0 be a Lie groupoi. Then we can prouce a simplicial manifol X as follows. For every p 0 we let X p be the manifol of composable sequences of elements of X 1 of length p. In other wors, X p = X 1 X0 X 1 X0... X0 X }{{} 1. p Then we have p + 1 ifferentiable maps i : X p X p 1, for i = 0,..., p, where i is given by leaving out the i-th object. Thus 0 leaves out the first

8 256 K. Behren arrow, p leaves out the last arrow, an 1,..., p 1 are given by composing two successive arrows. More precisely, i maps the element x 0 φ 1 x 1 φ 2... φ i 1 x i 1 φ i x i φ i+1 x i+1 φ i+2... φ p x p of X p to the element x 0 φ 1 x 1 φ 2... φ i 1 x i 1 φ i φ i+1 x i+1 φ i+2... φ p x p of X p 1. (There are also maps X p 1 X p, given by inserting ientity arrows, but they are less important for us.) Note that for the composition of maps X p X p 2 we have the relations i j = j 1 i, for all 0 i < j p. (2) We summarize this ata by the iagram of manifols Čech cohomology... X 2 X 1 X 0. (3) Let X 1 X 0 be a Lie groupoi an X the associate simplicial manifol. Letting Ω q be the sheaf of q-forms, we get an inuce cosimplicial set Ω q (X 0 ) Ω q (X 1 ) Ω q (X 2 )... (4) simply by pulling back q-forms. Since this is, in fact, a cosimplicial abelian group, we can associate a complex Ω q (X 0 ) Ω q (X 1 ) Ω q (X 2 )... Here : Ω q (X p 1 ) Ω q (X p ) is given by = p ( 1) i i. i=0 We call this complex the Čech complex associate to the sheaf Ωq an the Lie groupoi X. Its cohomology groups H k (X, Ω q ) are calle the Čech cohomology groups of the groupoi X = [X 1 X 0 ] with values in the sheaf of q-forms Ω q.

9 Cohomology of Stacks 257 Remark 1 (naturality) Given a morphism of Lie groupois f : X Y, we get an inuce homomorphism of Čech complexes It is given by the formula f : Č (Y, Ω q ) Č (X, Ω q ). f (ω)(φ 1... φ p ) = ω ( f(φ 1 )......f(φ p ) ), for ω Ω q (Y p ). Here φ 1... φ p abbreviates the element x 0 φ 1 x 1 φ 2... φ p x p in X p. This follows irectly from the presheaf property of Ω q an the functoriality of f. More interestingly, if we have a 2-isomorphism θ : f g between the two morphisms f, g : X Y, then we get an inuce homotopy θ : f g, between the two inuce homomorphisms f, g : Č (Y, Ω q ) Č (X, Ω q ). In fact, θ : Ω q (Y p+1 ) Ω q (X p ) is efine by the formula θ (ω)(φ 1... φ p ) = p ( 1) i ω ( f(φ 1 )... f(φ i )θ(x i )g(φ i+1 )... g(φ p ) ). i=0 One checks (this is straightforwar but teious) that θ + θ = g f. As consequences of these naturality properties we euce that groupoi morphisms inuce homomorphisms on Čech cohomology groups, 2-isomorphic groupoi morphisms inuce ientical homomorphisms on Čech cohomology groups, a Morita morphism amitting a section inuces isomorphisms on Čech cohomology groups. Note that these naturality properties follow formally from the presheaf properties of Ω q, an thus hol for any contravariant functor F : (manifols) (abelian groups).

10 258 K. Behren Proposition 2 If X 1 X 0 is the banal groupoi associate to a surjective submersion of manifols X 0 Y, then the Čech cohomology groups H k (X, Ω q ) vanish, for all k > 0 an all q 0. Moreover, H 0 (X, Ω q ) = Γ(Y, Ω q ). Proof. Recall that the banal groupoi associate to a submersion X 0 Y is efine by setting X 1 = X 0 Y X 0. Since X 1 X 0 X 0 is then an equivalence relation on X 0, we get a groupoi structure on X 1 X 0. Note that such a banal groupoi comes with a canonical Morita morphism X Y, where Y is consiere as a groupoi Y Y in the trivial way. For example, if {U i } is an open cover of Y, an X 0 = U i, then X 1 = Uij, where U ij = U i U j. In this case the proposition is a stanar fact, which follows essentially from the existence of partitions of unity. See for example Proposition 8.5 of [2], where this result is calle the generalize Mayer-Vietoris sequence. (Note that in [loc. cit.] alternating Čech cochains are use, whereas we o not make this restriction. The result is the same.) Another case where the proof is easy, is the case of a surjective submersion with a section. This is because a section s : Y X 0 inuces a section of the Morita morphism X Y. Thus by naturality we have H k (X, Ω q ) = H k (Y Y, Ω q ), which vanishes for k > 0, an equals Γ(Y, Ω q ), for k = 0. The general case now follows from these two special cases by a ouble fibration argument. Let {U i } be an open cover of Y over which X 0 Y amits local sections an let V 0 = U i. We consier the banal groupoi V given by V 0 Y. The key is to introuce W 00 = X 0 Y V 0. Thus W 00 V 0 is now a surjective submersion which amits a section. We efine W mn = X m Y V n, for all m, n 0. V n W mn X m Y Then W is a bisimplicial manifol. This means that we have an array as in Figure 1. It is important to notice that W n is the simplicial nerve of the banal groupoi associate to W 0n V n, an W m is the simplicial nerve of the banal groupoi associate to W m0 X m. All W 0n V n are submersions amitting sections an all W m0 X m are submersions coming from open covers. Thus we alreay know the proposition for all of these submersions.

11 Cohomology of Stacks W 22 W 12 W 02 V 2... W 21 W 11 W 01 V 1... W 20 W 10 W 00 V 0... X 2 X 1 X 0 Y Figure 1: The bisimplicial manifol We apply Ω q to this array to obtain a ouble complex Ω p (W ) mapping to the two complexes Ω q (X ) an Ω q (V ), see Figure 2. Passing to cohomology we get a commutative iagram H (W, Ω q ) H (V, Ω q ) H (X, Ω q ) H (Y Y, Ω q ) an noticing that the two arrows originating at H (W, Ω q ) are isomorphisms, which follows by calculating cohomology of the ouble complex in two ifferent ways, we get the require result. Corollary 3 Any Morita morphism of Lie groupois f : X Y inuces isomorphisms on Čech cohomology groups f : H k (Y, Ω q ) H k (X, Ω q ). Morita equivalent groupois have canonically isomorphic Čech cohomology groups with values in Ω q. Proof. Let X be the ifferentiable stack given by the groupoi Y. The compose morphism X X ientifies X as the stack given by X. Form =

12 260 K. Behren Ω q (W 22 ) Ω q (W 12 ) Ω q (W 02 ) Ω q (V 2 )... Ω q (W 21 ) Ω q (W 11 ) Ω q (W 01 ) Ω q (V 1 )... Ω q (W 20 ) Ω q (W 10 ) Ω q (W 00 ) Ω q (V 0 )... Ω q (X 2 ) Ω q (X 1 ) Ω q (X 0 ) Figure 2: The ouble complex the fibere prouct Z 00 = X 0 X Y 0. Then efine a bisimplicial manifol Z as in the previous proof an apply the same kin of ouble fibration argument to prouce isomorphisms H (Z, Ω q ) H (X, Ω q ) an H (Z, Ω q ) H (Y, Ω q ). Thus we can make the following efinition. Definition 4 Let X be a ifferentiable stack. Then H k (X, Ω q ) = H k (X 1 X 0, Ω q ), for any Lie groupoi X 1 X 0 giving an atlas for X. efines Γ(X, Ω q ) = H 0 (X, Ω q ). In particular, this Example 5 If G is a Lie group then H k (BG, Ω 0 ) is the group cohomology of G calculate with ifferentiable cochains. (Recall that BG is the ifferentiable stack given by the Lie group G itself, consiere as a Lie groupoi G.) Thus there are stacks for which these cohomology groups are non-trivial (see the following exercise).

13 Cohomology of Stacks 261 Exercise 6 Calculate H 1 (BR +, Ω 0 ). Remark 7 We can formalize the above constructions. Let F : (manifols) (abelian groups) be a contravariant functor such that (i) if we restrict F to any given manifol an its open subsets we get a sheaf on this manifol, (ii) for every manifol this sheaf has vanishing Čech cohomology groups. Then we can efine H k (X, F ) for any ifferentiable stack as the Čech cohomology of any groupoi presenting X. The above proof of well-efineness carries through. A contravariant functor F satisfying (i), is calle a big sheaf, on the category of manifols. Thus Ω q is an example of a big sheaf. The e Rham complex The exterior erivative : Ω q Ω q+1 connects the various Čech complexes of a Lie groupoi with each other. We thus get a ouble complex... Ω 2 (X 0 ) Ω 2 (X 1 ) Ω 2 (X 2 )... (5) Ω 1 (X 0 ) Ω 1 (X 1 ) Ω 1 (X 2 )... Ω 0 (X 0 ) Ω 0 (X 1 ) Ω 0 (X 2 )... We make a total complex out of this by setting CDR(X) n = Ω q (X p ) p+q=n an efining the total ifferential δ : C n DR (X) Cn+1 DR (X) by δ(ω) = (ω) + ( 1) p (ω), for ω Ω q (X p ). The sign change is introuce in orer that δ 2 = 0.

14 262 K. Behren Definition 8 The complex C DR (X) is calle the e Rham complex of the Lie groupoi X 1 X 0. Its cohomology groups H n DR(X) = h n( C DR(X) ) are calle the e Rham cohomology groups of X = [X 1 X 0 ]. If X 1 X 0 is the banal groupoi associate to an open cover X 0 Y of a manifol Y, then the e Rham complex of X 1 X 0 is just the usual Čech-e Rham complex as treate, for example, in Chapter II of [2]. Definition 9 One can use Proposition 2 an a ouble fibration argument to prove that e Rham cohomology is invariant uner Morita equivalence an hence well-efine for ifferentiable stacks: H n DR (X) = Hn DR (X 1 X 0 ), for any groupoi atlas X 1 X 0 of the stack X. Remark 10 Because the e Rham complex Ω of big sheaves resolves the big constant sheaf R, we may consier H DR (X) as the cohomology of X with values in R. Remark 11 Recall that any sheaf F on a topological space has a canonical flabby resolution. (This resolution starts out by embeing F into its associate sheaf of iscontinuous sections an continuing in like manner.) For example, enote by Z the canonical flabby resolution of Z, the constant sheaf. Because of the canonical nature of Z, we actually obtain for every q a big sheaf Z q, satisfying the two conitions of Remark 7. Now we can imitate the construction of e Rham cohomology: efine the cohomology of the stack X with values in Z as the cohomology of the total complex of the ouble complex Z (X ), where X is the nerve of any groupoi presenting X. The proof of well-efineness is ientical to the case of e Rham cohomology. For this construction, we on t even nee the stack X or the groupoi X 1 X 0 to be ifferentiable. It works for arbitrary topological stacks. In this way we can efine the cohomology of an arbitrary topological stack with values in an arbitrary (abelian) big sheaf.

15 Cohomology of Stacks 263 Equivariant Cohomology We will explain why the e Rham cohomology of a quotient stack is equal to equivariant cohomology. (Another proof follows from the results of the section on singular homology.) Let G be a Lie group with Lie algebra g an X a G-manifol. To fix ieas, assume that G always acts from the left. There is a well-known generalization of the e Rham complex to the equivariant case, namely the Cartan complex Ω G (X), efine by Ω n G(X) = 2k+i=n ( S k g Ω i (X) ) G. Here S g is the symmetric algebra on the ual of g, which can also be thought of as the space of polynomial functions on g. The group G acts by the ajoint representation on g an by pullback of ifferential forms on Ω (X). The Cartan complex consists of the G-invariants for the inuce action on S g Ω (X). The Cartan ifferential : Ω n G (x) Ωn+1 G (X) is given as the sum = DR ι, where DR : S k g Ω i (X) S k g Ω i+1 (X) is the e Rham ifferential an ι : S k g Ω i (X) S k+1 g Ω i 1 (X) is the tensor inuce by the vector bunle homomorphism g X T X coming from ifferentiating the action. Note that we have to pass to G-invariants in the Cartan complex before the Cartan ifferential satisfies 2 = 0. The functorial properties of the Cartan complex are analogous to the functorial properties of the e Rham complex: if f : X Y is an equivariant map of G-manifols, we get a natural morphism of complexes Ω G (Y ) Ω G (X). More generally, if G H is a morphism of Lie groups an X Y is equivariant from a G-manifol X to an H-manifol Y, we have a natural map Ω H (Y ) Ω G (X). In particular, there is always a quotient map Ω G (X) Ω (X). If G acts trivially on X, there is a canonical section Ω (X) Ω G (X), making Ω (X) a irect summan of Ω G (X). Recall that if G is compact, the cohomology groups HG i (X) = hi( Ω G (X)) are the equivariant cohomology groups of X. We will now formulate an equivariant analog of Proposition 2. Let X Y be a surjective submersion of G-manifols. Then the simplicial nerve of the associate banal groupoi X consists of G-manifols an equivariant maps. Thus we get an associate ouble complex Ω G (X ) together with an augmentation map Ω G (Y ) Ω G (X ).

16 264 K. Behren Lemma 12 If G is compact, the augmentation is a quasi-isomorphism, i.e., inuces isomorphisms for all i. H i G (Y ) hi( tot Ω G (X ) ), Proof. We cannot imitate the proof of Proposition 2, because X Y will, in general, not have sufficiently many local equivariant sections. This is also why we have to restrict to the case that G is compact. Recall that a compact Lie group amits a left invariant gauge form, i.e., a top egree form ω on G, which is left invariant an satisfies G ω = 1. Integrating against ω, we can efine a natural projection operator V Ω (X) ( V Ω (X) ) G, for any (finite-imensional) representation V of G. Now, to prove the lemma, we note that by Proposition 2, the augmentation Ω (Y ) Ω (X ) is a quasi-isomorphism. Tensoring with the representation S g, this remains a quasi-isomorphism. Moreover, because of the existence of the natural projection onto G-invariants, after taking G- invariants, we still have a quasi-isomorphism. Of course, so far, we are using only the ifferential DR on Ω G (X ). But there is a spectral sequence, starting with the cohomology of Ω G (X ) with respect to DR an abutting to the cohomology of Ω G (X ) with respect to = DR ι. There is a corresponing spectral sequence for Ω G (Y ), but it egenerates. Convergence of the spectral sequences now implies the result. We can use this lemma to efine a Morita-invariant notion of equivariant cohomology for groupois. This is not our goal here. Rather we are intereste in the following setup: Let X be a G-manifol. Consier the transformation groupoi G X X, which has projection an operation as structure maps. (The associate stack is the quotient stack [X/G].) We consier the pair (g, x) as a morphism with source gx an target x: gx (g,x) x Proposition 13 If G is compact, there is a natural isomorphism H i G (X) Hi DR (G X X) = Hi DR ([X/G]).

17 Cohomology of Stacks 265 Proof. Denote the simplicial nerve of the transformation groupoi G G G (left multiplication of G on itself) by EG. Then the simplicial manifol EG X is isomorphic to the simplicial nerve of the banal groupoi associate to the projection G X X. In particular, by Lemma 12, we have a quasi-isomorphism Ω G (X) Ω G (EG X). On the other han, we have a morphism of simplicial manifols π : EG X Γ, where Γ is the simplicial nerve of the transformation groupoi G X X. Thus Γ n = G n X. The morphism EG n X = G n+1 X Γ n = G n X is given by n+1, which maps (g 0,..., g n, x) to (g 0,..., g n 1, g n+1 x). Note that π is a (level-wise) principal G-bunle. Now we invoke the theorem of Cartan: if P M is a principal G- bunle, the canonical homomorphism Ω (M) Ω G (M) Ω G (P ) is a quasiisomorphism. Here we nee to use the fact that G is compact for the secon time. We apply the theorem of Cartan levelwise to the principal G-bunle π an obtain a quasi-isomorphism Ω (Γ ) Ω G (EG X). Corollary 14 For a compact Lie group G we have H DR (BG) = (S2 g ) G. Exercise 15 For example, H DR (BS1 ) = R[c]. Let t be a gauge form on S 1, i.e., S t = 1. This form efines a basis element in (s 1 ), which we 1 enote by c. Prove that t efines a 2-cocycle in the e Rham complex of the groupoi S 1. Fin the sign ɛ, such that uner the ientification of HDR 2 (S1 ) with HG 2 ( ), the class of t correspons to ɛc. Remark 16 If G is not compact, then H DR ([X/G]) is still equal to equivariant cohomology. This is not ifficult to believe, as the main ingreient in our proof of Proposition 13 was the fact that H G (P ) = H (M), for every principal G-bunle P over a manifol M. This fact hols for equivariant cohomology in general. Only, in general, the Cartan complex is insufficient to calculate equivariant cohomology. Remark 17 Most ifferentiable stacks occurring in nature are quotient stacks of a group action on a manifol (although not always quotients by a compact group). Thus, the cohomology of all such stacks is simply equivariant cohomology. The stack point of view as one essential insight: if transformation groupois G X X an H Y Y are Morita equivalent, then H G (X) = H H (Y ). In other wors, if G an H act freely an compatibly on a manifol Z, then H G (Z/H) = H H (Z/G).

18 266 K. Behren Exercise 18 Let G be a compact group acting on a manifol X. Prove that there exists a manifol Y with a U(n)-action, for some n, such that H G (X) = H U(n) (Y ). Multiplicative structure We efine a multiplication on the ouble complex (5) as follows. Let ω Ω q (X p ) an η Ω q (X p ). Then we set ω η = ( 1) qp π 1ω π 2η Ω q+q (X p+p ). (6) Here the map π 1 : X p+p X p projects the element φ 1... φ p φ p+1... φ p+p X p+p (7) to an π 2 : X p+p X p φ 1... φ p projects the same element (7) to φ p+1... φ p+p. One checks that δ(ω η) = δ(ω) η + ( 1) p+q ω δ(η), an so we get an inuce cup prouct H n DR (X ) H m DR (X ) H n+m DR (X ). The cup prouct is associative on the level of cochains. But note that this is not true for (skew) commutativity. The cup prouct is commutative only on the level of cohomology. Cohomology with compact supports As with cohomology, cohomology with compact supports is efine via a ouble complex. As usual, let X 1 X 0 be a Lie groupoi. But now we have to also assume that X 1 X 0 is oriente. This means that both the manifols X 1 an X 0 an the submersions s an t are oriente, in a

19 Cohomology of Stacks 267 compatible way. Moreover, assume that both X 0 an X 1 have constant imension. Define two numbers r, n by the formulas r = im X 1 im X 0, n = 2 im X 0 im X 1. Note that n is the imension of the stack efine by X 1 X 0 an r is the relative imension of X 1 over X 0. Let Ω q c(x p ) enote the space of ifferential forms on X p which have compact support. Note that Ω q c is not a sheaf. We consier the ouble complex...! Ωc n+3r (X 2 )! Ωc n+2r (X 1 )! Ωc n+r (X 0 )...! Ωc n+3r 1 (X 2 )! Ωc n+2r 1 (X 1 )! Ω n+r 1 c (X 0 ) (8)...! Ωc n+3r 2 (X 2 )! Ωc n+2r 2 (X 1 )! Ω n+r 2 c (X 0 )... For a form γ Ω n+(p+1)r j c (X p ) its horizontal egree is p an its vertical egree is n j. The vertical ifferential is the usual exterior erivative. The horizontal ifferential is efine in terms of! = i ( 1) i i!, where i! : Ωc q+r (X p ) Ω q c(x p 1 ) is the map obtaine from i : X p X p 1 by integration over the fiber. In fact, for γ Ω q c(x p ), the horizontal ifferential is efine as γ ( 1) p! γ. To make a single complex out of (8), we efine Cc ν (X) = Ω q c(x p ), an set the total ifferential equal to q rp p r=ν δ(γ) = ( 1) p(! γ + γ ), for γ Ω q c(x p ).

20 268 K. Behren Note that the total egree of an element of Ω q c(x p ) is equal to q r(p+1) p, which is the sum of its vertical an horizontal egrees. We also introuce notation for the horizontal cohomology of (8). Namely, we enote the k-th homology of ( Ω c ( +1)r q ) (X ), ±! by H k c (X, Ω q ). This efines Hc k (X, Ω q ) for k 0 an q n. We also enote Hc 0 (X, Ω q ) by Γ c (X, Ω). Moule structure Now we shall turn (8) into a moule over (5). Thus, given ω Ω q (X p ) an γ Ω q c (X p ), we set ω γ = ( 1) qp π 1! ( π 2 ω γ ), where π 1 an π 2 have similar meanings as in (6). More precisely, they are efine accoring to the cartesian iagram X p π 2 X p π 1 X p p (p p)-th projection X 0. 0-th projection Note that ω γ Ω q+q pr c (X p p) an hence we have eg(ω γ) = eg ω + eg γ. Of course, if p < p, then it is unerstoo that ω γ = 0. Using the projection formula f! (f η τ) = η f! (τ) it is not ifficult to check associativity (τ ω) γ = τ (ω γ). Using that integration over the fiber commutes with exterior erivative, one can also check the erivation property δ(ω γ) = δω γ + ( 1) eg ω ω δγ, which implies that the cap prouct passes to cohomology, an we have that H c (X) is a grae moule over the grae ring H (X).

21 Cohomology of Stacks 269 The integral, Poincaré uality We can efine an integral X : Hc n (X) R by noticing that the integral Ωc n+r (X 0 ) R vanishes on cobounaries of the total complex of C c (X ). Finally, we efine a pairing H DR(X) H c (X) R (9) ω γ ω γ, For Poincaré uality, let us assume that X 1 an X 0 are of finite type, i.e., that they both have a finite goo cover (see [2, 5]) an that these covers are compatible via s an t. Proposition 19 (Poincaré uality) Uner this assumption, the pairing (9) sets up a perfect pairing for all p 0. X H p (X) H n p c (X) R, Proof. Consier the homomorphism of complexes C (X) ( C c (X)[n] ) ω ω ( ). X 0 It suffices to prove that this is a quasi-isomorphism. But this we can check by consiering the associate spectral sequences whose E 1 -terms are given by H q (X p ) an Hc n q (X p ), respectively. Thus we conclue using usual Poincaré uality for manifols (see [ibi.]) Definition 20 A ifferentiable stack is of finite type, if there exists a Lie groupoi X 1 X 0 presenting it, where X 0 an X 1 are ifferentiable manifols of finite type amitting compatible finite goo covers.

22 270 K. Behren Let X be a finite type ifferentiable stack. Then Poincaré uality implies that cohomology with compact supports is inepenent of the groupoi chosen to present X. Thus we get well-efine Hc p (X) an an integral (X) R, Hc n X where n = im X. Poincaré uality hols: is a perfect pairing. H p (X) Hc n p (X) R Example 21 Recall that H (BS 1 ) = R[c] is a polynomial ring in one variable. By Poincaré uality, we have { Hc p (BS1 ) R if p is o an negative, = 0 otherwise. To exhibit the moule structure, let ψ i = ( 1) 1 2 i(i+1) 1 Ω 0( (S 1 ) i), which represents an element ψ i Hc 2i 1 (BS 1 ). Note that c ψ i = ψ i 1, for all i an hence c i ψ i = 1, BS 1 for all i, so that {ψ i } is the ual basis of {c i }. Note that the R[c]-moule Hc (BS1 ) is ivisible. Deligne-Mumfor stacks From now on, we will assume that our Lie groupois are étale, which means that s : X 1 X 0 an t : X 1 X 0 are étale (i.e., inuce isomorphisms on tangent spaces). We will also assume that X 1 X 0 X 0 is proper an unramifie, with finite fibers (unramifie means injective on tangent spaces). These conitions mean that the associate ifferentiable stack is of Deligne-Mumfor type. Definition 22 A partition of unity for the groupoi X 1 X 0 is an R- value C -function ρ on X 0 with the property that s ρ has proper support with respect to t : X 1 X 0 an t! s ρ 1.

23 Cohomology of Stacks 271 Partitions of unity may not exist, unless we pass to a Morita equivalent groupoi. This process works as follows. For a groupoi as above there always exists an open cover {U i } of X 0, with the property that the restricte groupoi V i U i (which is the restriction of X 1 X 0 via U i X 0 ) is a transformation groupoi associate to the action of a finite group G i on U i. Given such a cover, we let U = U i an V U be the restriction of X 1 X 0 via U X 0. Thus we have a Morita morphism from V U to X 1 X 0. Now we consier the coarse mouli space X of X, which is also the coarse mouli space of V U. One way to efine X is as the quotient of X 0 by the equivalence relation given by the image of X 1 X 0 X 0. Note that the coarse mouli space is Morita invariant an thus epens only on the stack efine by X. The open cover {U i } of X 0 inuces an open cover of X. Choose a ifferentiable partition of unity for X suborinate to this cover. Pull back to U. This gives over each U i a G i -invariant ifferentiable function ρ i. Define ρ : U R by setting ρ U i = 1 #G i ρ i. It is then straightforwar to check that ρ is, inee, a partition of unity for the groupoi V U. Proposition 23 Assume that the groupoi X 1 X 0 amits a partition of unity. Then for every q we have long exact sequences an...! Ω q c(x 1 ) 0 Γ(X, Ω q ) Ω q (X 0 )! Ω q c(x 0 ) Γ c (X, Ω q ) 0 Ω q (X 1 )... If we can fin a partition of unity with compact support, then there is a long exact sequence...! Ω q c(x 1 )! Ω q c(x 0 ) Ω q (X 0 ) Ω q (X 1 )... (10) Here the central map Ω q c(x 0 ) Ω q (X 0 ) is given by ω s! t ω = t! s ω. So in this latter case, we have a canonical isomorphism Γ c (X, Ω q ) Γ(X, Ω q ). Proof. To prove (10), let ρ : X 0 R be a partition of unity for X, such that ρ has compact support. We efine a contraction operator K : Ω q (X p ) Ω q (X p 1 ) ω 0! ( (π 0 ρ)ω ).

24 272 K. Behren Here π 0 : X p X 0 maps onto the zeroth object, 0 : X p X p 1 leaves out the zeroth object. This efinition is vali for p > 0. We also efine K : Ω q c (X p) Ω q c (X p+1) ω ( 1) p+1 π 0 ρ 0 ω. This efinition is vali for p 0. We finally efine K : Ω q (X 0 ) Ω q c(x 0 ) as multiplication by ρ. This efines a contraction operator for the total complex (10), i.e. we have Kδ + δk = i, where δ is the bounary operator of (10). The only place where we use properness was when we use multiplication by ρ to efine Ω q (X 0 ) Ω c (X 0 ). For this, ρ nees to have compact support, which is only true if X is proper. In this case, we may choose X 0 to come from a finite cover U i. The first two claims follow by just using part of K. Definition 24 A ifferentiable Deligne-Mumfor stack X is proper if (i) we can fin a groupoi presentation X 1 X 0 such that X 1 X 0 X 0 is proper (ii) the coarse mouli space of X (which is locally the quotient of a ifferentiable manifol by a finite group action) is proper. Corollary 25 For a ifferentiable Deligne-Mumfor stack X we have: the e Rham cohomology groups H k (X) can be calculate as the cohomology groups of the global e Rham complex ( Γ(X, Ω ), ). the compact support cohomology groups H k c (X) can be calculate using the global complex ( Γ c (X, Ω ), ). If X is proper, we also have: these two complexes are equal, i.e., for every q we have for every k we have Γ c (X, Ω q ) = Γ(X, Ω q ), H k (X) = H k c (X), in particular, there exists an integral : H n (X) R, (11) X

25 Cohomology of Stacks 273 the inuce pairing is perfect. H k (X) H n k (X) R Let us enote the structure map of an atlas X 0 amitting a partition of unity by π : X 0 X. With this notation, we may write the integral (11) as follows: ω = ρ π ω. X X 0 Example 26 Let us consier a finite type Deligne-Mumfor stack X of imension zero. We can present X by a groupoi X 1 X 0, where both X 1 an X 0 are zero-imensional, i.e., just finite collections of points. Then it is obvious, that X 1 X 0 is Morita equivalent to a isjoint union of groups: X 0 = { 1,..., n }, X 1 = n i=1 G i, for finite groups G i, an one-point manifols i. We have H 0 (X) = Hc 0 (X) = R n an all other cohomology groups vanish. There is the canonical element 1 H 0 (X), which together with the integral X : H0 (X) R, efines a canonical number X 1 R. To calculate X 1, note that ρ( i) = 1 #G i efines a partition of unity for X 1 X 0. Thus we have X 1 = n i=1 i 1 #G i = n i=1 1 #G i = #X. The number of points of an abstract finite groupoi X is efine as #X = 1 # Aut(x). x X/ = The class of a substack, intersection numbers Let f : Y X be a proper representable morphism of ifferentiable oriente Deligne-Mumfor stacks. In terms of presenting groupois, this means that we can fin X 1 X 0 for X, Y 1 Y 0 for Y an a morphism of groupois f : Y X presenting f, with the properties: (i) the iagram Y 1 s Y 0 f 1 f 0 s X 1 X 0

26 274 K. Behren is a pullback iagram of manifols (this is the representability of f) (ii) f 0 is proper. One checks that a proper representable morphism f amits a wrong way map Ω q c(x p ) Ω q c(y p ), by pulling back compactly supporte forms. This map passes to cohomology with compact supports an we enote the inuce map by f : Hc i(x) Hi c (Y). By uality, we get an inuce map on e Rham cohomology, which goes in the opposite irection. We enote it by f! : H i (Y) H i c (X), where c = im Y im X. Let im Y = k. We get a linear form Hc k (X) R γ Y f γ, an hence by uality an element cl(y) H n k (X), the class of Y. Alternatively, cl(y) = f! (1). Example 27 If E X is a vector bunle of rank r, then the class of the zero section in H r (E), pulls back (via the zero section) to an element e(e) H r (X), known as the Euler class of E. Every ifferentiable Deligne-Mumfor stack X has a tangent bunle T X. The Euler number of a compact X is efine as e(x) = e(t X ). Proposition 28 Consier a cartesian (i.e. pullback) iagram of ifferentiable stacks of Deligne-Mumfor type X W Z u Y X Assume that all maps are proper an representable. Moreover, assume that for all w W we have T W,w = T Y,w T Z,w T X,w (a conition which can be checke an efine by pulling back to an étale presentation X 0 X). Then we have cl(y) cl(z) = f! e(e),

27 Cohomology of Stacks 275 where E is the excess bunle E = u N Y/X /N W/Z, an f : W X is the structure morphism. (Strictly speaking, this formula shoul be a sum over the connecte components of W, because the excess bunle will not have constant rank, in general.) Proof. We have been very careful to state a proposition that oes not contain any global compactness assumptions on X. Thus the proof can be reuce to the case of manifols by using an étale presentation of X. If X is proper, we efine the intersection number of two proper representable stacks Y X an Z X to be cl(y) cl(z) R. X As applications of Proposition 28, we get (i) if Y an Z intersect transversally, (again a conition that can be checke an efine after pullback to any étale presentation of X) an im Y + im Z = im X, we have cl(y) cl(z) = #W. X (ii) if Y = Z we have the self-intersection formula X cl(y) 2 = Example: the stack of elliptic curves W e(u N Y/X ). Consier the stack M 1,2 of stable genus one curves with two marke points. If we consier M 1,2 as a stack over C, ignoring its arithmetic structure, we obtain a proper ifferentiable Deligne-Mumfor stack. Alternatively, we can think of M 1,2 as the stack of egenerate elliptic curves with a marke point (an elliptic curve is a genus one curve with a marke point serving as origin for the group law). The complex imension of M 1,2 is two an M 1,2 is generically a scheme. (Exercise: etermine the stacky points of M 1,2.)

28 276 K. Behren Recall that M 1,2 has a natural morphism π to M 1,1, exhibiting it as the universal family of (egenerate) elliptic curves over the stack M 1,1 of (egenerate) elliptic curves. Thus we can picture the surface M 1,2 as elliptically fibere over the curve M 1,1. There are two natural bounary ivisors on M 1,2. First there is the universal section of π : M 1,2 M 1,1. It maps every elliptic curve E in M 1,1 onto its base point (zero element) P E, where we ientify E with the fiber of π over E. Another way to think of the universal section is as the image of the morphism M 1,1 = M 0,3 M 1,1 M 1,2, which maps a genus one curve with one mark (E, P ) to the egenerate genus one curve with two marks obtaine by gluing P 1 with three marks to E, by ientifying the thir mark on P 1 with the mark P on E. By abuse of notation, we will enote this ivisor on M 1,2 by M 1,1. It is in fact a substack, an hence an honest ivisor. From this escription it follows that the normal bunle of M 1,1 in M 1,2 is the pullback of he relative tangent bunle of π to M 1,1. This may also be thought of as the bunle of Lie algebras associate to the family of groups π. Let us call this complex line bunle N. The other ivisor we are intereste in is the egenerate fiber of our elliptic fibration. Alreay, there is a lot of ambiguity in this statement. To be more precise, let us consier the morphism M 0,4 M 1,2 which takes a genus zero curve with four marks an glues together the marks labele 3 an 4 to obtain a egenerate elliptic curve (with a marke point). This morphism gives rise to the class cl(m 0,4 ) H 2 (M 1,2 ). The image of this morphism is a close substack of M 1,2. It is irreucible of coimension one, so a Weil ivisor on M 1,2, which we shall enote by W. Note that W is generically a scheme, but it has three stacky points: two smooth an one singular. (Exercise: what are the three corresponing egenerate curves with two marks?) Another way to escribe W is as the fiber of π over the close substack BZ 2 M 1,1 representing the egenerate curve. (Reuce close substacks are etermine by unerlying point sets.) This shows that the morphism P 1 = M 0,4 W factors through the fiber of π over { } M 1,1. This fiber is a noal elliptic curve, let us call it W. The morphism P 1 = M 0,4 W is the normalization map, the morphism W W is finite étale of egree 1 2. It is the quotient map by the action of the inverse map on the noal elliptic

29 Cohomology of Stacks 277 curve W. Yet another way to escribe W is as the zero locus of a section of a complex line bunle. Consier the coarse mouli map j : M 1,1 P 1 (the j- invariant). Pulling back, via j, the line bunle O( ) with its global section 1 Γ ( P 1, O( ) ), gives a line bunle L on M 1,1 with a section whose zero locus is BZ 2 M 1,1. Pulling back further to M 1,2, we get the line bunle π L, which has a global section, whose zero locus is W. To compute the self-intersection of M 0,4 in M 1,2 note that cl(m 0,4 ) = cl( W ) = 1 2 e(π L). Thus, we have cl(m 0,4 ) 2 = 1 π e(l) 2 M 1,2 4 M 1,2 = 1 e(l) 2 π! (1) 4 M 1,1 = 0, as expecte. Next, let us calculate the intersection of M 0,4 an M 1,1 insie M 1,2. We have a cartesian iagram { } M 0,4 M 1,1 M 1,2 To see that this intersection is transversal, note that the intersection point correpons to a curve with two noes an smoothing each noe gives a tangent irection to { } in M 1,2. Along M 1,1, one of the noes is smoothe, along M 0,4, the other. We conclue that M 1,2 cl(m 1,1 ) cl(m 0,4 ) = 1. More interesting is the self-intersection of M 1,1. Suppose given a family of egenerate elliptic curves E, parameterize by P 1. We then get an inuce cartesian iagram E M 1,2 π P 1 f M 1,1

30 278 K. Behren by the universal mapping property of π. It follows that f e(n) = e(n P 1 /E) an e(n P 1 /E) = f e(n) P 1 P 1 = f! f e(n) M 1,1 = eg(f) e(n), M 1,1 which we can solve for M 1,1 e(n). The egree of f is in fact twice the number of rational fibres of E P 1 (supposing that f is unramifie over j = ). If we call P e(n 1 P 1 /E) the egree of the fibration E P 1, enote eg(e), we see that eg E e(n) = M 1,2 2#{ratl. fibers}, for any elliptic fibration E/P 1. For example, we may consier the pencil of plane cubics through 8 generic points. In this case E is the blow up of P 2 in 9 points. Thus the number of rational fibres is equal to χ(e) = χ(p 2 ) + 9 = 12. Any of the exceptional lines of the blow up can be chosen as section of E P 1, proving that the egree of this elliptic fibration is 1. We conclue that cl(m 1,1 ) 2 = e(n) = 1 M 1,2 M 1,1 24. The Lefschetz trace formula Let f : X X be an enomorphism of a proper oriente ifferentiable Deligne-Mumfor stack X. Assume that f has non-egenerate fixe locus, which means that there exists a ifferentiable stack F fitting into a cartesian iagram F X Γ f X X X

31 Cohomology of Stacks 279 such that T F = T X T X. The usual proof applies an we get the Lefschetz trace formula tr f H (X) = e(f). If F is zero-imensional, then this says tr f H (X) = #F. For f the ientity of X, our assumption on F is automatically satisfie, an then the fixe stack F equals the inertia stack I X of X. We get χ DR (X) = e(i X ). In particular, the inertia stack has integer Euler number. The Euler number of the inertia stack is hence a cohomological invariant. This is not true for the Euler number of the stack itself. Example 29 Consier the trivial example of a finite group G acting on a finite set X, with zero-imensional quotient stack X. The Euler number of X is #X #G, the Euler number of the inertia stack of X is #(X/G), the number of orbits. Example 30 We can use these results to compute the Euler number of M 1,1. The inertia stack of M 1,1 is the isjoint union of two ientical copies of M 1,1, two copies of BZ 4 an four copies of BZ 6. Thus the Euler number of the inertia stack equals 2e(M 1,1 ) On the other han, the cohomological Euler characteristic of M 1,1 is the same as the cohomological Euler characteristic of the coarse mouli space P 1 (see Proposition 36), which is 2. Hence e(m 1,1 ) = Singular homology The e Rham theory has the rawback that it works only for ifferentiable stacks. Many algebraic stacks are singular, an hence o not have e Rham cohomology groups in the sense of the first section. That is why we nee to evelop a cohomology theory for topological stacks. In this part we will o this by generalizing singular homology an cohomology to stacks. We o not try to evelop the most general notion of topological stack here. A goo notion is given by those stacks on the category of topological

32 280 K. Behren spaces which can be presente by topological groupois X 1 X 0 satisfying the properties (i) both X 1 an X 0 are topological spaces, all structure maps are continuous (ii) the source an target maps s, t : X 1 X 0 are topological submersions. We will tacitly assume all topological groupois to satisfy these properties. This notion of topological stack inclues all stacks appearing in algebraic geometry, where one assumes s an t always to be at least smooth. The singular chain complex of a topological groupoi To set up notation, recall the singular chain complex of a topological space X. We shall enote it by C (X). Thus C q (X) is the abelian group of formal integer linear combinations of continuous maps q X. Let us enote the bounary maps by i : q 1 q, for i = 0,..., q. Then we have inuce maps i : Maps( q, X) Maps( q 1, X) an : C q (X) C q 1 (X) efine by (γ) = q j=0 ( 1)j j (γ). It is a stanar fact that 2 = 0, so that C (X) is, inee, a complex. The singular chain complex is covariant: if f : X Y is continuous, we get an inuce homomorphism of complexes f : C (X) C (Y ). Often we write f for f. As in e Rham theory, we stuy stacks via presenting groupois. Every topological groupoi efines a simplicial nerve... X 2 X 1 X 0, (12) because we can form fibere proucts liberally. Now applying C to (12), we get the iagram... C (X 2 ) C (X 1 ) C (X 0 ). (13) By efining = p i=0 ( 1)i j we get a morphism of complexes : C (X p )

33 Cohomology of Stacks 281 C (X p 1 ). Thus we have efine a ouble complex... C 0 (X 2 )... C 1 (X 2 )... C 2 (X 2 ). C 0 (X 1 ) C 1 (X 1 ) C 2 (X 1 ). C 0 (X 0 ) C 1 (X 0 ) C 2 (X 0 ). We efine the associate total complex C (X) C n (X) = C q (X p ) p+q=n with the ifferential δ : C n (X) C n 1 (X) given by δ(γ) = ( 1) p+q (γ) + ( 1) q (γ), if γ C q (X p ). It is immeiate that δ 2 = 0. Definition 31 The complex ( C (X), δ ) is calle the singular chain complex of the topological groupoi X = [X 1 X 0 ]. Its homology groups, enote H n (X, Z), are calle the singular homology groups of X 1 X 0. What o cycles look like? Typical examples of 1-cycles look like this: α ω α ω ω α (14) Here the soli lines are paths in X 0, in other wors paths of objects in the groupoi X 1 X 0. The otte lines are elements in X 1, in other wors

34 282 K. Behren morphisms in the groupoi X 1 X 0. The little circles represent elements of X 0, i.e., objects of X 1 X 0 Moreover, the cycle (14) has to be enowe with an orientation. This inuces an orientation on each of the eges. Thus a path of objects (labele ω) is then an oriente path in X 0. Each otte line correspons more precisely to an arrow an its inverse in the groupoi X 1 X 0. Among these two arrows we choose the one which points in the irection given by the orientation of (14). Thus there are two ways to travel aroun a circle such as (14), connecting several objects of our groupoi: we can continuously eform one object to the next, or we can use an isomorphism to move us along. 2-cycles are a little more ifficult to escribe. Picture an oriente close surface S. Assume that S has been tile with triangles an quarilaterals. The eges of this tiling come it two types: soli ones calle ω-eges an otte ones calle α-eges, just as in (14). The triangles in our tiling also come in two types, type α an type ω. Triangles of type α are always boune by three α-eges an triangles of type ω are always boune by three ω-eges. Finally, the quarilaterals are all boune by two α- an two ω-eges, in an alternating fashion. Given such a tile surface, every vertex will correspon to an object of X 1 X 0. Every ω-ege will represent a path of objects, every α-ege a morphism an its inverse in X 1 X 0. The ω-triangles correspon to continuous maps from 2 to X 0, the quarilaterals to paths in X 1 an the α-triangles to points in X 2. Thus the ω-triangles represent 2-simplices of objects, the quarilaterals paths of morphisms an the α-triangles represent commutative triangles in X 1 X 0. Here is an example of such a tile surface, but this one has a bounary, so it oes not give rise to a cycle. Instea, its bounary is the cycle given by (14). ω α ω ω This time we have labele the four triangles accoring to their type. (15)

35 Cohomology of Stacks 283 Let us be more precise about sign questions. Our tile surface is oriente, so it inuces an orientation on each ω-triangle an so every ω-triangle oes, in fact, give rise to a well-efine element of C 2 (X 0 ), at least an element which is well-efine up to a bounary in the complex C (X 0 ). For the quarilaterals, we make the convention that if we look at one in such a way that the α-eges are on the left an right, an the ω-eges on the top an bottom 1 0 then we choose the morphisms in the groupoi to point up an let the right arrow correspon to the value t = 0, an the left arrow to the value t = 1, where t is a coorinate on 1. (Exercise: check that the appearant ambiguity in this efinition leas to two choices which iffer by a bounary in the total complex C (X 1 X 0 ).) Finally, we also get inuce orientations on the α-triangles. For every α-triangle we choose arrows or their inverses in such a way that we en up with a commutative triangle in the groupoi X 1 X 0, whose orientation is compatible with the given one. In this case, we get an element of C 0 (X 2 ), which is well-efine up to a bounary in the complex ( C 0 (X ), ). One can now check that our close oriente tile surface, together with the aitional ata of singular triangles in X 0, paths in X 1 an points in X 2 oes, inee, give rise to a 2-cycle in the singular chain complex C (X 1 X 0 ). It is also true, that every 2-cycle is a linear combination of such tile surfaces. Note that a 1-cycle such as (14) represents 0 in H 1 (X 1 X 0 ), if an only if there exists a isc as in (15) (or of a more complicate type) whose bounary is the given 1-cycle. Examples Let us consier a transformation groupoi G X X, where the group G is iscrete. Then our ouble complex gives rise to a spectral sequence E 2 p,q = H q ( G, Hp (X) ) = H p+q (G X X). To see this, note that when taking vertical cohomology of the singular chain complex of G X X, we en up with a complex computing the group

36 284 K. Behren homology of G with values in the G-moule H (X). A simple case where this spectral sequence egenerates is the case of contractible X. In this case we get immeiately that H p (G X X) = H p (G, Z), so that the homology of the transformation groupoi is equal to the homology of the group G. For example, the stack of triangles up to similarity may be represente by the groupoi S Thus the homology of the stack of triangles is equal to the homology of the symmetric group S 3. Similarly, the stack of elliptic curves M 1,1 may be represente by the action of SL 2 (Z) by linear fractional transformations on the upper half plane in C. Thus the homology of the stack of elliptic curves is equal to the homology of SL 2 (Z). Invariance uner Morita equivalence We have alreay sai that the homology of a stack is efine via the homology of a topological groupoi presenting the stack. For this to make sense, the homology of a groupoi has to be invariant uner Morita equivalence. It is helpful to examine the 2-functorial properties of the singular chain complex of topological groupois. These are analogous to the properties of a contravariant functor of Remark 1. In fact, let F be any covariant functor from the category of topological spaces to the category of abelian groups. Then a morphism of groupois f : X Y inuces a homomorphism of homological complexes f : F (X ) F (Y ). A 2-morphism θ : f g between the two morphisms of groupois f, g : X Y inuces a homotopy θ : f g, efine as follows: The map θ : X 0 Y 1 extens to maps θ 0,..., θ p : X p Y p+1. Here θ i maps the element x 0 φ 1 x 1 φ 2... φ p x p of X p to the element f(x 0 ) f(φ 1) f(x 1 ) f(φ 2)... f(φ i ) f(x i ) θ(x i ) g(x i ) g(φ i+1)... g(φ p) g(x p )