Cohomology of Stacks

Transcription

1 Cohomology of Stacks K Behren University of British Columbia Lectures given at the: School an conference on Intersection theory an mouli September 9 27, 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 Keywors: Algebraic stacks, Lie groupois, e Rham cohomology, singular homology MSC: 14A20, 14D20, 22A22, 58H05

3 Contents Introuction 1 De Rham cohomology 1 The simplicial nerve of a Lie groupoi 3 Čech cohomology 3 The e Rham complex 7 Equivariant Cohomology 8 Multiplicative structure 10 Cohomology with compact supports 11 Moule structure 12 The integral, Poincaré uality 13 Deligne-Mumfor stacks 14 The class of a substack, intersection numbers 17 Example: the stack of elliptic curves 18 The Lefschetz trace formula 20 Singular homology 21 The singular chain complex of a topological groupoi 22 What o cycles look like? 23 Examples 24 Invariance uner Morita equivalence 25 Equivariant homology 27 Cohomology 27 Relation to e Rham cohomology 28 Relation to the cohomology of the coarse mouli space 29 Chern classes 30 References 32

4 1 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, ie, 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 Ω q (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 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

5 2 Cohomology of Stacks X 1 are ifferentiable (ie, 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 X 1 (s,t) X 0 X 0 f 1 f 0 f 0 (1) (s,t) Y 1 Y 0 Y 0 is cartesian, ie, 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 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

6 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 arrow, p leaves out the last arrow, an 1,, p 1 are given by composing two successive arrows More precisely, i maps the element of X p to the element x 0 φ 1 x1 φ 2 φ i 1 x i 1 φ i xi φ i+1 x i+1 φ i+2 φ p xp 3 x 0 φ 1 x1 φ 2 φ i 1 x i 1 φ i φ i+1 xi+1 φ i+2 φ p xp 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

7 4 Cohomology of Stacks 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 x1 φ 2 φ p xp 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 groups Čech cohomology 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) 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 Hk (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

8 5 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 For example, if {U i } is an open cover of Y, an X 0 = U i, then X 1 = U ij, 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 85 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 W mn V n 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 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

9 6 Cohomology of Stacks 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 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 Ωq 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 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 In particular, this efines = Γ(X, Ω q ) = H 0 (X, Ω q )

10 7 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) 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 an efining the total ifferential δ : C n DR C n DR(X) = p+q=n Ω q (X p ) (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

11 8 Cohomology of Stacks Definition 8 The complex C DR (X) is calle the e Rham complex of the Lie groupoi X 1 X 0 Its cohomology groups HDR(X) n = 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) = H n 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 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) = ( S k g Ω i (X) ) G 2k+i=n 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

12 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 ) 9 Lemma 12 If G is compact, the augmentation is a quasi-isomorphism, ie, inuces isomorphisms H i G(Y ) h i( tot Ω G(X ) ), for all i 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, ie, a top egree form ω on G, which is left invariant an satisfies ω = 1 Integrating G 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

13 10 Cohomology of Stacks Proposition 13 If G is compact, there is a natural isomorphism H i G(X) H i DR(G X X) = H i DR([X/G]) 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 quasi-isomorphism 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, ie, t = 1 S 1 This form efines a basis element in (s 1 ), which we 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) 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 η Ω q (X p ) Then we set Let ω Ω q (X p ) an ω η = ( 1) qp π 1ω π 2η Ω q+q (X p+p ) (6)

14 11 Here the map π 1 : X p+p X p projects the element φ 1 φp φ p+1 φ p+p X p+p (7) to an π 2 : X p+p X p φ 1 φp projects the same element (7) to One checks that an so we get an inuce cup prouct φ p+1 φ p+p δ(ω η) = δ(ω) η + ( 1) p+q ω δ(η), 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 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! Ω n+3r c (X 2 )! Ω n+2r c (X 1 )! Ωc n+r (X 0 )! Ω n+3r 1 c (X 2 )! Ω n+2r 1 c (X 1 )! Ω n+r 1 c (X 0 ) (8)! Ω n+3r 2 c (X 2 )! Ω n+2r 2 c (X 1 )! Ω n+r 2 c (X 0 )

15 12 Cohomology of Stacks 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! = ( 1) i i!, i where i! : Ω q+r c (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) = an set the total ifferential equal to q rp p r=ν Ω q c(x p ), δ(γ) = ( 1) p(! γ + γ ), for γ Ω q c(x p ) 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 ( +1)r q ) c (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 Xp 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)

16 13 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) The integral, Poincaré uality We can efine an integral X : H n c (X) R by noticing that the integral Ω n+r c (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, ie, 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 H p (X) Hc n p (X) R, for all p 0 Proof Consier the homomorphism of complexes X 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

17 14 Cohomology of Stacks 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 Hc n (X) R, where n = im X Poincaré uality hols: is a perfect pairing X 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 (BS 1 ) 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 } ivisible Note that the R[c]-moule H c (BS 1 ) is 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 (ie, 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 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

18 15 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! Ω q c(x 1 )! Ω q c(x 0 ) Γ c (X, Ω q ) 0 an 0 Γ(X, Ω q ) Ω q (X 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 ) 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 0ρ 0ω 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), ie 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

19 16 Cohomology of Stacks 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, ie, for every q we have for every k we have in particular, there exists an integral the inuce pairing is perfect i=1 Γ c (X, Ω q ) = Γ(X, Ω q ), H k (X) = H k c (X), X : H n (X) R, (11) 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, ie, 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 1 R X To calculate 1, note that ρ( X i) = 1 #G i efines a partition of unity for X 1 X 0 Thus we have n 1 n 1 1 = = = #X X i #G i #G i The number of points of an abstract finite groupoi X is efine as #X = 1 # Aut(x) x X/ = i=1

20 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 17 Y 1 s Y 0 f 1 f 0 X 1 s X 0 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 : H i c(x) H i 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 (ie pullback) iagram of ifferentiable stacks of Deligne-Mumfor type W Z u Y 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 X X cl(y) cl(z) = f! e(e),

21 18 Cohomology of Stacks 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 (ii) if Y = Z we have the self-intersection formula cl(y) 2 = e(u N Y/X ) Example: the stack of elliptic curves X 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 ) 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 W 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

22 19 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 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

23 20 Cohomology of Stacks 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 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 e(n P 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

24 21 ifferentiable stack F fitting into a cartesian iagram F X Γ f X X X such that T F = T X T X The usual proof applies an we get the Lefschetz trace formula If F is zero-imensional, then this says tr f H (X) = e(f) 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, the Euler number of the #G 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 4 6 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 ) = 5 12 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 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

25 22 Cohomology of Stacks 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 ) 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 It is immeiate that δ 2 = 0 δ(γ) = ( 1) p+q (γ) + ( 1) q (γ), if γ C q (X p ) 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

26 23 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 morphisms in the groupoi X 1 X 0 The little circles represent elements of X 0, ie, 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)

27 24 Cohomology of Stacks 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 wellefine 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 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 0 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 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)

28 25 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 x1 φ 2 φ p xp of X p to the element f(x 0 ) f(φ 1) f(x1 ) f(φ 2) f(φ i ) f(xi ) θ(x i ) g(x i ) g(φ i+1) g(φ p) g(xp ) of Y p+1 Then θ : F (X p ) F (Y p+1 ) is the alternating sum of the maps inuce by θ 0,, θ p Applying this to the singular chain complex functor F = C, we get that groupoi morphisms inuce homomorphisms of singular chain complexes an 2-isomorphic groupoi morphisms inuce homotopic chain maps We also get that groupoi morphisms inuce homomorphisms on singular homology groups, 2-isomorphic groupoi morphisms inuce ientical homomorphisms on homology an Morita morphisms with a section inuce isomorphisms on homology We nee to prove that this is true also for Morita morphisms amitting only local sections This will use ouble fibrations an a Mayer-Vietoris argument Let X be a topological stack an X 0 X an Y 0 X two presentations Let X an Y be the inuce topological groupois, or rather their inuce simplicial topological spaces We efine W mn = X m X Y n, for all m, n 0 W mn Y n X m X Then W is a bisimplicial topological space We apply C to W to obtain a triple complex C (W ) mapping to the two ouble complexes C (X ) an C (Y ), see Figure 3 We claim that both inuce maps on total complexes tot ( C (W ) ) tot ( C (X ) ) an tot ( C (W ) ) tot ( C (Y ) ) are quasi-isomorphisms But this follows immeiately from the following lemma