String topology for stacks 7 0 K. Behrend, G. Ginot, B. Noohi and P. Xu 0 2 February 2, 2008 c e D Abstract 2 2 Weestablishthegeneralmachineryofstringtopologyfordifferentiable stacks. This machinery allows us to treat on an equal footing free loops ] T in stacks and hidden loops. In particular, we give a good notion of a A free loop stack, and of a mapping stack Map(Y,X), where Y is a com- pactspaceandXatopological stack,whichisfunctorialbothinXandY . h and behaves well enough with respect to pushouts. We also construct a t bivariant (in the sense of Fulton and MacPherson) theory for topological a m stacks: itgivesusaflexibletheoryofGysinmapswhichareautomatically compatible with pullback, pushforward and products. We introduce ori- [ ented stacks, generalizing oriented manifolds, which are stacks on which 1 we can do string topology. We prove that the homology of the free loop v stack of an oriented stack is a BV-algebra and a Frobenius algebra, and 7 the homology of hidden loops is a Frobenius algebra. Using our gen- 5 eral machinery, we construct an intersection pairing for (non necessarily 8 compact) almost complex orbifolds which is in the same relation to the 3 intersection pairing for manifolds as Chen-Ruan orbifold cup-product is . 2 toordinarycup-productofmanifolds. Weshowthatthestringproductof 1 almost complex is isomorphic to the orbifold intersection pairing twisted 7 by a canonical class. 0 : v i Contents X r Introduction 2 a Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1 On topological stacks 8 1.1 Pretopologicalstacks . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.2 Geometric stacks . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3 Topological stacks . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2 Loop stacks 10 2.1 Mapping stacks and the free loop stack. . . . . . . . . . . . . . . 10 2.2 Groupoid presentation . . . . . . . . . . . . . . . . . . . . . . . . 12 1 3 Bivariant theory for topological stacks 15 3.1 Singular homology and cohomology . . . . . . . . . . . . . . . . . 15 3.2 Thom isomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.3 Bounded proper morphisms of pretopologicalstacks . . . . . . . 19 3.4 Some technical lemmas . . . . . . . . . . . . . . . . . . . . . . . . 20 3.5 Bivariant theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 4 Gysin maps 26 4.1 Normally nonsingular morphisms of stacks and oriented stacks . 26 4.2 Construction of the Gysin maps . . . . . . . . . . . . . . . . . . . 32 5 The loop product 34 5.1 Construction of the loop product . . . . . . . . . . . . . . . . . . 35 5.2 Proof of Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . 37 6 String product for family of groups over a stack 40 6.1 String product . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 6.2 Family of commutative groups and crossed modules . . . . . . . 44 7 Frobenius algebra structures 46 7.1 Quick review on Frobenius algebras. . . . . . . . . . . . . . . . . 46 7.2 Frobenius algebra structure for loop stacks. . . . . . . . . . . . . 47 7.3 Frobenius algebra structure for inertia stacks . . . . . . . . . . . 52 7.4 The canonical morphism ΛX→LX . . . . . . . . . . . . . . . . . 56 8 The BV-algebra on the homology of free loop stack 58 9 Orbifold intersection pairing 59 9.1 Poincar´e duality and Orbifolds . . . . . . . . . . . . . . . . . . . 60 9.2 Orbifold intersection pairing and string product . . . . . . . . . . 61 10 Examples 67 10.1 The case of manifolds . . . . . . . . . . . . . . . . . . . . . . . . 67 10.2 String (co)product for global quotient by a finite group. . . . . . 67 10.3 String topology of [S2n+1/(Z/2Z)n+1] . . . . . . . . . . . . . . . 70 10.4 String topology of L[∗/G] when G is a compact Lie group . . . . 73 11 Concluding Remarks 80 A Generalized Fulton-MacPherson bivariant theories 81 Introduction String topology is a term coined by Chas-Sullivan [14] to describe the rich al- gebraic structure on the homology of the free loop manifold LM of an oriented 2 manifold M. The algebraic structure in question is induced by geometric op- erations on loops such as glueing or pinching of loops. In particular, H (LM) • inherits a canonical product and coproduct yielding a structure of Frobenius algebra [14, 18]. Furthermore, the canonical action of S1 on LM together with the multiplicative structure make H (LM) a BV-algebra [14]. These algebraic • structures,especiallytheloopproduct,areknowntoberelatedtomanysubjects in mathematics and in particular mathematical physics [46, 13, 17, 2, 21]. Manyinterestinggeometricobjectsin(algebraicordifferential)geometryor mathematicalphysicsarenot manifolds. Thereare,forinstance,orbifolds,clas- sifying spaces of compact Lie groups, or, more generally, global quotients of a manifoldby a Lie group. All these examples belong to the realmof(geometric) stacks. A natural generalization of smooth manifolds, including the previous examples, is given by differentiable stacks [7] (on which one can still do differ- entiable geometry). Roughly speaking, differential stacks are Lie groupoids up to Morita equivalence. One important feature of differentiable stacks is that they are non-singular, whenviewedas stacks(even thoughtheir associatedcoarsespaces are typically singular). For this reason,differentiable stacks have an intersection product on their homology, and a loop product on the homology of their free loop stacks. Theaimofthispaperistoestablishthegeneralmachineryofstringtopology for differentiable stacks. This machinery allows us to treat on an equal footing free loops in stacks and hidden loops. The latter are loops inside the stack, which vanish on the associated coarse space. The stack of hidden loops in the stack X is the inertia stack of X, notation ΛX. The inertia stack ΛX → X is an example of a family of commutative (sic!) groups over the stack X, and the theory of hidden loops generalizes to arbitrary commutative families of groups over stacks. In the realm of stacks several new difficulties arise whose solutions should be of independent interest. First, we need a good notion of free loop stack LX of a stack X, and more generally of mapping stack Map(Y,X) (the stack of stack morphisms Y → X). Forthe generaltheoryofmapping stacks,we donotneeda differentiable struc- ture on X; we work with topological stacks. This is the content of Section 2.1. The issue here is to obtain a mapping stack with a topological structure which isfunctorialbothinXandY andbehaveswellenoughwithrespecttopushouts inordertogetgeometricoperationsonloops. Forinstance,akeypointinstring topologyis the identificationMap(S1∨S1,X)∼=LX×XLX. Since pushouts are adelicatematterintherealmofstacks,extracarehastobetakeninfindingthe correct definition of topological stack (Section 1 and [40]). Without restricting to this special class of topologial stacks, S1∨S1 would not be the pushout of two copies of S1, in the category of stacks. A crucial step in usual string topology is the existence of a canonical Gysin homomorphismH (LM×LM)→H (LM× LM)whenM isad-dimensional • •−d M 3 manifold. In fact, the loop product is the composition H (LM)⊗H (LM)→ p q →H (LM ×LM)→H (LM × LM)→H (LM), (0.1) p+q p+q−d M p+q−d where the last map is obtained by gluing two loops at their base point. Roughly speaking the Gysin map can be obtained as follows. The free loop manifold is equipped with a structure of Banach manifold such that the eval- uation map ev : LM → M which maps a loop f to f(0) is a surjective sub- mersion. The pullback along ev×ev of a tubular neighborhood of the diagonal M → M ×M in M ×M yields a normal bundle of codimension d for the em- bedding LM × LM →LM. The Gysin map can then be constructed using a M standard argument on Thom isomorphism and Thom collapse [19]. This approachdoes not havea straightforwardgeneralizationto stacks. For instance,thefreeloopstackofadifferentiablestackisnotaBanachstackingen- eral,andneitheristheinertiastack. InordertoobtainaflexibletheoryofGysin maps,weconstructabivarianttheory inthesenseofFulton-MacPherson[25]for topological stacks, whose underlying homology theory is singular homology. A bivarianttheoryis anefficienttoolencompassinginto aunified frameworkboth homologyandcohomologyaswellasmany(co)homologicaloperations,inpartic- ularGysinhomomorphisms. TheGysinmapsofabivarianttheoryareautomat- ically compatible with pullback, pushforward, cup and cap-products (see [25]). (Our bivariant theory is somewhat weaker than that of Fulton-MacPherson, in that products are not always defined.) InSection4.1weintroduceorientedstacks. Thesearethestacksoverwhich weareabletodostringtopology. Examplesoforientedstacksinclude: oriented manifolds, oriented orbifolds, and quotients of oriented manifolds by compact Lie groups (if the action is orientation preserving and of finite orbit type). A topological stack X is orientable if the diagonal map X→X×X factors as 0 X−→N−→E−→X×X, (0.2) whereNandEareorientablevectorbundlesoverXandX×Xrespectively,and N → E is an isomorphism onto an open substack (there is also the technical assumption that E is metrizable, and X → E factors through the unit disk bundle). The embedding N→E plays the role of a tubular neighborhood. The dimension of X is rkN−rkE. The factorization(0.2)givesriseto abivariantclassθ ∈H(X→X×X),the orientation of X. Sections 5-8 are devoted to the string topology operations, focusing on the Frobenius and BV-algebra structures. The bivariant formalism has the follow- ing consequence: if X is a an oriented stack of dimension d, then any cartesian 4 square Y // Z (cid:15)(cid:15) (cid:15)(cid:15) ∆ X // X×X defines a canonical Gysin map ∆! :H (Z)→H (Y). For example, the carte- • •−d sian square LX× LX // LX×LX X (cid:15)(cid:15) (cid:15)(cid:15) ∆ X // X×X Gives rise to a Gysin map ∆! :H (LX×LX)→H (LX× LX), and we can • •−d X construct a loop product ⋆:H (LX)⊗H (LX)→H (LX), • • •−d as in 0.1, or [14, 19, 18]. We also obtain a coproduct δ :H (LX)−→ H (LX)⊗H (LX). • i j i+jM=•−d Furthermore, LX admits a natural S1-action yielding the operator D : H (LX)→H (LX) which is the composition: • •+1 H (LX)−×→ω H (LX×S1)−→H (LX), • •+1 •+1 where ω ∈H (S1) is the fundamental class. Thus we prove that (H (LX),⋆,δ) 1 • is a Frobenius algebra and that the shifted homology (H (LX),⋆,D) is a •+d BV-algebra. Since the inertia stack can be considered as the stack of hidden loops, the general machinery of Gysin maps yields, for any oriented stack X, a product and a coproduct on the homology H (ΛX) of the inertia stack ΛX, making it • a Frobenius algebra, too. Moreover in Section 7.4, we construct a natural map Φ:ΛX→LX inducing a morphism of Frobenius algebras in homology. In Section 9, we consider almost complex orbifolds (not necessarely com- pact). Using Gysin maps and the obstruction bundle of Chen-Ruan [16], we constructthe orbifold intersection pairing onthe homologyof the inertia stack. Itis inthe same relationto the intersectionpairingonthe homologyofamani- fold as the Chen-Ruan orbifoldcup-product [16] is to the ordinary cup product on the cohomology of a manifold. The orbifold intersection pairing defines a structure of associative, graded commutativealgebraonHorb(X)foranyalmostcomplexorbifoldX. Asavector • 5 spacetheorbifoldhomologyHorb(X)coincideswiththehomologyoftheinertia • stack ΛX, but the grading is shifted according to the age as in [16, 23]. In the compact case, the orbifold intersection pairing is identified with the Chen-Ruan product, via orbifold Poincar´eduality. We also prove that the loop product, string product and intersection pair- ing (for almost complex orbifolds) can be twisted by a cohomology class in H (LX× LX)or H (ΛX× ΛX),satisfying the 2-cocylecondition(see Propo- • X • X sitions5.11,6.3,and9.6). Thenotionoftwistingprovidesaconnectionbetween the orbifoldintersectionpairingandthe stringproduct. Infact, weassociateto analmostcomplexorbifoldXacanonicalvectorbundleO ⊕N overΛX× ΛX X X X and prove that the orbifold intersection pairing, twisted by the Euler class of O ⊕N , is the string product of X. X X Parallel to our work, the string product for global quotient orbifolds was studied in [35, 27]. Furthermore, a nice interpretation of the string product in termsoftheChen-RuanproductofthecotangentbundlewasgivenbyGonz´alez et al. [27]. A loop product for global quotients of a manifold by a finite group was studied in [36, 35]. Conventions Topological spaces All topological spaces are compactly generated. The category of topological spaces endowed with the Grothendieck topology of open coverings is denoted Top. This is the site of topological spaces. Manifolds AllmanifoldsaresecondcountableandHausdorff. Inparticulartheyareregular Lindelo¨f and paracompact. Groupoids We will commit the usual abuse of notation and abbreviate a groupoid to Γ ⇉ Γ . A topological groupoid, is a groupoid Γ ⇉ Γ , where Γ and Γ 1 0 1 0 1 0 are topological spaces, but no further assumptions is made on the source and targetmaps,exceptcontinuity. AtopologicalgroupoidisaLiegroupoidifΓ ,Γ 1 0 are manifolds, all the structures maps are smooth and, in addition, the source and target maps are subjective submersions. Stacks For stacks, we use the words equivalent and isomorphic interchangeably. We will often omit 2-isomorphisms from the notation. For example, we may call morphisms equal if they are 2-isomorphic. The stack associated to a groupoid Γ ⇉Γ wedenoteby[Γ /Γ ],becausewethink ofitasthequotient. AlsoifG 1 0 0 1 6 isaLiegroupactingonaspaceY,wesimplydenote[Y/G]thestackassociated to the transformation groupoid Y ×G⇉Y. (Co)homology The coefficients of our (co)homology theories will be taken in a commutative unital ring k. All tensors products are over k unless otherwise specified. We will write both H(X), H (X) for the total homology groups H (X). • n We use the first notation when we deal with ungraded elements andLungraded maps,whileweusethesecondwhenwheredealingswithhomogeneoushomology classes and graded maps. Similarly, in Section 3.5, we use respectively the notations H(X →f Y) and H•(X →f Y) for the total bivariant cohomology groups when we want to deal with ungraded maps or with graded ones. Acknowledgements The authorswarmlythank GustavoGranjaandAndrewKreschforhelpful and inspiring discussion on the topological issues of this paper. The authors also thank Eckhard Meinrenken for his suggestions on the Cartan model. 7 1 On topological stacks 1.1 Pretopological stacks Apretopologicalstack([40],Definition7.1)isastackXoverTopwhichadmits a representable epimorphism p: X → X from a topological space X. Equiv- alently, X is isomorphic to the quotient stack (or the stack of torsors) of a topological groupoid X ⇉X . 1 0 Classifying space Many facts about pretopological stacks can be reduced to the case of topolog- ical spaces by means of the classifying space. If X is a pretopological stack, a classifying space for X is a topological space X, together with a morphism X → X, satisfying the property that for every morphism T → X from a topo- logical space T, the pull back T × X → T is a weak homotopy equivalence. X Such a classifying space always exists. Indeed a classifying space for X is given by the fat realization of the nerve of any groupoid Γ ⇉ Γ whose quotient 1 0 stack is X. For more details see [41]. 1.2 Geometric stacks We will encounter other types of stacks. A differentiable stack is a stack on thecategoryofC∞-manifolds,whichisisomorphictothequotientstackofaLie groupoid. Every differentiable stack has an underlying pretopological stack. If the Lie groupoidX ⇉X representsthe differentiable stack X, the underlying 1 0 topologicalgroupoid represents the underlying topologicalstack. Often we will tacitly pass from a differentiable stack to its underlying pretopological stack. For more on differentiable stacks, see [7]. An almost complex stack is a stack on the category of almost complex manifolds, which is isomorphic to the quotient stack of an almost complex Lie groupoid, i.e., a Lie groupoid X ⇉X , where X and X are almost complex 1 0 0 1 manifolds, and all structure maps respect the almost complex structure. Every almost complex stack has an underlying differentiable stack and hence also an underlying pretopologicalstack. 1.3 Topological stacks Inorderforloopstackstobehavewell,weneedtorestricttotopologicalstacks. Recall that a Hurewicz fibration is a map having the homotopy lifting property withrespecttoalltopologicalspaces. Amapf: X →Y oftopologicalspacesis alocalHurewicz fibrationifforeveryx∈X thereareopensx∈U andf(x)∈V suchthatf(U)⊆V andf| →V is a Hurewicz fibration. The most important U example for us is the case of a topological submersion: a map f :X →Y, such that locally U is homeomorphic to V ×Rn, for some n. Dually, we have the notion of local cofibration. It is known ([44]), that if A → Z is a closed embedding of topological spaces, it is a local cofibration 8 if and only if there exists and open neighborhood A ⊂ U ⊂ Z such that A is a strong deformation retract of U. If A → Z is a local cofibration, so is A×T →Z×T for every topologicalspace T. Moreover,the following result is essential for our purposes ([45]): Given a commutative diagram, with A→Z a local cofibration and X →Y a local fibration A //X (cid:15)(cid:15) (cid:15)(cid:15) Z // Y thenforeverypointa∈AthereexistsanopenneighborhoodZ′ ofainZ,such that there exists a lifting (the dotted arrow)giving two commutative triangles A′ //X >> (cid:15)(cid:15) (cid:15)(cid:15) Z′ // Y where A′ =A∩Z′. Definition 1.1 ApretopologicalstackXiscalledtopologicalifitisequivalent to the quotientstack [X /X ]of a topologicalgroupoidX ⇉X whose source 0 1 1 0 and target maps are local Hurewicz fibrations. Example 1.2 A topological space is a topological stack. The pretopological stack underlying any differentiable stack is a topological stack. In particular, any global quotient [M/G] of a manifold by a Lie group defines a topological stack. The following generalizes [[40], Theorem 16.2]. Proposition 1.3 Let A→Y be a closed embedding of Hausdorff spaces, which is a local cofibration. Let A → Z be a finite proper map of Hausdorff spaces. Suppose we are given a push-out diagram in the category of topological spaces (cid:31)(cid:127) A //Y (cid:15)(cid:15) (cid:15)(cid:15) Z // Z∨AY Then this diagram remains a push-out diagram in the 2-category of topological stacks. In other words, for every topological stack X, the morphism X(Z∨ Y)−→X(Z)× X(Y) A X(A) is an equivalence of groupoids. 9 Proof. Let us abbreviate the push-out by U =Z∨ Y. A The fully faithful property only uses that X is a pretopological stack and that U is a pushout. Let us concentrate on essential surjectivity. Because X is a stack and we already proved full faithfulness, the question is local in U. Assume given Z → X and Y → X, and an isomorphism over A. Let X ⇉ X 1 0 be a groupoid presenting X, whose source and target maps are local fibrations. Let us remark that both Z → U and Y → U are finite proper maps of Hausdorffspaces. Thus we cancover U by open subsets U , suchthat for every i i, both Z =U ∩Z and Y =Y ∩U admit liftings to X of their morphisms to i i i i 0 X. We thus reduce to the case that we have Z → X , Y → X , and A → X . 0 0 1 Next, we need to construct the dotted arrow in A // Y (cid:15)(cid:15) }} (cid:15)(cid:15) X // X 1 0 We can cover Y by opens over which this arrow exists, because A → Y is a local cofibration and X → X a local fibration. Then for a point u ∈ U we 1 0 choose an open neighborhood in U small enough such that the preimage in Y is a disjoint union of sets over with the dotted arrow exists. Passing to such a neighborhoodof u reduces to the case that the dotted arrowexists. Thenthere is nothing left to prove. (cid:3) Definition 1.4 A pretopological stack X is called regular Lindelo¨f if it is equivalent to the quotient stack [X /X ] of a topological groupoid X ⇉ X 0 1 1 0 such that X ,X are regular Lindelo¨f spaces. 1 0 Proposition 1.5 IfXisaregularLindelo¨fstack,thereexistsaclassifyingspace for X which is a regular Lindelo¨f space, in particular paracompact. Remark 1.6 Every differentiable stack is regular Lindelo¨f and hence has a paracompact classifying space. 2 Loop stacks 2.1 Mapping stacks and the free loop stack Let X and Y be stacks over Top. We define the stack Hom(Y,X), called the mapping stack from Y to X, by the rule T ∈Top 7→ Hom(T ×Y,X), where Hom denotes the groupoid of stack morphisms. This is easily seen to be astack. Itfollowsfromtheexponentiallawformappingspaces([48])thatwhen X andY arespaces,withY Hausdorff,thenHom(Y,X)isrepresentablebythe usual mapping space from Y to X (endowed with the compact-opentopology). The mapping stacks Hom(Y,X) are functorial in X and Y. 10