aa r X i v : . [ m a t h . A T ] N ov The Lambrechts–Stanley Model ofConfiguration Spaces
Najib Idrissi ∗ November 23, 2018
We prove the validity over R of a commutative differential graded algebramodel of configuration spaces for simply connected closed smooth manifolds,answering a conjecture of Lambrechts–Stanley. We get as a result that thereal homotopy type of such configuration spaces only depends on the realhomotopy type of the manifold. We moreover prove, if the dimension ofthe manifold is at least 4, that our model is compatible with the action ofthe Fulton–MacPherson operad (weakly equivalent to the little disks operad)when the manifold is framed. We use this more precise result to get a complexcomputing factorization homology of framed manifolds. Our proofs use thesame ideas as Kontsevich’s proof of the formality of the little disks operads. Contents
1. Background and recollections 6
CDGA s . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.2. (Co)operads and their right (co)modules . . . . . . . . . . . . . . . . . . . 81.3. Semi-algebraic sets and forms . . . . . . . . . . . . . . . . . . . . . . . . . 91.4. Little disks and related objects . . . . . . . . . . . . . . . . . . . . . . . . 101.5. Operadic twisting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.6. Formality of the little disks operad . . . . . . . . . . . . . . . . . . . . . . 131.7. Poincaré duality
CDGA models . . . . . . . . . . . . . . . . . . . . . . . . 181.8. The Lambrechts–Stanley
CDGA s . . . . . . . . . . . . . . . . . . . . . . . 20
2. The Hopf right comodule model G A ∗ Université Paris Diderot, Institut de Mathématiques de Jussieu-Paris Rive Gauche, Sorbonne ParisCité, CNRS, Sorbonne Université, F-75013 Paris, France. [email protected]
Gra R . . . . . . . . . . . . . . 233.3. The propagator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.4. Labeled graphs with internal and external vertices: Tw Gra R . . . . . . . . 263.5. The map ω : Tw Gra R → Ω ∗ PA ( FM M ) . . . . . . . . . . . . . . . . . . . . . . 273.6. Reduced labeled graphs: Graphs R . . . . . . . . . . . . . . . . . . . . . . 31
4. From the model to forms via graphs 34 G A . . . . . . . . . . . . . . . . . . . . . 344.2. The morphisms are quasi-isomorphisms . . . . . . . . . . . . . . . . . . . 384.3. Models for configurations on the 2- and 3-spheres . . . . . . . . . . . . . . 47
5. Factorization homology of universal enveloping E n -algebras 49
6. Outlook: The case of the 2-sphere and oriented manifolds 53 fFM n and potential model . . . . . . . . . . . . . . . . . . 546.3. Connecting fG A to Ω ∗ PA ( fFM S ) . . . . . . . . . . . . . . . . . . . . . . . . 55 References 56A. Glossary of notation 60
Introduction
Let M be a closed smooth n -manifold and consider the ordered configuration space of k points in M : Conf k ( M ) := { ( x , . . . , x k ) ∈ M k | x i = x j ∀ i = j } . Despite their apparent simplicity, configuration spaces remain intriguing. One of themost basic questions that can be asked about them is the following: if a manifold M ′ is obtained from M by continuous deformations, then can Conf k ( M ′ ) be obtained fromConf k ( M ) by continuous deformations? That is, does the homotopy type of M determinethe homotopy type of Conf k ( M )?Without any restriction, this is false: the point { } is homotopy equivalent to the line R , but Conf ( { } ) = ∅ is not homotopy equivalent to Conf ( R ) = ∅ . One might wonderif the conjecture becomes true if restricted to closed manifolds. In 2005, Longoni andSalvatore [LS05] found a counterexample: two closed 3-manifolds, given by lens spaces,which are homotopy equivalent but whose configuration spaces are not. This counterex-ample is not simply connected however. The question of the homotopy invariance ofConf k ( − ) for simply connected closed manifolds remains open to this day.Here, we do not work with the full homotopy type. Rather, we restrict ourselves tothe rational homotopy type. This amounts, in a sense, to forgetting all the torsion.Rational homotopy theory can be studied from an algebraic point of view [Sul77]. The2ational homotopy type of a simply connected space X is fully encoded in a “model” of X , i.e. a commutative differential graded algebra ( CDGA ) A which is quasi-isomorphicto the CDGA of piecewise polynomial forms A ∗ PL ( X ). Due to technical issues, we willin fact work over R . If M is a smooth manifold, then a real model is a CDGA which isquasi-isomorphic to the
CDGA of de Rham forms Ω ∗ dR ( M ). While this is slightly coarserthan the rational homotopy type of M , in terms of computations it is often enough.Thus, our goal is the following: given a model of M , deduce an explicit, small modelof Conf k ( M ). This explicit model only depends on the model of M . This shows the(real) homotopy invariance of Conf k ( − ) on the class of manifolds we consider. Moreover,this explicit model can be used to perform computations, e.g. the real cohomology ringof Conf k ( M ), etc.We focus on simply connected (thus orientable) closed manifolds. They satisfy Poincaréduality. Lambrechts and Stanley [LS08b] showed that any such manifold admits a model A which satisfies itself Poincaré duality, i.e. there is an “orientation” A n ε −→ R which in-duces non-degenerate pairings A k ⊗ A n − k → R for all k . Lambrechts and Stanley [LS08a]built a CDGA G A ( k ) out of such a Poincaré duality model (they denote it F ( A, k )). If weview H ∗ (Conf k ( R n )) as spanned by graphs modulo Arnold relations, then G A ( k ) consistsof similar graphs with connected components labeled by A , and the differential splitsedges. Lambrechts and Stanley proved that G A ( k ) is quasi-isomorphic to A ∗ PL (Conf k ( M ))as a dg-module. They conjectured that this quasi-isomorphism can be enhanced to givea quasi-isomorphism of CDGA s so that G A ( k ) defines a rational model of Conf k ( M ). Weanswer this conjecture by the affirmative in the real setting in the following theorem. Theorem 1 (Corollary 116) . Let M be a simply connected, closed, smooth manifold.Let A be any Poincaré duality model of M . Then for all k ≥ , G A ( k ) is a model for thereal homotopy type of Conf k ( M ) . Corollary 2 (Corollary 117) . For simply connected closed smooth manifolds, the realhomotopy type of M determines the real homotopy type of Conf k ( M ) . Over the past decades, attempts were made to solve the Lambrechts–Stanley conjec-ture, and results were obtained for special kinds of manifolds, or for low values of k .When M is a smooth complex projective variety, Kriz [Kri94] had previously shown that G H ∗ ( M ) ( k ) is actually a rational CDGA model for Conf k ( M ). The CDGA G H ∗ ( M ) ( k ) is the E page of a spectral sequence of Cohen–Taylor [CT78] that converges to H ∗ (Conf k ( M )).Totaro [Tot96] has shown that for a smooth complex compact projective variety, thespectral sequence only has one nonzero differential. When k = 2, then G A (2) was knownto be a model of Conf ( M ) either when M is 2-connected [LS04] or when dim M iseven [Cor15].Our approach is different than the ones used in these previous works. We use ideascoming from the theory of operads. In particular, we consider the operad of little n -disks, defined by Boardman–Vogt [BV73], which consists of configuration spaces of small n -disks (instead of points) embedded inside the unit n -disk. These spaces of little n -disks are equipped with composition products, which are basically defined by inserting aconfiguration of l little n -disks into the i th little disk of a configuration of k little n -disks,3esulting in a configuration of k + l − n -disks. The idea is that a configurationof little n -disks represents an operation acting on n -fold loop spaces, and the operadiccomposition products of little n -disks reflect the composition of such operations. Theconfiguration spaces of little n -disks are homotopy equivalent to the configurations spacesof points in the Euclidean n -space R n , but the operadic composition structure does notgo through this homotopy equivalence.In our work, we actually use another model of the little n -disk operads, definedusing the Fulton–MacPherson compactifications FM n ( k ) of the configurations spacesConf k ( R n ) [FM94; AS94; Sin04]. This compactification allows us to retrieve, on thiscollection of spaces FM n = { FM n ( k ) } , the operadic composition products which were lostin the configurations spaces Conf k ( R n ). We also use the Fulton–MacPherson compacti-fications FM M ( k ) of the configuration spaces Conf k ( M ) associated to a closed manifold M . When M is framed, these compactifications assemble into an operadic right module FM M over the Fulton–MacPherson operad FM n , which roughly means that we can insert aconfiguration in FM n into a configuration in FM M . We show that the Lambrechts–Stanleymodel is compatible with this action of the little disks operad, as we explain now.The little n -disks operads are formal [Kon99; Tam03; Pet14; LV14; FW15]. Kontse-vich’s proof [Kon99; LV14] of this theorem uses the spaces FM n . If we temporarily forgetabout operads, this formality theorem means in particular that each space FM n ( k ) is“formal”, i.e. the cohomology e ∨ n ( k ) := H ∗ ( FM n ( k )) (with a trivial differential) is a modelfor the real homotopy type of FM n ( k ). To prove Theorem 1, we generalize Kontsevich’sapproach to prove that G A ( k ) is a model of FM M ( k ).To establish his result, Kontsevich has to consider fiberwise integrations of forms alonga particular class of maps, which are not submersions, but represent the projection mapof “semi-algebraic bundles”. In order to define such fiberwise integration operations,Kontsevich uses CDGA s of piecewise semi-algebraic ( PA ) forms Ω ∗ PA ( − ) instead of theclassical CDGA s of de Rham forms. The theory of PA forms was developed in [KS00;HLTV11]. Any closed smooth manifold M is a semi-algebraic manifold [Nas52; Tog73],and the CDGA Ω ∗ PA ( M ) is a model for the real homotopy type of M . For the formalityof FM n , a descent argument [GNPR05] is available to show that formality over R impliesformality over Q . However, no such descent argument exists for models with a nontrivialdifferential such as G A . Therefore, although we conjecture that our results on real ho-motopy types descend to Q , we have no general argument ensuring that such a propertyholds.The cohomology e ∨ n = H ∗ ( FM n ) inherits a Hopf cooperad structure from FM n , i.e. it isa cooperad (the dual notion of operad) in the category of CDGA s. The
CDGA s of formsΩ ∗ PA ( FM n ( k )) also inherit a Hopf cooperad structure (up to homotopy). The formalityquasi-isomorphisms between the cohomology algebras e ∨ n ( k ) and the CDGA s of formson FM n ( k ) are compatible in a suitable sense with this structure. Therefore the Hopfcooperad e ∨ n fully encodes the rational homotopy type of the operad FM n .In this paper, we also prove that the Lambrechts-Stanley model G A determines thereal homotopy type of FM M as a right module over the operad FM n when M is a framedmanifold. To be precise, our result reads as follows.4 heorem 3 (Theorem 95) . Let M be a framed smooth simply connected closed manifoldwith dim M ≥ . Let A be any Poincaré duality model of M . Then the collection G A = { G A ( k ) } k ≥ forms a Hopf right e ∨ n -comodule. Moreover the Hopf right comodule ( G A , e ∨ n ) is weakly equivalent to (Ω ∗ PA ( FM M ) , Ω ∗ PA ( FM n )) . For dim M ≤
3, the proof fails (see Proposition 78). However, in this case, the onlyexamples of simply connected closed manifolds are spheres, thanks to Perelman’s proofof the Poincaré conjecture [Per02; Per03]. We can then directly prove that G A ( k ) is amodel for Conf k ( M ) (see Section 4.3).Our proof of Theorem 3, which is inspired by Kontsevich’s proof of the formality ofthe little disks operads, is radically different from the proofs of [LS08a]. It involves anintermediary Hopf right comodule of labeled graphs Graphs R . This comodule is simi-lar to a comodule recently studied by Campos–Willwacher [CW16], which correspondsto the case R = S ( ˜ H ∗ ( M )). Note however that the approach of Campos–Willwacherdiffers from ours. In comparison to their work, our main contribution is the defini-tion of the quasi-isomorphism between the CDGAs Ω ∗ PA ( FM M ( k )) and the small, explicitLambrechts-Stanley model G A ( k ), which has the advantage of being finite-dimensionaland much more computable than Graphs S ( ˜ H ( M )) ( k ). Applications.
Ordered configuration spaces appear in many places in topology and ge-ometry. Therefore, thanks to Theorems 1 and 3, the explicit model G A ( k ) provides anefficient computational tool in many concrete situations.To illustrate this, we show how to apply our results to compute factorization homology,an invariant of framed n -manifolds defined from an E n -algebra [AF15]. Let M be aframed manifold with Poincaré duality model A , and B be an n -Poisson algebras, i.e. analgebra over the operad H ∗ ( E n ). Our results shows that we can compute the factorizationhomology of M with coefficients in B just from G A and B . As an application, we computefactorization homology with coefficients in a higher enveloping algebra of a Lie algebra(Proposition 124), recovering a theorem of Knudsen [Knu16].The Taylor tower in the Goodwillie–Weiss calculus of embeddings may be computedin a similar manner [GW99; BW13]. It follows from a result of [Tur13, Section 5.1] that FM M may be used for this purpose. Therefore our theorem shows that G A may also beused for computing this Taylor tower. Roadmap.
In Section 1, we lay out our conventions and recall the necessary background.This includes dg-modules and
CDGA s, (co)operads and their (co)modules, semi-algebraicsets and PA forms. We also recall basic results on the Fulton–MacPherson compactifi-cations of configuration spaces FM n ( k ) and FM M ( k ), and the main ideas of Kontsevich’sproof of the formality of the little disks operads using the CDGA s of PA forms on thespaces FM n ( k ). We use the formalism of operadic twisting, which we recall, to dealwith signs more easily. Finally, we recollect the necessary background on Poincaré du-ality CDGA s and the Lambrechts–Stanley
CDGA s. In Section 2, we build out of theLambrechts–Stanley
CDGA s a Hopf right e ∨ n -comodule G A .In Section 3, we construct the labeled graph complex Graphs R which will be usedto connect G A to Ω ∗ PA ( FM M ). The construction is inspired by Kontsevich’s constructionof the unlabeled graph complex Graphs n . It is done in several steps. The first step5s to consider a graded module of labeled graphs, Gra R . In order to be able to map Gra R into Ω ∗ PA ( FM M ), we recall the construction of what is called a “propagator” in themathematical physics literature. We then “twist” Gra R to obtain a new object Tw Gra R ,which consists of graphs with two kinds of vertices: “external” and “internal”. Finally wemust reduce our graphs to obtain a new object, Graphs R , by removing all the connectedcomponents with only internals vertices in the graphs using a “partition function” (afunction which resembles the Chern–Simons invariants).In Section 4, we prove that the zigzag of Hopf right comodule morphisms between G A and Ω ∗ PA ( FM M ) is a weak equivalence. We first connect our graph complex Graphs R tothe Lambrechts–Stanley CDGA s G A . This requires vanishing results about the partitionfunction. Then we end the proof of the theorem by showing that all the morphisms arequasi-isomorphisms. Finally we study the cases S and S .In Section 5, we use our model to compute factorization homology of framed manifoldsand we compare the result to a complex obtained by Knudsen. In Section 6 we work outa variant of our theorem for the only simply connected surface using the formality ofthe framed little 2-disks operad, and we present a conjecture about higher dimensionaloriented manifolds.For convenience, we provide a glossary of our main notations at the end of this paper.
1. Background and recollections
CDGA s We consider differential graded modules (dg-modules) over the base field R . Unlessotherwise indicated, (co)homology of spaces is considered with real coefficients. All ourdg-modules will have a cohomological grading, V = L n ∈ Z V n . All the differentialsraise degrees by one: deg( dx ) = deg( x ) + 1. We say that a dg-module is of finite typeif it is finite dimensional in each degree. Let V [ k ] be the desuspension, defined by( V [ k ]) n = V n + k . For dg-modules V, W and homogeneous elements v ∈ V, w ∈ W , welet ( v ⊗ w ) := ( − (deg v )(deg w ) w ⊗ v and we extend this linearly to the tensor product.Moreover, given an element X ∈ V ⊗ W , we will often use a variant of Sweedler’s notationto express X as a sum of elementary tensors, X := P ( X ) X ′ ⊗ X ′′ ∈ V ⊗ W .We call CDGAs the (graded) commutative unital algebras in dg-modules. In general,for a
CDGA A , we let µ A : A ⊗ → A be its product. For a dg-module V , we let S ( V ) bethe free unital symmetric algebra on V .We will need a model category structure on the category of CDGA s. We use themodel category structure given by the general result of [Hin97] for categories of algebrasover operads. The weak equivalences are the quasi-isomorphisms, the fibrations are thesurjective morphisms, and the cofibrations are characterized by the left lifting propertywith respect to acyclic fibrations. A path object for the initial
CDGA R is given by A ∗ PL (∆ ) = S ( t, dt ), the CDGA of polynomials forms on the interval. It is equippedwith an inclusion R ֒ ∼ −→ A ∗ PL (∆ ), and two projections ev , ev : A ∗ PL (∆ ) ∼ −→ R givenby setting t = 0 or t = 1. Two morphisms f, g : A → B with cofibrant source are homotopic if there exists a homotopy h : A → B ⊗ A ∗ PL (∆ ) such that the following6iagram commutes: AB B ⊗ A ∗ PL (∆ ) B f h g id ⊗ ev ∼ id ⊗ ev ∼ . Many of the
CDGA s that appear in this paper are Z -graded. However, to deserve thename “model of X ”, a CDGA should be connected to A ∗ PL ( X ) only by N -graded CDGA s.The next proposition shows that considering this larger category does not change ourstatement.
Proposition 4.
Let A , B be two N -graded CDGA s which are homologically connected,i.e. H ( A ) = H ( B ) = R . If A and B are quasi-isomorphic as Z -graded CDGA s, thenthey also are as N -graded CDGA s.Proof.
This follows from the results of [Fre17, §II.6.2]. Let us temporarily denote cdga N the category of N -graded CDGA s ( dg ∗ Com in [Fre17]) and cdga Z the category of Z -graded CDGA s ( dg Com in [Fre17]). Note that in [Fre17], Z -graded CDGA s are homologicallygraded, but we can use the usual correspondence A i = A − i to keep our convention that alldg-modules are cohomologically graded. There is an obvious inclusion ι : cdga N → cdga Z ,which clearly defines a full functor that preserves and reflects quasi-isomorphisms.Let B m be the dg-module R concentrated in degree m , let E m be the dg-module givenby two copies of R in respective degree m − m such that d E m is the identity of R in these degrees (hence E m is acyclic), and let i : B m → E m be the obvious inclusion.The model category cdga N is equipped with a set of generating cofibrations given by themorphisms S ( i ) : S ( B m ) → S ( E m ) and of the morphism ε : S ( B ) → R . Recall that acellular complex of generating cofibrations is a CDGA obtained by a sequential colimit R = colim k R h k i , where R h i = R and R h k +1 i is obtained from R h k i by a pushout ofgenerating cofibrations along attaching maps h : S ( B m ) → R h k i . In [Fre17, §II.6.2], theexpression “connected generating cofibrations” is used for the generating cofibrations ofthe form S ( i ) : S ( B m ) → S ( E m ) with m > A is homologicallyconnected, then the attaching map h : S ( B ) → A associated to a generating cofibration ε : S ( B ) → R necessarily reduces to the augmentation ε : S ( B ) → R followed by theinclusion as the unit R ⊂ A . Thus a pushout of the generating cofibration ε : S ( B ) → R reduces to a neutral operation in this case. In the proof of [Fre17, Proposition II.6.2.8],it is deduced from this observation that any homologically connected algebra admits aresolution R A ∼ −→ A such that R A is a cellular complex of connected generating cofibra-tions. Connected generating cofibrations are also cofibrations in cdga Z after applying ι .Moreover ι preserves colimits. It follows that ιR A is cofibrant in cdga Z too.By hypothesis, ιA and ιB are weakly equivalent in cdga Z , hence ιR A and ιB are alsoweakly equivalent (because ι clearly preserves quasi-isomorphisms), through a zigzag ιR A ∼ ←− · ∼ −→ ιB . As ιR A is cofibrant (and all CDGA s are fibrant), we can find a directquasi-isomorphism ιR A ∼ −→ ιB and therefore a zigzag ιA ∼ ←− ιR A ∼ −→ ιB which onlyinvolves N -graded CDGA s. 7 .2. (Co)operads and their right (co)modules
We assume basic proficiency with Hopf (co)operads and (co)modules over (co)operads,see e.g. [Fre09; LV12; Fre17]. We index our (co)operads by finite sets instead of integersto ease the writing of some formulas. If W ⊂ U is a subset, we write the quotient U/W = ( U \ W ) ⊔ {∗} , where ∗ represents the class of W ; note that U/ ∅ = U ⊔ {∗} .An operad in dg-modules, for instance, is given by a functor from the category of finitesets and bijections (a symmetric collection) P : U P ( U ) to the category of dg-modules,together with a unit k → P ( {∗} ) and composition maps ◦ W : P ( U/W ) ⊗ P ( W ) → P ( U )for every pair of sets W ⊂ U , satisfying the usual associativity, unity and equivarianceconditions. Dually, a cooperad C is given by a symmetric collection, a counit C ( {∗} ) → k ,and cocomposition maps ◦ ∨ W : C ( U ) → C ( U/W ) ⊗ C ( W ) for every pair W ⊂ U .Let k = { , . . . , k } . We recover the usual notion of a cooperad indexed by the integersby considering the collection { C ( k ) } k ≥ , and the cocomposition maps ◦ ∨ i : C ( k + l − → C ( k ) ⊗ C ( l ) corresponds to ◦ ∨{ i,...,i + l − } .Following Fresse [Fre17, §II.9.3.1], a “Hopf cooperad” is a cooperad in the categoryof CDGA s. We do not assume that (co)operads are trivial in arity zero, but they willsatisfy P ( ∅ ) = k (resp. C ( ∅ ) = k ). Therefore we get (co)operad structures equivalentto the structure of Λ-(co)operads considered by Fresse [Fre17, §II.11], which he uses tomodel rational homotopy types of operads in spaces satisfying P (0) = ∗ (but we do notuse this formalism in the sequel).The result of Proposition 4 extends to Hopf cooperads (and to Hopf Λ-cooperads).To establish this result, we still use a description of generating cofibrations of N -gradedHopf cooperads, which are given by morphisms of symmetric algebras of cooperads S ( i ) : S ( C ) → S ( D ), where i : C → D is a dg-cooperad morphism that is injectivein positive degrees (see [Fre17, §II.9.3] for details). In the context of homologicallyconnected cooperads, we can check that the pushout of such a Hopf cooperad morphismalong an attaching map reduces to a pushout of a morphism of symmetric algebras ofcooperads S ( C / ker( i )) → S ( D ), where we mod out by the kernel of the map i : C → D indegree 0. We deduce from this observation that any homologically connected N -gradedHopf cooperad admits a resolution by a cellular complex of generating cofibrations ofthe form S ( i ) : S ( C ) → S ( D ), where the map i is injective in all degrees (we againcall such a generating cofibration connected). The category of Z -graded Hopf cooperadsinherits a model structure, like the category of N -graded Hopf cooperads considered in[Fre17, §II.9.3]. Cellular complexes of connected generating cofibrations of N -gradedHopf cooperads define cofibrations in the model category of Z -graded Hopf cooperadsyet, as in the proof of Proposition 4.Given an operad P , a right P -module is a symmetric collection M equipped with com-position maps ◦ W : M ( U/W ) ⊗ P ( W ) → M ( U ) satisfying the usual associativity, unity andequivariance conditions. A right comodule over a cooperad is defined dually. If C is aHopf cooperad, then a right Hopf C -comodule is a C -comodule N such that all the N ( U )are CDGA s and all the maps ◦ ∨ W are morphisms of CDGA s. Definition 5.
Let C (resp. C ′ ) be a Hopf cooperad and N (resp. N ′ ) be a Hopf rightcomodule over C (resp. C ′ ). A morphism of Hopf right comodules is a pair ( f N , f C )8onsisting of a morphism of Hopf cooperads f C : C → C ′ , and a map of Hopf right C ′ -comodules f N : N → N ′ , where N has the C -comodule structure induced by f C . It is a quasi-isomorphism if both f C and f N are quasi-isomorphisms in each arity. A Hopfright C -module N is said to be weakly equivalent to a Hopf right C ′ -module N ′ if thepair ( N , C ) can be connected to the pair ( N ′ , C ′ ) through a zigzag of quasi-isomorphisms.The next very general lemma can for example be found in [CW16, Section 5.2]. Let C be a cooperad, and see the CDGA A as an operad concentrated in arity 1. Recallthat C ◦ A = L i ≥ C ( i ) ⊗ Σ i A ⊗ i denotes the composition product of operads, where weview A as an operad concentrated in arity 1. Then the commutativity of A implies theexistence of a distributive law t : C ◦ A → A ◦ C , given in each arity by the morphism t : C ( n ) ⊗ A ⊗ n → A ⊗ C ( n ) given by x ⊗ a ⊗ . . . ⊗ a n a . . . a n ⊗ x . Lemma 6.
Let N be a right C -comodule, and see A as an operad concentrated in arity . Then N ◦ A is a right C -comodule through the map: N ◦ A ∆ N ◦ −−−→ N ◦ C ◦ A ◦ t −−→ N ◦ A ◦ C . Kontsevich’s proof of the formality of the little disks operads [Kon99] uses the theory ofsemi-algebraic sets, as developed in [KS00; HLTV11]. A semi-algebraic set is a subset of R N defined by finite unions of finite intersections of zero sets of polynomials and poly-nomial inequalities. By the Nash–Tognoli Theorem [Nas52; Tog73], any closed smoothmanifold is algebraic hence semi-algebraic.There is a functor Ω ∗ PA of “piecewise semi-algebraic (PA) differential forms”, analogousto de Rham forms. If X is a compact semi-algebraic set, then Ω ∗ PA ( X ) ≃ A ∗ PL ( X ) ⊗ Q R ,i.e. the CDGA Ω ∗ PA ( X ) models the real homotopy type of X [HLTV11, Theorem 6.1].A key feature of PA forms is that it is possible to compute integrals of “minimal forms”along fibers of “ PA bundles”, i.e. maps with local semi-algebraic trivializations [HLTV11,Section 8]. A minimal form is of the type f df ∧ . . . ∧ df k where f i : M → R are semi-algebraic maps. Given such a minimal form λ and a PA bundle p : M → B with fibersof dimension r , there is a new form (which is not minimal in general), also called thepushforward of λ along p : p ∗ λ := Z p : M → B λ ∈ Ω k − r PA ( B ) . In what follows, we use an extension of the fiberwise integration of minimal forms tothe sub-
CDGA of “trivial forms” given in [CW16, Appendix C]. Briefly recall that trivialforms are integrals of minimal forms along fibers of a trivial PA bundle (see [CW16,Definition 81]). In fact, in Section 3.3, we consider a certain form, the “propagator”,which is not minimal but trivial in this sense, and we apply the extension of the fiberwiseintegration to this form.The functor Ω ∗ P A is monoidal, but not strongly monoidal, and contravariant. Thus,given an operad P in semi-algebraic sets, Ω ∗ PA ( P ) is an “almost” Hopf cooperad and9atisfies a slightly modified version of the cooperad axioms, as explained in [LV14,Definition 3.1]. Cooperadic structure maps are replaced by zigzags Ω ∗ PA ( P ( U )) ◦ ∗ W −−→ Ω ∗ PA ( P ( U/W ) × P ( W )) ∼ ←− Ω ∗ PA ( P ( U/W )) × Ω ∗ PA ( P ( W )) (where the second map is theKünneth morphism). If C is a Hopf cooperad, an “almost” morphism f : C → Ω ∗ PA ( P )is a collection of CDGA morphisms f U : C ( U ) → Ω ∗ PA ( P ( U )) for all U , such that thefollowing diagrams commute: C ( U ) C ( U/W ) ⊗ C ( W )Ω ∗ PA ( P ( U )) Ω ∗ PA ( P ( U/W ) × P ( W )) Ω ∗ PA ( P ( U/W )) ⊗ Ω ∗ PA ( P ( W )) ◦ ∨ W f U f U/W ⊗ f W ◦ ∗ W ∼ Similarly, if M is a P -module, then Ω ∗ PA ( M ) is an “almost” Hopf right comodule overΩ ∗ PA ( P ). If N is a Hopf right C -comodule, where C is a cooperad equipped with an“almost” morphism f : C → Ω ∗ PA ( P ), then an “almost” morphism g : N → Ω ∗ PA ( M ) is acollection of CDGA morphisms g U : N ( U ) → Ω ∗ PA ( M ( U )) that make the following diagramscommute: N ( U ) N ( U/W ) ⊗ C ( W )Ω ∗ PA ( M ( U )) Ω ∗ PA ( M ( U/W ) × P ( W )) Ω ∗ PA ( M ( U/W )) ⊗ Ω ∗ PA ( P ( W )) ◦ ∨ W g U g U/W ⊗ f W ◦ ∗ W ∼ We will generally omit the adjective “almost”, keeping in mind that some commutativediagrams are a bit more complicated than at first glance.
Remark . There is a construction Ω ∗ ♯ that turns a simplicial operad P into a Hopf co-operad and such that a morphism of Hopf cooperads C → Ω ∗ ♯ ( P ) is the same thing as an“almost” morphism C → A ∗ PL ( P ), where A ∗ PL is the functor of Sullivan forms [Fre17, Sec-tion II.10.1]. Moreover there is a canonical collection of maps (Ω ∗ ♯ ( P ))( U ) → A ∗ PL ( P ( U )),which are weak equivalences if P is a cofibrant operad. This functor is built by consider-ing the right adjoint of the functor on operads induced by the Sullivan realization functor,which is monoidal. A similar construction can be extended to Ω ∗ PA and to modules overoperads. This construction allows us to make sure that the cooperads and comodules weconsider truly encode the rational or real homotopy type of the initial operad or module(see [Fre17, §II.10.2]). The little disks operad E n is a topological operad initially introduced by May andBoardman–Vogt [May72; BV73] to study iterated loop spaces. Its homology e n := H ∗ ( E n ) is described by a theorem of Cohen [Coh76]: it is either the operad governingassociative algebras for n = 1, or n -Poisson algebras for n ≥
2. We also consider thelinear dual e ∨ n := H ∗ ( E n ), which is a Hopf cooperad.In fact, we use the Fulton–MacPherson operad FM n , which is an operad in spacesweakly equivalent to the little disks operad E n . The components FM n ( k ) are compactifi-cations of the configuration spaces Conf k ( R n ), defined by using a real analogue due to10xelrod–Singer [AS94] of the Fulton–MacPherson compactifications [FM94]. The ideaof this compactification is to allow configurations where points become “infinitesimallyclose”. Then one uses insertion of such infinitesimal configurations to define operadiccomposition products on the spaces FM n ( k ). We refer to [Sin04] for a detailed treatmentand to [LV14, Sections 5.1–5.2] for a clear summary. In both references, the name C [ k ]is used for what we call FM n ( k ).The first two spaces FM n ( ∅ ) = FM n (1) = ∗ are singletons, and FM n (2) = S n − is asphere. We let the volume form of FM n (2) be:vol n − ∈ Ω n − ( S n − ) = Ω n − ( FM n (2)) (8)The space FM n ( k ) is a semi-algebraic stratified manifold, of dimension nk − n − k ≥
2, and of dimension 0 otherwise. For u = v ∈ U , we can define the projectionmaps that forget all but two points in the configuration, p uv : FM n ( U ) → FM n (2). Theseprojections are semi-algebraic bundles.If M is a manifold, the configuration space Conf k ( M ) can similarly be compactifiedto give a space FM M ( k ). By forgetting points, we again obtain projection maps, for u, v ∈ U : p u : FM M ( U ) ։ FM M (1) = M, p uv : FM M ( U ) ։ FM M (2) . (9)The two projections p and p are equal when restricted ∂ FM M (2), and they define asphere bundle of rank n − p : ∂ FM M (2) ։ M. (10)When M is framed, the collection of spaces FM M assemble to form a topological rightmodule over FM n , with composition products defined by insertion of infinitesimal con-figurations. Moreover in this case, the sphere bundle p : ∂ FM M (2) → M is trivializedby: M × S n − ∼ = FM M (1) × FM n (2) ◦ −→ ∂ FM M (2) . (11)Recall from Section 1.3 that we can endow M with a semi-algebraic structure. It isimmediate that FM M ( k ) is a stratified semi-algebraic manifold of dimension nk . Moreover,the proofs of [LV14, Section 5.9] can be adapted to show that the projections p U : FM M ( U ⊔ V ) → FM M ( U ) are PA bundles. We will make use of the “operadic twisting” procedure in what follows [DW15]. Let usnow recall this procedure, in the case of cooperads.Let
Lie n be the operad governing shifted Lie algebras. A Lie n -algebra is a dg-module g equipped with a Lie bracket [ − , − ] : g ⊗ → g [1 − n ] of degree 1 − n , i.e. we have[ g i , g j ] ⊂ g i + j +(1 − n ) . Remark . The degree convention is such that there is an embedding of operads
Lie n → H ∗ ( FM n ), i.e. Poisson n -algebras are Lie n -algebras. The usual Lie operad is Lie . Thisconvention is consistent with [Wil14]. However in [Wil16], the notation is Lie ( n ) = Lie n +1 . In [DW15], only the unshifted operad Lie = Lie is considered.11he operad Lie n is quadratic Koszul (see e.g. [LV12, Section 13.2.6]), and as suchadmits a cofibrant resolution hoLie n := Ω( K ( Lie n )), where Ω is the cobar constructionand K ( Lie n ) is the Koszul dual cooperad of Lie . Algebras over hoLie n are (shifted) L ∞ -algebras, also known as homotopy Lie algebras, i.e. dg-modules g equipped withhigher brackets [ − , . . . , − ] k : g ⊗ k → g [3 − k − n ] (for k ≥
1) satisfying the classical L ∞ equations.Let C be a cooperad (with finite-type components in each arity) equipped with a mapto the dual of hoLie n . This map can equivalently be seen as a Maurer–Cartan elementin the following dg-Lie algebra [LV12, Section 6.4.2]:Hom Σ ( K ( Lie n ) , C ∨ ) := (cid:18)Y i ≥ (cid:0) C ∨ ( i ) ⊗ R [ − n ] ⊗ i (cid:1) Σ i [ n ] , ∂, [ − , − ] (cid:19) , (13)where we used the explicit description of the Koszul dual K ( Lie n ) as a shifted versionof the cooperad encoding cocommutative coalgebras. Given f, g ∈ Hom Σ ( K ( Lie n ) , C ∨ ),their bracket is [ f, g ] = f ⋆ g ∓ g ⋆ f , where ⋆ is given by: f ⋆ g : K ( Lie n ) cooperad −−−−−→ K ( Lie n ) ◦ K ( Lie n ) f ◦ g −−→ C ∨ ◦ C ∨ operad −−−−→ C ∨ . An element µ ∈ Hom Σ ( K ( Lie n ) , C ∨ ) is said to satisfy the Maurer–Cartan equation if ∂µ + µ ⋆ µ = 0. Such an element is called a twisting morphism in [LV12, Section 6.4.3],and the equivalence with morphisms hoLie n → C ∨ (or dually C → hoLie ∨ n ) is [LV12, The-orem 6.5.7]. In the sequel, we will alternate between the two points of view, morphismsor Maurer–Cartan elements.There is an action of the symmetric group Σ i on i = { , . . . , i } . As a graded module,the twist of C with respect to µ is given by:Tw C ( U ) := M i ≥ (cid:0) C ( U ⊔ i ) ⊗ R [ n ] ⊗ i (cid:1) Σ i . (14)The symmetric collection Tw C inherits a cooperad structure from C . The differentialof Tw C is the sum of the internal differential of C with a differential coming from theaction of µ that we now explain. The action of µ is threefold, and the total differentialof Tw C ( U ) can be expressed as: d Tw C := d C + ( − · µ ) + ( − · µ ) + ( µ · − ) . (15)Let us now explain these notations. Let i ≥ µ on C ( U ⊔ i ) ⊂ Tw C ( U ) (up to degree shifts). In what follows, for a set J ⊂ i , we let j := J , and i/J ∼ = i + j − µ is a formal sum of elements C ( j ) ∨ for j ≥
0. The first action ( − · µ ) isthe sum over all subsets J ⊂ i of the following cocompositions: C ( U ⊔ i ) ◦ ∨ J −→ C ( U ⊔ i/J ) ⊗ C ( J ) id ⊗ µ −−−→ C ( U ⊔ i/J ) ⊗ R ∼ = C ( U ⊔ i + j − . (16)For the two other terms, we need the element µ ∈ Q j ≥ C ( j ⊔ {∗} ) ∨ . It is the sumover all possible ways of distinguishing one input of µ in each arity. (Distinguishing one12nput does not respect the invariants in the definition of Equation (13), but taking thesum over all possible ways does.)The second action ( − · µ ) is then the sum of the following cocompositions, over allsubsets J ⊂ i and over all ∗ ∈ U (where we use the obvious bijection U/ {∗} ∼ = U ): C ( U ⊔ i ) ◦ ∨{ u }⊔ J −−−−→ C (cid:0) ( U ⊔ i ) / ( {∗} ⊔ J ) (cid:1) ⊗ C ( {∗} ⊔ J ) id ⊗ µ −−−−→ C ( U ⊔ i + j − , (17)Finally, the third action ( µ ·− ) is the sum over all subsets J ⊂ i of the cocompositions(where we use the obvious bijection ( U ⊔ I ) / ( U ⊔ J ) = {∗} ⊔ I \ J ): C ( U ⊔ i ) ◦ ∨ U ⊔ J −−−→ C ( {∗} ⊔ i \ J ) ⊗ C ( U ⊔ J ) µ ⊗ id −−−−→ C ( U ⊔ J ) , (18) Lemma 19. If C is a Hopf cooperad satisfying C ( ∅ ) = k , then Tw C inherits a Hopfcooperad structure.Proof. To multiply an element of C ( U ⊔ I ) ⊂ Tw C ( U ) with an element of C ( U ⊔ J ) ⊂ Tw C ( U ), we use the maps C ( V ) ◦ ∨ ∅ −→ C ( V / ∅ ) ⊗ C ( ∅ ) ∼ = C ( V ⊔ {∗} ) iterated several times,to obtain elements in C ( U ⊔ I ⊔ J ) and the product.Moreover, we will need to twist right comodules over cooperads. This construction isfound (for operads) in [Wil16, Appendix C.1]. Let us fix a cooperad C and a twist Tw C with respect to µ as above. Given a right C -comodule M , we can also twist it with respectto µ , as follows. As a graded module, the object Tw M ( U ) is defined by:Tw M ( U ) := Y i ≥ (cid:0) M ( U ⊔ i ) ⊗ ( R [ n ]) ⊗ i (cid:1) Σ i . The comodule structure is inherited from M . The total differential is the sum: d Tw M := d M + ( − · µ ) + ( − · µ ) , (20)where ( − · µ ) and ( − · µ ) are as in Equations (16) and (17) but using the comodulestructure. Note that M is only a right module, so there can be no term ( µ · − ) in thisdifferential. Lemma 19 has an immediate extension: Lemma 21. If C is a Hopf cooperad satisfying C ( ∅ ) = k and M is a Hopf right C -comodule,then Tw M inherits a Hopf right (Tw C ) -comodule structure. Kontsevich’s proof of the formality of the little disks operads [Kon99, Section 3], can besummarized by the fact that Ω ∗ PA ( FM n ) is weakly equivalent to e ∨ n as a Hopf cooperad.For detailed proofs, we refer to [LV14].We outline this proof here as we will mimic its pattern for our theorem. The ideaof the proof is to construct a Hopf cooperad Graphs n . The elements of Graphs n are13ormal linear combinations of special kinds of graphs, with two types of vertices, num-bered “external” vertices and unnumbered “internal” vertices. The differential is definedcombinatorially by edge contraction. It is built in such a way that there exists a zigzag e ∨ n ∼ ←− Graphs n ∼ −→ Ω ∗ PA ( FM n ). The first map is the quotient by the ideal of graphscontaining internal vertices. The second map is defined using integrals along fibers ofthe PA bundles FM n ( U ⊔ I ) → FM n ( U ) which forget some points in the configuration. Aninduction argument shows that the first map is a quasi-isomorphism, and the secondmap is easily seen to be surjective on cohomology.In order to deal with signs more easily, we use (co)operadic twisting (Section 1.5).Thus the Hopf cooperad Graphs n is not the same as the Hopf cooperad D from [LV14],see Remark 33. The cohomology of E n . The cohomology e ∨ n ( U ) = H ∗ ( E n ( U )) has a classical presentationdue to Arnold [Arn69] and Cohen [Coh76]. We have e ∨ n ( U ) = S ( ω uv ) u,v ∈ U /I, (22)where the generators ω uv have cohomological degree n −
1, and the ideal I encoding therelations is generated by the polynomials (called Arnold relations): ω uu = 0; ω vu = ( − n ω uv ; ω uv = 0; ω uv ω vw + ω vw ω wu + ω wu ω uv = 0 . (23)The cooperadic structure maps are given by (where [ u ] , [ v ] ∈ U/W are the classes of u and v in the quotient): ◦ ∨ W : e ∨ n ( U ) → e ∨ n ( U/W ) ⊗ e ∨ n ( W ) , ω uv ( ⊗ ω uv , if u, v ∈ W ; ω [ u ][ v ] ⊗ , otherwise . (24) Graphs with only external vertices.
The intermediary cooperad of graphs,
Graphs n , isbuilt in several steps. In the first step, define a cooperad of graphs with only externalvertices, with generators e uv of degree n − Gra n ( U ) = (cid:0) S ( e uv ) u,v ∈ U / ( e uv = e uu = 0 , e vu = ( − n e uv ) , d = 0 (cid:1) . (25)The definition of Gra n ( U ) is almost identical to the definition of e ∨ n ( U ), except that wedo not kill the Arnold relations.The CDGA
Gra n ( U ) is spanned by words of the type e u v . . . e u r v r . Such a word can beviewed as a graph with U as the set of vertices, and an edge between u i and v i for eachfactor e u i v i . For example, e uv is a graph with a single edge from u to v (see Equation (26)for another example). Edges are oriented, but for even n an edge is identified with itsmirror (so we can forget orientations), while for odd n it is identified with the opposite ofits mirror. In pictures, we do not draw orientations, keeping in mind that for odd n , theyare necessary to get precise signs. Graphs with double edges or edges between a vertexand itself are set to zero. Given such a graph, its set of edges E Γ ⊂ (cid:0) U (cid:1) is well-defined.The vertices of these graphs are called “external”, in contrast with the internal verticesthat are going to appear in the next part. 14 e e = 1 23 4 56 ∈ Gra n (6) (26)The multiplication of the CDGA
Gra n ( U ), from this point of view, consists of glu-ing two graphs along their vertices. The cooperadic structure map ◦ ∨ W : Gra n ( U ) → Gra n ( U/W ) ⊗ Gra n ( W ) maps a graph Γ to ± Γ U/W ⊗ Γ W such that Γ W is the full sub-graph of Γ with vertices W and Γ U/W collapses this full subgraph to a single vertex.On generators, ◦ ∨ W is defined by a formula which is in fact identical to Equation (24),replacing ω ?? by e ?? . This implies that the cooperad Gra n maps to e ∨ n by sending e uv to ω uv .There is a morphism ω ′ : Gra n → Ω ∗ PA ( FM n ) given on generators by: ω ′ : Gra n ( U ) → Ω ∗ PA ( FM n ( U )) , Γ ^ ( u,v ) ∈ E Γ p ∗ uv (vol n − ) , (27)where p uv : FM n ( U ) → FM n (2) is the projection map defined in Section 1.4, and vol n − isthe volume form of FM n (2) ∼ = S n − from Equation (8). Twisting.
The second step of the construction is cooperadic twisting, using the procedureoutlined in Section 1.5. The Hopf cooperad
Gra n maps into Lie ∨ n as follows. Thecooperad Lie ∨ n is cogenerated by Lie ∨ n (2), and on cogenerators the cooperad map isgiven by sending e ∈ Gra n (2) to the cobracket in Lie ∨ n (2) and all the other graphsto zero. This map to Lie ∨ n yields a map to hoLie ∨ n by composition with the canonicalmap Lie ∨ n ∼ −→ hoLie ∨ n . In the dual basis, the corresponding Maurer–Cartan element µ isgiven by: µ := e ∨ = 1 2 ∈ Gra ∨ n (2) (28)The cooperad Gra n satisfies Gra n ( ∅ ) = R . Thus by Lemma 19, Tw Gra n inherits a Hopfcooperad structure, which we now explicitly describe.The dg-module Tw Gra n ( U ) is spanned by graphs with two types of vertices: externalvertices, which correspond to elements of U and that we will picture as circles with thename of the label in U inside, and indistinguishable internal vertices, corresponding tothe elements of i in Equation (14) and that we will draw as black points. For example,the graph inside the differential in the left hand side of Figure 29 represents an elementof Tw Gra n ( U ) with U = { , , } and i = 1. The degree of an edge is still n −
1, thedegree of an external vertex is still 0, and the degree of an internal vertex is − n .The product of Tw Gra n ( U ) glues graphs along their external vertices only. Comparedto Lemma 19, this coincides with adding isolated internal vertices (by iterating thecooperad structure map ◦ ∨ ∅ ) and gluing along all vertices.Let us now describe the differential adapted from [LV14, Section 6.4] (see Remark 32for the differences). We first give the final result, then we explain how it is obtainedfrom the description in Section 1.5. An edge is said to be contractible if it connectsany vertex to an internal vertex, except if it connects a univalent internal vertices to a15ertex which is not a univalent internal vertex. The differential of a graph Γ is the sum: d Γ = X e ∈ E Γ contractible ± Γ /e, where Γ /e is Γ with e collapsed, and e ranges over all contractible edges.12 3 d
12 3 ±
12 3 ±
12 3Figure 29: The differential of Tw
Gra n . This particular example shows that the Arnoldrelation (the RHS) is killed up to homotopy.Let us now explain how to compare this with the description in Section 1.5, seealso [Wil14, Appendix I.3] for a detailed description. Recall that the Maurer–Cartanelement µ (Equation (28)) is equal to 1 on the graph with exactly two vertices and oneedge, and vanishes on all other graphs. Recall from Equation (15) that the differentialof Tw Gra n has three terms: ( − · µ ) + ( − · µ ) + ( µ · − ), plus the differential of Gra n which vanishes. Let Γ be some graph. Then d Γ = Γ · µ + Γ · µ + µ · Γ where: • The element Γ · µ is the sum over all ways of collapsing a subgraph Γ ′ ⊂ Γ withonly internal vertices, the result being µ (Γ ′ )Γ / Γ ′ . This is nonzero only if Γ ′ hasexactly two vertices and one edge. Thus this summand corresponds to contractingall edges between two (possibly univalent) internal vertices in Γ. • The element Γ · µ is the sum over all ways of collapsing a subgraph Γ ′ ⊂ Γwith exactly one external vertex (and any number of internal vertices), with re-sult µ (Γ ′ )Γ / Γ ′ . This summands corresponds to contracting all edges between oneexternal vertex and one internal (possibly univalent) vertex. • The element µ · Γ is the sum over all ways of collapsing a subgraph Γ ′ ⊂ Γcontaining all the external vertices, with result µ (Γ / Γ ′ )Γ ′ . The coefficient µ (Γ / Γ ′ )can only be nonzero if Γ is obtained from Γ ′ by adding a univalent internal vertex.A careful analysis of the signs [Wil14, Appendix I.3] shows that this cancels outwith the contraction of edges connected to univalent internal vertices from theother two summands, unless both endpoints of the edge are univalent and internal(and hence disconnected from the rest of the graph), in which cases the same termappears three times, and only two cancel out (see [Wil14, Fig. 3] for the dualpicture). Definition 30.
A graph is internally connected if it remains connected when the ex-ternal vertices are deleted. It is easily checked that as a commutative algebra, Tw
Gra n ( U )is freely generated by such graphs. 16he morphisms e ∨ n ← Gra n ω ′ −→ Ω ∗ PA ( FM n ) extend along the inclusion Gra n ⊂ Tw Gra n as follows. The extended morphism Tw Gra n → e ∨ n simply sends a graph with internalvertices to zero. We need to check that this commutes with the differential. We thus needto determine when a graph with internal vertices (sent to zero) can have a differentialwith no internal vertices (possibly sent to a nonzero element in e ∨ n ). The differentialdecreases the number of internal vertices by exactly one. So by looking at generators(internally connected graphs) we can look at the case of graphs with a single internalvertex connected to some external vertices. Either the internal vertex is univalent, butthen the edge is not contractible and the differential vanishes. Or the internal vertex isconnected to more than one external vertices. In this case, one check that the differentialof the graph is zero modulo the Arnold relations, (see [LV14, Introduction] and Figure 29for an example).The extended morphism ω : Tw Gra n → Ω ∗ PA ( FM n ) (see [Kon99, Definition 14] and [LV14,Chapter 9] where the analogous integral is denoted b I ) sends a graph Γ ∈ Gra n ( U ⊔ I ) ⊂ Tw Gra n ( U ) to: ω (Γ) := Z FM n ( U ⊔ I ) pU −−→ FM n ( U ) ω ′ (Γ) = ( p U ) ∗ ( ω ′ (Γ)) , (31)where p U is the projection that forgets the points of the configuration corresponding to I , and the integral is an integral along the fiber of this PA bundle (see Section 1.3). Notethat the volume form on the sphere is minimal, hence ω ′ (Γ) is minimal and therefore wecan compute this integral. Remark . This Hopf cooperad is different from the module of diagrams b D introducedin [LV14, Section 6.2]: Tw Gra n is the quotient of b D by graphs with multiple edges andloops. The analogous integral b I : b D → Ω ∗ PA ( FM n ) is from [LV14, Chapter 9]. It vanisheson graphs with multiple edges and loops by [LV14, Lemmas 9.3.5, 9.3.6], so ω is well-defined. Moreover the differential is slightly different. In [LV14] some kind of edges,called “dead ends” [LV14, Definition 6.1.1], are not contractible. When restricted tographs without multiple edges or loops, these are edges connected to univalent internalvertices. But in Tw Gra n , edges connecting two internal vertices that are both univalentare contractible (see [Wil14, Fig. 3] for the dual picture). This does not change b I , whichvanishes on graphs with univalent internal vertices anyway [LV14, Lemma 9.3.8]. Notethat b D is not a Hopf cooperad [LV14, Example 7.3.2] due to multiple edges. Reduction.
The cooperad Tw
Gra n does not have the homotopy type of the cooperadΩ ∗ PA ( FM n ). It is reduced by quotienting out all the graphs with connected componentsconsisting exclusively of internal vertices. This is a bi-ideal generated by Tw Gra n ( ∅ ),thus the resulting quotient is a Hopf cooperad: Graphs n := Tw Gra n / (cid:0) Tw Gra n ( ∅ ) (cid:1) . Remark . This Hopf cooperad is not isomorphic to the Hopf cooperad D from [LV14,Section 6.5]. We allow internal vertices of any valence, whereas in D internal verticesmust be at least trivalent. There is a quotient map Graphs n → D , which is a quasi-isomorphism by [Wil14, Proposition 3.8]. The statement of [Wil14, Proposition 3.8] is17ctually about the dual operads, but as we work over a field and the spaces we considerhave finite-type cohomology, this is equivalent. The notation is also different: the couple( Graphs n , fGraphs n,c ) in [Wil14] denotes ( D ∨ , Graphs ∨ n ) in [LV14].One checks that the two morphisms e ∨ n ← Tw Gra n → Ω ∗ PA ( FM n ) factor through thequotient (the first one because graphs with internal vertices are sent to zero, the secondone because ω vanishes on graphs with only internal vertices by [LV14, Lemma 9.3.7]).The resulting zigzag e ∨ n ← Graphs n → Ω ∗ PA ( FM n ) is then a zigzag of weak equivalence ofHopf cooperads thanks to the proof of [Kon99, Theorem 2] (or [LV14, Theorem 8.1] andthe discussion at the beginning of [LV14, Chapter 10]), combined with the comparisonbetween D and Graphs n from [Wil14, Proposition 3.8] (see Remark 33). CDGA models
The model for Ω ∗ PA ( FM M ) relies on a Poincaré duality model of M . We mostly borrowthe terminology and notation from [LS08b].Fix an integer n and let A be a connected CDGA (i.e. A = R ⊕ A ≥ ). An orientation on A is a linear map A n → R satisfying ε ◦ d = 0 (which we often view as a chain map A → R [ − n ]) such that the induced pairing h− , −i : A k ⊗ A n − k → R , a ⊗ b ε ( ab ) (34)is non-degenerate for all k . This implies that A = A ≤ n , and that ε : A n → R is anisomorphism. The pair ( A, ε ) is called a
Poincaré duality
CDGA . If A is such aPoincaré duality CDGA , then so is its cohomology. The following “converse” has beenshown by Lambrechts–Stanley.
Theorem 35 (Direct corollary of Lambrechts–Stanley [LS08b, Theorem 1.1]) . Let M be a simply connected semi-algebraic closed oriented manifold. Then there exists a zigzagof quasi-isomorphisms of CDGA s A ρ ←− R σ −→ Ω ∗ PA ( M ) , such that A is a Poincaré duality CDGA of dimension n , R is a quasi-free CDGA generatedin degrees ≥ , σ factors through the sub- CDGA of trivial forms.Proof.
We refer to Section 1.3 for a reminder on trivial forms. We pick a minimalmodel R of the manifold M (over R ) and a quasi-isomorphism from R to the sub-CDGA of trivial forms in Ω ∗ P A ( M ), which exists because the sub-CDGA of trivial formsis quasi-isomorphic to Ω ∗ P A ( M ) (see Section 1.3), and hence, is itself a real model for M . We compose this new quasi-isomorphism this the inclusion to eventually get aquasi-isomorphism σ : R → Ω ∗ P A ( M ) which factors through the sub-CDGA of trivialforms, and we set ε = R M σ ( − ) : R → R [ − n ]. The CDGA R is of finite type because M is a closed manifold. Hence, we can apply the Lambrechts-Stanley Theorem [LS08b,Theorem 1.1] to the pair ( R, ǫ ) to get the Poincaré duality algebra A of our statement.18et A be a Poincaré duality CDGA of finite type and let { a i } be a homogeneous basisof A . Consider the dual basis { a ∗ i } with respect to the duality pairing, i.e. ε ( a i a j ) = δ ij is given by the Kronecker symbol. Then the diagonal cocycle is defined by thefollowing formula and is independent of the chosen homogeneous basis (see e.g. [FOT08,Definition 8.16]: ∆ A := X i ( − | a i | a i ⊗ a ∗ i ∈ A ⊗ A. (36)The element ∆ A is a cocycle of degree n (this follows from ε ◦ d = 0). It satisfies∆ A = ( − n ∆ A (where ( − ) is defined in Section 1.1). For all a ∈ A it satisfies theequation ( a ⊗ A = (1 ⊗ a )∆ A . There is a volume form,vol A := ε − (1 R ) ∈ A n . The product µ A : A ⊗ A → A sends ∆ A to χ ( A ) · vol A , where χ ( A ) is the Euler charac-teristic of A . We will need the following technical result later. Proposition 37.
One can choose the zigzag of Theorem 35 such there exists a symmetriccocycle ∆ R ∈ R ⊗ R of degree n satisfying ( ρ ⊗ ρ )(∆ R ) = ∆ A . If χ ( M ) = 0 we canmoreover choose it so that µ R (∆ R ) = 0 .Proof. We follow closely the proof of [LS08b] to obtain the result. Recall that the proofof [LS08b] has two different cases: n ≤
6, where the manifold is automatically formal andhence A = H ∗ ( M ), and n ≥
7, where the
CDGA is built out of an inductive argument.We split our proof along these two cases.Let us first deal with the case n ≥
7. When n ≥
7, the proof of Lambrechts and Stanleybuilds a zigzag of weak equivalences A ρ ←− R ← R ′ → Ω ∗ PA ( M ), where R ′ is the minimalmodel of M , the CDGA R is obtained from R ′ by successively adjoining generators ofdegree ≥ n/ CDGA A is a quotient of R by an ideal of“orphans”. We let ε : R ′ → R [ − n ] be the composite R ′ → Ω ∗ PA ( M ) R M −−→ R [ − n ].The minimal model R ′ is quasi-free, and since M is simply connected it is generated indegrees ≥
2. The
CDGA R is obtained from R ′ by a cofibrant cellular extension, adjoiningcells of degree greater than 2. It follows that R is cofibrant and quasi-freely generated indegrees ≥
2. Composing with R ′ → Ω ∗ PA ( M ) yields a morphism σ : R → Ω ∗ PA ( M ) andwe therefore get a zigzag A ← R → Ω ∗ PA ( M ).The morphism ρ is a quasi-isomorphism, so there exists some cocycle ˜∆ ∈ R ⊗ R suchthat ρ ( ˜∆) = ∆ A + dα for some α . By surjectivity of ρ (it is a quotient map) there issome β such that ρ ( β ) = α ; we let ∆ ′ = ˜∆ − dβ , and now ρ (∆ ′ ) = ∆ A .Let us assume for the moment that χ ( M ) = 0. Then the cocycle µ R (∆ ′ ) ∈ R satisfies ρ ( µ R (∆ ′ )) = µ A (∆ A ) = 0, i.e. it is in the kernel of ρ . It follows that the cocycle∆ ′′ = ∆ ′ − µ R (∆ ′ ) ⊗ A by ρ , and satisfies µ R (∆ ′′ ) = 0. If χ ( M ) = 0we just let ∆ ′′ = ∆ ′ . Finally we symmetrize ∆ ′′ to get the ∆ R of the lemma, whichsatisfies all the requirements.Let us now deal with the case n ≤
6. The
CDGA Ω ∗ PA ( M ) is formal [NM78, Proposition4.6]. We choose A = ( H ∗ ( M ) , d A = 0), and R to be the minimal model of M , whichmaps into both A and Ω ∗ PA ( M ) by quasi-isomorphisms. The rest of the proof is nowidentical to the previous case. 19 .8. The Lambrechts–Stanley CDGA s We now give the definition of the
CDGA G A ( k ) from [LS08a, Definition 3.4], where it iscalled F ( A, k ).Let A be a Poincaré duality CDGA of dimension n and let k be an integer. For1 ≤ i = j ≤ k , let ι i : A → A ⊗ k be defined by ι i ( a ) = 1 ⊗ i − ⊗ a ⊗ ⊗ k − i − , and let ι ij : A ⊗ A → A ⊗ k be given by ι ij ( a ⊗ b ) = ι i ( a ) · ι j ( b ). Recalling the description of e ∨ n in Equation (22), the CDGA G A ( k ) is defined by: G A ( k ) := (cid:0) A ⊗ k ⊗ e ∨ n ( k ) / ( ι i ( a ) · ω ij = ι j ( a ) · ω ij ) , dω ij = ι ij (∆ A ) (cid:1) . (38)The fact that this is well-defined is proved in [LS08a, Lemma 3.2]. We will callthese CDGA s the Lambrechts–Stanley
CDGA s, or
LS CDGAs for short. For example G A (0) = R , G A (1) = A , and G A (2) is isomorphic to: G A (2) ∼ = (cid:0) ( A ⊗ A ) ⊕ ( A ⊗ ω ) , d ( a ⊗ ω ) = ( a ⊗ · ∆ A = (1 ⊗ a ) · ∆ A ) . Recall that there always exists a Poincaré duality model of M (Section 1.7). When M is a simply connected closed manifold, a theorem of Lambrechts–Stanley [LS08a,Theorem 10.1] implies that for any such A , H ∗ ( G A ( k ); Q ) ∼ = H ∗ ( FM M ( k ); Q ) as graded modules . (39)
2. The Hopf right comodule model G A In this section we describe the Hopf right e ∨ n -comodule derived from the LS CDGA sof Section 1.8. From now on we fix a simply connected smooth closed manifold M .Following Section 1.4, we endow M with a fixed semi-algebraic structure. Note that fornow, we do not impose any further conditions on M , but a key argument (Proposition 78)will require dim M ≥
4. We also fix a arbitrary Poincaré duality
CDGA model A of M . Wethen define the right comodule structure of G A as follows, using the cooperad structureof e ∨ n given by Equation (24): Proposition 40. If χ ( M ) = 0 , then the following maps are well-defined on G A = { G A ( k ) } k ≥ and endow it with a Hopf right e ∨ n -comodule structure: ◦ ∨ W : A ⊗ U ⊗ e ∨ n ( U ) → (cid:0) A ⊗ ( U/W ) ⊗ e ∨ n ( U/W ) (cid:1) ⊗ e ∨ n ( W ) , ( a u ) u ∈ U ⊗ ω (( a u ) u ∈ U \ W ⊗ Q w ∈ W a w ) | {z } ∈ A ⊗ ( U/W ) ⊗ ◦ ∨ W ( ω ) | {z } ∈ e ∨ n ( U/W ) ⊗ e ∨ n ( W ) . (41)In informal terms, ◦ ∨ W multiplies together all the elements of A indexed by W on the A ⊗ U factor and indexes the result by ∗ ∈ U/W , while it applies the cooperadic structuremap of e ∨ n on the other factor. Note that if W = ∅ , then ◦ ∨ W adds a factor of 1 A (theempty product) indexed by ∗ ∈ U/ ∅ = U ⊔ {∗} .20 roof. We split the proof in three parts: factorization of the maps through the quotient,compatibility with the differential, and compatibility of the maps with the cooperadicstructure of e ∨ n .Let us first prove that the comodule structure maps we wrote factor through thequotient. Since A is commutative and e ∨ n is a Hopf cooperad, the maps of the propositioncommute with multiplication. The ideals defining G A ( U ) are multiplicative ideals. Henceit suffices to show that the maps (41) take the generators ( ι u ( a ) − ι v ( a )) · ω uv of the idealto elements of the ideal in the target. We simply check each case, using Equations (24)and (41): • If u, v ∈ W , then ◦ ∨ W ( ι u ( a ) ω uv ) = ι ∗ ( a ) ⊗ ω uv , which is also equal to ◦ ∨ W ( ι v ( a ) ω uv ). • Otherwise, we have ◦ ∨ W ( ι u ( a ) ω uv ) = ι [ u ] ( a ) ω [ u ][ v ] ⊗
1, which is equal to ι [ v ] ( a ) ω [ u ][ v ] ⊗ ◦ ∨ W ( ι v ( a ) ω uv ) modulo the relations.Let us now prove that they are compatible with the differential. It is again sufficientto prove this on generators. The equality ◦ ∨ W ( d ( ι u ( a ))) = d ( ◦ ∨ W ( ι u ( a ))) is immediate.For ω uv we again check the three cases. Recall that since our manifold has vanishingEuler characteristic, µ A (∆ A ) = 0. • If u, v ∈ W , then ◦ ∨ W ( dω uv ) = ι ∗ ( µ A (∆ A )) = 0, while by definition d ( ◦ ∨ W ( ω uv )) = d (1 ⊗ ω uv ) = 0. • Otherwise, ◦ ∨ W ( dω uv ) = ι [ u ][ v ] (∆ A ) ⊗
1, which is equal to d ( ◦ ∨ W ( ω uv )) = d ( ω [ u ][ v ] ⊗ e ∨ n . Let Com ∨ be the cooperad governing cocommutative coalgebras. It follows fromLemma 6 that Com ∨ ◦ A = { A ⊗ k } k ≥ inherits a Com ∨ -comodule structure. Therefore thearity-wise tensor product (see [LV12, Section 5.1.12], where this operation is called theHadamard product) ( Com ∨ ◦ A ) ⊠ e ∨ n := { A ⊗ k ⊗ e ∨ n ( k ) } k ≥ is a ( Com ∨ ⊠ e ∨ n )-comodule. Thecooperad Com ∨ is the unit of ⊠ . Hence the ( Com ∨ ◦ A ) ⊠ e ∨ n is an e ∨ n -comodule. It remainsto make the easy check that the resulting comodule maps are given by Equation (41).
3. Labeled graph complexes
In this section we construct the intermediary comodule,
Graphs R , used to prove ourtheorem, where R is a suitable cofibrant CDGA quasi-isomorphic to A and Ω ∗ PA ( M )(Theorem 35). We will construct a zigzag of CDGA s of the form: G A ← Graphs R → Ω ∗ PA ( FM M ) . The construction of
Graphs R follows the same pattern as the construction of Graphs n in Section 1.6, but with the vertices of the graph labeled by elements of R . The differen-tial moreover mimics the definition of the differential of G A , together with a differentialthat mimics the one of Graphs n .If χ ( M ) = 0, then the collections G A and Graphs R are Hopf right comodules re-spectively over e ∨ n and over Graphs n , and the left arrow is a morphism of comodules21etween ( G A , e ∨ n ) and ( Graphs R , Graphs n ). When M is moreover framed, Ω ∗ PA ( FM M ) isa Hopf right comodule over Ω ∗ PA ( FM n ), and the right arrow is then a morphism from( Graphs R , Graphs n ) to (Ω ∗ PA ( FM M ) , Ω ∗ PA ( FM n )).In order to deal with signs more easily and make sure that the differential squares tozero, we want to use the formalism of operadic twisting, as in the definition of Graphs n .But when χ ( M ) = 0 there is no comodule structure, so we make a detour through graphswith loops (Section 3.1 below), see Remark 60. We first define a variant
Graphs (cid:9) n of Graphs n , where graphs are allowed to have “loops”(also sometimes known as “tadpoles”) and multiple edges, see [Wil14, Section 3]. For afinite set U , the CDGA
Gra (cid:9) n ( U ) is presented by (where the generators have degree n − Gra (cid:9) n ( U ) := (cid:0) S ( e uv ) u,v ∈ U / ( e vu = ( − n e uv ) , d = 0 (cid:1) . The difference with Equation (25) is that we no longer set e uu = e uv = 0. Note that Gra (cid:9) n ( U ) is actually free as a CDGA : given an arbitrary linear order on U , Gra (cid:9) n ( U ) isfreely generated by the generators { e uv } u ≤ v ∈ U . Remark . When n is even, e uv = 0 since deg e uv = n − n is odd,the relation e uu = ( − n e uu implies e uu = 0. In other words, for even n , there are nomultiple edges, and for odd n , there are no loops [Wil14, Remark 3.1].The dg-modules Gra (cid:9) n ( U ) form a Hopf cooperad, like Gra n , with cocomposition givenby a formula similar to the definition of Equation (24): ◦ ∨ W : Gra (cid:9) n ( U ) → Gra (cid:9) n ( U/W ) ⊗ Gra (cid:9) n ( W ) ,e uv ( e ∗∗ ⊗ ⊗ e uv , if u, v ∈ W ; e [ u ][ v ] ⊗ , otherwise . (43)This new cooperad has a graphical description similar to Gra n . The cooperad Gra n isthe quotient of Gra (cid:9) n by the ideal generated by the loops and the multiple edges. Thedifference in the cooperad structure is that when we collapse a subgraph, we sum overall ways of choosing whether edges are in the subgraph or not; if they are not, then theyyield a loop. For example:1 23 ◦ ∨{ , } ∗ ⊗ + ∗ ⊗ (44)The element µ := e ∨ ∈ ( Gra (cid:9) n ) ∨ (2) still defines a morphism Gra (cid:9) n → hoLie ∨ n , whichallows us to define the twisted Hopf cooperad Tw Gra (cid:9) n . It has a graphical descriptionsimilar to Tw Gra n with internal and external vertices. Finally we can quotient by graphscontaining connected component consisting exclusively of internal vertices to get a Hopfcooperad: Graphs (cid:9) n := Tw Gra (cid:9) n / (connected components with only internal vertices) . emark . The Hopf cooperad Tw
Gra (cid:9) n is slightly different from b D from [LV14, Section6]. First the cocomposition is different, and the first term of the RHS in Equation (44)would not appear in b D . The differential is also slightly different: an edge connectedto two univalent internal vertices – hence disconnected from the rest of the graph – iscontractible here (see [Wil14, Section 3] and Remark 32). This fixes the failure of b D tobe a cooperad [LV14, Example 7.3.2]. Gra R We construct a collection of
CDGA s Gra R , corresponding to the first step in the construc-tion of Graphs n of Section 1.6. We first apply the formalism of Section 1.7 to Ω ∗ PA ( M )in order to obtain a Poincaré duality CDGA out of M , thanks to Theorem 35. We thusfix a zigzag of quasi-isomorphisms A ρ ←− R σ −→ Ω ∗ PA ( M ), where A is a Poincaré duality CDGA , R is a cofibrant CDGA , and σ factors through the sub- CDGA of trivial forms (seeSection 1.3).Recall the definition of the diagonal cocycle ∆ A ∈ ( A ⊗ A ) n from Equation (36).Recall also Proposition 37, where we fixed a symmetric cocycle ∆ R ∈ ( R ⊗ R ) n suchthat ( ρ ⊗ ρ )(∆ R ) = ∆ A . Moreover recall that if χ ( M ) = 0, then µ A (∆ A ) = 0, and wechoose ∆ R such that µ R (∆ R ) = 0 too. Definition 46.
Let
CDGA of labeled graphs with loops on the set U be: Gra (cid:9) R ( U ) := (cid:0) R ⊗ U ⊗ Gra (cid:9) n ( U ) , de uv = ι uv (∆ R ) (cid:1) . This
CDGA is well-defined because
Gra (cid:9) n ( U ) is free as a CDGA , hence
Gra (cid:9) R ( U ) is arelative Sullivan algebra in the terminology of [FHT01, Section 14]. Remark . This definition is valid for any
CDGA R and any symmetric cocycle ∆ R . Weneed R as in Proposition 37 to connect Gra (cid:9) R with G A and Ω ∗ PA ( FM M ). Remark . It follows that the differential of a loop is de uu = ι uu (∆ R ) = ι u ( µ R (∆ R )),which is zero when χ ( M ) = 0. Proposition 49.
The collection
Gra (cid:9) R ( U ) forms a Hopf right Gra (cid:9) n -comodule. This is true even if χ ( M ) = 0 thanks to the introduction of the loops.. Proof.
The proof of this proposition is almost identical to the proof of Proposition 40.If we forget the extra differential (keeping only the internal differential of R ), then Gra (cid:9) R is the arity-wise tensor product ( Com ∨ ◦ R ) ⊠ Gra (cid:9) n , which is automatically a Hopf Gra (cid:9) n -right comodule. Checking the compatibility with the differential involves almost exactlythe same equations as Proposition 40, except that when u, v ∈ W we have: ◦ ∨ W ( d ( e uv )) = ι ∗ ( µ R (∆ R )) ⊗ d ( e ∗∗ ⊗ ⊗ e uv ) = d ( ◦ ∨ W ( e uv )) , where de ∗∗ = ι ∗ ( µ R (∆ R )) by Remark 48, and d (1 ⊗ e uv ) = 0 by definition.23e now give a graphical interpretation of Definition 46, in the spirit of Section 3.1.We view Gra (cid:9) R ( U ) as spanned by graphs with U as set of vertices, and each vertex hasa label which is an element of R . The Gra (cid:9) n -comodule structure collapses subgraphsas before, and the label of the collapsed vertex is the product of all the labels in thesubgraph. An example of graph in Gra (cid:9) R (3) is given by (where x, y, z ∈ R ):1 x x x (50)The product glues two graphs along their vertices, multiplying the labels in the process.The differential of Γ, as defined in Definition 46, is the sum of d R , the internal differentialof R acting on each label (one at a time), together with the sum over the edges e ∈ E Γ of the graph Γ \ e with that edge removed and the labels of the endpoints multipliedby the factors of ∆ R = P (∆ R ) ∆ ′ R ⊗ ∆ ′′ R ∈ R ⊗ R , where we use Sweedler’s notation(Section 1.1). We will often write d split for this differential, to contrast it with thedifferential that contracts edges which will occur in the complex Tw Gra (cid:9) R defined lateron. If e is a loop, then in the corresponding term of d Γ the vertex incident to e hasits label multiplied by µ R (∆ R ), while the loop is removed. For example, we have (grayvertices can be either internal or external and x, y ∈ R ): x y d X (∆ R ) x ∆ ′ R y ∆ ′′ R . If χ ( M ) = 0, we cannot directly map Gra (cid:9) R to Ω ∗ PA ( FM M ), as the Euler class in Ω ∗ PA ( M )would need to be the boundary of the image of the loop e ∈ Gra (cid:9) R (1). We thus definea sub- CDGA which will map to Ω ∗ PA ( FM M ) whether χ ( M ) vanishes or not. Definition 51.
For a given finite set U , let Gra R ( U ) be the submodule of Gra (cid:9) R ( U )spanned by graphs without loops.One has to be careful with the notation. While Gra (cid:9) R ( U ) = R ⊗ U ⊗ Gra (cid:9) n , it is not truethat Gra R ( U ) = R ⊗ U ⊗ Gra n ( U ): in Gra n ( U ), multiple edges are forbidden, whereasthey are allowed in Gra R ( U ). Proposition 52.
The space
Gra R ( U ) is a sub- CDGA of Gra (cid:9) R ( U ) . If χ ( M ) = 0 thecollection Gra R assembles to form a Hopf right Gra n -comodule.Proof. Clearly, neither the splitting part of the differential nor the internal differentialcoming from R can create new loops, nor can the product of two graphs without loopscontain a loop, thus Gra R ( U ) is indeed a sub- CDGA of Gra (cid:9) R ( U ). If χ ( M ) = 0, theproof that Gra R is a Gra n -comodule is almost identical to the proof of Proposition 49,except that we need to use µ R (∆ R ) = 0 to check that d ( ◦ ∨ W ( e uv )) = ◦ ∨ W ( d ( e uv )) when u, v ∈ W . 24 .3. The propagator To define ω ′ : Gra R → Ω PA ( FM M ), we need a “propagator” ϕ ∈ Ω n − ( FM M (2)), for whicha reference is [CM10, Section 4].Recall from Equation (9) the projections p u : FM M ( U ) → M and p uv : FM M ( U ) → FM M (2). Recall moreover the sphere bundle p : ∂ FM M (2) → M defined in Equation (10),which is trivial when M is framed, with the isomorphism M × S n − ◦ −→ ∂ FM M (2) fromEquation (11). We denote by ( p , p ) : FM M (2) → M × M the product of the twocanonical projections. Proposition 53 ([CW16, Propositions 7 and 87]) . There exists a form ϕ ∈ Ω n − ( FM M (2)) such that ϕ = ( − n ϕ , dϕ = ( p , p ) ∗ (( σ ⊗ σ )(∆ R )) and such that the restriction of ϕ to ∂ FM M (2) is a global angular form, i.e. it is a volume form of S n − when restrictedto each fiber. When M is framed one can moreover choose ϕ | ∂ FM M (2) = 1 × vol S n − ∈ Ω n − P A ( M × S n − ) . This propagator can moreover be chosen to be a trivial form ((seeSection 1.3). The proofs of [CW16] relies on earlier computations given in [CM10], where thispropagator is studied in detail. One can see from the proofs of [CM10, Section 4] that dϕ can in fact be chosen to be any pullback of a form cohomologous to the diagonalclass ∆ M ∈ Ω n PA ( M × M ). We will make further adjustments to the propagator ϕ inProposition 75. Recall p u , p uv from Equation (9). Proposition 54.
There is a morphism of collections of
CDGA s given by: ω ′ : Gra R → Ω PA ( FM M ) , (N u ∈ U x u ∈ R ⊗ U V u ∈ U p ∗ u ( σ ( x u )) ,e uv p ∗ uv ( ϕ ) . Moreover, if M is framed, then ω ′ defines a morphism of comodules, where ω ′ : Gra n → Ω ∗ PA ( FM n ) was defined in Section 1.6: ( Gra R , Gra n ) ( ω ′ ,ω ′ ) −−−−→ (Ω ∗ PA ( FM M ) , Ω ∗ PA ( FM n )) Proof.
The property dϕ = ( p , p ) ∗ (( σ ⊗ σ )(∆ R )) shows that the map ω ′ preserves thedifferential. Let us now assume that M is framed to prove that this is a morphism ofright comodules. Cocomposition commutes with ω ′ on the generators coming from A ⊗ U ,since the comodule structure of Ω ∗ PA ( FM M ) multiplies together forms that are pullbacksof forms on M : ◦ ∨ W ( p ∗ u ( x )) = ( p ∗ u ( x ) ⊗ u W ; p ∗∗ ( x ) ⊗ u ∈ W. We now check the compatibility of the cocomposition ◦ ∨ W with ω ′ on the generator ω uv , for some W ⊂ U . • If one of u, v , or both, is not in W , then the equality ◦ ∨ W ( ω ′ ( e uv )) = ( ω ′ ⊗ ω ′ )( ◦ ∨ W ( e uv )) . is clear. 25 Otherwise suppose { u, v } ⊂ W . We may assume that U = W = 2 (it suffices topull back the result along p uv to get the general case), so that we are consideringthe insertion of an infinitesimal configuration M × FM n (2) → FM M (2). This insertionfactors through the boundary ∂ FM M (2). We have (see Definition 53): ◦ ∨ ( ϕ ) = 1 ⊗ vol S n − ∈ Ω ∗ PA ( M ) ⊗ Ω ∗ PA ( FM n (2)) = Ω ∗ PA ( M ) ⊗ Ω ∗ PA ( S n − ) . Going back to the general case, we find: ◦ ∨ W ( ω ′ ( e uv )) = ◦ ∨ W ( p ∗ uv ( ϕ )) = 1 ⊗ p ∗ uv (vol S n − ) , which is indeed the image of ◦ ∨ W ( ω uv ) = 1 ⊗ ω uv by ω ′ ⊗ ω ′ . Tw Gra R The general framework of operadic twisting, recalled in Section 1.5, shows that to twista right (co)module, one only needs to twist the (co)operad. Since our cooperad isone-dimensional in arity zero, the comodule inherits a Hopf comodule structure too(Lemma 21).
Definition 55.
The twisted labeled graph comodule Tw Gra (cid:9) R is a Hopf right(Tw Gra (cid:9) n )-comodule obtained from Gra (cid:9) R by twisting with respect to the Maurer–Cartanelement µ ∈ ( Gra (cid:9) n ) ∨ (2) of Section 1.6.We now explicitly describe this comodule in terms of graphs. The dg-module Tw Gra (cid:9) R ( U )is spanned by graphs with two kinds of vertices, external vertices corresponding to ele-ments of U , and indistinguishable internal vertices (usually drawn in black). The degreeof an edge is n −
1, the degree of an external vertex is 0, while the degree of an internalvertex is − n . All the vertices are labeled by elements of R , and their degree is added tothe degree of the graph.The Hopf structure glues two graphs along their external vertices, multiplying labelsin the process. The differential is a sum of three terms d = d R + d split + d contr . The first part is the internal differential coming from R , acting on each label separately.The second part comes from Gra (cid:9) R and splits edges, multiplying by ∆ R the labels ofthe endpoints. The third part is similar to the differential of Tw Gra (cid:9) n : it contracts all contractible edges, i.e. edges connecting an internal vertex to another vertex of eitherkind. When an edge is contracted, the label of the resulting vertex is the product of thelabels of of the endpoints of the former edge (see Figure 29). This result comes from thetwisting construction (see the definition in Equation (20)). For example, we have: (cid:18) x (cid:19) d (cid:18) d R x (cid:19) ± X (∆ R ) (cid:18) ∆ ′ R x ∆ ′′ R (cid:19) ± x ( x ∈ R ) . (56)26 emark . An edge connected to a univalent internal vertex is contractible in Tw
Gra (cid:9) R ,though this is not the case in Tw Gra (cid:9) n . Indeed, if we go back to the definition ofthe differential in a twisted comodule (Equation (20)), we see that the Maurer–Cartanelement µ (Equation (28)) only acts on the right of the graph. Therefore, there is noterm to cancel out the contraction of such edges, as was the case in Tw Gra n (see thediscussion in Section 1.6 about the differential). In Equation (56), the only edge wouldnot be considered as contractible in Tw Gra n if we forgot the labels, but it is in Tw Gra R .Finally, the comodule structure is similar to the cooperad structure of Tw Gra (cid:9) n : forΓ ∈ Gra (cid:9) R ( U ⊔ I ) ⊂ Tw Gra (cid:9) R ( U ), the cocomposition ◦ ∨ W (Γ) is the sum over tensors of thetype ± Γ U/W ⊗ Γ W , where Γ U/W ∈ Gra (cid:9) R ( U/W ⊔ J ), Γ W ∈ Gra n ( W ⊔ J ′ ), J ⊔ J ′ = I , andthere exists a way of inserting Γ W in the vertex ∗ of Γ U/W and reconnecting edges to getΓ back. See the following example of cocomposition ◦ ∨{ } : Tw Gra R (1) → Tw Gra R (1) ⊗ Tw Gra R (1), where x, y ∈ R :1 xy ◦ ∨{ } ∗ xy ⊗ ± ∗ xy ⊗ ± ∗ xy ⊗ Lemma 58.
The subspace Tw Gra R ( U ) ⊂ Tw Gra (cid:9) R ( U ) spanned by graphs with no loopsis a sub- CDGA .Proof.
It is clear that this defines a subalgebra. We need to check that it is preservedby the differential, i.e. that the differential cannot create new loops if there are none ina graph. This is clear for the internal differential coming from R and for the splittingpart of the differential. The contracting part of the differential could create a loop froma double edge. However for even n multiple edges are zero for degree reasons, and forodd n loops are zero because of the antisymmetry relation (see Remark 48).Note that despite the notation, Tw Gra R is a priori not defined as the twisting of the Gra n -comodule Gra R : when χ ( M ) = 0, the collection Gra R is not even a Gra n -comodule.However, the following proposition is clear and shows that we can get away with thisabuse of notation: Proposition 59. If χ ( M ) = 0 , then Tw Gra R assembles to a right Hopf (Tw Gra n ) -comodule, isomorphic to the twisting of the right Hopf Gra n -comodule Gra R of Defini-tion 51.Remark . We could have defined the algebra Tw
Gra R explicitly in terms of graphs,and defined the differential d using an ad-hoc formula. The difficult part would havethen been to check that d = 0 (involving difficult signs), which is a consequence of thegeneral operadic twisting framework. ω : Tw Gra R → Ω ∗ PA ( FM M ) This section is dedicated to the proof of the following proposition.27 roposition 61.
There is a morphism of collections of
CDGA s ω : Tw Gra R → Ω ∗ PA ( FM M ) extending ω ′ , given on a graph Γ ∈ Gra R ( U ⊔ I ) ⊂ Tw Gra R ( U ) by: ω (Γ) := Z p U : FM M ( U ⊔ I ) → FM M ( U ) ω ′ (Γ) = ( p U ) ∗ ( ω ′ (Γ)) . Moreover, if M is framed, then this defines a morphism of Hopf right comodules: ( ω, ω ) : (Tw Gra R , Tw Gra n ) → (Ω ∗ PA ( FM M ) , Ω ∗ PA ( FM n )) . Recall that in general, it is not possible to consider integrals along fibers of arbitrary PA forms, see [HLTV11, Section 9.4]. However, here, the image of σ is included inthe sub- CDGA of trivial forms in Ω ∗ PA ( M ), and the propagator is a trivial form (seeProposition 53), therefore the integral ( p U ) ∗ ( ω ′ (Γ)) exists.The proof of the compatibility with the Hopf structure and, in the framed case, the co-module structure, is formally similar to the proof of the same facts about ω : Tw Gra n → Ω ∗ PA ( FM n ). We refer to [LV14, Sections 9.2, 9.5]. The proof is exactly the same proof, butwriting FM M or FM n instead of C [ − ] and ϕ instead of vol S n − in every relevant sentence,and recalling that when M is framed, we choose ϕ such that ◦ ∨ ( ϕ ) = 1 ⊗ vol S n − .The proof that ω is a chain map is different albeit similar. We recall Stokes’ formulafor integrals along fibers of semi-algebraic bundles. If π : E → B is a semi-algebraicbundle, the fiberwise boundary π ∂ : E ∂ → B is the bundle with E ∂ := [ b ∈ B ∂π − ( b ) . Remark . The space E ∂ is neither ∂E nor S b ∈ B π − ( b ) ∩ ∂E in general. (Consider forexample the projection on the first coordinate [0 , × → [0 , d (cid:18)Z π : E → B α (cid:19) = Z π : E → B dα ± Z π ∂ : E ∂ → B α | E ∂ . If we apply this formula to compute dω (Γ), we find that the first term is: Z p U dω ′ (Γ) = Z p U ω ′ ( d R Γ + d split Γ) = ω ( d R Γ + d split Γ) , (63)since ω ′ was a chain map. It thus remain to check that the second term satisfies: Z p ∂U : FM ∂M ( U ⊔ I ) → FM M ( U ) ω ′ (Γ) = Z p U ω ′ ( d contr Γ) = ω ( d contr Γ) . The fiberwise boundary of the projection p U : FM n ( U ⊔ I ) → FM n ( U ) is rather com-plex [LV14, Section 5.7], essentially due to the quotient by the affine group in the defini-tion of FM n which lowers dimensions. We will not repeat its explicit decomposition intocells as we do not need it here. 28he fiberwise boundary of p U : FM M ( U ⊔ I ) → FM M ( U ) is simpler. Our definitionsmimick the description of [LV14, Section 5.7]. Let V = U ⊔ I . The interior of FM M ( U ) isthe space Conf U ( M ), and thus FM ∂M ( V ) is the closure of ( ∂ FM M ( V )) ∩ π − (Conf U ( M )).Let the set of “boundary faces” be given by: BF M ( V, U ) = { W ⊂ V | W ≥ W ∩ U ≤ } . This set indexes the strata of the fiberwise boundary of p U . The idea is that a configu-ration is in the fiberwise boundary iff it is obtained by an insertion map ◦ W with W ∈BF M ( V, U ). In the description of FM ∂n ( V ), similar boundary faces, denoted BF ( V, U ),appear. But there, there was an additional part which corresponds to U ⊂ W . Unlikethe case of FM n , for FM M the image of p U ( − ◦ W − ) is always included in the boundary of FM M ( U ) when U ⊂ W . We follow a pattern similar to the one used in the proof of [LV14,Proposition 5.7.1]. Lemma 64.
The subspace FM ∂M ( V ) ⊂ FM M ( V ) is equal to: [ W ∈BF M ( V,U ) im (cid:0) ◦ W : FM M ( V /W ) × FM n ( W ) → FM M ( V ) (cid:1) . Proof.
Let cls denote the closure operator. Since Conf U ( M ) is the interior of FM M ( U )and p : FM M ( V ) → FM M ( U ) is a bundle, it follows that the fiberwise boundary FM ∂M isobtained as the closure of the preimage of the interior (see the corresponding statementin the proof of [LV14, Proposition 5.7.1]), i.e.: FM ∂M ( V ) = cls (cid:16) FM ∂M ( V ) ∩ p − (Conf U ( M )) (cid:17) = cls (cid:0) ∂ FM M ( V ) ∩ p − (Conf U ( M )) (cid:1) . The boundary ∂ FM M ( V ) is the union of the subsets im( ◦ W ) for W ≥ W = V is included, unlike for FM n ). If W ∩ U ≥
2, which is equivalent to W
6∈ BF M ( V, U ), then im( p U ( − ◦ W − )) ⊂ ∂ FM M ( U ), because if a configuration belongsto this image then at least two points of U are infinitesimally close. Therefore:cls (cid:0) ∂ FM M ( V ) ∩ p − (Conf U ( M )) (cid:1) = cls (cid:18)[ W ≥ im( ◦ W ) ∩ p − (Conf U ( M )) (cid:19) = cls (cid:18)[ W ∈BF M ( V,U ) im( ◦ W ) ∩ p − (Conf U ( M )) (cid:19) = [ W ∈BF M ( V,U ) cls (cid:0) im( ◦ W ) ∩ p − (Conf U ( M )) (cid:1) = [ W ∈BF M ( V,U ) im( ◦ W ) . Lemma 65.
For a given graph Γ ∈ Tw Gra R ( U ) , the integral over the fiberwise boundaryis given by: Z p ∂U ω ′ (Γ) | FM ∂M ( V ) = ω ( d contr Γ) . roof. The maps ◦ W : FM M ( V /W ) × FM n ( W ) → FM M ( V ) are smooth injective maps andtheir domains are compact, thus they are homeomorphisms onto their images. Recallthat W ≥ W ∈ BF M ( V, U ); hence dim FM n ( W ) = n W − n −
1. The dimensionof the image of ◦ W is then:dim im( ◦ W ) = dim FM M ( V /W ) + dim FM n ( W )= n V /W ) + ( n W − n − n V − , i.e. the image is of codimension 1 in FM M ( V ). It is also easy to check that if W = W ′ ,then im( ◦ W ) ∩ im( ◦ W ′ ) is of codimension strictly bigger than 1.We now fix W ∈ BF M ( V, U ). Since W ∩ U ≤
1, the composition U ⊂ V → V /W is injective and identifies U with a subset of V /W . There is then a forgetful map p ′ U : FM M ( V /W ) → FM M ( U ). We then have a commutative diagram: FM M ( V /W ) × FM n ( W ) FM M ( V /W ) FM M ( V ) FM M ( U ) ◦ W p p ′ U p U . (66)It follows that p U ( − ◦ W − ) = p ′ U ◦ p is the composite of two semi-algebraic bundles,hence it is a semi-algebraic bundle itself [HLTV11, Proposition 8.5]. Combined with thefact about codimensions above, we can therefore apply the summation formula [HLTV11,Proposition 8.11]: Z p ∂U ω ′ (Γ) = X W ∈BF M ( V,U ) Z p U ( −◦ W − ) ω ′ (Γ) | FM M ( V/W ) × FM n ( W ) . (67)Now we can directly adapt the proof of Lambrechts and Volić. For a fixed W ,by [HLTV11, Proposition 8.13], the corresponding summand is equal to ± ω (Γ V/W ) · R FM n ( W ) ω ′ (Γ W ), where • Γ V/W ∈ Tw Gra R ( U ) is the graph with W collapsed to a vertex and U ֒ → V /W isidentified with its image; • Γ W ∈ Tw Gra n ( W ) is the full subgraph of Γ with vertices W and the labels re-moved.The vanishing lemmas in the proof of Lambrechts and Volić then imply that theintegral R FM n ( W ) ω ′ (Γ W ) is zero unless Γ W is the graph with exactly two vertices and oneedge, in which case the integral is equal to 1. In this case, Γ V/W is the graph Γ with oneedge connecting an internal vertex to some other vertex collapsed. The sum runs overall such edges, and dealing with signs carefully we see that Equation (67) is preciselyequal to ω ( d contr Γ).We can now finish proving Proposition 61. We combine Equation (63) and Lemma 65,and apply Stokes’ formula to dω (Γ) to show that it is equal to ω ( d Γ) = ω ( d R Γ+ d split Γ)+ ω ( d contr Γ). 30 .6. Reduced labeled graphs:
Graphs R The last step in the construction of
Graphs R is the reduction of Tw Gra R so that it hasthe right cohomology. We borrow the terminology of Campos–Willwacher [CW16] forthe next two definitions. Definition 68.
The full graph complex fGC R is the CDGA Tw Gra R ( ∅ ). It consistsof labeled graphs with only internal vertices, and the product is disjoint union of graphs. Remark . The adjective “full” refers to the fact that graphs are possibly disconnectedand have vertices of any valence in fGC R .As an algebra, fGC R is free and generated by connected graphs. In general we will call internal components the connected components of a graph that only contain internalvertices. The full graph complex naturally acts on Tw Gra R ( U ) by adding extra internalcomponents. Definition 70.
The partition function Z ϕ : fGC R → R is the restriction of ω :Tw Gra R → Ω ∗ PA ( FM M ) to fGC R = Tw Gra R ( ∅ ) → Ω ∗ PA ( FM M ( ∅ )) = Ω ∗ PA (pt) = R . Remark . The expression “partition function” comes from the mathematical physicsliterature, more specifically Chern–Simons invariant theory, where it refers to the parti-tion function of a quantum field theory.By the double-pushforward formula [HLTV11, Proposition 8.13] and Fubini’s theo-rem [HLTV11, Proposition 8.15], Z ϕ is an algebra morphism and ∀ γ ∈ fGC R , ∀ Γ ∈ Tw Gra R ( U ) , ω ( γ · Γ) = Z ϕ ( γ ) · ω (Γ) . (72) Definition 73.
Let R ϕ be the fGC R -module of dimension 1 induced by Z ϕ : fGC R → R .The reduced graph comodule Graphs ϕR is the tensor product: Graphs ϕR ( U ) := R ϕ ⊗ fGC R Tw Gra R ( U ) . In other words, a graph of the type Γ ⊔ γ containing an internal component γ ∈ fGC R is identified with Z ϕ ( γ ) · Γ. It is spanned by representative classes of graphs with nointernal connected component; we call such graphs reduced . The notation is meantto evoke the fact that
Graphs ϕR depends on the choice of the propagator ϕ , unlike thecollection Graphs εR that will appear in Section 4.1. Proposition 74.
The map ω : Tw Gra R ( U ) → Ω ∗ PA ( FM M ( U )) defined in Proposition 61factors through the quotient defining Graphs ϕR .If χ ( M ) = 0 , then Graphs ϕR forms a Hopf right Graphs n -comodule. If moreover M isframed, then the map ω defines a Hopf right comodule morphism.Proof. Equation (72) immediately implies that ω factors through the quotient.The vanishing lemmas shows that if Γ ∈ Tw Gra n ( U ) has internal components, then ω (Γ) vanishes by [LV14, Lemma 9.3.7], so it is straightforward to check that if χ ( M ) = 0,then Graphs ϕR becomes a Hopf right comodule over Graphs n . It is also clear that for M framed, the quotient map ω remains a Hopf right comodule morphism.31 roposition 75 ([CM10, Lemma 3]) . The propagator ϕ can be chosen such that thefollowing additional property (P4) holds: Z p : FM M (2) → FM M (1)= M p ∗ ( σ ( x )) ∧ ϕ = 0 , ∀ x ∈ R. (P4)From now on and until the end, we assume that ϕ satisfies (P4). Remark . The additional property (P5) of the paper mentioned above would be helpfulin order to get a direct morphism
Graphs ϕR → G A , because then the partition functionwould vanish on all connected graphs with at least two vertices. However we run intodifficulties when trying to adapt the proof in the setting of PA forms, mainly due to thelack of an operator d M acting on Ω ∗ PA ( M × N ) differentiating “only in the first slot”. Corollary 77.
The morphism ω vanishes on graphs containing univalent internal ver-tices.Proof. Let Γ ∈ Gra R ( U ⊔ I ) ⊂ Tw Gra R ( U ) be a graph with a univalent internal vertex u ∈ I , labeled by x , and let v be the only vertex connected to u . Let ˜Γ be the fullsubgraph of Γ on the set of vertices U ⊔ I \ { u } . Then using [HLTV11, Propositions 8.10and 8.15] (in a way similar to the end of the proof of [LV14, Lemma 9.3.8]), we find: ω (Γ) = Z FM M ( U ⊔ I ) → FM M ( U ) ω ′ (Γ)= Z FM M ( U ⊔ I ) → FM M ( U ) ω ′ (˜Γ) p ∗ uv ( ϕ ) p ∗ u ( σ ( x ))= Z FM M ( U ⊔ I \{ u } ) → FM M ( U ) ω ′ (˜Γ) ∧ p ∗ v Z FM M ( { u,v } ) → FM M ( { v } ) p ∗ uv ( ϕ ) p ∗ u ( σ ( x )) ! , which vanishes by (P4) in Proposition 75.Almost everything we have done so far works in full generality. We now prove a factwhich sets a class of manifolds apart. Proposition 78.
Assume that M is simply connected and that dim M ≥ . Then thepartition function Z ϕ vanishes on any connected graph with no bivalent vertices labeledby R and containing at least two vertices.Remark . If γ ∈ fGC R has only one vertex, labeled by x , then Z ϕ ( γ ) = R M σ ( x ) whichcan be nonzero. Proof.
Let γ ∈ fGC R be a connected graph with at least two vertices and no bivalentvertices labeled by 1 R . By Corollary 77, we can assume that all the vertices of γ are atleast bivalent. By hypothesis, if a vertex is bivalent then it is labeled by an element of R > = R ≥ .Let k = i + j be the number of vertices of γ , with i vertices that are at least trivalentand j vertices that are bivalent and labeled by R ≥ . It follows that γ has at least32 (3 i + 2 j ) edges, all of degree n −
1. Since bivalent vertices are labeled by R ≥ , theirlabels contribute by at least 2 j to the degree of γ . The (internal) vertices contribute by − kn to the degree, and the other labels have a nonnegative contribution. Thus:deg γ ≥ (cid:18) i + j (cid:19) ( n −
1) + 2 j − kn = (cid:18) k − j + j (cid:19) ( n −
1) + 2 j − kn = 12 (cid:0) k ( n − − j ( n − (cid:1) . This last number is always positive for 0 ≤ j ≤ k : it is an affine function of j , and it ispositive when j = 0 and j = k (recall that n ≥ γ ∈ fGC R must bezero for the integral defining Z ϕ ( γ ) to be the integral of a top form of FM M ( k ) and hencepossibly nonzero. But by the above computation, deg γ > ⇒ Z ϕ ( γ ) = 0. Remark . When n = 3, the manifold M is the 3-sphere S by Perelman’s proof ofthe Poincaré conjecture [Per02; Per03]. The partition function Z ϕ is conjectured to betrivial on S for a proper choice of framing, thus bypassing the need for the above degreecounting argument. See also Proposition 118.We will also need the following technical property of fGC R . Lemma 81.
The
CDGA fGC R is cofibrant.Proof. We filter fGC R by the number of edges, defining F s fGC R to be the submodule offGC R spanned by graphs of γ such that all the connected components γ have at most s edges. The differential of fGC R can only decrease ( d split and d contr ) or leave constant ( d R )the number of edges. Moreover F s fGC R is clearly stable under products (disjoint unions),hence F s fGC R is a sub- CDGA of fGC R . It is also clear that fGC R = colim s F s fGC R . Wewill prove that F fGC R is cofibrant, and that each F s fGC R ⊂ F s +1 fGC R is a cofibration.The CDGA F fGC R is the free CDGA on graphs with a single vertex labeled by R . Inother words, F fGC R = S ( R, d R ) is the free symmetric algebra on the dg-module R , andany free symmetric algebra on a dg-module is cofibrant.Let us now show that F s fGC R ⊂ F s +1 fGC R is a cofibration for any s ≥
0. We willshow that it is in fact a “relative Sullivan algebra” [FHT01, Section 14]. As a
CDGA , wehave F s +1 fGC R = ( F s fGC R ⊗ S ( V s +1 ) , d ), where V s +1 is the graded module of connectedgraphs with exactly s + 1 edges. Let us now show the Sullivan condition.Recall that R is obtained from the minimal model of M by a relative Sullivan extension,hence it is itself a Sullivan algebra [FHT01, Section 12]. In other words, R = ( S ( W ) , d )where W is increasingly and exhaustively filtered by W ( −
1) = 0 ⊂ . . . ⊂ W ( t ) ⊂ . . . ⊂ W such that d ( W ( t )) ⊂ S ( W ( t − R by defining R ( t ) := L t + ... + t r = t (cid:0) V ( t ) ⊗ . . . ⊗ V ( t r ) (cid:1) Σ r .This in turns induces an increasing and exhaustive filtration on V s +1 by submodules V s +1 ( t ) as follows. A connected graph γ ∈ V s +1 is in V s +1 ( t ) if each label x i ∈ R of avertex of γ belongs to the filtration R ( t i ) such that P t i = t . It is then immediate tocheck that d ( V s +1 ( t + 1)) ⊂ V s ⊗ S ( V s +1 ( t )). Indeed, if γ ∈ V s +1 ( t + 1), then d split γ and d contr γ ∈ V s , because both strictly decrease the number of edges. And d R γ ∈ V s +1 ( t )because the internal differential of R decreases the filtration of R .33 . From the model to forms via graphs In this section we connect G A to Ω ∗ PA ( FM M ) and we prove that the connecting morphismsare quasi-isomorphisms. We assume that M is a simply connected closed smooth mani-fold with dim M ≥ G A Proposition 82.
For each finite set U , there is a CDGA morphism ρ ′∗ : Gra R ( U ) → G A ( U ) given by ρ on the R ⊗ U factor and sending the generators e uv to ω uv on the Gra n factor. When χ ( M ) = 0 , this defines a Hopf right comodule morphism ( Gra R , Gra n ) → ( G A , e ∨ n ) . If we could find a propagator for which property (P5) held (see Remark 76), then wecould just send all graphs containing internal vertices to zero and obtain an extension
Graphs ϕR → G A . Since we cannot assume that (P5) holds, the definition of the extensionis more complex. However we still have Proposition 78, and homotopically speaking,graphs with bivalent vertices are irrelevant. Definition 83.
Let fGC R be the quotient of fGC R defined by identifying a disconnectedvertex labeled by x with the number ε A ( ρ ( x )). Lemma 84.
The subspace I ⊂ fGC R spanned by graphs with at least one univalentvertex, or at least one bivalent vertex labeled by R , or at least one label in ker( ρ : R → A ) ,is a CDGA ideal.Proof.
It is clear that I is an algebra ideal. Let us prove that it is a differential ideal. Ifone of the labels of Γ is in ker ρ , then so do all the summands of d Γ, because ker ρ is a CDGA ideal of R .If Γ contains a bivalent vertex u labeled by 1 R , then so does d R Γ. In d split Γ, splittingone of the two edges connected to u produces a univalent vertex and hence vanishes infGC R because the label is 1 R . In d contr Γ, the contraction of the two edges connected to u cancel each other.Finally let us prove that if Γ has a univalent vertex u , then d Γ lies in I . It is clearthat d R Γ ∈ I . Contracting or splitting the only edge connected to the univalent vertexcould remove the univalent vertex. Let us prove that these two summands cancel eachother up to ker ρ .It is helpful to consider the case pictured in Equation (56). Let y be the label of theunivalent vertex u , and let x be the label of the only vertex incident to u . Contractingthe edge yields a new vertex labeled by xy . Due to the definition of fGC R , splittingthe edge yields a new vertex labeled by α := P (∆ R ) ε ( ρ ( x ∆ ′′ R )) y ∆ ′ R . We thus have ρ ( α ) = ρ ( x ) · P (∆ A ) ± ε A ( ρ ( y )∆ ′′ A )∆ ′ A .It is a standard property of the diagonal class that P (∆ A ) ± ε A ( a ∆ ′′ A )∆ ′ A = a for all a ∈ A (this property is a direct consequence of the definition in Equation (36)). Appliedto a = ρ ( y ), it follows from the previous equation that ρ ( α ) = ± ρ ( xy ); examining34he signs, this summand cancels from the summand that comes from contracting theedge. Definition 85.
The algebra fGC ′ R is the quotient of fGC R by the ideal I .Note that fGC ′ R is also free as an algebra, with generators given by connected graphswith no isolated vertices, nor univalent vertices, nor bivalent vertices labeled by 1 R , andwhere the labels lie in R/ ker( ρ ) = A . Definition 86.
A circular graph is a graph in the shape of a circle and where all verticesare labeled by 1 R , i.e. graphs of the type e e . . . e ( k − k e k . Let fLoop R ⊂ fGC R bethe submodule spanned by graphs whose connected components either have univalentvertices or are equal to a circular graphs. Lemma 87.
The submodule fLoop R is a sub- CDGA of fGC R .Proof. The submodule fLoop R is stable under products (disjoint union) by definition,so we just need to check that it is stable under the differential. Thanks to the proof ofLemma 84, in fGC R , if a graph contains a univalent vertex, then so do all the summandsof its differential. On a circular graph, the internal differential of R vanish, because alllabels are equal to 1 R . Contracting an edge in a circular graph yields another circulargraph, and splitting an edge yields a graph with univalent vertices, which belongs tofLoop R . Proposition 88.
The sequence fLoop R → fGC R → fGC ′ R is a homotopy cofiber sequenceof CDGA s.Proof.
The
CDGA fGC R is freely generated by connected labeled graphs with at leasttwo vertices. It is a quasi-free extension of fLoop R by the algebra generated by graphsthat are not circular and that do not contain any univalent vertices. The homotopycofiber of the inclusion fLoop R → fGC R is this algebra fGC ′′ R , generated by graphs thatare not circular and do not contain any univalent vertices, together with a differentialinduced by the quotient fGC R / (fLoop R ).Let us note that the quotient map fGC R → fGC ′ R = fGC R /I vanishes on fLoop R ,because fLoop R is included in R . Thus we have a diagram:0 fLoop R fGC R fGC ′′ R := fGC R / fLoop R I fGC R fGC ′ R := fGC R /I ′′ R → fGC ′ R is a quasi-isomorphism. Define an in-creasing filtration on both algebras by letting F s fGC ′ R (resp. F s fGC ′′ R ) be the submodulespanned by graphs Γ such that − ≤ s . The splitting part of the differ-ential strictly decreases the filtration, so only d R and d contr remain on the first page ofthe associated spectral sequences. 35ne can then filter by the number of edges. On the first page of the spectral sequenceassociated to this new filtration, there is only the internal differential d R . Thus on thesecond page, the vertices are labeled by H ∗ ( R ) = H ∗ ( M ). The contracting part of thedifferential decreases the new filtration by exactly one, and so on the second page we seeall of d contr .We can now adapt the proof of [Wil14, Proposition 3.4] to show that on the part of thecomplex with bivalent vertices, only the circular graphs contribute to the cohomology(we work dually so we consider a quotient instead of an ideal, but the idea is the same).To adapt the proof, one must see the labels of positive degree as formally adding oneto the valence of the vertex, thus “breaking” a line of bivalent vertices. These labelsbreak the symmetry (recall the coinvariants in the definition of the twisting) that allowcohomology classes to be produced. Corollary 89.
The morphism Z ϕ : fGC R → R factors through fGC ′ R in the homotopycategory of CDGA s.Proof.
Let us show that Z ϕ is homotopic to zero when restricted to the ideal definingfGC ′ R = fGC R /I as a quotient of fGC R . Up to rescaling ε A by a real coefficient, wemay assume that ε A ρ ( − ) and R M σ ( − ) are homotopic, which induces a homotopy (byderivations) on the sub- CDGA of graphs with no edges. Hence Z ϕ is homotopic to zerowhen restricted to the ideal defining fGC R from fGC R . Moreover the map Z ϕ vanisheson graphs with univalent vertices by Corollary 77. The degree of a circular graph with k vertices is − k < R in a circular graph), but Z ϕ vanishes on graphs of nonzero degree. Hence Z ϕ vanishes on the connected graphsappearing in the definition of fLoop R . Therefore, in the homotopy category of CDGA s, Z ϕ factors through the homotopy cofiber of the inclusion fLoop R → fGC R , which isquasi-isomorphic to fGC ′ R by Proposition 88.The statement of the corollary is not concrete, as the “factorization” could go through azigzag of maps. However, the CDGA s fGC R and fGC ′ R are both cofibrant (see Lemma 81for fGC R , whose proof can easily be adapted to fGC ′ R ). Recall from Section 1.1 thefollowing definition of homotopy. Let π : fGC R → fGC ′ R be the quotient map. Recallthat A ∗ PL (∆ ) = S ( t, dt ) is a path object for the CDGA R , and ev , ev : A ∗ PL (∆ ) → R are evaluation at t = 0 and t = 1. There exists some morphism Z ′ ϕ : fGC ′ R → R andsome homotopy h : fGC R → A ∗ PL (∆ ) such that the following diagram commutes:fGC R R A ∗ PL (∆ ) R Z ϕ Z ′ ϕ πh ∼ ev ev ∼ Definition 90.
Let A ∗ PL (∆ ) h be the fGC R -module induced by h , and let Graphs ′ R ( U ) = A ∗ PL (∆ ) h ⊗ fGC R Tw Gra R ( U ) . efinition 91. Let Z ε : fGC R → R be the algebra morphism that sends a graph γ with a single vertex labeled by x ∈ R to ε A ( ρ ( x )), and that vanishes on all the otherconnected graphs. Let R ε be the one-dimensional fGC R -module induced by Z ε , and let Graphs εR ( U ) = R ε ⊗ fGC R Tw Gra R ( U ) . Explicitly, in
Graphs εR , all internal components with at least two vertices are identifiedwith zero, whereas an internal component with a single vertex labeled by x ∈ R isidentified with the number ε A ( ρ ( x )). Lemma 92.
The morphism Z ′ ϕ π is equal to Z ε .Proof. This is a rephrasing of Proposition 78. Using the same degree counting argument,all the connected graphs with more than one vertex in fGC ′ R are of positive degree. Since R is concentrated in degree zero, Z ′ ϕ π must vanish on these graphs, just like Z ε . Moreoverthe morphism π : fGC R → fGC ′ R = fGC R /I factors through fGC R , where graphs γ witha single vertex are already identified with the numbers Z ε ( γ ). Proposition 93.
For each finite set U , we have a zigzag of quasi-isomorphisms of CDGA s: Graphs εR ( U ) ∼ ←− Graphs ′ R ( U ) ∼ −→ Graphs ϕR ( U ) . If χ ( M ) = 0 , then Graphs ′ R and Graphs εR are right Hopf Graphs n -comodules, and thezigzag defines a zigzag of Hopf right comodule morphisms.Proof. We have a commutative diagram:
Graphs εR ( U ) Graphs ′ R ( U ) Graphs ϕR ( U )Tw Gra R ( U ) ⊗ fGC R R ε Tw Gra R ( U ) ⊗ fGC R A ∗ PL (∆ ) h Tw Gra R ( U ) ⊗ fGC R R ϕ = =1 ⊗ ev ⊗ ev = The fGC R -module Tw Gra R ( U ) is cofibrant. Indeed, it is quasi-free, because Tw Gra R ( U )is freely generated as a graded fGC R -module by reduced graphs. Moreover, we can adaptthe proof of Lemma 81 to filter the space of generators in an appropriate manner andshow that Tw Gra R ( U ) is cofibrant.Therefore the functor Tw Gra R ( U ) ⊗ fGC R ( − ) preserves quasi-isomorphisms. The twoevaluation maps ev , ev : A ∗ PL (∆ ) → R are quasi-isomorphisms. It follows that all themaps in the diagram are quasi-isomorphisms.If χ ( M ) = 0, the proof that Graphs ′ R and Graphs εR assemble to Graphs n -comodulesis identical to the proof for Graphs ϕR (see Proposition 74). It is also clear that thetwo zigzags define morphisms of comodules: in Graphs n , as all internal components areidentified with zero anyway. Proposition 94.
The
CDGA morphisms ρ ′∗ : Gra R ( U ) → G A ( U ) extend to CDGA mor-phisms ρ ∗ : Graphs εR ( U ) → G A ( U ) by sending all reduced graphs containing internalvertices to zero. If χ ( M ) = 0 this extension defines a Hopf right comodule morphism. roof. The submodule of reduced graphs containing internal vertices is a multiplicativeideal and a cooperadic coideal, so all we are left to prove is that ρ ∗ is compatible withdifferentials. Since ρ ′∗ was a chain map, we must only prove that if Γ is a reduced graphwith internal vertices, then ρ ∗ ( d Γ) = 0.If a summand of d Γ still contains an internal vertex, then it is mapped to zero bydefinition of ρ ∗ . So we need to look for the summands of the differential that can removeall internal vertices at once.The differential of R leaves the number of internal vertices constant, therefore if Γalready had an internal vertex, so do all the summands of d R Γ. The contracting part d contr of the differential decreases the number of internal vertices by exactly one, so letus assume that Γ has exactly one internal vertex. This vertex is at least univalent, aswe consider reduced graphs. Then there are several cases to consider, depending of thevalence of the internal vertex: • if it is univalent, then the argument of Lemma 84 shows that contracting theincident edge cancels with the splitting part of the differential; • if it is bivalent, the contracting part has two summands, and both cancel by thesymmetry relation ι u ( a ) ω uv = ι v ( a ) ω uv in Equation (38); • if it is at least trivalent, then we can use the symmetry relation ι u ( a ) ω uv = ι v ( a ) ω uv to push all the labels on a single vertex, and we see that the sum of graphs thatappear is obtained by the Arnold relation (see Figure 29 for an example in the caseof Graphs n → e ∨ n ).Finally, the splitting part of the differential leaves the number of internal verticesconstant, unless it splits off a whole connected component with only internal vertices, inwhich case the component is evaluated using the partition function Z ε . If that connectedcomponent consists of a single internal vertex, then we saw in the previous item thatsplitting the edge connecting this univalent vertex to the rest of the graph cancels withthe contraction of that edge. Otherwise, if the graph has more than one vertex, then bydefinition Z ε vanishes on that graph. In this section we prove that the morphisms constructed in Proposition 74 and Proposi-tion 94 are quasi-isomorphisms, completing the proof of Theorem 3.Let us recall our hypotheses and constructions. Let M be a simply connected closedsmooth manifold of dimension at least 4. We endow M with a semi-algebraic structure(Section 1.3) and we consider the CDGA Ω ∗ PA ( M ) of PA forms on M , which is a modelfor the real homotopy type of M . Recall that we fix a zigzag of quasi-isomorphisms of CDGA s A ρ ←− R σ −→ Ω ∗ PA ( M ), where A is a Poincaré duality CDGA (Theorem 35), and σ factors through the quasi-isomorphic sub- CDGA of trivial forms.Recall that ϕ ∈ Ω n − ( FM M (2)) is an (anti-)symmetric trivial form on the compacti-fication of the configuration space of two points in M , whose restriction to the spherebundle ∂ FM M (2) is a global angular form, and whose differential dϕ is a representative38f the diagonal class of M (Proposition 53). Recall that we defined the graph complex Graphs ϕR ( U ) using reduced labeled graphs with internal and external vertices (Defini-tion 73) and a partition function built from ϕ (Definition 70). We also defined thevariants Graphs εR and Graphs ′ R (Definitions 90 and 91). Theorem 95 (Precise version of Theorem 3) . Let M be a simply connected closed smoothmanifold of dimension at least . Using the notation recalled above, the following zigzag,where the maps were constructed in Proposition 74, Proposition 93, and Proposition 94,is a zigzag of quasi-isomorphisms of Z -graded CDGA s for all finite sets U : G A ( U ) ∼ ←− Graphs εR ( U ) ∼ ←− Graphs ′ R ( U ) ∼ −→ Graphs ϕR ( U ) ∼ −→ Ω ∗ PA ( FM M ( U )) . If χ ( M ) = 0 , then the left-pointing maps form a quasi-isomorphism of Hopf rightcomodules: ( G A , e ∨ n ) ∼ ←− ( Graphs εR , Graphs n ) ∼ ←− ( Graphs ′ R , Graphs n ) . If moreover M is framed, then the right-pointing maps also form a quasi-isomorphismof Hopf right comodules: ( Graphs ′ R , Graphs n ) ∼ −→ ( Graphs ϕR , Graphs n ) ∼ −→ (Ω ∗ PA ( FM M ) , Ω ∗ PA ( FM n )) . The rest of the section is dedicated to the proof of this theorem. Let us give a roadmapof this proof. We first prove that
Graphs εR ( U ) → G A ( U ) is a quasi-isomorphism by an in-ductive argument on U (Proposition 97). This involves setting up a spectral sequenceso that we can reduce the argument to connected graphs. Then we use explicit homo-topies in order to show that both complexes have cohomology of the same dimension,and we show that the morphism is surjective on cohomology by describing a section byexplicit arguments. Then we prove that Graphs ϕR ( U ) → Ω ∗ PA ( FM M ( U )) is surjective oncohomology explicitly (Proposition 112). Since we know that G A ( U ) and FM M ( U ) havethe same cohomology by the theorem of Lambrechts–Stanley [LS08a, Theorem 10.1],this completes the proof that all the maps are quasi-isomorphisms. Compatibility withthe various comodules structures was already shown in Section 3. Lemma 96.
The morphisms
Graphs εR ( U ) → G A ( U ) factor through quasi-isomorphisms Graphs εR ( U ) → Graphs εA ( U ) , where Graphs εA ( U ) is the CDGA obtained by modding graphswith a label in ker( ρ : R → A ) in Graphs εR ( U ) .Proof. The morphism
Graphs εR → Graphs εA simply applies the surjective map ρ : R → A to all the labels. Hence Graphs εR → G A factors through the quotient.We can consider the spectral sequences associated to the filtrations of both Graphs εR and Graphs εA by the number of edges, and we obtain a morphism E Graphs εR → E Graphs εA .On both E pages, only the internal differentials coming from R and A remain. The chainmap R → A is a quasi-isomorphism; hence we obtain an isomorphism on the E page.By standard spectral sequence arguments, it follows that Graphs εR → Graphs εA is aquasi-isomorphism. 39he CDGA
Graphs εA ( U ) has the same graphical description as the CDGA
Graphs εR ( U ),except that now vertices are labeled by elements of A . An internal component with asingle vertex labeled by a ∈ A is identified with ε ( a ), and an internal component withmore than one vertex is identified with zero. Proposition 97.
The morphism
Graphs εA → G A is a quasi-isomorphism. Before starting to prove this proposition, let us outline the different steps. We filter ourcomplex in such a way that on the E page, only the contracting part of the differentialremains (such a technique was already used in the proof of Proposition 88). Using asplitting result, we can focus on connected graphs. Finally, we use a “trick” (Figure 109)for moving labels around in a connected component, reducing ourselves to the case whereonly one vertex is labeled. We then get a chain map A ⊗ Graphs n → A ⊗ e ∨ n ( U ), whichis a quasi-isomorphism thanks to the formality theorem.Let us start with the first part of the outlined program, removing the splitting part ofthe differential from the picture. We now define an increasing filtration on Graphs εA . Thesubmodule F s Graphs εA is spanned by reduced graphs such that − ≤ s . Lemma 98.
The above submodules define a filtration of
Graphs εA by subcomplexes, satis-fying F − U − Graphs εA ( U ) = 0 for each finite set U . The E page of the spectral sequenceassociated to this filtration is isomorphic as a module to Graphs εA . Under this isomor-phism the differential d is equal to d A + d ′ contr , where d A is the internal differentialcoming from A and d ′ contr is the part of the differential that contracts all edges but edgesconnected to a univalent internal vertex.Proof. Let Γ be an internally connected (Definition 30) reduced graph. If Γ ∈ Graphs εA ( U )is the graph with no edges and no internal vertices, then it lives in filtration level − U .Adding edges can only increase the filtration. Since we consider reduced graphs (i.e. nointernal components), each time we add an internal vertex (decreasing the filtration) wemust add at least one edge (bringing it back up). By induction on the number of internalvertices, each graph is of filtration at least − U .Let us now prove that the differential preserves the filtration and check which partsremain on the associated graded complex. The internal differential d A does not changeeither the number of edges nor the number of vertices and so keeps the filtration constant.The contracting part d contr of the differential decreases both by exactly one, and so keepsthe filtration constant too.The splitting part d split of the differential removes one edge. If the resulting graph isstill connected, then nothing else changes and the filtration is decreased exactly by 1. Ifthe resulting graph is not connected, then we get an internal component γ which wasconnected to the rest of the graph by a single edge, and was then split off and identifiedwith a number in the process. If γ has a single vertex labeled by a (i.e. we split an edgeconnected to a univalent vertex), then this number is ε ( a ), and the filtration is keptconstant. Otherwise, the summand is zero (and so the filtration is obviously preserved).In all cases, the differential preserves the filtration, and so we get a filtered chaincomplex. On the associated graded complex, the only remaining parts of the differential40re d A , d contr , and the part that splits off edges connected to univalent vertices. But bythe proof of Proposition 94 this last part cancels out with the part that contracts theseedges connected to univalent vertices.The symmetric algebra S ( ω uv ) u = v ∈ U has a weight grading by the word-length on thegenerators ω uv . This induces a weight grading on e ∨ n ( U ), because the ideal definingthe relations is compatible with the weight grading. This grading in turn induces anincreasing filtration F ′ s G A on G A (the extra differential strictly decreases the weight).Define a shifted filtration on G A by: F s G A ( U ) := F ′ s + U G A ( U ) . Lemma 99.
The E page of the spectral sequence associated to F ∗ G A is isomorphic as amodule to G A . Under this isomorphism the d differential is just the internal differentialof A . Lemma 100.
The morphism
Graphs εA → G A preserves the filtration and induces achain map E Graphs εA ( U ) → E G A ( U ) , for each U . It maps reduced graphs with internalvertices to zero, an edge e uv between external vertices to ω uv , and a label a of an externalvertex u to ι u ( a ) .Proof. The morphism
Graphs εA ( U ) → G A ( U ) preserves the filtration by construction.If a graph has internal vertices, then its image in G A ( U ) is of strictly lower filtrationunless the graph is a forest (i.e. a product of trees). But trees have leaves, therefore byCorollary 77 and the formula defining Graphs εA → G A they are mapped to zero in G A ( U )anyway. It is clear that the rest of the morphism preserves filtrations exactly, and so isgiven on the associated graded complex as stated in the lemma.We now use arguments similar to [LV14, Lemma 8.3]. For a partition π of U , definethe submodule Graphs εA h π i ⊂ E Graphs εA ( U ) spanned by reduced graphs Γ such thatthe partition of U induced by the connected components of Γ is exactly π . In particularlet Graphs εA h{ U }i be the submodule of connected graphs, where { U } is the indiscretepartition of U consisting of a single element. Lemma 101.
For each partition π of U , Graphs εA h π i is a subcomplex of E Graphs εA ( U ) ,and E Graphs εA ( U ) splits as the sum over all partitions π : E Graphs εA ( U ) = M π O V ∈ π Graphs εA h{ V }i . Proof.
Since there is no longer any part of the differential that can split off connectedcomponents in E Graphs εA , it is clear that Graphs εA h{ U }i is a subcomplex. The splittingresult is immediate.The complex E G A ( U ) splits in a similar fashion. For a monomial in S ( ω uv ) u = v ∈ U ,say that u and v are “connected” if the term ω uv appears in the monomial. Considerthe equivalence relation generated by “ u and v are connected”. The monomial induces41n this way a partition π of U , and this definition factors through the quotient defining e ∨ n ( U ) (draw a picture of the 3-term relation). Finally, for a given monomial in G A ( U ),the induced partition of U is still well-defined.Thus for a given partition π of U , we can define e ∨ n h π i and G A h π i to be the submodulesof e ∨ n ( U ) and E G A ( U ) spanned by monomials inducing the partition π . It is a standardfact that e ∨ n h{ U }i = Lie ∨ n ( U ), see [Sin07]. The proof of the following lemma is similarto the proof of the previous lemma: Lemma 102.
For each partition π of U , G A h π i is a subcomplex of E G A ( U ) , and E G A ( U ) splits as the sum over all partitions π of U : E G A ( U ) = M π O V ∈ π G A h{ V }i . Lemma 103.
The map E Graphs εA ( U ) → E G A ( U ) preserves the splitting. We can now focus on connected graphs to prove Proposition 97.
Lemma 104.
The complex G A h{ U }i is isomorphic to A ⊗ e ∨ n h{ U }i .Proof. We define explicit isomorphisms in both directions.Define A ⊗ U ⊗ e ∨ n h{ U }i → A ⊗ e ∨ n h{ U }i using the multiplication of A . This construc-tions induces a map on the quotient E G A ( U ) → A ⊗ e ∨ n h{ U }i , which restricts to a map G A h{ U }i → e ∨ n h{ U }i . Since d A is a derivation, this is a chain map.Conversely, define A ⊗ e ∨ n h{ U }i → A ⊗ U ⊗ e ∨ n h{ U }i by a ⊗ x ι u ( a ) ⊗ x for somefixed u ∈ U (it does not matter which one since x ∈ e ∨ n h{ U }i is “connected”). Thisconstruction gives a map A ⊗ e ∨ n h{ U }i → G A h{ U }i , and it is straightforward to checkthat this map is the inverse isomorphism of the previous map.We have a commutative diagram of complexes: Graphs εA h{ U }i A ⊗ Graphs ′ n h{ U }i G A h{ U }i A ⊗ e ∨ n h{ U }i ∼∼ = Here
Graphs ′ n ( U ) is defined similarly to Graphs n ( U ) except that multiple edges areallowed. It is known that the quotient map Graphs ′ n ( U ) → e ∨ n ( U ) (which factorsthrough Graphs n ( U )) is a quasi-isomorphism [Wil14, Proposition 3.9]. The subcom-plex Graphs ′ n h{ U }i is spanned by connected graphs. The upper horizontal map in thediagram multiplies all the labels of a graph.The right vertical map is the tensor product of id A and Graphs n h{ U }i ∼ −→ e ∨ n h{ U }i (see 1.6). The bottom row is the isomorphism of the previous lemma.It then remains to prove that Graphs εA h{ U }i → A ⊗ Graphs ′ n h{ U }i is a quasi-isomorphismto prove Proposition 97. If U = ∅ , then Graphs ′ A ( ∅ ) = R = G A ( ∅ ) and the morphismis the identity, so there is nothing to do. From now on we assume that U ≥ emma 105. The morphism
Graphs εA h{ U }i → A ⊗ Graphs ′ n h{ U }i is surjective oncohomology.Proof. Choose some u ∈ U . There is an explicit chain-level section of the morphism,sending x ⊗ Γ to Γ u,x , the same graph with the vertex u labeled by x and all the othervertices labeled by 1 R . It is a well-defined chain map, which is clearly a section of themorphism in the lemma, hence the morphism of the lemma is surjective on cohomology.We now use a proof technique similar to the proof of [LV14, Lemma 8.3], working byinduction. The dimension of H ∗ ( Graphs ′ n h{ U }i ) = e ∨ n h{ U }i = Lie ∨ n ( U ) is well-known:dim H i ( Graphs ′ n h{ U }i ) = ( ( U − , if i = ( n − U − , otherwise. (106) Lemma 107.
For all sets U with U ≥ , the dimension of H i ( Graphs εA h{ U }i ) is thesame as the dimension: dim H i ( A ⊗ Graphs ′ n h{ U }i ) = ( U − · dim H i − ( n − U − ( A ) . The proof will be by induction on the cardinality of U . Before proving this lemma,we will need two additional sub-lemmas. Lemma 108.
The complex
Graphs εA h i has the same cohomology as A .Proof. Let I be the subcomplex spanned by graphs with at least one internal vertex.We will show that I is acyclic; as Graphs εA h i / I ∼ = A , this will prove the lemma.There is an explicit homotopy h that shows that I is acyclic. Given a graph Γ with asingle external vertex and at least one internal vertex, define h (Γ) to be the same graphwith the external vertex replaced by an internal vertex, a new external vertex labeledby 1 A , and an edge connecting the external vertex to the new internal vertex: u x h u A x The differential in
Graphs εA h i only retains the internal differential of A and the con-tracting part of the differential. Contracting the new edge in h (Γ) gives Γ back, and itis now straightforward to check that dh (Γ) = Γ ± h ( d Γ).Now let U be a set with at least two elements, and fix some element u ∈ U . Let Graphs uA h{ U }i ⊂ Graphs εA h{ U }i be the subcomplex spanned by graphs Γ such that u has valence 1, is labeled by 1 A , and is connected to another external vertex.We now get to the core of the proof of Lemma 107. The idea (adapted from [LV14,Lemma 8.3]) is to “push” the labels of positive degree away from the chosen vertex u through a homotopy. Roughly speaking, we use Figure 109 to move labels around up tohomotopy. 43 contr (cid:18) x (cid:19) = x − x Figure 109: Trick for moving labels around (gray vertices are either internal or external)
Lemma 110.
The inclusion
Graphs uA h{ U }i ⊂ Graphs εA h{ U }i is a quasi-isomorphism.Proof. Let Q be the quotient. We will prove that it is acyclic. The module Q furtherdecomposes into a direct sum of modules (but the differential does not preserve thedirect sum): • The module Q spanned by graphs where u is of valence 1, labeled by 1 A , andconnected to an internal vertex; • The module Q spanned by graphs where u is of valence ≥ A > .We now filter Q as follows. For s ∈ Z , let F s Q be the submodule of Q spanned bygraphs with at most s + 1 edges, and let F s Q be the submodule spanned by graphswith at most s edges. This filtration is preserved by the differential of Q .Consider the E page of the spectral sequence associated to this filtration. Then thedifferential d is a morphism E Q → E Q (count the number of edges and use thecrucial fact that edges connected to univalent vertices are not contractible when lookingat reduced graphs). This differential contracts the only edge incident to u . It is anisomorphism, with an inverse similar to the homotopy defined in Lemma 108, “blowingup” the point u into a new edge connecting u to a new internal vertex that replaces u .This shows that ( E Q , d ) is acyclic, hence E Q = 0. It follows that Q itself isacyclic. Proof of Lemma 107.
The case U = 0 is obvious, and the case U = 1 of the lemmawas covered in Lemma 108. We now work by induction and assume the claim proved for U ≤ k , for some k ≥ U be of cardinality k + 1. Choose some u ∈ U and define Graphs uA h{ U }i as before.By Lemma 110 we only need to show that this complex has the right cohomology. Itsplits as: Graphs uA h{ U }i ∼ = M v ∈ U \{ u } e uv · Graphs εA h{ U \ { u }}i , (111)and therefore using the induction hypothesis:dim H i ( Graphs uA h{ U }i ) = k · dim H i − ( n − ( Graphs εA h{ U \ { u }}i )= k ! · dim H i − k ( n − ( A ) . Proof of Proposition 97.
By Lemma 105, the morphism induced by
Graphs εA → G A onthe E page is surjective on cohomology. By Lemma 107 and Equation (106), both E pages have the same cohomology, and so the induced morphism is a quasi-isomorphism.Standard spectral arguments imply the proposition.44 roposition 112. The morphism ω : Graphs ′ R ( U ) → Ω ∗ PA ( FM M ( U )) is a quasi-isomor-phism.Proof. By Equation (39), Proposition 93, Lemma 96, and Proposition 97, both
CDGA shave the same cohomology of finite type, so it will suffice to show that the map issurjective on cohomology to prove that it is a quasi-isomorphism.We work by induction. The case U = ∅ is immediate, as Graphs ′ R ( ∅ ) ∼ −→ Graphs ϕR ( ∅ ) =Ω ∗ PA ( FM M ( ∅ )) = R and the last map is the identity.Suppose that U = { u } is a singleton. Since ρ is a quasi-isomorphism, for every co-cycle α ∈ Ω ∗ PA ( FM M ( U )) = Ω ∗ PA ( M ) there is some cocycle x ∈ R such that ρ ( x ) iscohomologous to α . Then the graph γ x with a single (external) vertex labeled by x is a cocycle in Graphs ′ R ( U ), and ω ( γ x ) = ρ ( x ) is cohomologous to α . This provesthat Graphs ′ R ( { u } ) → Ω ∗ PA ( M ) is surjective on cohomology, and hence is a quasi-isomorphism.Now assume that U = { u } ⊔ V , where V ≥
1, and assume that the claim is provenfor sets of vertices of size at most V = U −
1. By Equation (39), we may representany cohomology class of FM M ( U ) by an element z ∈ G A ( U ) satisfying dz = 0. Using therelations defining G A ( U ), we may write z as z = z ′ + X v ∈ V ω uv z v , where z ′ ∈ A ⊗ G A ( V ) and z v ∈ G A ( V ). The relation dz = 0 is equivalent to dz ′ + X v ∈ V ( p u × p v ) ∗ (∆ A ) · z v = 0 , (113)and dz v = 0 for all v. (114)By the induction hypothesis, for all v ∈ V there exists a cocycle γ v ∈ Graphs ′ R ( V )such that ω ( γ v ) represents the cohomology class of the cocycle z v in H ∗ ( FM M ( V )), andsuch that σ ∗ ( γ v ) is equal to z v up to a coboundary.By Equation (113), the cocycle˜ γ = X v ∈ V ( p u × p v ) ∗ (∆ R ) · γ v ∈ R ⊗ Graphs ′ R ( V )is mapped to a coboundary in A ⊗ G A ( V ). The map σ ∗ : R ⊗ Graphs ′ R ( V ) → A ⊗ G A ( V )is a quasi-isomorphism, hence ˜ γ = d ˜ γ is a coboundary too.It follows that z ′ − σ ∗ (˜ γ ) ∈ A ⊗ G A ( V ) is a cocycle. Thus by the induction hypothesisthere exists some ˜ γ ∈ R ⊗ Graphs ′ R ( V ) whose cohomology class represents the samecohomology class as z ′ − σ ∗ (˜ γ ) in H ∗ ( A ⊗ G A ( V )) = H ∗ ( M × FM M ( V )).We now let γ ′ = − ˜ γ + ˜ γ , hence dγ ′ = − ˜ γ + 0 = − ˜ γ and σ ∗ ( γ ′ ) is equal to z ′ up toa coboundary. By abuse of notation we still let γ ′ be the image of γ ′ under the obviousmap R ⊗ Graphs ′ R ( V ) → Graphs ′ R ( U ), x ⊗ Γ ι u ( x ) · Γ. Then γ = γ ′ + X v ∈ V e uv · γ v ∈ Graphs ′ R ( U )45s a cocycle, and ω ( γ ) represents the cohomology class of z in Ω ∗ PA ( FM M ( U )). We haveshown that the morphism Graphs ′ R ( U ) → Ω ∗ PA ( FM M ( U )) is surjective on cohomology,and hence it is a quasi-isomorphism. Proof of Theorem 95.
The zigzag of the theorem becomes, after factorizing the first mapthrough
Graphs εA : G A ( U ) ← Graphs εA ( U ) ← Graphs εR ( U ) ← Graphs ′ R ( U ) → Graphs ϕR ( U ) → Ω ∗ PA ( FM M ( U ))All these maps are quasi-isomorphisms by Lemma 96, Proposition 93, Proposition 97,and Proposition112. Their compatibility with the comodule structures (under the rele-vant hypotheses) are due to Proposition 74, Proposition 93, and Proposition 94.The last thing we need to check is the following proposition, which shows that thatwe can choose any Poincaré duality model. Proposition 115. If A and A ′ are two quasi-isomorphic simply connected Poincaréduality CDGA s, then there is a weak equivalence of symmetric collections G A ≃ G A ′ . Ifmoreover χ ( A ) = 0 then this weak equivalence is a weak equivalence of right Hopf e ∨ n -comodules.Proof. The
CDGA s A and A ′ are quasi-isomorphic, hence there exists some cofibrant S and quasi-isomorphisms f : S ∼ −→ A and f ′ : S ∼ −→ A ′ . This yields two chain maps ε = ε A ◦ f, ε ′ = ε A ′ ◦ f ′ : S → R [ − n ]. Mimicking the proof of Proposition 37, we canalso find (anti-)symmetric cocycles ∆ , ∆ ′ ∈ S ⊗ S and such that ( f ⊗ f )∆ = ∆ A and( f ′ ⊗ f ′ )∆ ′ = ∆ A ′ .We can then build symmetric collections Graphs ε, ∆ S and a quasi-isomorphism f ∗ : Graphs ε, ∆ S → G A similarly to Section 3. The differential of an edge e uv in Graphs ε, ∆ S is ι uv (∆), and an isolated internal vertex labeled by x ∈ S is identified with ε ( x ). Inparallel, we can build f ′∗ : Graphs ε ′ , ∆ ′ S ∼ −→ G A ′ .If moreover χ ( A ) = 0, then we can choose ∆ , ∆ ′ such that both graph complexesbecome right Hopf Graphs n -comodules, and f ∗ , f ′∗ are compatible with the comodulestructure. It thus suffices to find a quasi-isomorphism Graphs ε, ∆ S ≃ Graphs ε ′ , ∆ ′ S to provethe proposition.We first have an isomorphism Graphs ε ′ , ∆ ′ S ∼ = Graphs ε ′ , ∆ S (with the obvious notation).Indeed, the two cocycles ∆ and ∆ ′ are cohomologous, say ∆ ′ − ∆ = dα for some α ∈ S ⊗ S of degree n −
1. If we replace α by ( α + ( − n α ) /
2, then we can assume that α = ( − n α . Moreover if χ ( A ) = 0, then we can replace α by α − ( µ S ( α ) ⊗ − n ⊗ µ S ( α )) / µ S ( α ) = 0. We then obtain an isomorphism by mapping anedge e uv to e uv ± ι uv ( α ) (the sign depending on the direction of the isomorphism). Thismap is compatible with differentials, with products, and with the comodule structuresif χ ( A ) = 0.The dg-module S is cofibrant and R [ − n ] is fibrant (like all dg-modules). We can as-sume that ε and ε ′ induce the same map on cohomology (it suffices to rescale one map, say ε ′ , and there is an automorphism of Graphs ε ′ , ∆ S which takes care of this rescaling). Thus46he two maps ε, ε ′ : S → R [ − n ] are homotopic, i.e. there exists some h : S [1] → R [ − n ]such that ε ( x ) − ε ′ ( x ) = h ( dx ) for all x ∈ S . This homotopy induces a homotopy betweenthe two morphisms Z ε , Z ε ′ : fGC S → R . Because Tw Gra ∆ S ( U ) and Tw Gra ∆ ′ S ( U ) are cofi-brant as modules over fGC S , we obtain quasi-isomorphisms Graphs ε, ∆ S ≃ Graphs ε ′ , ∆ S (compare with Proposition 93). Corollary 116.
Let M be a smooth simply connected closed manifold and A be anyPoincaré duality model of M . Then G A ( k ) is a real model for Conf k ( M ) .Proof. The corollary follows from Theorem 95 in the case where dim M ≥ A in our constructions). Note that the graph complexes are, in general, nonzero evenin negative degrees, but by Proposition 4 this does not change the result. In dimensionat most 3, the only examples of simply connected closed manifolds are S and S . Weaddress these examples in Section 4.3. Corollary 117.
The real homotopy types of the configuration spaces of a smooth simplyconnected closed manifold only depends on the real homotopy type of the manifold.Proof.
When dim M ≥
3, the Fadell–Neuwirth fibrations [FN62] Conf k − ( M \ ∗ ) ֒ → Conf k ( M ) → M show by induction that if M is simply connected, then so is Conf k ( M )for all k ≥
1. Hence the real model G A ( k ) completely encodes the real homotopy typeof Conf k ( M ). The degree-counting argument of Proposition 78 does not work in dimension less than 4,so we have to use other means to prove that the Lambrechts–Stanley
CDGA s are modelsfor the configuration spaces.There are no simply connected closed manifolds of dimension 1. In dimension 2, theonly simply connected closed manifold is the 2-sphere, S . This manifold is a complexprojective variety: S = CP . Hence the result of Kriz [Kri94] shows that G H ∗ ( S ) ( k )(denoted E ( k ) there) is a rational model for Conf k ( S ). The 2-sphere S is studied ingreater detail in Section 6, where we study the action of the framed little 2-disks operadon a framed version of FM S .In dimension 3, the only simply connected smooth closed manifold is the 3-sphere byPerelman’s proof of the Poincaré conjecture [Per02; Per03]. we also the following partialresult, communicated to us by Thomas Willwacher: Proposition 118.
The
CDGA G A ( k ) , where A = H ∗ ( S ; Q ) , is a rational model of Conf k ( S ) for all k ≥ .Proof. The claim is clear for k = 0. Since S is a Lie group, the Fadell–Neuwirthfibration is trivial [FN62, Theorem 4]:Conf k ( R ) ֒ → Conf k +1 ( S ) → S k +1 ( S ) is thus identified with S × Conf k ( R ), which is rationally formalwith cohomology H ∗ ( S ) ⊗ e ∨ ( k ). It thus suffices to build a quasi-isomorphism between G A ( k + 1) and H ∗ ( S ) ⊗ e ∨ n ( k ).To simplify notation, we consider G A ( k + ) (where k + = { , . . . , k } ), which is obviouslyisomorphic to G A ( k + 1). Let us denote by υ ∈ H ( S ) = A the volume form of S , andrecall that the diagonal class ∆ A is given by 1 ⊗ υ − υ ⊗
1. We have an explicit map f : H ∗ ( S ) → e ∨ ( k ) given on generators by f ( ν ⊗
1) = ι ( ν ) and f (1 ⊗ ω ij ) = ω ij + ω i − ω j .The Arnold relations show that this is a well-defined algebra morphism. Let us provethat d ◦ f = 0 on the generator ω ij (the vanishing on υ ⊗ k = 2 and ( i, j ) = (1 , ι ij to get the general case. Then we have:( d ◦ f )( ω ) = (1 ⊗ ⊗ υ − ⊗ υ ⊗
1) + (1 ⊗ υ ⊗ − υ ⊗ ⊗ − (1 ⊗ ⊗ υ − υ ⊗ ⊗
1) = 0We know that both
CDGA s have the same cohomology, so to check that f is a quasi-isomorphism it suffices to check that it is surjective in cohomology. The cohomology H ∗ ( G A ( k + )) ∼ = H ∗ ( S ) ⊗ e ∨ ( k ) is generated in degrees 2 (by the ω ij ’s) and 3 (by the ι i ( υ )’s), so it suffices to check surjectivity in these degrees.In degree 3, the cocycle υ ⊗ H ( G A ( k + )) ∼ = H ( S ) = Q .Indeed, assume ι ( υ ) = dω , where ω is a linear combination of the ω ij for degree reasons.In dω , the sum of the coefficients of each ι i ( υ ) is zero, because they all come in pairs( dω ij = ι j ( υ ) − ι i ( υ )). We want the coefficient of ι ( υ ) to be 1, so at least one of theother coefficient must be nonzero to compensate, hence dω = ι ( υ ).It remains to prove that H ( f ) is surjective. We consider the quotient map p : G A ( k + ) → e ∨ ( k ) that maps ι i ( υ ) and ω i to zero for all 1 ≤ i ≤ k . We also con-sider the quotient map q : H ∗ ( S ) ⊗ e ∨ ( k ) → e ∨ ( k ) sending υ ⊗ q H ∗ ( S ) ⊗ e ∨ ( k ) e ∨ ( k ) 00 ker p G A ( k ) e ∨ ( k ) 0 qf = p We consider part of the long exact sequence in cohomology induced by these shortexact sequences of complexes: e ∨ ( k ) H (ker q ) H ( H ∗ ( S ) ⊗ e ∨ ( k )) = e ∨ ( k ) e ∨ ( k ) e ∨ ( k ) H (ker p ) H ( G A ( k + )) e ∨ ( k )
2= (1) H ( f ) = For degree reasons, H (ker q ) = 0 and so the map (1) is injective. By the four lemma,it follows that H ( f ) is injective. Since both domain and codomain have the same finitedimension, it follows that H ( f ) is an isomorphism.48 . Factorization homology of universal enveloping E n -algebras The manifold R n is framed. Let U be a finite set and consider the space of framedembeddings (i.e. such that the differential at each point preserves the given trivializationsof the tangent bundles) of U copies of R n in itself, with the compact open topology: Disk fr n ( U ) := Emb fr ( R n × U, R n ) ⊂ Map( R n × U, R n ) . (119)Using composition of embeddings, these spaces assemble to form a topological operad Disk fr n . This operad is weakly equivalent to the operad of little n -disks [AF15, Remark2.10], and the application that takes f ∈ Disk fr n ( U ) to { f (0 × u ) } u ∈ U ∈ Conf U ( R n ) is ahomotopy equivalence.Similarly if M is a framed manifold, then the spaces Emb fr ( R n × − , M ) assemble toform a topological right Disk fr n -module, again given by composition of embeddings. Wecall it Disk fr M . If B is a Disk fr n -algebra, factorization homology is given by a derivedcomposition product [AF15, Definition 3.2]: Z M B := Disk fr M ◦ L Disk fr n B := hocoeq (cid:0) Disk fr M ◦ Disk fr n ◦ B ⇒ Disk fr M ◦ B (cid:1) . (120)Using [Tur13, Section 2], the pair ( FM M , FM n ) is weakly equivalent to the pair ( Disk fr M , Disk fr n ).So if B is an FM n -algebra, then its factorization homology is: Z M B ≃ FM M ◦ L FM n B := hocoeq (cid:0) FM M ◦ FM n ◦ B ⇒ FM M ◦ B (cid:1) . (121)We now work in the category of chain complexes over R . We use the formality theorem(Section 1.6) and the fact that weak equivalences of operads induce Quillen equivalencebetween categories of right modules (resp. categories of algebras) by [Fre09, Theorems16.A, 16.B]. Thus, to any homotopy class [ B ] of E n -algebras in the category of chaincomplexes, there corresponds a homotopy class [ ˜ B ] of e n -algebras (which is generallynot easy to describe).Using Theorem 95, a game of adjunctions [Fre09, Theorems 15.1.A and 15.2.A] showsthat: Z M B ≃ G ∨ A ◦ L e n ˜ B, (122)where A is the Poincaré duality model of M mentioned in the theorem, and G ∨ A is theright e n -module dual to G A viewed as a chain complex. Knudsen [Knu16, Theorem A] considers a higher enveloping algebra functor U n fromhomotopy Lie algebras to nonunital E n -algebras. This functor generalizes the standardenveloping algebra functor from the category of Lie algebras to the category of associativealgebras. 49et n be at least 2. We can again use Kontsevich’s theorem on the formality of thelittle disks operads to identity the category of non-unital E n -algebras with the categoryof e n -algebras in homotopy. We also use that a homotopy Lie algebra is equivalent, inhomotopy, to an ordinary Lie algebra. Then we get that Knudsen’s higher envelopingalgebra functor is equivalent to the left adjoint of the obvious forgetful functor e n -Alg → Lie -Alg, which maps an n -Poisson algebra B to its underlying shifted Lie algebra B [1 − n ].This model ˜ U n : Lie -Alg → e n -Alg maps a Lie algebra g to the n -Poisson algebra givenby ˜ U n ( g ) = S ( g [ n − g is a Lie algebra, then so is A ⊗ g for any CDGA A .Then the factorization homology of U n ( g ) on M is given by: Z M U n ( g ) ≃ C CE ∗ ( A −∗ PL ( M ) ⊗ g ) (123)where C CE ∗ is the Chevalley–Eilenberg complex and A −∗ PL ( M ) is the CDGA of rationalpiecewise polynomial differential forms, with the usual grading reversed.
Proposition 124.
Let A be a Poincaré duality CDGA . Then we have a quasi-isomorphismof chain complexes: G ∨ A ◦ L e n S ( g [ n − ∼ −→ C CE ∗ ( A −∗ ⊗ g ) . If A is a Poincaré duality model of M , we have A ≃ Ω ∗ PA ( M ) ≃ A ∗ PL ( M ) ⊗ Q R [HLTV11,Theorem 6.1]. It follows that the Chevalley–Eilenberg complex of the previous propo-sition is weakly equivalent to the Chevalley–Eilenberg complex of Equation (123). ByEquation (121), the derived circle product over e n computes the factorization homologyof U n ( g ) on M , and so we recover Knudsen’s theorem (over the reals) for closed framedsimply connected manifolds.Let I be the unit of the composition product, defined by I (1) = R and I ( U ) = 0 for U = 1. Let Λ be the suspension of operads, satisfyingΛ P ◦ ( X [ − P ◦ X )[ −
1] = I [ − ◦ ( P ◦ X ) . As as symmetric collection, Λ P is simply given by Λ P = I [ − ◦ P ◦ I [1]. Recall that welet Lie n = Λ − n Lie . The symmetric collection L n := Lie ◦ I [1 − n ] = I [1 − n ] ◦ Lie n (125)is a ( Lie , Lie n )-bimodule, i.e. a Lie -algebra in the category of
Lie n -right modules. Wehave L n ( U ) = ( Lie n ( U ))[1 − n ]. This bimodule satisfies, for any Lie algebra g , L n ◦ Lie n g [ n − ∼ = g as Lie algebras. (126)We can view the CDGA A −∗ as a symmetric collection concentrated in arity 0, andas such it is a commutative algebra in the category of symmetric collections. Thus thetensor product A −∗ ⊗ L n = { A −∗ ⊗ L n ( k ) } k ≥ Lie -algebra in right
Lie n -modules, where the right Lie n -module structurecomes from L n and the Lie algebra structure combines the Lie algebra structure of L n and the CDGA structure of A −∗ . Its Chevalley–Eilenberg complex C CE ∗ ( A −∗ ⊗ L n ) iswell-defined, and by functoriality of C CE ∗ , it is a right Lie n -module.The proof of the following lemma is essentially found (in a different language) in [FT04,Section 2]. Lemma 127.
The right
Lie n -modules G ∨ A and C CE ∗ ( A −∗ ⊗ L n ) are isomorphic.Proof. We will actually define a non-degenerate pairing h− , −i : G A ( U ) ⊗ C CE ∗ ( A −∗ ⊗ L n )( U ) → R , for each finite set U , compatible with differentials and the right Lie n -(co)module struc-tures. As both complexes are finite-dimensional in each degree, this is sufficient to provethat they are isomorphic.Recall that the Chevalley–Eilenberg complex C CE ∗ ( g ) is given by the cofree cocommu-tative conilpotent coalgebra S c ( g [ − − Com ∨ → Lie . It follows that as a module, C CE ∗ ( A −∗ ⊗ L n )( U ) isgiven by:C CE ∗ ( A −∗ ⊗ L n )( U ) = M r ≥ M π ∈ Part r ( U ) A −∗ ⊗ L n ( U )[ − ⊗ . . . ⊗ A −∗ ⊗ L n ( U r )[ − Σ r = M r ≥ M π ∈ Part r ( U ) ( A n −∗ ) ⊗ r ⊗ Lie n ( U ) ⊗ . . . ⊗ Lie n ( U r ) Σ r (128)where the sums run over all partitions π = { U ⊔ . . . ⊔ U r } of U and A n −∗ = A −∗ [ − n ](which is a CDGA , Poincaré dual to A ).Fix some r ≥ π = { U ⊔ . . . ⊔ U r } . We define a first pairing: (cid:0) A ⊗ U ⊗ e ∨ n ( U ) (cid:1) ⊗ (cid:0) ( A n −∗ ) ⊗ r ⊗ Lie n ( U ) ⊗ . . . ⊗ Lie n ( U r ) (cid:1) → R (129)as follows: • On the A factors, the pairing uses the Poincaré duality pairing ε A . It is given bythe following formula (where a U i = Q u ∈ U i a u ):( a u ) u ∈ U ⊗ ( a ′ ⊗ . . . ⊗ a ′ r )
7→ ± ε A ( a U · a ′ ) . . . ε A ( a U r · a ′ r ) , • On the factor e ∨ n ( U ) ⊗ N ri =1 Lie n ( U i ), it uses the duality pairing on e ∨ n ( U ) ⊗ e n ( U )(recalling that e n = Com ◦ Lie n so that we can view N ri =1 Lie n ( U i ) as a submoduleof e n ( U )). 51he pairing in Equation (129) is the product of the two pairings we just defined. It isextended linearly on all of ( A ⊗ U ⊗ e ∨ n ( U )) ⊗ C CE ∗ ( A −∗ ⊗ L n )( U ), and it factors throughthe quotient defining G A ( U ) from A ⊗ U ⊗ e ∨ n ( U ).To check the non-degeneracy of this pairing, we use the vector subspaces G A h π i ofLemma 102, which are well-defined even though they are not preserved by the differentialif we do not consider the graded space E G A . Fix some partition π = { U , . . . , U r } of U ,then we have an isomorphism of vector spaces: G A h π i ∼ = A ⊗ r ⊗ Lie ∨ n ( U ) ⊗ . . . ⊗ Lie ∨ n ( U r ) . It is clear that G A h π i is paired with the factor corresponding to π in Equation (128),using the Poincaré duality pairing of A and the pairing between Lie n and its dual; and iftwo elements correspond to different partitions, then their pairing is equal to zero. Sinceboth ε A and the pairing between Lie n and its dual are non-degenerate, the total pairingis non-degenerate.The pairing is compatible with the Lie n -(co)module structures, i.e. the following dia-gram commutes (a relatively easy but notationally tedious check): G A ( U ) ⊗ C CE ∗ ( A −∗ ⊗ L n )( U/W ) ⊗ Lie n ( W ) G A ( U ) ⊗ C CE ∗ ( A −∗ ⊗ L n )( U ) G A ( U/W ) ⊗ C CE ∗ ( A −∗ ⊗ L n )( U/W ) Lie ∨ n ( W ) ⊗ Lie n ( W ) R ⊗◦ W ◦ ∨ W ⊗ h− , −ih− , −ih− , −i Lie n Finally, we easily check, using the identity ε A ( aa ′ ) = P (∆ A ) ± ε A ( a ∆ ′ A ) ε A ( a ′ ∆ ′′ A )(which in turns follows from the definition of ∆ A ) that the pairing commutes with dif-ferentials (i.e. h d ( − ) , −i = ±h− , d ( − ) i ). Proof of Proposition 124.
The operad e n is given by the composition product Com ◦ Lie n equipped with a distributive law that encodes the Leibniz rule. We get the followingisomorphism (natural in g ): G ∨ A ◦ e n S ( g [ n − G ∨ A ◦ e n ( Com ◦ g [ n − ∼ = G ∨ A ◦ e n ( e n ◦ Lie n g [ n − ∼ = G ∨ A ◦ Lie n g [ n − . According to Lemma 127, the right
Lie n -module G ∨ A is isomorphic to C CE ∗ ( A −∗ ⊗ L n ).The functoriality of A −∗ ⊗ − and C CE ∗ ( − ), as well as Equation (126), imply that we havethe following isomorphism (natural in g ): G ∨ A ◦ Lie n g [ n − ∼ = C CE ∗ ( A −∗ ⊗ L n ) ◦ Lie n g [ n − ∼ = C CE ∗ (cid:0) A −∗ ⊗ (( L n ) ◦ Lie n g [ n − (cid:1) ∼ = C CE ∗ ( A −∗ ⊗ g ) . S ( g [ n − Q g ∼ −→ g be a cofibrant resolution of the Lie algebra g . Then S ( Q g [ n − e n -algebra, and by Künneth’s formula S ( Q g [ n − → S ( g [ n − G ∨ A ◦ L e n S ( g [ n − G ∨ A ◦ e n S ( Q g [ n − . We therefore have a commutative diagram: G ∨ A ◦ L e n S ( g [ n − G ∨ A ◦ e n S ( g [ n − CE ∗ ( A −∗ ⊗ Q g ) C CE ∗ ( A −∗ ⊗ g ) ∼ = ∼ = The functor C CE ∗ preserves quasi-isomorphisms of Lie algebras, hence the bottom mapis a quasi-isomorphism. The proposition follows.
6. Outlook: The case of the 2-sphere and oriented manifolds
Up to now, we were considering framed manifolds M in order to define the action of the(unframed) Fulton–MacPherson FM n on FM M . When M is not framed, it is not possibleto coherently define insertion of infinitesimal configurations from FM n into the tangentspace of M , because we lack a coherent identification of the tangent space at every pointwith R n . Instead, for an oriented (but not necessarily framed) manifold M , there existsan action of the framed Fulton–MacPherson operad obtained by considering infinitesimalconfigurations together with rotations of SO( n ) (see below for precise definitions).In dimension 2, the formality of FM was extended to a proof of the formality of theframed version of FM in [GS10] (see also [Šev10] for an alternative proof and [KW17]for a generalization for even n ). We now provide a generalization of the previous workfor the 2-sphere, and we formulate a conjecture for higher dimensional closed manifoldsthat are not necessarily framed. Following Salvatore–Wahl [SW03, Definition 2.1], we describe the framed little disksoperad as a semi-direct product. If G is a topological group and P is an operad in G -spaces, the semi-direct product P ⋊ G is the topological operad defined by ( P ⋊ G )( n ) = P ( n ) × G n and explicit formulas for the composition. If H is a commutative Hopf algebraand C is a Hopf cooperad in H -comodules, then the semi-direct product C ⋊ H is definedby formally dual formulas.The operad FM n is an operad in SO( n )-spaces, the action rotating configurations. Thuswe can form an operad fFM n = FM n ⋊ SO( n ), the framed Fulton–MacPherson operad,weakly equivalent to the standard framed little disks operad.Given an oriented n -manifold M , there is a corresponding right module over fFM n ,which we call fFM M [Tur13, Section 2]. The space fFM M ( U ) is a principal SO( n ) × U -bundle over FM M ( U ). Since SO( n ) is an algebraic group, fFM n and fFM M ( U ) are respec-tively an operad and a module in semi-algebraic spaces.53 .2. Cohomology of fFM n and potential model The cohomology of SO( n ) is classically given by Pontryagin and Euler classes: H ∗ (SO(2 n ); Q ) = S ( β , . . . , β n − , α n − ) (deg α n − = 2 n − H ∗ (SO(2 n + 1)) = S ( β , . . . , β n ) (deg β i = 4 i − fe ∨ n ( U ) = e ∨ n ( U ) ⊗ H ∗ (SO( n )) ⊗ U . We now provide explicitformulas for the cocomposition [SW03]. If x ∈ H ∗ (SO( n )) and u ∈ U , then denote asbefore ι u ( x ) ∈ H ∗ (SO( n )) ⊗ U . Let W ⊂ U . If x is either β i or α n − in the even case,then we have: ◦ ∨ W ( ι u ( x )) = ( ι ∗ ( x ) ⊗ ⊗ ι u ( x ) , if u ∈ W ; ι u ( x ) ⊗ , otherwise. (130)The formula for ◦ ∨ W ( ω uv ) depends on the parity of n . If n is odd, then ◦ ∨ W ( ω uv ) is stillgiven by Equation (24). Otherwise, in fe ∨ n we have: ◦ ∨ W ( ω uv ) = ( ι ∗ ( α n − ) ⊗ ⊗ ω uv , if u, v ∈ W ; ω [ u ][ v ] ⊗ , otherwise . (131)From now on, we focus on oriented surfaces. The only simply connected compactsurface is M = S . We can choose A = H ∗ ( S ) = S ( υ ) / ( υ ) as its Poincaré dualitymodel. The Euler class of A is e A = χ ( S )vol A = 2 υ , and the diagonal class is ∆ A = υ ⊗ ⊗ υ . Recall that µ A (∆ A ) = e A . Definition 132.
The framed
LS CDGA fG A ( U ) is given by: fG A ( U ) = ( A ⊗ U ⊗ fe ∨ ( U ) / ( ι u ( a ) · ω uv = ι v ( a ) · ω uv ) , d ) , where the differential is given by dω uv = ι uv (∆ A ) and dι u ( α ) = ι u ( e A ). Proposition 133.
The collection { fG A ( U ) } U is a Hopf right fe ∨ -comodule, with cocom-position given by the same formula as Equation (41) .Proof. The proofs that the cocomposition is compatible with the cooperad structureof fe ∨ , and that this is compatible with the quotient, is the same as in the proof ofProposition 40. It remains to check compatibility with differentials.We check this compatibility on generators. The internal differential of A = H ∗ ( S )is zero, so it is easy to check that ◦ ∨ W ( d ( ι u ( a ))) = d ( ◦ ∨ W ( ι u ( a ))) = 0. Similarly, byEquation (130), checking the equality on α is immediate. As before there are severalcases to check for ω uv . If u, v ∈ W , then by Equation (131), d ( ◦ ∨ W ( ω uv )) = d ( ι ∗ ( α ) ⊗ ⊗ ω uv ) = ι ∗ ( e A ) ⊗ ι ∗ ( µ A (∆ A )) ⊗ ◦ ∨ W ( dω uv ) , and otherwise the proof is identical to the proof of Proposition 40.54 .3. Connecting fG A to Ω ∗ PA ( fFM S ) The framed little 2-disks operad is formal [GS10; Šev10]. We focus on the proof ofGiansiracusa–Salvatore [GS10], which goes along the same line as the proof of Kontsevichof the formality of FM n . To simplify notations, let H = H ∗ ( S ), which is a Hopf algebra.The operad Graphs is an operad in H -comodules, so there is a semi-direct product Graphs ⋊ H . Giansiracusa and Salvatore construct a zigzag: fe ∨ ∼ ←− Graphs ⋊ H ∼ −→ Ω PA ( fFM ) . (134)The first map is the tensor product of Graphs ∼ −→ e ∨ and the identity of H . Thesecond map is given by the Kontsevich integral on Graphs and by sending the generator α ∈ H to the volume form of Ω ∗ PA ( S ) (pulled back by the relevant projection). Theycheck that both maps are maps of Hopf (almost) cooperads, and they use the Künnethformula to conclude that these maps are quasi-isomorphisms. Theorem 135.
The Hopf right comodule ( fG A , fe ∨ ) , where A = H ∗ ( S ; R ) , is quasi-isomorphic to the Hopf right comodule (Ω ∗ PA ( fFM S ) , Ω ∗ PA ( fFM )) .Proof. It is now straightforward to adapt the proof of Theorem 3 to this setting, reusingthe proof of Giansiracusa–Salvatore [GS10]. We build the zigzag: fG A ← Graphs εA ⋊ H → Ω ∗ PA ( fFM S ) . We simply choose R = A = H ∗ ( S ), mapping υ ∈ H ( S ) to the volume form of S .Note that the propagator can be made completely explicit on S , and it can be checkedthat Z ϕ vanishes on all connected graphs with more than one vertex [CW16, Proposition80]. The middle term is a Hopf right ( Graphs ⋊ H )-comodule built out of Graphs εA and H , using formulas similar to the formulas defining Graphs ⋊ H out of Graphs and H .The first map is given by the tensor product of Graphs R → G A and the identity of H .The second map is given by the morphism of Proposition 74 on the Graphs εA factor,composed with the pullback along the projection fFM S → FM S . The generator α ∈ H issent to a pullback of a global angular form ψ of the principal SO(2)-bundle fFM S (1) → FM S (1) = S induced by the orientation of S . This form satisfies dψ = χ ( S )vol S .The proof of Giansiracusa–Salvatore [GS10] then adapts itself to prove that these twomaps are maps of Hopf right comodules. The Künneth formula implies that the firstmap is a quasi-isomorphism, and the second map induces an isomorphism on the E -pageof the Serre spectral sequence associated to the bundle fFM S → FM S and hence is itselfa quasi-isomorphism. Corollary 136.
The
CDGA fG H ∗ ( S ) ( k ) of Definition 132 is a real model for Conf or k ( S ) ,the SO(2) × k -principal bundle over Conf k ( S ) induced by the orientation of S . If M is an oriented n -manifold with n >
2, Definition 132 readily adapts to define fG H ∗ ( M ) , by setting dα to be the Euler class of M (when n is even), and dβ i to be the i thPontryagin class of M . The proof of Proposition 133 adapts easily to this new setting,and fG H ∗ ( M ) becomes a Hopf right fe ∨ n -comodule.55 onjecture . If M is a formal, simply connected, oriented closed n -manifold and ifthe framed little n -disks operad fe n is formal, then the pair ( fG H ∗ ( M ) , fe ∨ n ) is quasi-isomorphic to the pair (Ω ∗ PA ( fFM M ) , Ω ∗ PA ( fFM n )).To directly adapt our proof for the conjecture, the difficulty would be the same asencountered by Giansiracusa–Salvatore [GS10], namely finding forms in Ω ∗ PA ( fFM n ) cor-responding to the generators of H ∗ (SO( n )) and compatible with the Kontsevich integral.It was recently proved that the framed Fulton–MacPherson is formal for even n andnot formal for odd n ≥ fFM n is formal foreven n ≥
4, due to Khoroshkin and Willwacher [KW17], is much more involved thanthe proof of the formality of fFM . In particular, the zigzag of maps is not completelyexplicit and relies on obstruction-theoretical arguments. It would be interesting to tryand adapt the conjecture in this setting.If M itself is not formal then it is also not clear how to define Pontryagin classes insome Poincaré duality model of M (the Euler class is canonically given by χ ( A )vol A ).Nevertheless, for any oriented manifold M we get invariants of fe n -algebras by con-sidering the functor fG ∨ H ∗ ( M ) ◦ L fe n ( − ). Despite not necessarily computing factorizationhomology, they could prove interesting. Acknowledgments
I would like to thank several people: my (now former) advisorBenoit Fresse for giving me the opportunity to study this topic and for numerous helpfuldiscussions regarding the content of this paper; Thomas Willwacher and Ricardo Camposfor helpful discussions about their own model for configuration spaces ktheir explanationof propagators and partition functors, and for several helpful remarks; Ben Knudsen forexplaining the relationship between the
LS CDGA s and the Chevalley–Eilenberg complex;Pascal Lambrechts for several helpful discussions; Ivo Dell’Ambrogio, Julien Ducoulom-bier, Matteo Felder, and Antoine Touzé for their comments; and finally the anonymousreferee, for a thorough and detailed report with numerous suggestions that greatly im-proved this paper. The author was supported by ERC StG 678156–GRAPHCPX duringpart of the completion of this manuscript.
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DG-modules and
CDGA s V [ k ] = L n ∈ Z V n + k : desuspension of a dg-module (Section 1.1)( v ⊗ w ) := ± w ⊗ v (Section 1.1) X = P ( X ) X ′ ⊗ X ′′ ∈ V ⊗ W : Sweedler’s notation (Section 1.1) Cooperads and comodules k = { , . . . , k } (Section 1.2) ◦ ∨ W : C ( U ) → C ( U/W ) ⊗ C ( W ): cooperadic cocomposition (Section 1.2) ◦ ∨ W : N ( U ) → N ( U/W ) ⊗ C ( W ): right comodule structure map (Section 1.2) Semi-algebraic sets and PA forms Ω ∗ PA ( − ): CDGA of piecewise semi-algebraic (PA) forms (Section 1.3) p ∗ ( − ) = R p : E → B ( − ): integral along the fibers of the pa bundle p (Section 1.3) Little disks and related objects FM n ( k ): Fulton–MacPherson compactification of Conf k ( R n ) (Section 1.4) e n := H ∗ ( FM n ), e ∨ n := H ∗ ( FM n ) homology and cohomology of FM n (Section 1.4) ol n − ∈ Ω n − ( FM n (2)) volume form (Section 1.4) FM M ( k ): Fulton–MacPherson compactification of Conf k ( M ) (Section 1.4) p : ∂ FM M (2) → M sphere bundle of rank n − Poincaré duality
CDGA s ( A, ε A ): Poincaré duality CDGA with its orientation (Section 1.7)vol A ∈ A n : volume form (Section 1.7)∆ A ∈ ( A ⊗ A ) n : diagonal cocycle (Section 1.7) G A ( k ): Lambrechts–Stanley CDGA s (Section 1.8)
Graph complexes for R n Gra n : graphs with only external vertices (Section 1.6)Tw Gra n : graphs with external and internal vertices (Section 1.6) Graphs n : reduced graphs with external and internal vertices (Section 1.6) Gra (cid:9) n , Graphs (cid:9) n : variants with loops and multiple edges (Section 3.1) µ = e ∨ : Maurer–Cartan element used to twist the graphs cooperad (Section 1.6) ω : Graphs n → Ω ∗ PA ( FM n ): Kontsevich’s integrals (Section 1.6) Graph complexes for a closed manifold M Gra R : labeled graphs with only external vertices (Section 3.2) Gra (cid:9) R : variant with loops and multiple edges (Section 3.2)Tw Gra R : labeled graphs with internal and external vertices (Section 3.4) ϕ ∈ Ω n − ( FM M (2)): propagator (Section 3.3)fGC R : full labeled graph complex (Definition 68) Z ϕ : fGC R → R : partition function (Section 3) Graphs ϕR : reduced labeled graphs with internal and external vertices (Section 3.6) ω : Graphs ϕR : integrals (Section 3.6) Z ε : fGC R → R : almost trivial partition function (Definition 91) Graphs εR : reduced labeled graphs with internal and external vertices (Definition 91) Factorization homology
Disk fr n : operad of framed embeddings (Section 5) Disk fr M : module of framed embeddings for a framed M (Section 5) R M A := Disk fr M ◦ L Disk fr n A : factorization homology (Section 5)C CE ∗ : Chevalley–Eilenberg complex (Section 5) Framed case fFM n = FM n ⋊ SO( n ) framed Fulton–MacPherson operad (Section 6) fFM M : framed Fulton–MacPherson compactification (Section 6) fG A ( k ): framed Lambrechts–Stanley CDGA s (Section 6)s (Section 6)