The Lambrechts–Stanley model of configuration spaces
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Abstract
We prove the validity over \({\mathbb {R}}\) of a commutative differential graded algebra model of configuration spaces for simply connected closed smooth manifolds, answering a conjecture of Lambrechts–Stanley. We get as a result that the real homotopy type of such configuration spaces only depends on the real homotopy type of the manifold. We moreover prove, if the dimension of the manifold is at least 4, that our model is compatible with the action of the Fulton–MacPherson operad (weakly equivalent to the little disks operad) when the manifold is framed. We use this more precise result to get a complex computing factorization homology of framed manifolds. Our proofs use the same ideas as Kontsevich’s proof of the formality of the little disks operads.
Mathematics Subject Classification
55R80 55P62 18D501 Introduction
Without any restriction, this is false: the point \(\{0\}\) is homotopy equivalent to the line \({\mathbb {R}}\), but \(\mathrm {Conf}_{2}(\{0\}) = \varnothing \) is not homotopy equivalent to \(\mathrm {Conf}_{2}({\mathbb {R}}) \ne \varnothing \). One might wonder if the conjecture becomes true if restricted to closed manifolds. In 2005, Longoni and Salvatore [36] found a counterexample: two closed 3manifolds, given by lens spaces, which are homotopy equivalent but whose configuration spaces are not. This counterexample is not simply connected however. The question of the homotopy invariance of \(\mathrm {Conf}_{k}()\) for simply connected closed manifolds remains open to this day.
Here, we do not work with the full homotopy type. Rather, we restrict ourselves to the 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 [48]. The rational 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 quasiisomorphic to the CDGA of piecewise polynomial forms \(A_{\mathrm {PL}}^{*}(X)\). Due to technical issues, we will in fact work over \({\mathbb {R}}\). If M is a smooth manifold, then a real model is a CDGA which is quasiisomorphic to the CDGA of de Rham forms \(\varOmega ^{*}_{\mathrm {dR}}(M)\). While this is slightly coarser than 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 model of \(\mathrm {Conf}_{k}(M)\). This explicit model only depends on the model of M. This shows the (real) homotopy invariance of \(\mathrm {Conf}_{k}()\) on the class of manifolds we consider. Moreover, this explicit model can be used to perform computations, e.g. the real cohomology ring of \(\mathrm {Conf}_{k}(M)\), etc.
We focus on simply connected (thus orientable) closed manifolds. They satisfy Poincaré duality. Lambrechts and Stanley [32] showed that any such manifold admits a model A which satisfies itself Poincaré duality, i.e. there is an “orientation” \(A^{n} \xrightarrow {\varepsilon } {\mathbb {R}}\) which induces nondegenerate pairings \(A^{k} \otimes A^{nk} \rightarrow {\mathbb {R}}\) for all k. Lambrechts and Stanley [33] built a CDGA \(\texttt {G}_{A}(k)\) out of such a Poincaré duality model (they denote it F(A, k)). If we view \(H^{*}(\mathrm {Conf}_{k}({\mathbb {R}}^{n}))\) as spanned by graphs modulo Arnold relations, then \(\texttt {G}_{A}(k)\) consists of similar graphs with connected components labeled by A, and the differential splits edges. Lambrechts and Stanley proved that \(\texttt {G}_{A}(k)\) is quasiisomorphic to \(A_{\mathrm {PL}}^{*}(\mathrm {Conf}_{k}(M))\) as a dgmodule. They conjectured that this quasiisomorphism can be enhanced to give a quasiisomorphism of CDGAs so that \(\texttt {G}_{A}(k)\) defines a rational model of \(\mathrm {Conf}_{k}(M)\). We answer this conjecture by the affirmative in the real setting in the following theorem.
Theorem 1
(Corollary 78) Let M be a simply connected, closed, smooth manifold. Let A be any Poincaré duality model of M. Then for all \(k \ge 0\), \(\texttt {G}_{A}(k)\) is a model for the real homotopy type of \(\mathrm {Conf}_{k}(M)\).
Corollary 2
(Corollary 79) For simply connected closed smooth manifolds, the real homotopy type of M determines the real homotopy type of \(\mathrm {Conf}_{k}(M)\).
Over the past decades, attempts were made to solve the Lambrechts–Stanley conjecture, and results were obtained for special kinds of manifolds, or for low values of k. When M is a smooth complex projective variety, Kriz [30] had previously shown that \(\texttt {G}_{H^{*}(M)}(k)\) is actually a rational CDGA model for \(\mathrm {Conf}_{k}(M)\). The CDGA \(\texttt {G}_{H^{*}(M)}(k)\) is the \({\mathsf {E}}^{2}\) page of a spectral sequence of Cohen–Taylor [9] that converges to \(H^{*}(\mathrm {Conf}_{k}(M))\). Totaro [51] has shown that for a smooth complex compact projective variety, the spectral sequence only has one nonzero differential. When \(k = 2\), then \(\texttt {G}_{A}(2)\) was known to be a model of \(\mathrm {Conf}_{2}(M)\) either when M is 2connected [31] or when \(\dim M\) is even [10].
Our approach is different than the ones used in these previous works. We use ideas coming from the theory of operads. In particular, we consider the operad of little ndisks, defined by Boardman–Vogt [4], which consists of configuration spaces of small ndisks (instead of points) embedded inside the unit ndisk. These spaces of little ndisks are equipped with composition products, which are basically defined by inserting a configuration of l little ndisks into the ith little disk of a configuration of k little ndisks, resulting in a configuration of \(k+l1\) little ndisks. The idea is that a configuration of little ndisks represents an operation acting on nfold loop spaces, and the operadic composition products of little ndisks reflect the composition of such operations. The configuration spaces of little ndisks are homotopy equivalent to the configurations spaces of points in the Euclidean nspace \({\mathbb {R}}^{n}\), but the operadic composition structure does not go through this homotopy equivalence.
In our work, we actually use another model of the little ndisk operads, defined using the Fulton–MacPherson compactifications \(\texttt {FM}_{n}(k)\) of the configurations spaces \(\mathrm {Conf}_k({\mathbb {R}}^n)\) [2, 19, 46]. This compactification allows us to retrieve, on this collection of spaces \(\texttt {FM}_{n} = \{ \texttt {FM}_{n}(k) \}\), the operadic composition products which were lost in the configurations spaces \(\mathrm {Conf}_k({\mathbb {R}}^n)\). We also use the Fulton–MacPherson compactifications \(\texttt {FM}_{M}(k)\) of the configuration spaces \(\mathrm {Conf}_{k}(M)\) associated to a closed manifold M. When M is framed, these compactifications assemble into an operadic right module \(\texttt {FM}_{M}\) over the Fulton–MacPherson operad \(\texttt {FM}_{n}\), which roughly means that we can insert a configuration in \(\texttt {FM}_{n}\) into a configuration in \(\texttt {FM}_{M}\). We show that the Lambrechts–Stanley model is compatible with this action of the little disks operad, as we explain now.
The little ndisks operads are formal [18, 28, 34, 43, 49]. Kontsevich’s proof [28, 34] of this theorem uses the spaces \(\texttt {FM}_{n}\). If we temporarily forget about operads, this formality theorem means in particular that each space \(\texttt {FM}_{n}(k)\) is “formal”, i.e. the cohomology \({\texttt {e}_{n}^{\vee }}(k) {:}{=}H^{*}(\texttt {FM}_{n}(k))\) (with a trivial differential) is a model for the real homotopy type of \(\texttt {FM}_{n}(k)\). To prove Theorem 1, we generalize Kontsevich’s approach to prove that \(\texttt {G}_{A}(k)\) is a model of \(\texttt {FM}_{M}(k)\).
To establish his result, Kontsevich has to consider fiberwise integrations of forms along a particular class of maps, which are not submersions, but represent the projection map of “semialgebraic bundles”. In order to define such fiberwise integration operations, Kontsevich uses CDGAs of piecewise semialgebraic (PA) forms \(\varOmega _{\mathrm {PA}}^{*}()\) instead of the classical CDGAs of de Rham forms. The theory of PA forms was developed in [23, 29]. Any closed smooth manifold M is a semialgebraic manifold [39, 50], and the CDGA \(\varOmega _{\mathrm {PA}}^{*}(M)\) is a model for the real homotopy type of M. For the formality of \(\texttt {FM}_{n}\), a descent argument [22] is available to show that formality over \({\mathbb {R}}\) implies formality over \({\mathbb {Q}}\). However, no such descent argument exists for models with a nontrivial differential such as \(\texttt {G}_{A}\). Therefore, although we conjecture that our results on real homotopy types descend to \({\mathbb {Q}}\), we have no general argument ensuring that such a property holds.
The cohomology \({\texttt {e}_{n}^{\vee }}= H^{*}(\texttt {FM}_{n})\) inherits a Hopf cooperad structure from \(\texttt {FM}_{n}\), i.e. it is a cooperad (the dual notion of operad) in the category of CDGAs. The CDGAs of forms \(\varOmega _{\mathrm {PA}}^{*}(\texttt {FM}_{n}(k))\) also inherit a Hopf cooperad structure (up to homotopy). The formality quasiisomorphisms between the cohomology algebras \({\texttt {e}_{n}^{\vee }}(k)\) and the CDGAs of forms on \(\texttt {FM}_{n}(k)\) are compatible in a suitable sense with this structure. Therefore the Hopf cooperad \({\texttt {e}_{n}^{\vee }}\) fully encodes the rational homotopy type of the operad \(\texttt {FM}_{n}\).
In this paper, we also prove that the Lambrechts–Stanley model \(\texttt {G}_{A}\) determines the real homotopy type of \(\texttt {FM}_{M}\) as a right module over the operad \(\texttt {FM}_{n}\) when M is a framed manifold. To be precise, our result reads as follows.
Theorem 3
(Theorem 62) Let M be a framed smooth simply connected closed manifold with \(\dim M \ge 4\). Let A be any Poincaré duality model of M. Then the collection \(\texttt {G}_{A} = \{ \texttt {G}_{A}(k) \}_{k \ge 0}\) forms a Hopf right \({\texttt {e}_{n}^{\vee }}\)comodule. Moreover the Hopf right comodule \((\texttt {G}_{A}, {\texttt {e}_{n}^{\vee }})\) is weakly equivalent to \((\varOmega _{\mathrm {PA}}^{*}(\texttt {FM}_{M}), \varOmega _{\mathrm {PA}}^{*}(\texttt {FM}_{n}))\).
For \(\dim M \le 3\), the proof fails (see Proposition 45). However, in this case, the only examples of simply connected closed manifolds are spheres, thanks to Perelman’s proof of the Poincaré conjecture [41, 42]. We can then directly prove that \(\texttt {G}_{A}(k)\) is a model for \(\mathrm {Conf}_{k}(M)\) (see Sect. 4.3).
Our proof of Theorem 3, which is inspired by Kontsevich’s proof of the formality of the little disks operads, is radically different from the proofs of [33]. It involves an intermediary Hopf right comodule of labeled graphs \(\texttt {Graphs}_{R}\). This comodule is similar to a comodule recently studied by Campos–Willwacher [6], which corresponds to the case \(R = S({\tilde{H}}^{*}(M))\). Note however that the approach of Campos–Willwacher differs from ours. In comparison to their work, our main contribution is the definition of the quasiisomorphism between the CDGAs \(\varOmega _{\mathrm {PA}}^{*}(\texttt {FM}_{M}(k))\) and the small, explicit Lambrechts–Stanley model \(\texttt {G}_{A}(k)\), which has the advantage of being finitedimensional and much more computable than \(\texttt {Graphs}_{S({\tilde{H}}(M))}(k)\).
Applications. Ordered configuration spaces appear in many places in topology and geometry. Therefore, thanks to Theorems 1 and 3, the explicit model \(\texttt {G}_{A}(k)\) provides an efficient computational tool in many concrete situations.
To illustrate this, we show how to apply our results to compute factorization homology, an invariant of framed nmanifolds defined from an \(\texttt {E}_{n}\)algebra [3]. Let M be a framed manifold with Poincaré duality model A, and B be an nPoisson algebras, i.e. an algebra over the operad \(H_{*}(\texttt {E}_{n})\). Our results shows that we can compute the factorization homology of M with coefficients in B just from \(\texttt {G}_{A}\) and B. As an application, we compute factorization homology with coefficients in a higher enveloping algebra of a Lie algebra (Proposition 81), recovering a theorem of Knudsen [27].
The Taylor tower in the Goodwillie–Weiss calculus of embeddings may be computed in a similar manner [5, 21]. It follows from a result of [52, Section 5.1] that \(\texttt {FM}_{M}\) may be used for this purpose. Therefore our theorem shows that \(\texttt {G}_{A}\) may also be used for computing this Taylor tower.
Roadmap. In Sect. 1, we lay out our conventions and recall the necessary background. This includes dgmodules and CDGAs, (co)operads and their (co)modules, semialgebraic sets and PA forms. We also recall basic results on the Fulton–MacPherson compactifications of configuration spaces \(\texttt {FM}_{n}(k)\) and \(\texttt {FM}_{M}(k)\), and the main ideas of Kontsevich’s proof of the formality of the little disks operads using the CDGAs of PA forms on the spaces \(\texttt {FM}_{n}(k)\). We use the formalism of operadic twisting, which we recall, to deal with signs more easily. Finally, we recollect the necessary background on Poincaré duality CDGAs and the Lambrechts–Stanley CDGAs. In Sect. 2, we build out of the Lambrechts–Stanley CDGAs a Hopf right \({\texttt {e}_{n}^{\vee }}\)comodule \(\texttt {G}_{A}\).
In Sect. 3, we construct the labeled graph complex \(\texttt {Graphs}_{R}\) which will be used to connect \(\texttt {G}_{A}\) to \(\varOmega _{\mathrm {PA}}^{*}(\texttt {FM}_{M})\). The construction is inspired by Kontsevich’s construction of the unlabeled graph complex \(\texttt {Graphs}_{n}\). It is done in several steps. The first step is to consider a graded module of labeled graphs, \(\texttt {Gra}_{R}\). In order to be able to map \(\texttt {Gra}_{R}\) into \(\varOmega _{\mathrm {PA}}^{*}(\texttt {FM}_{M})\), we recall the construction of what is called a “propagator” in the mathematical physics literature. We then “twist” \(\texttt {Gra}_{R}\) to obtain a new object \({{\mathrm{Tw}}}\texttt {Gra}_{R}\), which consists of graphs with two kinds of vertices: “external” and “internal”. Finally we must reduce our graphs to obtain a new object, \(\texttt {Graphs}_{R}\), by removing all the connected components with only internals vertices in the graphs using a “partition function” (a function which resembles the Chern–Simons invariants).
In Sect. 4, we prove that the zigzag of Hopf right comodule morphisms between \(\texttt {G}_{A}\) and \(\varOmega _{\mathrm {PA}}^{*}(\texttt {FM}_{M})\) is a weak equivalence. We first connect our graph complex \(\texttt {Graphs}_{R}\) to the Lambrechts–Stanley CDGAs \(\texttt {G}_{A}\). This requires vanishing results about the partition function. Then we end the proof of the theorem by showing that all the morphisms are quasiisomorphisms. Finally we study the cases \(S^{2}\) and \(S^{3}\).
In Sect. 5, we use our model to compute factorization homology of framed manifolds and we compare the result to a complex obtained by Knudsen. In Sect. 6 we work out a variant of our theorem for the only simply connected surface using the formality of the framed little 2disks operad, and we present a conjecture about higher dimensional oriented manifolds.
For convenience, we provide a glossary of our main notations at the end of this paper.
2 Background and recollections
2.1 DGmodules and CDGAs
We consider differential graded modules (dgmodules) over the base field \({\mathbb {R}}\). Unless otherwise indicated, (co)homology of spaces is considered with real coefficients. All our dgmodules will have a cohomological grading, \(V = \bigoplus _{n \in {\mathbb {Z}}} V^{n}\). All the differentials raise degrees by one: \(\deg (dx) = \deg (x) + 1\). We say that a dgmodule is of finite type if it is finite dimensional in each degree. Let V[k] be the desuspension, defined by \((V[k])^{n} = V^{n+k}\). For dgmodules V, W and homogeneous elements \(v \in V, \, w \in W\), we let \((v \otimes w)^{21} {:}{=}(1)^{(\deg v)(\deg w)} w \otimes v\) and we extend this linearly to the tensor product. Moreover, given an element \(X \in V \otimes W\), we will often use a variant of Sweedler’s notation to express X as a sum of elementary tensors, \(X {:}{=}\sum _{(X)} X' \otimes X'' \in V \otimes W\).
We call CDGAs the (graded) commutative unital algebras in dgmodules. In general, for a CDGA A, we let \(\mu _{A} : A^{\otimes 2} \rightarrow A\) be its product. For a dgmodule V, we let S(V) be the free unital symmetric algebra on V.
Many of the CDGAs that appear in this paper are \({\mathbb {Z}}\)graded. However, to deserve the name “model of X”, a CDGA should be connected to \(A_{\mathrm {PL}}^{*}(X)\) only by \({\mathbb {N}}\)graded CDGAs. The next proposition shows that considering this larger category does not change our statement.
Proposition 4
Let A, B be two \({\mathbb {N}}\)graded CDGAs which are homologically connected, i.e. \(H^{0}(A) = H^{0}(B) = {\mathbb {R}}\). If A and B are quasiisomorphic as \({\mathbb {Z}}\)graded CDGAs, then they also are as \({\mathbb {N}}\)graded CDGAs.
Proof
This follows from the results of [17, §II.6.2]. Let us temporarily denote \(\mathsf {cdga}_{{\mathbb {N}}}\) the category of \({\mathbb {N}}\)graded CDGAs (\(dg^{*}\texttt {Com}\) in [17]) and \(\mathsf {cdga}_{{\mathbb {Z}}}\) the category of \({\mathbb {Z}}\)graded CDGAs (\(dg\texttt {Com}\) in [17]). Note that in [17], \({\mathbb {Z}}\)graded CDGAs are homologically graded, but we can use the usual correspondence \(A^{i} = A_{i}\) to keep our convention that all dgmodules are cohomologically graded. There is an obvious inclusion \(\iota : \mathsf {cdga}_{{\mathbb {N}}} \rightarrow \mathsf {cdga}_{{\mathbb {Z}}}\), which clearly defines a full functor that preserves and reflects quasiisomorphisms.
Let \({\mathbb {B}}^{m}\) be the dgmodule \({\mathbb {R}}\) concentrated in degree m, let \({\mathbb {E}}^{m}\) be the dgmodule given by two copies of \({\mathbb {R}}\) in respective degree \(m1\) and m such that \(d_{{\mathbb {E}}^{m}}\) is the identity of \({\mathbb {R}}\) in these degrees (hence \({\mathbb {E}}^{m}\) is acyclic), and let \(i : {\mathbb {B}}^{m} \rightarrow {\mathbb {E}}^{m}\) be the obvious inclusion. The model category \(\mathsf {cdga}_{{\mathbb {N}}}\) is equipped with a set of generating cofibrations given by the morphisms \(S(i) : S({\mathbb {B}}^{m}) \rightarrow S({\mathbb {E}}^{m})\) and of the morphism \(\varepsilon : S({\mathbb {B}}^{0}) \rightarrow {\mathbb {R}}\). Recall that a cellular complex of generating cofibrations is a CDGA obtained by a sequential colimit \(R = {\text {colim}}_{k} R_{\langle k \rangle }\), where \(R_{\langle 0 \rangle } = {\mathbb {R}}\) and \(R_{\langle k+1 \rangle }\) is obtained from \(R_{\langle k \rangle }\) by a pushout of generating cofibrations along attaching maps \(h: S({\mathbb {B}}^m)\rightarrow R_{\langle k \rangle }\). In [17, §II.6.2], the expression “connected generating cofibrations” is used for the generating cofibrations of the form \(S(i): S({\mathbb {B}}^m)\rightarrow S({\mathbb {E}}^m)\) with \(m>0\).
In the proof of [17, Proposition II.6.2.8], it is observed that, if A is homologically connected, then the attaching map \(h: S({\mathbb {B}}^0)\rightarrow A\) associated to a generating cofibration \(\varepsilon : S({\mathbb {B}}^0)\rightarrow {\mathbb {R}}\) necessarily reduces to the augmentation \(\varepsilon : S({\mathbb {B}}^0)\rightarrow {\mathbb {R}}\) followed by the inclusion as the unit \({\mathbb {R}}\subset A\). Thus a pushout of the generating cofibration \(\varepsilon : S({\mathbb {B}}^0)\rightarrow {\mathbb {R}}\) reduces to a neutral operation in this case. In the proof of [17, Proposition II.6.2.8], it is deduced from this observation that any homologically connected algebra admits a resolution \(R_A \xrightarrow {\sim } A\) such that \(R_A\) is a cellular complex of connected generating cofibrations. Connected generating cofibrations are also cofibrations in \(\mathsf {cdga}_{{\mathbb {Z}}}\) after applying \(\iota \). Moreover \(\iota \) preserves colimits. It follows that \(\iota R_{A}\) is cofibrant in \(\mathsf {cdga}_{{\mathbb {Z}}}\) too.
By hypothesis, \(\iota A\) and \(\iota B\) are weakly equivalent in \(\mathsf {cdga}_{{\mathbb {Z}}}\), hence \(\iota R_{A}\) and \(\iota B\) are also weakly equivalent (because \(\iota \) clearly preserves quasiisomorphisms), through a zigzag \(\iota R_{A} \xleftarrow {\sim } \cdot \xrightarrow {\sim } \iota B\). As \(\iota R_{A}\) is cofibrant (and all CDGAs are fibrant), we can find a direct quasiisomorphism \(\iota R_{A} \xrightarrow {\sim } \iota B\) and therefore a zigzag \(\iota A \xleftarrow {\sim } \iota R_{A} \xrightarrow {\sim } \iota B\) which only involves \({\mathbb {N}}\)graded CDGAs. \(\square \)
2.2 (Co)operads and their right (co)modules
We assume basic proficiency with Hopf (co)operads and (co)modules over (co)operads, see e.g. [16, 17, 35]. We index our (co)operads by finite sets instead of integers to ease the writing of some formulas. If \(W \subset U\) is a subset, we write the quotient \(U/W = (U {\setminus } W) \sqcup \{*\}\), where \(*\) represents the class of W; note that \(U/\varnothing = U \sqcup \{ * \}\). An operad in dgmodules, for instance, is given by a functor from the category of finite sets and bijections (a symmetric collection) \(\texttt {P}: U \mapsto \texttt {P}(U)\) to the category of dgmodules, together with a unit \(\Bbbk \rightarrow \texttt {P}(\{*\})\) and composition maps \(\circ _{W} : \texttt {P}(U/W) \otimes \texttt {P}(W) \rightarrow \texttt {P}(U)\) for every pair of sets \(W \subset U\), satisfying the usual associativity, unity and equivariance conditions. Dually, a cooperad \(\texttt {C}\) is given by a symmetric collection, a counit \(\texttt {C}(\{*\}) \rightarrow \Bbbk \), and cocomposition maps \(\circ _{W}^{\vee } : \texttt {C}(U) \rightarrow \texttt {C}(U/W) \otimes \texttt {C}(W)\) for every pair \(W \subset U\).
Let \({\underline{k}} = \{1,\dots ,k\}\). We recover the usual notion of a cooperad indexed by the integers by considering the collection \(\{\texttt {C}({\underline{k}})\}_{k \ge 0}\), and the cocomposition maps \(\circ _{i}^{\vee } : \texttt {C}(\underline{k+l1}) \rightarrow \texttt {C}({\underline{k}}) \otimes \texttt {C}({\underline{l}})\) corresponds to \(\circ _{\{i, \dots , i+l1\}}^{\vee }\).
Following Fresse [17, §II.9.3.1], a “Hopf cooperad” is a cooperad in the category of CDGAs. We do not assume that (co)operads are trivial in arity zero, but they will satisfy \(\texttt {P}(\varnothing ) = \Bbbk \) (resp. \(\texttt {C}(\varnothing ) = \Bbbk \)). Therefore we get (co)operad structures equivalent to the structure of \(\varLambda \)(co)operads considered by Fresse [17, §II.11], which he uses to model rational homotopy types of operads in spaces satisfying \(\texttt {P}(0) = *\) (but we do not use this formalism in the sequel).
The result of Proposition 4 extends to Hopf cooperads (and to Hopf \(\varLambda \)cooperads). To establish this result, we still use a description of generating cofibrations of Ngraded Hopf cooperads, which are given by morphisms of symmetric algebras of cooperads \(S(i): S(\texttt {C})\rightarrow S(\texttt {D})\), where \(i: \texttt {C}\rightarrow \texttt {D}\) is a dgcooperad morphism that is injective in positive degrees (see [17, §II.9.3] for details). In the context of homologically connected cooperads, we can check that the pushout of such a Hopf cooperad morphism along an attaching map reduces to a pushout of a morphism of symmetric algebras of cooperads \(S(\texttt {C}/\ker (i)) \rightarrow S(\texttt {D})\), where we mod out by the kernel of the map \(i: \texttt {C}\rightarrow \texttt {D}\) in degree 0. We deduce from this observation that any homologically connected Ngraded Hopf cooperad admits a resolution by a cellular complex of generating cofibrations of the form \(S(i): S(C)\rightarrow S(D)\), where the map i is injective in all degrees (we again call such a generating cofibration connected). The category of \({\mathbb {Z}}\)graded Hopf cooperads inherits a model structure, like the category of \({\mathbb {N}}\)graded Hopf cooperads considered in [17, §II.9.3]. Cellular complexes of connected generating cofibrations of \({\mathbb {N}}\)graded Hopf cooperads define cofibrations in the model category of \({\mathbb {Z}}\)graded Hopf cooperads yet, as in the proof of Proposition 4.
Given an operad \(\texttt {P}\), a right \(\texttt {P}\)module is a symmetric collection \(\texttt {M}\) equipped with composition maps \(\circ _{W} : \texttt {M}(U/W) \otimes \texttt {P}(W) \rightarrow \texttt {M}(U)\) satisfying the usual associativity, unity and equivariance conditions. A right comodule over a cooperad is defined dually. If \(\texttt {C}\) is a Hopf cooperad, then a right Hopf \(\texttt {C}\)comodule is a \(\texttt {C}\)comodule \(\texttt {N}\) such that all the \(\texttt {N}(U)\) are CDGAs and all the maps \(\circ _{W}^{\vee }\) are morphisms of CDGAs.
Definition 5
Let \(\texttt {C}\) (resp. \(\texttt {C}'\)) be a Hopf cooperad and \(\texttt {N}\) (resp. \(\texttt {N}'\)) be a Hopf right comodule over \(\texttt {C}\) (resp. \(\texttt {C}'\)). A morphism of Hopf right comodules is a pair \((f_{\texttt {N}}, f_{\texttt {C}})\) consisting of a morphism of Hopf cooperads \(f_{\texttt {C}} : \texttt {C}\rightarrow \texttt {C}'\), and a map of Hopf right \(\texttt {C}'\)comodules \(f_{\texttt {N}} : \texttt {N}\rightarrow \texttt {N}'\), where \(\texttt {N}\) has the \(\texttt {C}\)comodule structure induced by \(f_{\texttt {C}}\). It is a quasiisomorphism if both \(f_{\texttt {C}}\) and \(f_{\texttt {N}}\) are quasiisomorphisms in each arity. A Hopf right \(\texttt {C}\)module \(\texttt {N}\) is said to be weakly equivalent to a Hopf right \(\texttt {C}'\)module \(\texttt {N}'\) if the pair \((\texttt {N}, \texttt {C})\) can be connected to the pair \((\texttt {N}', \texttt {C}')\) through a zigzag of quasiisomorphisms.
The next very general lemma can for example be found in [6, Section 5.2]. Let \(\texttt {C}\) be a cooperad, and see the CDGA A as an operad concentrated in arity 1. Recall that \(\texttt {C}\circ A = \bigoplus _{i \ge 0} \texttt {C}(i) \otimes _{\varSigma _{i}} A^{\otimes i}\) denotes the composition product of operads, where we view A as an operad concentrated in arity 1. Then the commutativity of A implies the existence of a distributive law \(t : \texttt {C}\circ A \rightarrow A \circ \texttt {C}\), given in each arity by the morphism \(t : \texttt {C}({\underline{n}}) \otimes A^{\otimes n} \rightarrow A \otimes \texttt {C}({\underline{n}})\) given by \(x \otimes a_{1} \otimes \dots \otimes a_{n} \mapsto a_{1} \dots a_{n} \otimes x\).
Lemma 6
2.3 Semialgebraic sets and forms
Kontsevich’s proof of the formality of the little disks operads [28] uses the theory of semialgebraic sets, as developed in [23, 29]. A semialgebraic set is a subset of \({\mathbb {R}}^{N}\) defined by finite unions of finite intersections of zero sets of polynomials and polynomial inequalities. By the Nash–Tognoli Theorem [39, 50], any closed smooth manifold is algebraic hence semialgebraic.
There is a functor \(\varOmega _{\mathrm {PA}}^{*}\) of “piecewise semialgebraic (PA) differential forms”, analogous to de Rham forms. If X is a compact semialgebraic set, then \(\varOmega _{\mathrm {PA}}^{*}(X) \simeq A_{\mathrm {PL}}^{*}(X) \otimes _{{\mathbb {Q}}} {\mathbb {R}}\), i.e. the CDGA \(\varOmega _{\mathrm {PA}}^{*}(X)\) models the real homotopy type of X [23, Theorem 6.1].
Remark 7
There is a construction \(\varOmega ^{*}_{\sharp }\) that turns a simplicial operad \(\texttt {P}\) into a Hopf cooperad and such that a morphism of Hopf cooperads \(\texttt {C}\rightarrow \varOmega ^{*}_{\sharp }(\texttt {P})\) is the same thing as an “almost” morphism \(\texttt {C}\rightarrow A_{\mathrm {PL}}^{*}(\texttt {P})\), where \(A_{\mathrm {PL}}^{*}\) is the functor of Sullivan forms [17, Section II.10.1]. Moreover there is a canonical collection of maps \((\varOmega ^{*}_{\sharp }(\texttt {P}))(U) \rightarrow A_{\mathrm {PL}}^{*}(\texttt {P}(U))\), which are weak equivalences if \(\texttt {P}\) is a cofibrant operad. This functor is built by considering the right adjoint of the functor on operads induced by the Sullivan realization functor, which is monoidal. A similar construction can be extended to \(\varOmega _{\mathrm {PA}}^{*}\) and to modules over operads. This construction allows us to make sure that the cooperads and comodules we consider truly encode the rational or real homotopy type of the initial operad or module (see [17, §II.10.2]).
2.4 Little disks and related objects
The little disks operad \(\texttt {E}_{n}\) is a topological operad initially introduced by May and Boardman–Vogt [4, 37] to study iterated loop spaces. Its homology \(\texttt {e}_{n} := H_{*}(\texttt {E}_{n})\) is described by a theorem of Cohen [8]: it is either the operad governing associative algebras for \(n = 1\), or nPoisson algebras for \(n \ge 2\). We also consider the linear dual \({\texttt {e}_{n}^{\vee }}{:}{=}H^{*}(\texttt {E}_{n})\), which is a Hopf cooperad.
In fact, we use the Fulton–MacPherson operad \(\texttt {FM}_{n}\), which is an operad in spaces weakly equivalent to the little disks operad \(\texttt {E}_{n}\). The components \(\texttt {FM}_{n}(k)\) are compactifications of the configuration spaces \(\mathrm {Conf}_{k}({\mathbb {R}}^{n})\), defined by using a real analogue due to Axelrod–Singer [2] of the Fulton–MacPherson compactifications [19]. The idea of this compactification is to allow configurations where points become “infinitesimally close”. Then one uses insertion of such infinitesimal configurations to define operadic composition products on the spaces \(\texttt {FM}_n(k)\). We refer to [46] for a detailed treatment and to [34, Sections 5.1–5.2] for a clear summary. In both references, the name C[k] is used for what we call \(\texttt {FM}_{n}(k)\).
2.5 Operadic twisting
We will make use of the “operadic twisting” procedure in what follows [11]. Let us now recall this procedure, in the case of cooperads.
Let \(\texttt {Lie}_{n}\) be the operad governing shifted Lie algebras. A \(\texttt {Lie}_{n}\)algebra is a dgmodule \({\mathfrak {g}}\) equipped with a Lie bracket \([,] : {\mathfrak {g}}^{\otimes 2} \rightarrow {\mathfrak {g}}[1n]\) of degree \(1n\), i.e. we have \([{\mathfrak {g}}^{i}, {\mathfrak {g}}^{j}] \subset {\mathfrak {g}}^{i+j+(1n)}\).
Remark 8
The degree convention is such that there is an embedding of operads \(\texttt {Lie}_{n} \rightarrow H_{*}(\texttt {FM}_{n})\), i.e. Poisson nalgebras are \(\texttt {Lie}_{n}\)algebras. The usual Lie operad is \(\texttt {Lie}_{1}\). This convention is consistent with [53]. However in [54], the notation is \(\texttt {Lie}^{(n)} = \texttt {Lie}_{n+1}\). In [11], only the unshifted operad \(\texttt {Lie}= \texttt {Lie}_{1}\) is considered.
The operad \(\texttt {Lie}_{n}\) is quadratic Koszul (see e.g. [35, Section 13.2.6]), and as such admits a cofibrant resolution \(\texttt {hoLie}_{n} {:}{=}\varOmega (K(\texttt {Lie}_{n}))\), where \(\varOmega \) is the cobar construction and \(K(\texttt {Lie}_{n})\) is the Koszul dual cooperad of \(\texttt {Lie}\). Algebras over \(\texttt {hoLie}_{n}\) are (shifted) \(L_{\infty }\)algebras, also known as homotopy Lie algebras, i.e. dgmodules \({\mathfrak {g}}\) equipped with higher brackets \([,\dots ,]_{k} : {\mathfrak {g}}^{\otimes k} \rightarrow {\mathfrak {g}}[3kn]\) (for \(k \ge 1\)) satisfying the classical \(L_{\infty }\) equations.
Lemma 9
If \(\texttt {C}\) is a Hopf cooperad satisfying \(\texttt {C}(\varnothing ) = \Bbbk \), then \({{\mathrm{Tw}}}\texttt {C}\) inherits a Hopf cooperad structure.
Proof
To multiply an element of \(\texttt {C}(U \sqcup I) \subset {{\mathrm{Tw}}}\texttt {C}(U)\) with an element of \(\texttt {C}(U \sqcup J) \subset {{\mathrm{Tw}}}\texttt {C}(U)\), we use the maps \(\texttt {C}(V) \xrightarrow {\circ _{\varnothing }^{\vee }} \texttt {C}(V / \varnothing ) \otimes \texttt {C}(\varnothing ) \cong \texttt {C}(V \sqcup \{*\})\) iterated several times, to obtain elements in \(\texttt {C}(U \sqcup I \sqcup J)\) and the product. \(\square \)
Lemma 10
If \(\texttt {C}\) is a Hopf cooperad satisfying \(\texttt {C}(\varnothing ) = \Bbbk \) and \(\texttt {M}\) is a Hopf right \(\texttt {C}\)comodule, then \({{\mathrm{Tw}}}\texttt {M}\) inherits a Hopf right \(({{\mathrm{Tw}}}\texttt {C})\)comodule structure. \(\square \)
2.6 Formality of the little disks operad
Kontsevich’s proof of the formality of the little disks operads [28, Section 3], can be summarized by the fact that \(\varOmega _{\mathrm {PA}}^{*}(\texttt {FM}_{n})\) is weakly equivalent to \({\texttt {e}_{n}^{\vee }}\) as a Hopf cooperad. For detailed proofs, we refer to [34].
We outline this proof here as we will mimic its pattern for our theorem. The idea of the proof is to construct a Hopf cooperad \(\texttt {Graphs}_{n}\). The elements of \(\texttt {Graphs}_{n}\) are formal linear combinations of special kinds of graphs, with two types of vertices, numbered “external” vertices and unnumbered “internal” vertices. The differential is defined combinatorially by edge contraction. It is built in such a way that there exists a zigzag \({\texttt {e}_{n}^{\vee }}\xleftarrow {\sim } \texttt {Graphs}_{n} \xrightarrow {\sim } \varOmega _{\mathrm {PA}}^{*}(\texttt {FM}_{n})\). The first map is the quotient by the ideal of graphs containing internal vertices. The second map is defined using integrals along fibers of the PA bundles \(\texttt {FM}_{n}(U \sqcup I) \rightarrow \texttt {FM}_{n}(U)\) which forget some points in the configuration. An induction argument shows that the first map is a quasiisomorphism, and the second map is easily seen to be surjective on cohomology.
In order to deal with signs more easily, we use (co)operadic twisting (Sect. 1.5). Thus the Hopf cooperad \(\texttt {Graphs}_{n}\) is not the same as the Hopf cooperad \({\mathcal {D}}\) from [34], see Remark 13.
The dgmodule \({{\mathrm{Tw}}}\texttt {Gra}_{n}(U)\) is spanned by graphs with two types of vertices: external vertices, which correspond to elements of U and that we will picture as circles with the name of the label in U inside, and indistinguishable internal vertices, corresponding to the elements of \({\underline{i}}\) in Eq. (6) and that we will draw as black points. For example, the graph inside the differential in the left hand side of Fig. 1 represents an element of \({{\mathrm{Tw}}}\texttt {Gra}_{n}(U)\) with \(U = \{1,2,3\}\) and \(i = 1\). The degree of an edge is still \(n1\), the degree of an external vertex is still 0, and the degree of an internal vertex is \(n\).
The product of \({{\mathrm{Tw}}}\texttt {Gra}_{n}(U)\) glues graphs along their external vertices only. Compared to Lemma 9, this coincides with adding isolated internal vertices (by iterating the cooperad structure map \(\circ _{\varnothing }^{\vee }\)) and gluing along all vertices.

The element \(\varGamma \cdot \mu \) is the sum over all ways of collapsing a subgraph \(\varGamma ' \subset \varGamma \) with only internal vertices, the result being \(\mu (\varGamma ') \varGamma / \varGamma '\). This is nonzero only if \(\varGamma '\) has exactly two vertices and one edge. Thus this summand corresponds to contracting all edges between two (possibly univalent) internal vertices in \(\varGamma \).

The element \(\varGamma \cdot \mu _{1}\) is the sum over all ways of collapsing a subgraph \(\varGamma ' \subset \varGamma \) with exactly one external vertex (and any number of internal vertices), with result \(\mu (\varGamma ') \varGamma / \varGamma '\). This summands corresponds to contracting all edges between one external vertex and one internal (possibly univalent) vertex.

The element \(\mu _{1} \cdot \varGamma \) is the sum over all ways of collapsing a subgraph \(\varGamma ' \subset \varGamma \) containing all the external vertices, with result \(\mu _{1}(\varGamma /\varGamma ') \varGamma '\). The coefficient \(\mu _{1}(\varGamma /\varGamma ')\) can only be nonzero if \(\varGamma \) is obtained from \(\varGamma '\) by adding a univalent internal vertex. A careful analysis of the signs [53, Appendix I.3] shows that this cancels out with the contraction of edges connected to univalent internal vertices from the other 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 term appears three times, and only two cancel out (see [53, Fig. 3] for the dual picture).
Definition 11
A graph is internally connected if it remains connected when the external vertices are deleted. It is easily checked that as a commutative algebra, \({{\mathrm{Tw}}}\texttt {Gra}_{n}(U)\) is freely generated by such graphs.
The morphisms \({\texttt {e}_{n}^{\vee }}\leftarrow \texttt {Gra}_{n} \xrightarrow {\omega '} \varOmega _{\mathrm {PA}}^{*}(\texttt {FM}_{n})\) extend along the inclusion \(\texttt {Gra}_{n} \subset {{\mathrm{Tw}}}\texttt {Gra}_{n}\) as follows. The extended morphism \({{\mathrm{Tw}}}\texttt {Gra}_{n} \rightarrow {\texttt {e}_{n}^{\vee }}\) simply sends a graph with internal vertices to zero. We need to check that this commutes with the differential. We thus need to determine when a graph with internal vertices (sent to zero) can have a differential with no internal vertices (possibly sent to a nonzero element in \({\texttt {e}_{n}^{\vee }}\)). The differential decreases 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 internal vertex connected to some external vertices. Either the internal vertex is univalent, but then the edge is not contractible and the differential vanishes. Or the internal vertex is connected to more than one external vertices. In this case, one check that the differential of the graph is zero modulo the Arnold relations, (see [34, Introduction] and Fig. 1 for an example).
Remark 12
This Hopf cooperad is different from the module of diagrams \(\widehat{{\mathcal {D}}}\) introduced in [34, Section 6.2]: \({{\mathrm{Tw}}}\texttt {Gra}_{n}\) is the quotient of \(\widehat{{\mathcal {D}}}\) by graphs with multiple edges and loops. The analogous integral \({\widehat{I}} : \widehat{{\mathcal {D}}} \rightarrow \varOmega _{\mathrm {PA}}^{*}(\texttt {FM}_{n})\) is from [34, Chapter 9]. It vanishes on graphs with multiple edges and loops by [34, Lemmas 9.3.5, 9.3.6], so \(\omega \) is welldefined. Moreover the differential is slightly different. In [34] some kind of edges, called “dead ends” [34, Definition 6.1.1], are not contractible. When restricted to graphs without multiple edges or loops, these are edges connected to univalent internal vertices. But in \({{\mathrm{Tw}}}\texttt {Gra}_{n}\), edges connecting two internal vertices that are both univalent are contractible (see [53, Fig. 3] for the dual picture). This does not change \({\widehat{I}}\), which vanishes on graphs with univalent internal vertices anyway [34, Lemma 9.3.8]. Note that \(\widehat{{\mathcal {D}}}\) is not a Hopf cooperad [34, Example 7.3.2] due to multiple edges.
Remark 13
This Hopf cooperad is not isomorphic to the Hopf cooperad \({\mathcal {D}}\) from [34, Section 6.5]. We allow internal vertices of any valence, whereas in \({\mathcal {D}}\) internal vertices must be at least trivalent. There is a quotient map \(\texttt {Graphs}_{n} \rightarrow {\mathcal {D}}\), which is a quasiisomorphism by [53, Proposition 3.8]. The statement of [53, Proposition 3.8] is actually about the dual operads, but as we work over a field and the spaces we consider have finitetype cohomology, this is equivalent. The notation is also different: the couple \((\texttt {Graphs}_{n}, \texttt {fGraphs}_{n,c})\) in [53] denotes \(({\mathcal {D}}^{\vee }, \texttt {Graphs}_{n}^{\vee })\) in [34].
One checks that the two morphisms \({\texttt {e}_{n}^{\vee }}\leftarrow {{\mathrm{Tw}}}\texttt {Gra}_{n} \rightarrow \varOmega _{\mathrm {PA}}^{*}(\texttt {FM}_{n})\) factor through the quotient (the first one because graphs with internal vertices are sent to zero, the second one because \(\omega \) vanishes on graphs with only internal vertices by [34, Lemma 9.3.7]). The resulting zigzag \({\texttt {e}_{n}^{\vee }}\leftarrow \texttt {Graphs}_{n} \rightarrow \varOmega _{\mathrm {PA}}^{*}(\texttt {FM}_{n})\) is then a zigzag of weak equivalence of Hopf cooperads thanks to the proof of [28, Theorem 2] (or [34, Theorem 8.1] and the discussion at the beginning of [34, Chapter 10]), combined with the comparison between \({\mathcal {D}}\) and \(\texttt {Graphs}_{n}\) from [53, Proposition 3.8] (see Remark 13).
2.7 Poincaré duality CDGA models
The model for \(\varOmega _{\mathrm {PA}}^{*}(\texttt {FM}_{M})\) relies on a Poincaré duality model of M. We mostly borrow the terminology and notation from [32].
Theorem 14
Proof
We refer to Sect. 1.3 for a reminder on trivial forms. We pick a minimal model R of the manifold M (over \({\mathbb {R}}\)) and a quasiisomorphism from R to the subCDGA of trivial forms in \(\varOmega _{PA}^*(M)\), which exists because the subCDGA of trivial forms is quasiisomorphic to \(\varOmega _{PA}^*(M)\) (see Sect. 1.3), and hence, is itself a real model for M. We compose this new quasiisomorphism this the inclusion to eventually get a quasiisomorphism \(\sigma : R\rightarrow \varOmega _{PA}^*(M)\) which factors through the subCDGA of trivial forms, and we set \(\varepsilon = \int _M\sigma (): R\rightarrow {\mathbb {R}}[n]\). The CDGA R is of finite type because M is a closed manifold. Hence, we can apply the Lambrechts–Stanley Theorem [32, Theorem 1.1] to the pair \((R,\epsilon )\) to get the Poincaré duality algebra A of our statement. \(\square \)
Proposition 15
One can choose the zigzag of Theorem 14 such there exists a symmetric cocycle \(\varDelta _{R} \in R \otimes R\) of degree n satisfying \((\rho \otimes \rho )(\varDelta _{R}) = \varDelta _{A}\). If \(\chi (M) = 0\) we can moreover choose it so that \(\mu _{R}(\varDelta _{R}) = 0\).
Proof
We follow closely the proof of [32] to obtain the result. Recall that the proof of [32] has two different cases: \(n \le 6\), where the manifold is automatically formal and hence \(A = H^{*}(M)\), and \(n \ge 7\), 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 \(n \ge 7\). When \(n \ge 7\), the proof of Lambrechts and Stanley builds a zigzag of weak equivalences \(A \xleftarrow {\rho } R \leftarrow R' \rightarrow \varOmega _{\mathrm {PA}}^{*}(M)\), where \(R'\) is the minimal model of M, the CDGA R is obtained from \(R'\) by successively adjoining generators of degree \(\ge n/2+1\), and the Poincaré duality CDGA A is a quotient of R by an ideal of “orphans”. We let \(\varepsilon : R' \rightarrow {\mathbb {R}}[n]\) be the composite \(R' \rightarrow \varOmega _{\mathrm {PA}}^{*}(M) \xrightarrow {\int _{M}} {\mathbb {R}}[n]\).
The minimal model \(R'\) is quasifree, and since M is simply connected it is generated in degrees \(\ge 2\). The CDGA R is obtained from \(R'\) by a cofibrant cellular extension, adjoining cells of degree greater than 2. It follows that R is cofibrant and quasifreely generated in degrees \(\ge 2\). Composing with \(R' \rightarrow \varOmega _{\mathrm {PA}}^{*}(M)\) yields a morphism \(\sigma : R \rightarrow \varOmega _{\mathrm {PA}}^{*}(M)\) and we therefore get a zigzag \(A \leftarrow R \rightarrow \varOmega _{\mathrm {PA}}^{*}(M)\).
The morphism \(\rho \) is a quasiisomorphism, so there exists some cocycle \({{\tilde{\varDelta }}} \in R \otimes R\) such that \(\rho ({{\tilde{\varDelta }}}) = \varDelta _{A} + d\alpha \) for some \(\alpha \). By surjectivity of \(\rho \) (it is a quotient map) there is some \(\beta \) such that \(\rho (\beta ) = \alpha \); we let \(\varDelta ' = {{\tilde{\varDelta }}}  d\beta \), and now \(\rho (\varDelta ') = \varDelta _{A}\).
Let us assume for the moment that \(\chi (M) = 0\). Then the cocycle \(\mu _{R}(\varDelta ') \in R\) satisfies \(\rho (\mu _{R}(\varDelta ')) = \mu _{A}(\varDelta _{A}) = 0\), i.e. it is in the kernel of \(\rho \). It follows that the cocycle \(\varDelta '' = \varDelta '  \mu _{R}(\varDelta ') \otimes 1\) is still mapped to \(\varDelta _{A}\) by \(\rho \), and satisfies \(\mu _{R}(\varDelta '') = 0\). If \(\chi (M) \ne 0\) we just let \(\varDelta '' = \varDelta '\). Finally we symmetrize \(\varDelta ''\) to get the \(\varDelta _{R}\) of the lemma, which satisfies all the requirements.
Let us now deal with the case \(n \le 6\). The CDGA \(\varOmega _{\mathrm {PA}}^{*}(M)\) is formal [40, Proposition 4.6]. We choose \(A = (H^{*}(M), d_{A} = 0)\), and R to be the minimal model of M, which maps into both A and \(\varOmega _{\mathrm {PA}}^{*}(M)\) by quasiisomorphisms. The rest of the proof is now identical to the previous case. \(\square \)
2.8 The Lambrechts–Stanley CDGAs
We now give the definition of the CDGA \(\texttt {G}_{A}(k)\) from [33, Definition 3.4], where it is called F(A, k).
3 The Hopf right comodule model \(\texttt {G}_{A}\)
In this section we describe the Hopf right \({\texttt {e}_{n}^{\vee }}\)comodule derived from the LS CDGAs of Sect. 1.8. From now on we fix a simply connected smooth closed manifold M. Following Sect. 1.4, we endow M with a fixed semialgebraic structure. Note that for now, we do not impose any further conditions on M, but a key argument (Proposition 45) will require \(\dim M \ge 4\). We also fix a arbitrary Poincaré duality CDGA model A of M. We then define the right comodule structure of \(\texttt {G}_{A}\) as follows, using the cooperad structure of \({\texttt {e}_{n}^{\vee }}\) given by Eq. (14):
Proposition 16
In informal terms, \(\circ _{W}^{\vee }\) multiplies together all the elements of A indexed by W on the \(A^{\otimes U}\) factor and indexes the result by \(* \in U/W\), while it applies the cooperadic structure map of \({\texttt {e}_{n}^{\vee }}\) on the other factor. Note that if \(W = \varnothing \), then \(\circ _{W}^{\vee }\) adds a factor of \(1_{A}\) (the empty product) indexed by \(* \in U/\varnothing = U \sqcup \{*\}\).
Proof
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 cooperadic structure of \({\texttt {e}_{n}^{\vee }}\).

If \(u,v \in W\), then \(\circ _{W}^{\vee }(\iota _{u}(a) \omega _{uv}) = \iota _{*}(a) \otimes \omega _{uv}\), which is also equal to \(\circ _{W}^{\vee }(\iota _{v}(a) \omega _{uv})\).

Otherwise, we have \(\circ _{W}^{\vee }(\iota _{u}(a) \omega _{uv}) = \iota _{[u]}(a) \omega _{[u][v]} \otimes 1\), which is equal to \(\iota _{[v]}(a) \omega _{[u][v]} \otimes 1 = \circ _{W}^{\vee }(\iota _{v}(a) \omega _{uv})\) modulo the relations.

If \(u,v \in W\), then \(\circ _{W}^{\vee }(d\omega _{uv}) = \iota _{*}(\mu _{A}(\varDelta _{A})) = 0\), while by definition \(d(\circ _{W}^{\vee }(\omega _{uv})) = d(1 \otimes \omega _{uv}) = 0\).

Otherwise, \(\circ _{W}^{\vee }(d \omega _{uv}) = \iota _{[u][v]}(\varDelta _{A}) \otimes 1\), which is equal to \(d (\circ _{W}^{\vee }(\omega _{uv})) = d(\omega _{[u][v]} \otimes 1)\).
4 Labeled graph complexes
If \(\chi (M) = 0\), then the collections \(\texttt {G}_{A}\) and \(\texttt {Graphs}_{R}\) are Hopf right comodules respectively over \({\texttt {e}_{n}^{\vee }}\) and over \(\texttt {Graphs}_{n}\), and the left arrow is a morphism of comodules between \((\texttt {G}_{A}, {\texttt {e}_{n}^{\vee }})\) and \((\texttt {Graphs}_{R}, \texttt {Graphs}_{n})\). When M is moreover framed, \(\varOmega _{\mathrm {PA}}^{*}(\texttt {FM}_{M})\) is a Hopf right comodule over \(\varOmega _{\mathrm {PA}}^{*}(\texttt {FM}_{n})\), and the right arrow is then a morphism from \((\texttt {Graphs}_{R}, \texttt {Graphs}_{n})\) to \((\varOmega _{\mathrm {PA}}^{*}(\texttt {FM}_{M}), \varOmega _{\mathrm {PA}}^{*}(\texttt {FM}_{n}))\).
In order to deal with signs more easily and make sure that the differential squares to zero, we want to use the formalism of operadic twisting, as in the definition of \(\texttt {Graphs}_{n}\). But when \(\chi (M) \ne 0\) there is no comodule structure, so we make a detour through graphs with loops (Sect. 3.1 below), see Remark 31.
4.1 Graphs with loops and multiple edges
Remark 17
When n is even, \(e_{uv}^{2} = 0\) since \(\deg e_{uv} = n1\) is odd; and when n is odd, the relation \(e_{uu} = (1)^{n} e_{uu}\) implies \(e_{uu} = 0\). In other words, for even n, there are no multiple edges, and for odd n, there are no loops [53, Remark 3.1].
Remark 18
The Hopf cooperad \({{\mathrm{Tw}}}\texttt {Gra}_{n}^{\circlearrowleft }\) is slightly different from \(\widehat{{\mathcal {D}}}\) from [34, Section 6]. First the cocomposition is different, and the first term of the RHS in Eq. (26) would not appear in \(\widehat{{\mathcal {D}}}\). The differential is also slightly different: an edge connected to two univalent internal vertices – hence disconnected from the rest of the graph – is contractible here (see [53, Section 3] and Remark 12). This fixes the failure of \(\widehat{{\mathcal {D}}}\) to be a cooperad [34, Example 7.3.2].
4.2 Labeled graphs with only external vertices: \(\texttt {Gra}_{R}\)
We construct a collection of CDGAs \(\texttt {Gra}_{R}\), corresponding to the first step in the construction of \(\texttt {Graphs}_{n}\) of Sect. 1.6. We first apply the formalism of Sect. 1.7 to \(\varOmega _{\mathrm {PA}}^{*}(M)\) in order to obtain a Poincaré duality CDGA out of M, thanks to Theorem 14. We thus fix a zigzag of quasiisomorphisms \(A \xleftarrow {\rho } R \xrightarrow {\sigma } \varOmega _{\mathrm {PA}}^{*}(M)\), where A is a Poincaré duality CDGA, R is a cofibrant CDGA, and \(\sigma \) factors through the subCDGA of trivial forms (see Sect. 1.3).
Recall the definition of the diagonal cocycle \(\varDelta _{A} \in (A \otimes A)^{n}\) from Eq. (21). Recall also Proposition 15, where we fixed a symmetric cocycle \(\varDelta _{R} \in (R \otimes R)^{n}\) such that \((\rho \otimes \rho )(\varDelta _{R}) = \varDelta _{A}\). Moreover recall that if \(\chi (M) = 0\), then \(\mu _{A}(\varDelta _{A}) = 0\), and we choose \(\varDelta _{R}\) such that \(\mu _{R}(\varDelta _{R}) = 0\) too.
Definition 19
This CDGA is welldefined because \(\texttt {Gra}_{n}^{\circlearrowleft }(U)\) is free as a CDGA, hence \(\texttt {Gra}^{\circlearrowleft }_{R}(U)\) is a relative Sullivan algebra in the terminology of [13, Section 14].
Remark 20
This definition is valid for any CDGA R and any symmetric cocycle \(\varDelta _{R}\). We need R as in Proposition 15 to connect \(\texttt {Gra}_{R}^{\circlearrowleft }\) with \(\texttt {G}_{A}\) and \(\varOmega _{\mathrm {PA}}^{*}(\texttt {FM}_{M})\).
Remark 21
It follows that the differential of a loop is \(d e_{uu} = \iota _{uu}(\varDelta _{R}) = \iota _{u}(\mu _{R}(\varDelta _{R}))\), which is zero when \(\chi (M) = 0\).
Proposition 22
The collection \(\texttt {Gra}^{\circlearrowleft }_{R}(U)\) forms a Hopf right \(\texttt {Gra}^{\circlearrowleft }_{n}\)comodule.
This is true even if \(\chi (M) \ne 0\) thanks to the introduction of the loops..
Proof
If \(\chi (M) \ne 0\), we cannot directly map \(\texttt {Gra}^{\circlearrowleft }_{R}\) to \(\varOmega _{\mathrm {PA}}^{*}(\texttt {FM}_{M})\), as the Euler class in \(\varOmega _{\mathrm {PA}}^{*}(M)\) would need to be the boundary of the image of the loop \(e_{11} \in \texttt {Gra}^{\circlearrowleft }_{R}({\underline{1}})\). We thus define a subCDGA which will map to \(\varOmega _{\mathrm {PA}}^{*}(\texttt {FM}_{M})\) whether \(\chi (M)\) vanishes or not.
Definition 23
For a given finite set U, let \(\texttt {Gra}_{R}(U)\) be the submodule of \(\texttt {Gra}_{R}^{\circlearrowleft }(U)\) spanned by graphs without loops.
One has to be careful with the notation. While \(\texttt {Gra}_{R}^{\circlearrowleft }(U) = R^{\otimes U} \otimes \texttt {Gra}_{n}^{\circlearrowleft }\), it is not true that \(\texttt {Gra}_{R}(U) = R^{\otimes U} \otimes \texttt {Gra}_{n}(U)\): in \(\texttt {Gra}_{n}(U)\), multiple edges are forbidden, whereas they are allowed in \(\texttt {Gra}_{R}(U)\).
Proposition 24
The space \(\texttt {Gra}_{R}(U)\) is a subCDGA of \(\texttt {Gra}_{R}^{\circlearrowleft }(U)\). If \(\chi (M) = 0\) the collection \(\texttt {Gra}_{R}\) assembles to form a Hopf right \(\texttt {Gra}_{n}\)comodule.
Proof
Clearly, neither the splitting part of the differential nor the internal differential coming from R can create new loops, nor can the product of two graphs without loops contain a loop, thus \(\texttt {Gra}_{R}(U)\) is indeed a subCDGA of \(\texttt {Gra}_{R}^{\circlearrowleft }(U)\). If \(\chi (M) = 0\), the proof that \(\texttt {Gra}_{R}\) is a \(\texttt {Gra}_{n}\)comodule is almost identical to the proof of Proposition 22, except that we need to use \(\mu _{R}(\varDelta _{R}) = 0\) to check that \(d(\circ _{W}^{\vee }(e_{uv})) = \circ ^{\vee }_{W}(d(e_{uv}))\) when \(u,v \in W\). \(\square \)
4.3 The propagator
To define \(\omega ' : \texttt {Gra}_{R} \rightarrow \varOmega _{\mathrm {PA}}(\texttt {FM}_{M})\), we need a “propagator” \(\varphi \in \varOmega _{\mathrm {PA}}^{n1}(\texttt {FM}_{M}(2))\), for which a reference is [7, Section 4].
Recall from Eq. (2) the projections \(p_{u} : \texttt {FM}_{M}(U) \rightarrow M\) and \(p_{uv} : \texttt {FM}_{M}(U) \rightarrow \texttt {FM}_{M}({\underline{2}})\). Recall moreover the sphere bundle \(p : \partial \texttt {FM}_{M}({\underline{2}}) \rightarrow M\) defined in Eq. (3), which is trivial when M is framed, with the isomorphism \(M \times S^{n1} \xrightarrow {\circ _{1}} \partial \texttt {FM}_{M}({\underline{2}})\) from Eq. (4). We denote by \((p_{1}, p_{2}) : \texttt {FM}_{M}({\underline{2}}) \rightarrow M \times M\) the product of the two canonical projections.
Proposition 25
([6, Propositions 7 and 87]) There exists a form \(\varphi \in \varOmega _{\mathrm {PA}}^{n1}(\texttt {FM}_{M}({\underline{2}}))\) such that \(\varphi ^{21} = (1)^{n} \varphi \), \(d \varphi = (p_{1}, p_{2})^{*}((\sigma \otimes \sigma )(\varDelta _{R}))\) and such that the restriction of \(\varphi \) to \(\partial \texttt {FM}_{M}(2)\) is a global angular form, i.e. it is a volume form of \(S^{n1}\) when restricted to each fiber. When M is framed one can moreover choose \(\varphi _{\partial \texttt {FM}_{M}(2)} = 1 \times \mathrm {vol}_{S^{n1}} \in \varOmega ^{n1}_{PA}(M \times S^{n1})\). This propagator can moreover be chosen to be a trivial form (see Sect. 1.3).
The proofs of [6] relies on earlier computations given in [7], where this propagator is studied in detail. One can see from the proofs of [7, Section 4] that \(d\varphi \) can in fact be chosen to be any pullback of a form cohomologous to the diagonal class \(\varDelta _{M} \in \varOmega _{\mathrm {PA}}^{n}(M \times M)\). We will make further adjustments to the propagator \(\varphi \) in Proposition 42. Recall \(p_{u}\), \(p_{uv}\) from Eq. (2).
Proposition 26
Proof

If one of u, v, or both, is not in W, then the equality \(\circ _{W}^{\vee }(\omega '(e_{uv})) = (\omega ' \otimes \omega ')(\circ _{W}^{\vee }(e_{uv})).\) is clear.
 Otherwise suppose \(\{u,v\} \subset W\). We may assume that \(U = W = {\underline{2}}\) (it suffices to pull back the result along \(p_{uv}\) to get the general case), so that we are considering the insertion of an infinitesimal configuration \(M \times \texttt {FM}_{n}({\underline{2}}) \rightarrow \texttt {FM}_{M}({\underline{2}})\). This insertion factors through the boundary \(\partial \texttt {FM}_{M}({\underline{2}})\). We have (see Definition 25):Going back to the general case, we find:$$\begin{aligned}&\circ _{{\underline{2}}}^{\vee }(\varphi ) = 1 \otimes \mathrm {vol}_{S^{n1}} \in \varOmega _{\mathrm {PA}}^{*}(M) \otimes \varOmega _{\mathrm {PA}}^{*}(\texttt {FM}_{n}({\underline{2}})) \\&\qquad = \varOmega _{\mathrm {PA}}^{*}(M) \otimes \varOmega _{\mathrm {PA}}^{*}(S^{n1}). \end{aligned}$$which is indeed the image of \(\circ _{W}^{\vee }(\omega _{uv}) = 1 \otimes \omega _{uv}\) by \(\omega ' \otimes \omega '\). \(\square \)$$\begin{aligned} \circ _{W}^{\vee }(\omega '(e_{uv})) = \circ _{W}^{\vee }(p_{uv}^{*}(\varphi )) = 1 \otimes p_{uv}^{*}(\mathrm {vol}_{S^{n1}}), \end{aligned}$$
4.4 Labeled graphs with internal and external vertices: \({{\mathrm{Tw}}}\texttt {Gra}_{R}\)
The general framework of operadic twisting, recalled in Sect. 1.5, shows that to twist a right (co)module, one only needs to twist the (co)operad. Since our cooperad is onedimensional in arity zero, the comodule inherits a Hopf comodule structure too (Lemma 10).
Definition 27
The twisted labeled graph comodule \({{\mathrm{Tw}}}\texttt {Gra}^{\circlearrowleft }_{R}\) is a Hopf right \(({{\mathrm{Tw}}}\texttt {Gra}^{\circlearrowleft }_{n})\)comodule obtained from \(\texttt {Gra}^{\circlearrowleft }_{R}\) by twisting with respect to the Maurer–Cartan element \(\mu \in (\texttt {Gra}^{\circlearrowleft }_{n})^{\vee }({\underline{2}})\) of Sect. 1.6.
We now explicitly describe this comodule in terms of graphs. The dgmodule \({{\mathrm{Tw}}}\texttt {Gra}^{\circlearrowleft }_{R} (U)\) is spanned by graphs with two kinds of vertices, external vertices corresponding to elements of U, and indistinguishable internal vertices (usually drawn in black). The degree of an edge is \(n1\), the degree of an external vertex is 0, while the degree of an internal vertex is \(n\). All the vertices are labeled by elements of R, and their degree is added to the degree of the graph.
Remark 28
An edge connected to a univalent internal vertex is contractible in \({{\mathrm{Tw}}}\texttt {Gra}^{\circlearrowleft }_{R}\), though this is not the case in \({{\mathrm{Tw}}}\texttt {Gra}_{n}^{\circlearrowleft }\). Indeed, if we go back to the definition of the differential in a twisted comodule (Eq. (11)), we see that the Maurer–Cartan element \(\mu \) (Eq. (18)) only acts on the right of the graph. Therefore, there is no term to cancel out the contraction of such edges, as was the case in \({{\mathrm{Tw}}}\texttt {Gra}_{n}\) (see the discussion in Sect. 1.6 about the differential). In Eq. (28), the only edge would not be considered as contractible in \({{\mathrm{Tw}}}\texttt {Gra}_{n}\) if we forgot the labels, but it is in \({{\mathrm{Tw}}}\texttt {Gra}_{R}\).
Lemma 29
The subspace \({{\mathrm{Tw}}}\texttt {Gra}_{R}(U) \subset {{\mathrm{Tw}}}\texttt {Gra}^{\circlearrowleft }_{R}(U)\) spanned by graphs with no loops is a subCDGA.
Proof
It is clear that this defines a subalgebra. We need to check that it is preserved by the differential, i.e. that the differential cannot create new loops if there are none in a graph. This is clear for the internal differential coming from R and for the splitting part of the differential. The contracting part of the differential could create a loop from a double edge. However for even n multiple edges are zero for degree reasons, and for odd n loops are zero because of the antisymmetry relation (see Remark 21). \(\square \)
Note that despite the notation, \({{\mathrm{Tw}}}\texttt {Gra}_{R}\) is a priori not defined as the twisting of the \(\texttt {Gra}_{n}\)comodule \(\texttt {Gra}_{R}\): when \(\chi (M) \ne 0\), the collection \(\texttt {Gra}_{R}\) is not even a \(\texttt {Gra}_{n}\)comodule. However, the following proposition is clear and shows that we can get away with this abuse of notation:
Proposition 30
If \(\chi (M) = 0\), then \({{\mathrm{Tw}}}\texttt {Gra}_{R}\) assembles to a right Hopf \(({{\mathrm{Tw}}}\texttt {Gra}_{n})\)comodule, isomorphic to the twisting of the right Hopf \(\texttt {Gra}_{n}\)comodule \(\texttt {Gra}_{R}\) of Definition 23. \(\square \)
Remark 31
We could have defined the algebra \({{\mathrm{Tw}}}\texttt {Gra}_{R}\) explicitly in terms of graphs, and defined the differential d using an adhoc formula. The difficult part would have then been to check that \(d^{2} = 0\) (involving difficult signs), which is a consequence of the general operadic twisting framework.
4.5 The map \(\omega : {{\mathrm{Tw}}}\texttt {Gra}_{R} \rightarrow \varOmega _{\mathrm {PA}}^{*}(\texttt {FM}_{M})\)
This section is dedicated to the proof of the following proposition.
Proposition 32
Recall that in general, it is not possible to consider integrals along fibers of arbitrary PA forms, see [23, Section 9.4]. However, here, the image of \(\sigma \) is included in the subCDGA of trivial forms in \(\varOmega _{\mathrm {PA}}^{*}(M)\), and the propagator is a trivial form (see Proposition 25), therefore the integral \((p_{U})_{*}(\omega '(\varGamma ))\) exists.
The proof of the compatibility with the Hopf structure and, in the framed case, the comodule structure, is formally similar to the proof of the same facts about \(\omega : {{\mathrm{Tw}}}\texttt {Gra}_{n} \rightarrow \varOmega _{\mathrm {PA}}^{*}(\texttt {FM}_{n})\). We refer to [34, Sections 9.2, 9.5]. The proof is exactly the same proof, but writing \(\texttt {FM}_{M}\) or \(\texttt {FM}_{n}\) instead of \(C[]\) and \(\varphi \) instead of \(\mathrm {vol}_{S^{n1}}\) in every relevant sentence, and recalling that when M is framed, we choose \(\varphi \) such that \(\circ _{{\underline{2}}}^{\vee }(\varphi ) = 1 \otimes \mathrm {vol}_{S^{n1}}\).
Remark 33
The space \(E^{\partial }\) is neither \(\partial E\) nor \(\bigcup _{b \in B} \pi ^{1}(b) \cap \partial E\) in general. (Consider for example the projection on the first coordinate \([0,1]^{\times 2} \rightarrow [0,1]\).)
Lemma 34
Proof
Lemma 35
Proof

\(\varGamma _{V/W} \in {{\mathrm{Tw}}}\texttt {Gra}_{R}(U)\) is the graph with W collapsed to a vertex and \(U \hookrightarrow V/W\) is identified with its image;

\(\varGamma _{W} \in {{\mathrm{Tw}}}\texttt {Gra}_{n}(W)\) is the full subgraph of \(\varGamma \) with vertices W and the labels removed.
We can now finish proving Proposition 32. We combine Eq. (29) and Lemma 35, and apply Stokes’ formula to \(d \omega (\varGamma )\) to show that it is equal to \(\omega (d \varGamma ) = \omega (d_{R} \varGamma + d_{\mathrm {split}} \varGamma ) + \omega (d_{\mathrm {contr}} \varGamma )\).
4.6 Reduced labeled graphs: \(\texttt {Graphs}_R\)
The last step in the construction of \(\texttt {Graphs}_{R}\) is the reduction of \({{\mathrm{Tw}}}\texttt {Gra}_{R}\) so that it has the right cohomology. We borrow the terminology of Campos–Willwacher [6] for the next two definitions.
Definition 36
The full graph complex \(\mathrm {fGC}_{R}\) is the CDGA \({{\mathrm{Tw}}}\texttt {Gra}_{R}(\varnothing )\). It consists of labeled graphs with only internal vertices, and the product is disjoint union of graphs.
Remark 37
The adjective “full” refers to the fact that graphs are possibly disconnected and have vertices of any valence in \(\mathrm {fGC}_{R}\).
As an algebra, \(\mathrm {fGC}_{R}\) is free and generated by connected graphs. In general we will call internal components the connected components of a graph that only contain internal vertices. The full graph complex naturally acts on \({{\mathrm{Tw}}}\texttt {Gra}_{R}(U)\) by adding extra internal components.
Definition 38
The partition function \(Z_{\varphi }: \mathrm {fGC}_{R} \rightarrow {\mathbb {R}}\) is the restriction of \(\omega : {{\mathrm{Tw}}}\texttt {Gra}_{R} \rightarrow \varOmega _{\mathrm {PA}}^{*}(\texttt {FM}_{M})\) to \(\mathrm {fGC}_{R} = {{\mathrm{Tw}}}\texttt {Gra}_{R}(\varnothing ) \rightarrow \varOmega _{\mathrm {PA}}^{*}(\texttt {FM}_{M}(\varnothing )) = \varOmega _{\mathrm {PA}}^{*}(\mathrm {pt}) = {\mathbb {R}}\).
Remark 39
The expression “partition function” comes from the mathematical physics literature, more specifically Chern–Simons invariant theory, where it refers to the partition function of a quantum field theory.
Definition 40
In other words, a graph of the type \(\varGamma \sqcup \gamma \) containing an internal component \(\gamma \in \mathrm {fGC}_{R}\) is identified with \(Z_{\varphi }(\gamma ) \cdot \varGamma \). It is spanned by representative classes of graphs with no internal connected component; we call such graphs reduced. The notation is meant to evoke the fact that \(\texttt {Graphs}^{\varphi }_{R}\) depends on the choice of the propagator \(\varphi \), unlike the collection \(\texttt {Graphs}^{\varepsilon }_{R}\) that will appear in Sect. 4.1.
Proposition 41
The map \(\omega : {{\mathrm{Tw}}}\texttt {Gra}_{R}(U) \rightarrow \varOmega _{\mathrm {PA}}^{*}(\texttt {FM}_{M}(U))\) defined in Proposition 32 factors through the quotient defining \(\texttt {Graphs}^{\varphi }_{R}\).
If \(\chi (M) = 0\), then \(\texttt {Graphs}^{\varphi }_{R}\) forms a Hopf right \(\texttt {Graphs}_{n}\)comodule. If moreover M is framed, then the map \(\omega \) defines a Hopf right comodule morphism.
Proof
Equation (32) immediately implies that \(\omega \) factors through the quotient.
The vanishing lemmas shows that if \(\varGamma \in {{\mathrm{Tw}}}\texttt {Gra}_{n}(U)\) has internal components, then \(\omega (\varGamma )\) vanishes by [34, Lemma 9.3.7], so it is straightforward to check that if \(\chi (M) = 0\), then \(\texttt {Graphs}^{\varphi }_{R}\) becomes a Hopf right comodule over \(\texttt {Graphs}_{n}\). It is also clear that for M framed, the quotient map \(\omega \) remains a Hopf right comodule morphism. \(\square \)
Proposition 42
From now on and until the end, we assume that \(\varphi \) satisfies (P4).
Remark 43
The additional property (P5) of the paper mentioned above would be helpful in order to get a direct morphism \(\texttt {Graphs}^{\varphi }_{R} \rightarrow \texttt {G}_{A}\), because then the partition function would vanish on all connected graphs with at least two vertices. However we run into difficulties when trying to adapt the proof in the setting of PA forms, mainly due to the lack of an operator \(d_{M}\) acting on \(\varOmega _{\mathrm {PA}}^{*}(M \times N)\) differentiating “only in the first slot”.
Corollary 44
The morphism \(\omega \) vanishes on graphs containing univalent internal vertices.
Proof
Almost everything we have done so far works in full generality. We now prove a fact which sets a class of manifolds apart.
Proposition 45
Assume that M is simply connected and that \(\dim M \ge 4\). Then the partition function \(Z_{\varphi }\) vanishes on any connected graph with no bivalent vertices labeled by \(1_{R}\) and containing at least two vertices.
Remark 46
If \(\gamma \in \mathrm {fGC}_{R}\) has only one vertex, labeled by x, then \(Z_{\varphi }(\gamma ) = \int _{M} \sigma (x)\) which can be nonzero.
Proof
Let \(\gamma \in \mathrm {fGC}_{R}\) be a connected graph with at least two vertices and no bivalent vertices labeled by \(1_{R}\). By Corollary 44, we can assume that all the vertices of \(\gamma \) are at least bivalent. By hypothesis, if a vertex is bivalent then it is labeled by an element of \(R^{> 0} = R^{\ge 2}\).
Remark 47
When \(n = 3\), the manifold M is the 3sphere \(S^{3}\) by Perelman’s proof of the Poincaré conjecture [41, 42]. The partition function \(Z_{\varphi }\) is conjectured to be trivial on \(S^{3}\) for a proper choice of framing, thus bypassing the need for the above degree counting argument. See also Proposition 80.
We will also need the following technical property of \(\mathrm {fGC}_{R}\).
Lemma 48
The CDGA \(\mathrm {fGC}_{R}\) is cofibrant.
Proof
We filter \(\mathrm {fGC}_{R}\) by the number of edges, defining \(F_{s}\mathrm {fGC}_{R}\) to be the submodule of \(\mathrm {fGC}_{R}\) spanned by graphs of \(\gamma \) such that all the connected components \(\gamma \) have at most s edges. The differential of \(\mathrm {fGC}_{R}\) can only decrease (\(d_{\mathrm {split}}\) and \(d_{\mathrm {contr}}\)) or leave constant (\(d_{R}\)) the number of edges. Moreover \(F_{s}\mathrm {fGC}_{R}\) is clearly stable under products (disjoint unions), hence \(F_{s}\mathrm {fGC}_{R}\) is a subCDGA of \(\mathrm {fGC}_{R}\). It is also clear that \(\mathrm {fGC}_{R} = {\text {colim}}_{s} F_{s}\mathrm {fGC}_{R}\). We will prove that \(F_{0}\mathrm {fGC}_{R}\) is cofibrant, and that each \(F_{s}\mathrm {fGC}_{R} \subset F_{s+1}\mathrm {fGC}_{R}\) is a cofibration.
The CDGA \(F_{0}\mathrm {fGC}_{R}\) is the free CDGA on graphs with a single vertex labeled by R. In other words, \(F_{0}\mathrm {fGC}_{R} = S(R, d_{R})\) is the free symmetric algebra on the dgmodule R, and any free symmetric algebra on a dgmodule is cofibrant.
Let us now show that \(F_{s}\mathrm {fGC}_{R} \subset F_{s+1}\mathrm {fGC}_{R}\) is a cofibration for any \(s \ge 0\). We will show that it is in fact a “relative Sullivan algebra” [13, Section 14]. As a CDGA, we have \(F_{s+1}\mathrm {fGC}_{R} = (F_{s}\mathrm {fGC}_{R} \otimes S(V_{s+1}), d)\), where \(V_{s+1}\) is the graded module of connected graphs 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 [13, Section 12]. In other words, \(R = (S(W), d)\) where W is increasingly and exhaustively filtered by \(W(1) = 0 \subset \dots \subset W(t) \subset \dots \subset W\) such that \(d(W(t)) \subset S(W(t1))\). This induces a filtration on R by defining \(R(t) {:}{=}\bigoplus _{t_{1} + \dots + t_{r} = t} \bigl ( V(t_{1}) \otimes \dots \otimes V(t_{r}) \bigr )_{\varSigma _{r}}\).
This in turns induces an increasing and exhaustive filtration on \(V_{s+1}\) by submodules \(V_{s+1}(t)\) as follows. A connected graph \(\gamma \in V_{s+1}\) is in \(V_{s+1}(t)\) if each label \(x_{i} \in R\) of a vertex of \(\gamma \) belongs to the filtration \(R(t_{i})\) such that \(\sum t_{i} = t\). It is then immediate to check that \(d(V_{s+1}(t+1)) \subset V_{s} \otimes S(V_{s+1}(t))\). Indeed, if \(\gamma \in V_{s+1}(t+1)\), then \(d_{\mathrm {split}}\gamma \) and \(d_{\mathrm {contr}}\gamma \in V_{s}\), because both strictly decrease the number of edges. And \(d_{R}\gamma \in V_{s+1}(t)\) because the internal differential of R decreases the filtration of R. \(\square \)
5 From the model to forms via graphs
In this section we connect \(\texttt {G}_{A}\) to \(\varOmega _{\mathrm {PA}}^{*}(\texttt {FM}_{M})\) and we prove that the connecting morphisms are quasiisomorphisms. We assume that M is a simply connected closed smooth manifold with \(\dim M \ge 4\) (see Proposition 45).
5.1 Construction of the morphism to \(\texttt {G}_{A}\)
Proposition 49
For each finite set U, there is a CDGA morphism \(\rho '_{*} : \texttt {Gra}_{R}(U) \rightarrow \texttt {G}_{A}(U)\) given by \(\rho \) on the \(R^{\otimes U}\) factor and sending the generators \(e_{uv}\) to \(\omega _{uv}\) on the \(\texttt {Gra}_{n}\) factor. When \(\chi (M) = 0\), this defines a Hopf right comodule morphism \((\texttt {Gra}_{R}, \texttt {Gra}_{n}) \rightarrow (\texttt {G}_{A}, {\texttt {e}_{n}^{\vee }})\). \(\square \)
If we could find a propagator for which property (P5) held (see Remark 43), then we could just send all graphs containing internal vertices to zero and obtain an extension \(\texttt {Graphs}^{\varphi }_{R} \rightarrow \texttt {G}_{A}\). Since we cannot assume that (P5) holds, the definition of the extension is more complex. However we still have Proposition 45, and homotopically speaking, graphs with bivalent vertices are irrelevant.
Definition 50
Let \(\mathrm {fGC}^{0}_{R}\) be the quotient of \(\mathrm {fGC}_{R}\) defined by identifying a disconnected vertex labeled by x with the number \(\varepsilon _{A}(\rho (x))\).
Lemma 51
The subspace \(I \subset \mathrm {fGC}^{0}_{R}\) spanned by graphs with at least one univalent vertex, or at least one bivalent vertex labeled by \(1_{R}\), or at least one label in \(\ker (\rho : R \rightarrow A)\), is a CDGA ideal.
Proof
It is clear that I is an algebra ideal. Let us prove that it is a differential ideal. If one of the labels of \(\varGamma \) is in \(\ker \rho \), then so do all the summands of \(d \varGamma \), because \(\ker \rho \) is a CDGA ideal of R.
If \(\varGamma \) contains a bivalent vertex u labeled by \(1_{R}\), then so does \(d_{R} \varGamma \). In \(d_{\mathrm {split}} \varGamma \), splitting one of the two edges connected to u produces a univalent vertex and hence vanishes in \(\mathrm {fGC}_{R}^{0}\) because the label is \(1_{R}\). In \(d_{\mathrm {contr}} \varGamma \), the contraction of the two edges connected to u cancel each other.
Finally let us prove that if \(\varGamma \) has a univalent vertex u, then \(d\varGamma \) lies in I. It is clear that \(d_{R} \varGamma \in I\). Contracting or splitting the only edge connected to the univalent vertex could remove the univalent vertex. Let us prove that these two summands cancel each other up to \(\ker \rho \).
It is helpful to consider the case pictured in Eq. (28). Let y be the label of the univalent vertex u, and let x be the label of the only vertex incident to u. Contracting the edge yields a new vertex labeled by xy. Due to the definition of \(\mathrm {fGC}^{0}_{R}\), splitting the edge yields a new vertex labeled by \(\alpha {:}{=}\sum _{(\varDelta _{R})} \varepsilon (\rho (x \varDelta ''_{R})) y \varDelta '_{R}\). We thus have \(\rho (\alpha ) = \rho (x) \cdot \sum _{(\varDelta _{A})} \pm \varepsilon _{A}(\rho (y) \varDelta _{A}'') \varDelta _{A}'\).
It is a standard property of the diagonal class that \(\sum _{(\varDelta _{A})} \pm \varepsilon _{A}(a \varDelta _{A}'') \varDelta _{A}' = a\) for all \(a \in A\) (this property is a direct consequence of the definition in Eq. (21)). Applied to \(a = \rho (y)\), it follows from the previous equation that \(\rho (\alpha ) = \pm \rho (xy)\); examining the signs, this summand cancels from the summand that comes from contracting the edge. \(\square \)
Definition 52
The algebra \(\mathrm {fGC}'_{R}\) is the quotient of \(\mathrm {fGC}^{0}_{R}\) by the ideal I.
Note that \(\mathrm {fGC}'_{R}\) is also free as an algebra, with generators given by connected graphs with no isolated vertices, nor univalent vertices, nor bivalent vertices labeled by \(1_{R}\), and where the labels lie in \(R / \ker (\rho ) = A\).
Definition 53
A circular graph is a graph in the shape of a circle and where all vertices are labeled by \(1_{R}\), i.e. graphs of the type \(e_{12} e_{23} \dots e_{(k1)k} e_{k1}\). Let \(\mathrm {fLoop}_{R} \subset \mathrm {fGC}^{0}_{R}\) be the submodule spanned by graphs whose connected components either have univalent vertices or are equal to a circular graphs.
Lemma 54
The submodule \(\mathrm {fLoop}_{R}\) is a subCDGA of \(\mathrm {fGC}_{R}^{0}\).
Proof
The submodule \(\mathrm {fLoop}_{R}\) 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 of Lemma 51, in \(\mathrm {fGC}_{R}^{0}\), if a graph contains a univalent vertex, then so do all the summands of its differential. On a circular graph, the internal differential of R vanish, because all labels are equal to \(1_{R}\). Contracting an edge in a circular graph yields another circular graph, and splitting an edge yields a graph with univalent vertices, which belongs to \(\mathrm {fLoop}_{R}\). \(\square \)
Proposition 55
The sequence \(\mathrm {fLoop}_{R} \rightarrow \mathrm {fGC}^{0}_{R} \rightarrow \mathrm {fGC}'_{R}\) is a homotopy cofiber sequence of CDGAs.
Proof
The CDGA \(\mathrm {fGC}^{0}_{R}\) is freely generated by connected labeled graphs with at least two vertices. It is a quasifree extension of \(\mathrm {fLoop}_{R}\) by the algebra generated by graphs that are not circular and that do not contain any univalent vertices. The homotopy cofiber of the inclusion \(\mathrm {fLoop}_{R} \rightarrow \mathrm {fGC}^{0}_{R}\) is this algebra \(\mathrm {fGC}''_{R}\), generated by graphs that are not circular and do not contain any univalent vertices, together with a differential induced by the quotient \(\mathrm {fGC}^{0}_{R} / (\mathrm {fLoop}_{R})\).
One can then filter by the number of edges. On the first page of the spectral sequence associated to this new filtration, there is only the internal differential \(d_{R}\). Thus on the second page, the vertices are labeled by \(H^{*}(R) = H^{*}(M)\). The contracting part of the differential decreases the new filtration by exactly one, and so on the second page we see all of \(d_{\mathrm {contr}}\).
We can now adapt the proof of [53, Proposition 3.4] to show that on the part of the complex 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 one to the valence of the vertex, thus “breaking” a line of bivalent vertices. These labels break the symmetry (recall the coinvariants in the definition of the twisting) that allow cohomology classes to be produced. \(\square \)
Corollary 56
The morphism \(Z_{\varphi }: \mathrm {fGC}_{R} \rightarrow {\mathbb {R}}\) factors through \(\mathrm {fGC}'_{R}\) in the homotopy category of CDGAs.
Proof
Let us show that \(Z_{\varphi }\) is homotopic to zero when restricted to the ideal defining \(\mathrm {fGC}'_{R} = \mathrm {fGC}_{R}^{0} / I\) as a quotient of \(\mathrm {fGC}_{R}\). Up to rescaling \(\varepsilon _{A}\) by a real coefficient, we may assume that \(\varepsilon _{A} \rho ()\) and \(\int _{M} \sigma ()\) are homotopic, which induces a homotopy (by derivations) on the subCDGA of graphs with no edges. Hence \(Z_{\varphi }\) is homotopic to zero when restricted to the ideal defining \(\mathrm {fGC}_{R}^{0}\) from \(\mathrm {fGC}_{R}\). Moreover the map \(Z_{\varphi }\) vanishes on graphs with univalent vertices by Corollary 44. The degree of a circular graph with k vertices is \(k < 0\) (recall that all the labels are \(1_{R}\) in a circular graph), but \(Z_{\varphi }\) vanishes on graphs of nonzero degree. Hence \(Z_{\varphi }\) vanishes on the connected graphs appearing in the definition of \(\mathrm {fLoop}_{R}\). Therefore, in the homotopy category of CDGAs, \(Z_{\varphi }\) factors through the homotopy cofiber of the inclusion \(\mathrm {fLoop}_{R} \rightarrow \mathrm {fGC}_{R}^{0}\), which is quasiisomorphic to \(\mathrm {fGC}_{R}'\) by Proposition 55. \(\square \)
Definition 57
Definition 58
Explicitly, in \(\texttt {Graphs}^{\varepsilon }_{R}\), all internal components with at least two vertices are identified with zero, whereas an internal component with a single vertex labeled by \(x \in R\) is identified with the number \(\varepsilon _{A}(\rho (x))\).
Lemma 59
The morphism \(Z_{\varphi }' \pi \) is equal to \(Z_{\varepsilon }\).
Proof
This is a rephrasing of Proposition 45. Using the same degree counting argument, all the connected graphs with more than one vertex in \(\mathrm {fGC}'_{R}\) are of positive degree. Since \({\mathbb {R}}\) is concentrated in degree zero, \(Z_{\varphi }' \pi \) must vanish on these graphs, just like \(Z_{\varepsilon }\). Moreover the morphism \(\pi : \mathrm {fGC}_{R} \rightarrow \mathrm {fGC}'_{R} = \mathrm {fGC}^{0}_{R} / I\) factors through \(\mathrm {fGC}^{0}_{R}\), where graphs \(\gamma \) with a single vertex are already identified with the numbers \(Z_{\varepsilon }(\gamma )\). \(\square \)
Proposition 60
Proof
Therefore the functor \({{\mathrm{Tw}}}\texttt {Gra}_{R}(U) \otimes _{\mathrm {fGC}_{R}} ()\) preserves quasiisomorphisms. The two evaluation maps \({{\mathrm{ev}}}_{0}, {{\mathrm{ev}}}_{1} : A_{\mathrm {PL}}^{*}(\varDelta ^{1}) \rightarrow {\mathbb {R}}\) are quasiisomorphisms. It follows that all the maps in the diagram are quasiisomorphisms.
If \(\chi (M) = 0\), the proof that \(\texttt {Graphs}'_{R}\) and \(\texttt {Graphs}^{\varepsilon }_{R}\) assemble to \(\texttt {Graphs}_{n}\)comodules is identical to the proof for \(\texttt {Graphs}^{\varphi }_{R}\) (see Proposition 41). It is also clear that the two zigzags define morphisms of comodules: in \(\texttt {Graphs}_{n}\), as all internal components are identified with zero anyway. \(\square \)
Proposition 61
The CDGA morphisms \(\rho '_{*} : \texttt {Gra}_{R}(U) \rightarrow \texttt {G}_{A}(U)\) extend to CDGA morphisms \(\rho _{*} : \texttt {Graphs}^{\varepsilon }_{R}(U) \rightarrow \texttt {G}_{A}(U)\) by sending all reduced graphs containing internal vertices to zero. If \(\chi (M) = 0\) this extension defines a Hopf right comodule morphism.
Proof
The submodule of reduced graphs containing internal vertices is a multiplicative ideal and a cooperadic coideal, so all we are left to prove is that \(\rho _{*}\) is compatible with differentials. Since \(\rho '_{*}\) was a chain map, we must only prove that if \(\varGamma \) is a reduced graph with internal vertices, then \(\rho _{*}(d \varGamma ) = 0\).
If a summand of \(d\varGamma \) still contains an internal vertex, then it is mapped to zero by definition of \(\rho _{*}\). So we need to look for the summands of the differential that can remove all internal vertices at once.

if it is univalent, then the argument of Lemma 51 shows that contracting the incident edge cancels with the splitting part of the differential;

if it is bivalent, the contracting part has two summands, and both cancel by the symmetry relation \(\iota _{u}(a) \omega _{uv} = \iota _{v}(a) \omega _{uv}\) in Eq. (22);

if it is at least trivalent, then we can use the symmetry relation \(\iota _{u}(a) \omega _{uv} = \iota _{v}(a) \omega _{uv}\) to push all the labels on a single vertex, and we see that the sum of graphs that appear is obtained by the Arnold relation (see Fig. 1 for an example in the case of \(\texttt {Graphs}_{n} \rightarrow {\texttt {e}_{n}^{\vee }}\)).
5.2 The morphisms are quasiisomorphisms
In this section we prove that the morphisms constructed in Proposition 41 and Proposition 61 are quasiisomorphisms, completing the proof of Theorem 3.
Let us recall our hypotheses and constructions. Let M be a simply connected closed smooth manifold of dimension at least 4. We endow M with a semialgebraic structure (Sect. 1.3) and we consider the CDGA \(\varOmega _{\mathrm {PA}}^{*}(M)\) of PA forms on M, which is a model for the real homotopy type of M. Recall that we fix a zigzag of quasiisomorphisms of CDGAs \(A \xleftarrow {\rho } R \xrightarrow {\sigma } \varOmega _{\mathrm {PA}}^{*}(M)\), where A is a Poincaré duality CDGA (Theorem 14), and \(\sigma \) factors through the quasiisomorphic subCDGA of trivial forms.
Recall that \(\varphi \in \varOmega _{\mathrm {PA}}^{n1}(\texttt {FM}_{M}({\underline{2}}))\) is an (anti)symmetric trivial form on the compactification of the configuration space of two points in M, whose restriction to the sphere bundle \(\partial \texttt {FM}_{M}({\underline{2}})\) is a global angular form, and whose differential \(d\varphi \) is a representative of the diagonal class of M (Proposition 25). Recall that we defined the graph complex \(\texttt {Graphs}_{R}^{\varphi }(U)\) using reduced labeled graphs with internal and external vertices (Definition 40) and a partition function built from \(\varphi \) (Definition 38). We also defined the variants \(\texttt {Graphs}_{R}^{\varepsilon }\) and \(\texttt {Graphs}'_{R}\) (Definitions 57 and 58).
Theorem 62
The rest of the section is dedicated to the proof of this theorem. Let us give a roadmap of this proof. We first prove that \(\texttt {Graphs}^{\varepsilon }_{R}(U) \rightarrow \texttt {G}_{A}(U)\) is a quasiisomorphism by an inductive argument on \(\#U\) (Proposition 64). This involves setting up a spectral sequence so that we can reduce the argument to connected graphs. Then we use explicit homotopies 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 by explicit arguments. Then we prove that \(\texttt {Graphs}^{\varphi }_{R}(U) \rightarrow \varOmega _{\mathrm {PA}}^{*}(\texttt {FM}_{M}(U))\) is surjective on cohomology explicitly (Proposition 76). Since we know that \(\texttt {G}_{A}(U)\) and \(\texttt {FM}_{M}(U)\) have the same cohomology by the theorem of Lambrechts–Stanley [33, Theorem 10.1], this completes the proof that all the maps are quasiisomorphisms. Compatibility with the various comodules structures was already shown in Sect. 3.
Lemma 63
The morphisms \(\texttt {Graphs}^{\varepsilon }_{R}(U) \rightarrow \texttt {G}_{A}(U)\) factor through quasiisomorphisms \(\texttt {Graphs}^{\varepsilon }_{R}(U) \rightarrow \texttt {Graphs}^{\varepsilon }_{A}(U)\), where \(\texttt {Graphs}^{\varepsilon }_{A}(U)\) is the CDGA obtained by modding graphs with a label in \(\ker (\rho : R \rightarrow A)\) in \(\texttt {Graphs}^{\varepsilon }_{R}(U)\).
Proof
The morphism \(\texttt {Graphs}^{\varepsilon }_{R} \rightarrow \texttt {Graphs}^{\varepsilon }_{A}\) simply applies the surjective map \(\rho : R \rightarrow A\) to all the labels. Hence \(\texttt {Graphs}^{\varepsilon }_{R} \rightarrow \texttt {G}_{A}\) factors through the quotient.
We can consider the spectral sequences associated to the filtrations of both \(\texttt {Graphs}^{\varepsilon }_{R}\) and \(\texttt {Graphs}^{\varepsilon }_{A}\) by the number of edges, and we obtain a morphism \({\mathsf {E}}^{0} \texttt {Graphs}^{\varepsilon }_{R} \rightarrow {\mathsf {E}}^{0} \texttt {Graphs}^{\varepsilon }_{A}\). On both \({\mathsf {E}}^{0}\) pages, only the internal differentials coming from R and A remain. The chain map \(R \rightarrow A\) is a quasiisomorphism; hence we obtain an isomorphism on the \({\mathsf {E}}^{1}\) page. By standard spectral sequence arguments, it follows that \(\texttt {Graphs}^{\varepsilon }_{R} \rightarrow \texttt {Graphs}^{\varepsilon }_{A}\) is a quasiisomorphism. \(\square \)
The CDGA \(\texttt {Graphs}^{\varepsilon }_{A}(U)\) has the same graphical description as the CDGA \(\texttt {Graphs}^{\varepsilon }_{R}(U)\), except that now vertices are labeled by elements of A. An internal component with a single vertex labeled by \(a \in A\) is identified with \(\varepsilon (a)\), and an internal component with more than one vertex is identified with zero.
Proposition 64
The morphism \(\texttt {Graphs}^{\varepsilon }_{A} \rightarrow \texttt {G}_{A}\) is a quasiisomorphism.
Before starting to prove this proposition, let us outline the different steps. We filter our complex in such a way that on the \({\mathsf {E}}^{0}\) page, only the contracting part of the differential remains (such a technique was already used in the proof of Proposition 55). Using a splitting result, we can focus on connected graphs. Finally, we use a “trick” (Fig. 2) for moving labels around in a connected component, reducing ourselves to the case where only one vertex is labeled. We then get a chain map \(A \otimes \texttt {Graphs}_{n} \rightarrow A \otimes {\texttt {e}_{n}^{\vee }}(U)\), which is a quasiisomorphism thanks to the formality theorem.
Let us start with the first part of the outlined program, removing the splitting part of the differential from the picture. We now define an increasing filtration on \(\texttt {Graphs}^{\varepsilon }_{A}\). The submodule \(F_{s} \texttt {Graphs}^{\varepsilon }_{A}\) is spanned by reduced graphs such that \(\#\text {edges}  \#\text {vertices} \le s\).
Lemma 65
The above submodules define a filtration of \(\texttt {Graphs}_{A}^{\varepsilon }\) by subcomplexes, satisfying \(F_{\#U1} \texttt {Graphs}^{\varepsilon }_{A}(U) = 0\) for each finite set U. The \({\mathsf {E}}^{0}\) page of the spectral sequence associated to this filtration is isomorphic as a module to \(\texttt {Graphs}^{\varepsilon }_{A}\). Under this isomorphism the differential \(d^{0}\) is equal to \(d_{A} + d'_{\mathrm {contr}}\), where \(d_{A}\) is the internal differential coming from A and \(d'_{\mathrm {contr}}\) is the part of the differential that contracts all edges but edges connected to a univalent internal vertex.
Proof
Let \(\varGamma \) be an internally connected (Definition 11) reduced graph. If \(\varGamma \in \texttt {Graphs}^{\varepsilon }_{A}(U)\) 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. no internal components), each time we add an internal vertex (decreasing the filtration) we must add at least one edge (bringing it back up). By induction on the number of internal vertices, each graph is of filtration at least \(\#U\).
Let us now prove that the differential preserves the filtration and check which parts remain on the associated graded complex. The internal differential \(d_{A}\) does not change either the number of edges nor the number of vertices and so keeps the filtration constant. The contracting part \(d_{\mathrm {contr}}\) of the differential decreases both by exactly one, and so keeps the filtration constant too.
The splitting part \(d_{\mathrm {split}}\) of the differential removes one edge. If the resulting graph is still connected, then nothing else changes and the filtration is decreased exactly by 1. If the resulting graph is not connected, then we get an internal component \(\gamma \) which was connected to the rest of the graph by a single edge, and was then split off and identified with a number in the process. If \(\gamma \) has a single vertex labeled by a (i.e. we split an edge connected to a univalent vertex), then this number is \(\varepsilon (a)\), and the filtration is kept constant. 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 chain complex. On the associated graded complex, the only remaining parts of the differential are \(d_{A}\), \(d_{\mathrm {contr}}\), and the part that splits off edges connected to univalent vertices. But by the proof of Proposition 61 this last part cancels out with the part that contracts these edges connected to univalent vertices. \(\square \)
Lemma 66
The \({\mathsf {E}}^{0}\) page of the spectral sequence associated to \(F_{*} \texttt {G}_{A}\) is isomorphic as a module to \(\texttt {G}_{A}\). Under this isomorphism the \(d^{0}\) differential is just the internal differential of A. \(\square \)
Lemma 67
The morphism \(\texttt {Graphs}^{\varepsilon }_{A} \rightarrow \texttt {G}_{A}\) preserves the filtration and induces a chain map \({\mathsf {E}}^{0} \texttt {Graphs}^{\varepsilon }_{A}(U) \rightarrow {\mathsf {E}}^{0} \texttt {G}_{A}(U)\), for each U. It maps reduced graphs with internal vertices to zero, an edge \(e_{uv}\) between external vertices to \(\omega _{uv}\), and a label a of an external vertex u to \(\iota _{u}(a)\).
Proof
The morphism \(\texttt {Graphs}^{\varepsilon }_{A}(U) \rightarrow \texttt {G}_{A}(U)\) preserves the filtration by construction. If a graph has internal vertices, then its image in \(\texttt {G}_{A}(U)\) is of strictly lower filtration unless the graph is a forest (i.e. a product of trees). But trees have leaves, therefore by Corollary 44 and the formula defining \(\texttt {Graphs}^{\varepsilon }_{A} \rightarrow \texttt {G}_{A}\) they are mapped to zero in \(\texttt {G}_{A}(U)\) anyway. It is clear that the rest of the morphism preserves filtrations exactly, and so is given on the associated graded complex as stated in the lemma. \(\square \)
We now use arguments similar to [34, Lemma 8.3]. For a partition \(\pi \) of U, define the submodule \(\texttt {Graphs}^{\varepsilon }_{A}\langle \pi \rangle \subset {\mathsf {E}}^{0} \texttt {Graphs}^{\varepsilon }_{A}(U)\) spanned by reduced graphs \(\varGamma \) such that the partition of U induced by the connected components of \(\varGamma \) is exactly \(\pi \). In particular let \(\texttt {Graphs}^{\varepsilon }_{A}\langle \{U\} \rangle \) be the submodule of connected graphs, where \(\{U\}\) is the indiscrete partition of U consisting of a single element.
Lemma 68
Proof
Since there is no longer any part of the differential that can split off connected components in \({\mathsf {E}}^{0} \texttt {Graphs}^{\varepsilon }_{A}\), it is clear that \(\texttt {Graphs}^{\varepsilon }_{A}\langle \{U\} \rangle \) is a subcomplex. The splitting result is immediate. \(\square \)
The complex \({\mathsf {E}}^{0} \texttt {G}_{A}(U)\) splits in a similar fashion. For a monomial in \(S(\omega _{uv})_{u \ne v \in U}\), say that u and v are “connected” if the term \(\omega _{uv}\) appears in the monomial. Consider the equivalence relation generated by “u and v are connected”. The monomial induces in this way a partition \(\pi \) of U, and this definition factors through the quotient defining \({\texttt {e}_{n}^{\vee }}(U)\) (draw a picture of the 3term relation). Finally, for a given monomial in \(\texttt {G}_{A}(U)\), the induced partition of U is still welldefined.
Thus for a given partition \(\pi \) of U, we can define \({\texttt {e}_{n}^{\vee }}\langle \pi \rangle \) and \(\texttt {G}_{A}\langle \pi \rangle \) to be the submodules of \({\texttt {e}_{n}^{\vee }}(U)\) and \({\mathsf {E}}^{0} \texttt {G}_{A}(U)\) spanned by monomials inducing the partition \(\pi \). It is a standard fact that \({\texttt {e}_{n}^{\vee }}\langle \{U\} \rangle = \texttt {Lie}_{n}^{\vee }(U)\), see [47]. The proof of the following lemma is similar to the proof of the previous lemma:
Lemma 69
Lemma 70
The map \({\mathsf {E}}^{0} \texttt {Graphs}^{\varepsilon }_{A}(U) \rightarrow {\mathsf {E}}^{0} \texttt {G}_{A}(U)\) preserves the splitting.
We can now focus on connected graphs to prove Proposition 64.
Lemma 71
The complex \(\texttt {G}_{A} \langle \{U\} \rangle \) is isomorphic to \(A \otimes {\texttt {e}_{n}^{\vee }}\langle \{U\} \rangle \).
Proof
We define explicit isomorphisms in both directions.
Define \(A^{\otimes U} \otimes {\texttt {e}_{n}^{\vee }}\langle \{U\} \rangle \rightarrow A \otimes {\texttt {e}_{n}^{\vee }}\langle \{U\} \rangle \) using the multiplication of A. This constructions induces a map on the quotient \({\mathsf {E}}^{0} \texttt {G}_{A}(U) \rightarrow A \otimes {\texttt {e}_{n}^{\vee }}\langle \{U\} \rangle \), which restricts to a map \(\texttt {G}_{A} \langle \{U\} \rangle \rightarrow {\texttt {e}_{n}^{\vee }}\langle \{U\} \rangle \). Since \(d_{A}\) is a derivation, this is a chain map.
Conversely, define \(A \otimes {\texttt {e}_{n}^{\vee }}\langle \{U\} \rangle \rightarrow A^{\otimes U} \otimes {\texttt {e}_{n}^{\vee }}\langle \{U\} \rangle \) by \(a \otimes x \mapsto \iota _{u}(a) \otimes x\) for some fixed \(u \in U\) (it does not matter which one since \(x \in {\texttt {e}_{n}^{\vee }}\langle \{U\} \rangle \) is “connected”). This construction gives a map \(A \otimes {\texttt {e}_{n}^{\vee }}\langle \{U\} \rangle \rightarrow \texttt {G}_{A} \langle \{U\} \rangle \), and it is straightforward to check that this map is the inverse isomorphism of the previous map. \(\square \)
Here \(\texttt {Graphs}'_{n}(U)\) is defined similarly to \(\texttt {Graphs}_{n}(U)\) except that multiple edges are allowed. It is known that the quotient map \(\texttt {Graphs}'_{n}(U) \rightarrow {\texttt {e}_{n}^{\vee }}(U)\) (which factors through \(\texttt {Graphs}_{n}(U)\)) is a quasiisomorphism [53, Proposition 3.9]. The subcomplex \(\texttt {Graphs}'_{n} \langle \{U\} \rangle \) is spanned by connected graphs. The upper horizontal map in the diagram multiplies all the labels of a graph.
The right vertical map is the tensor product of \({\text {id}}_{A}\) and \(\texttt {Graphs}_{n} \langle \{U\} \rangle \xrightarrow {\sim } {\texttt {e}_{n}^{\vee }}\langle \{U\} \rangle \) (see 1.6). The bottom row is the isomorphism of the previous lemma.
It then remains to prove that \(\texttt {Graphs}^{\varepsilon }_{A} \langle \{U\} \rangle \rightarrow A \otimes \texttt {Graphs}'_{n} \langle \{U\} \rangle \) is a quasiisomorphism to prove Proposition 64. If \(U = \varnothing \), then \(\texttt {Graphs}'_{A}(\varnothing ) = {\mathbb {R}}= \texttt {G}_{A}(\varnothing )\) and the morphism is the identity, so there is nothing to do. From now on we assume that \(\# U \ge 1\).
Lemma 72
The morphism \(\texttt {Graphs}^{\varepsilon }_{A} \langle \{U\} \rangle \rightarrow A \otimes \texttt {Graphs}'_{n} \langle \{U\} \rangle \) is surjective on cohomology.
Proof
Choose some \(u \in U\). There is an explicit chainlevel section of the morphism, sending \(x \otimes \varGamma \) to \(\varGamma _{u,x}\), the same graph with the vertex u labeled by x and all the other vertices labeled by \(1_{R}\). It is a welldefined chain map, which is clearly a section of the morphism in the lemma, hence the morphism of the lemma is surjective on cohomology. \(\square \)
Lemma 73
The proof will be by induction on the cardinality of U. Before proving this lemma, we will need two additional sublemmas.
Lemma 74
The complex \(\texttt {Graphs}^{\varepsilon }_{A} \langle {\underline{1}} \rangle \) has the same cohomology as A.
Proof
Let \({\mathcal {I}}\) be the subcomplex spanned by graphs with at least one internal vertex. We will show that \({\mathcal {I}}\) is acyclic; as \(\texttt {Graphs}^{\varepsilon }_{A} \langle {\underline{1}} \rangle / {\mathcal {I}} \cong A\), this will prove the lemma.
Now let U be a set with at least two elements, and fix some element \(u \in U\). Let \(\texttt {Graphs}^{u}_{A} \langle \{U\} \rangle \subset \texttt {Graphs}^{\varepsilon }_{A} \langle \{U\} \rangle \) be the subcomplex spanned by graphs \(\varGamma \) such that u has valence 1, is labeled by \(1_{A}\), and is connected to another external vertex.
Lemma 75
The inclusion \(\texttt {Graphs}^{u}_{A} \langle \{U\} \rangle \subset \texttt {Graphs}^{\varepsilon }_{A} \langle \{U\} \rangle \) is a quasiisomorphism.
Proof

The module \({\mathcal {Q}}_{1}\) spanned by graphs where u is of valence 1, labeled by \(1_{A}\), and connected to an internal vertex;

The module \({\mathcal {Q}}_{2}\) spanned by graphs where u is of valence \(\ge 2\) or has a label in \(A^{> 0}\).
Consider the \({\mathsf {E}}^{0}\) page of the spectral sequence associated to this filtration. Then the differential \(d^{0}\) is a morphism \({\mathsf {E}}^{0} {\mathcal {Q}}_{1} \rightarrow {\mathsf {E}}^{0} {\mathcal {Q}}_{2}\) (count the number of edges and use the crucial fact that edges connected to univalent vertices are not contractible when looking at reduced graphs). This differential contracts the only edge incident to u. It is an isomorphism, with an inverse similar to the homotopy defined in Lemma 74, “blowing up” the point u into a new edge connecting u to a new internal vertex that replaces u.
This shows that \(({\mathsf {E}}^{0} {\mathcal {Q}}, d^{0})\) is acyclic, hence \({\mathsf {E}}^{1} {\mathcal {Q}} = 0\). It follows that \({\mathcal {Q}}\) itself is acyclic. \(\square \)
Proof of Lemma 73
The case \(\# U = 0\) is obvious, and the case \(\# U = 1\) of the lemma was covered in Lemma 74. We now work by induction and assume the claim proved for \(\# U \le k\), for some \(k \ge 1\).
Proof of Proposition 64
By Lemma 72, the morphism induced by \(\texttt {Graphs}^{\varepsilon }_{A} \rightarrow \texttt {G}_{A}\) on the \({\mathsf {E}}^{0}\) page is surjective on cohomology. By Lemma 73 and Eq. (33), both \({\mathsf {E}}^{0}\) pages have the same cohomology, and so the induced morphism is a quasiisomorphism. Standard spectral arguments imply the proposition. \(\square \)
Proposition 76
The morphism \(\omega : \texttt {Graphs}'_{R}(U) \rightarrow \varOmega _{\mathrm {PA}}^{*}(\texttt {FM}_{M}(U))\) is a quasiisomorphism.
Proof
By Eq. (23), Proposition 60, Lemma 63, and Proposition 64, both CDGAs have the same cohomology of finite type, so it will suffice to show that the map is surjective on cohomology to prove that it is a quasiisomorphism.
We work by induction. The case \(U = \varnothing \) is immediate, as \(\texttt {Graphs}'_{R}(\varnothing ) \xrightarrow {\sim } \texttt {Graphs}^{\varphi }_{R}(\varnothing ) = \varOmega _{\mathrm {PA}}^{*}(\texttt {FM}_{M}(\varnothing )) = {\mathbb {R}}\) and the last map is the identity.
Suppose that \(U = \{ u \}\) is a singleton. Since \(\rho \) is a quasiisomorphism, for every cocycle \(\alpha \in \varOmega _{\mathrm {PA}}^{*}(\texttt {FM}_{M}(U)) = \varOmega _{\mathrm {PA}}^{*}(M)\) there is some cocycle \(x \in R\) such that \(\rho (x)\) is cohomologous to \(\alpha \). Then the graph \(\gamma _{x}\) with a single (external) vertex labeled by x is a cocycle in \(\texttt {Graphs}'_{R}(U)\), and \(\omega (\gamma _{x}) = \rho (x)\) is cohomologous to \(\alpha \). This proves that \(\texttt {Graphs}'_{R}(\{u\}) \rightarrow \varOmega _{\mathrm {PA}}^{*}(M)\) is surjective on cohomology, and hence is a quasiisomorphism.
It follows that \(z'  \sigma _{*}({{\tilde{\gamma }}}_{1}) \in A \otimes \texttt {G}_{A}(V)\) is a cocycle. Thus by the induction hypothesis there exists some \({{\tilde{\gamma }}}_{2} \in R \otimes \texttt {Graphs}'_{R}(V)\) whose cohomology class represents the same cohomology class as \(z'  \sigma _{*}({{\tilde{\gamma }}}_{1})\) in \(H^{*}(A \otimes \texttt {G}_{A}(V)) = H^{*}(M \times \texttt {FM}_{M}(V))\).
Proof of Theorem 62
The last thing we need to check is the following proposition, which shows that that we can choose any Poincaré duality model.
Proposition 77
If A and \(A'\) are two quasiisomorphic simply connected Poincaré duality CDGAs, then there is a weak equivalence of symmetric collections \(\texttt {G}_{A} \simeq \texttt {G}_{A'}\). If moreover \(\chi (A) = 0\) then this weak equivalence is a weak equivalence of right Hopf \({\texttt {e}_{n}^{\vee }}\)comodules.
Proof
The CDGAs A and \(A'\) are quasiisomorphic, hence there exists some cofibrant S and quasiisomorphisms \(f : S \xrightarrow {\sim } A\) and \(f' : S \xrightarrow {\sim } A'\). This yields two chain maps \(\varepsilon = \varepsilon _{A} \circ f,\, \varepsilon ' = \varepsilon _{A'} \circ f' : S \rightarrow {\mathbb {R}}[n]\). Mimicking the proof of Proposition 15, we can also find (anti)symmetric cocycles \(\varDelta , \varDelta ' \in S \otimes S\) and such that \((f \otimes f)\varDelta = \varDelta _{A}\) and \((f' \otimes f')\varDelta ' = \varDelta _{A'}\).
We can then build symmetric collections \(\texttt {Graphs}_{S}^{\varepsilon , \varDelta }\) and a quasiisomorphism \(f_{*} : \texttt {Graphs}_{S}^{\varepsilon , \varDelta } \rightarrow \texttt {G}_{A}\) similarly to Sect. 3. The differential of an edge \(e_{uv}\) in \(\texttt {Graphs}_{S}^{\varepsilon ,\varDelta }\) is \(\iota _{uv}(\varDelta )\), and an isolated internal vertex labeled by \(x \in S\) is identified with \(\varepsilon (x)\). In parallel, we can build \(f_{*}' : \texttt {Graphs}_{S}^{\varepsilon ',\varDelta '} \xrightarrow {\sim } \texttt {G}_{A'}\).
If moreover \(\chi (A) = 0\), then we can choose \(\varDelta , \, \varDelta '\) such that both graph complexes become right Hopf \(\texttt {Graphs}_{n}\)comodules, and \(f_{*}\), \(f'_{*}\) are compatible with the comodule structure. It thus suffices to find a quasiisomorphism \(\texttt {Graphs}_{S}^{\varepsilon ,\varDelta } \simeq \texttt {Graphs}_{S}^{\varepsilon ',\varDelta '}\) to prove the proposition.
We first have an isomorphism \(\texttt {Graphs}_{S}^{\varepsilon ',\varDelta '} \cong \texttt {Graphs}_{S}^{\varepsilon ',\varDelta }\) (with the obvious notation). Indeed, the two cocycles \(\varDelta \) and \(\varDelta '\) are cohomologous, say \(\varDelta '  \varDelta = d\alpha \) for some \(\alpha \in S \otimes S\) of degree \(n1\). If we replace \(\alpha \) by \((\alpha + (1)^{n} \alpha ^{21})/2\), then we can assume that \(\alpha ^{21} = (1)^{n} \alpha \). Moreover if \(\chi (A) = 0\), then we can replace \(\alpha \) by \(\alpha  (\mu _{S}(\alpha ) \otimes 1 + (1)^{n} 1 \otimes \mu _{S}(\alpha ))/2\) to get \(\mu _{S}(\alpha ) = 0\). We then obtain an isomorphism by mapping an edge \(e_{uv}\) to \(e_{uv} \pm \iota _{uv}(\alpha )\) (the sign depending on the direction of the isomorphism). This map is compatible with differentials, with products, and with the comodule structures if \(\chi (A) = 0\).
The dgmodule S is cofibrant and \({\mathbb {R}}[n]\) is fibrant (like all dgmodules). We can assume that \(\varepsilon \) and \(\varepsilon '\) induce the same map on cohomology (it suffices to rescale one map, say \(\varepsilon '\), and there is an automorphism of \(\texttt {Graphs}_{S}^{\varepsilon ',\varDelta }\) which takes care of this rescaling). Thus the two maps \(\varepsilon , \varepsilon ' : S \rightarrow {\mathbb {R}}[n]\) are homotopic, i.e. there exists some \(h : S[1] \rightarrow {\mathbb {R}}[n]\) such that \(\varepsilon (x)  \varepsilon '(x) = h(dx)\) for all \(x \in S\). This homotopy induces a homotopy between the two morphisms \(Z_{\varepsilon }, Z_{\varepsilon '} : \mathrm {fGC}_{S} \rightarrow {\mathbb {R}}\). Because \({{\mathrm{Tw}}}\texttt {Gra}_{S}^{\varDelta }(U)\) and \({{\mathrm{Tw}}}\texttt {Gra}_{S}^{\varDelta '}(U)\) are cofibrant as modules over \(\mathrm {fGC}_{S}\), we obtain quasiisomorphisms \(\texttt {Graphs}_{S}^{\varepsilon ,\varDelta } \simeq \texttt {Graphs}_{S}^{\varepsilon ',\varDelta }\) (compare with Proposition 60). \(\square \)
Corollary 78
Let M be a smooth simply connected closed manifold and A be any Poincaré duality model of M. Then \(\texttt {G}_{A}({\underline{k}})\) is a real model for \(\mathrm {Conf}_{k}(M)\).
Proof
The corollary follows from Theorem 62 in the case where \(\dim M \ge 4\) (together with the previous proposition to ensure that we can choose any Poincaré duality model A in our constructions). Note that the graph complexes are, in general, nonzero even in negative degrees, but by Proposition 4 this does not change the result. In dimension at most 3, the only examples of simply connected closed manifolds are \(S^{2}\) and \(S^{3}\). We address these examples in Sect. 4.3. \(\square \)
Corollary 79
The real homotopy types of the configuration spaces of a smooth simply connected closed manifold only depends on the real homotopy type of the manifold.
Proof
When \(\dim M \ge 3\), the Fadell–Neuwirth fibrations [12] \(\mathrm {Conf}_{k1}(M {\setminus } *) \hookrightarrow \mathrm {Conf}_{k}(M) \rightarrow M\) show by induction that if M is simply connected, then so is \(\mathrm {Conf}_{k}(M)\) for all \(k \ge 1\). Hence the real model \(\texttt {G}_{A}({\underline{k}})\) completely encodes the real homotopy type of \(\mathrm {Conf}_{k}(M)\). \(\square \)
5.3 Models for configurations on the 2 and 3spheres
The degreecounting argument of Proposition 45 does not work in dimension less than 4, so we have to use other means to prove that the Lambrechts–Stanley CDGAs are models for the configuration spaces.
There are no simply connected closed manifolds of dimension 1. In dimension 2, the only simply connected closed manifold is the 2sphere, \(S^{2}\). This manifold is a complex projective variety: \(S^{2} = {{\mathbb {C}}}{{\mathbb {P}}}^{1}\). Hence the result of Kriz [30] shows that \(\texttt {G}_{H^{*}(S^{2})}({\underline{k}})\) (denoted E(k) there) is a rational model for \(\mathrm {Conf}_{k}(S^{2})\). The 2sphere \(S^{2}\) is studied in greater detail in Sect. 6, where we study the action of the framed little 2disks operad on a framed version of \(\texttt {FM}_{S^{2}}\).
In dimension 3, the only simply connected smooth closed manifold is the 3sphere by Perelman’s proof of the Poincaré conjecture [41, 42]. we also the following partial result, communicated to us by Thomas Willwacher:
Proposition 80
The CDGA \(\texttt {G}_{A}({\underline{k}})\), where \(A = H^{*}(S^{3}; {\mathbb {Q}})\), is a rational model of \(\mathrm {Conf}_{k}(S^{3})\) for all \(k \ge 0\).
Proof
To simplify notation, we consider \(\texttt {G}_{A}({\underline{k}}_{+})\) (where \({\underline{k}}_{+} = \{ 0, \dots , k \}\)), which is obviously isomorphic to \(\texttt {G}_{A}(\underline{k+1})\). Let us denote by \(\upsilon \in H^{3}(S^{3}) = A^{3}\) the volume form of \(S^{3}\), and recall that the diagonal class \(\varDelta _{A}\) is given by \(1 \otimes \upsilon  \upsilon \otimes 1\). We have an explicit map \(f : H^{*}(S^{3}) \rightarrow {\texttt {e}_{3}^{\vee }}({\underline{k}})\) given on generators by \(f(\nu \otimes 1) = \iota _{0}(\nu )\) and \(f(1 \otimes \omega _{ij}) = \omega _{ij} + \omega _{0i}  \omega _{0j}\).
In degree 3, the cocycle \(\upsilon \otimes 1\) is sent to a generator of \(H^{3}(\texttt {G}_{A}({\underline{k}}_{+})) \cong H^{3}(S^{3}) = {\mathbb {Q}}\). Indeed, assume \(\iota _{0}(\upsilon ) = d\omega \), where \(\omega \) is a linear combination of the \(\omega _{ij}\) for degree reasons. In \(d\omega \), the sum of the coefficients of each \(\iota _{i}(\upsilon )\) is zero, because they all come in pairs (\(d\omega _{ij} = \iota _{j}(\upsilon )  \iota _{i}(\upsilon )\)). We want the coefficient of \(\iota _{0}(\upsilon )\) to be 1, so at least one of the other coefficient must be nonzero to compensate, hence \(d\omega \ne \iota _{0}(\upsilon )\).
For degree reasons, \(H^{2}(\ker q) = 0\) and so the map (1) is injective. By the four lemma, it follows that \(H^{2}(f)\) is injective. Since both domain and codomain have the same finite dimension, it follows that \(H^{2}(f)\) is an isomorphism. \(\square \)
6 Factorization homology of universal enveloping \(E_{n}\)algebras
6.1 Factorization homology and formality
6.2 Higher enveloping algebras
Knudsen [27, Theorem A] considers a higher enveloping algebra functor \(U_n\) from homotopy Lie algebras to nonunital \(E_n\)algebras. This functor generalizes the standard enveloping algebra functor from the category of Lie algebras to the category of associative algebras.
Let n be at least 2. We can again use Kontsevich’s theorem on the formality of the little disks operads to identity the category of nonunital \(\texttt {E}_{n}\)algebras with the category of \(\texttt {e}_{n}\)algebras in homotopy. We also use that a homotopy Lie algebra is equivalent, in homotopy, to an ordinary Lie algebra. Then we get that Knudsen’s higher enveloping algebra functor is equivalent to the left adjoint of the obvious forgetful functor \(\texttt {e}_{n}\text {Alg} \rightarrow \texttt {Lie}\text {Alg}\), which maps an nPoisson algebra B to its underlying shifted Lie algebra \(B[1n]\). This model \({\tilde{U}}_{n} : \texttt {Lie}\text {Alg} \rightarrow \texttt {e}_{n}\text {Alg}\) maps a Lie algebra \({\mathfrak {g}}\) to the nPoisson algebra given by \({\tilde{U}}_{n}({\mathfrak {g}}) = S({\mathfrak {g}}[n1])\), with the shifted Lie bracket defined using the Leibniz rule.
Proposition 81
If A is a Poincaré duality model of M, we have \(A \simeq \varOmega _{\mathrm {PA}}^{*}(M) \simeq A_{\mathrm {PL}}^{*}(M) \otimes _{{\mathbb {Q}}} {\mathbb {R}}\) [23, Theorem 6.1]. It follows that the Chevalley–Eilenberg complex of the previous proposition is weakly equivalent to the Chevalley–Eilenberg complex of Eq. (41). By Eq. (39), the derived circle product over \(\texttt {e}_{n}\) computes the factorization homology of \(U_{n}({\mathfrak {g}})\) on M, and so we recover Knudsen’s theorem (over the reals) for closed framed simply connected manifolds.
The proof of the following lemma is essentially found (in a different language) in [15, Section 2].
Lemma 82
The right \(\texttt {Lie}_{n}\)modules \(\texttt {G}_{A}^{\vee }\) and \(\mathrm {C}_{*}^{\mathrm {CE}}(A^{*} \otimes \texttt {L}_{n})\) are isomorphic.
Proof
 On the A factors, the pairing uses the Poincaré duality pairing \(\varepsilon _{A}\). It is given by the following formula (where \(a_{U_{i}} = \prod _{u \in U_{i}} a_{u}\)):$$\begin{aligned} (a_{u})_{u \in U} \otimes (a'_{1} \otimes \dots \otimes a'_{r}) \mapsto \pm \varepsilon _{A} (a_{U_{1}} \cdot a'_{1}) \dots \varepsilon _{A} (a_{U_{r}} \cdot a'_{r}), \end{aligned}$$

On the factor \({\texttt {e}_{n}^{\vee }}(U) \otimes \bigotimes _{i=1}^{r} \texttt {Lie}_{n}(U_{i})\), it uses the duality pairing on \({\texttt {e}_{n}^{\vee }}(U) \otimes \texttt {e}_{n}(U)\) (recalling that \(\texttt {e}_{n} = \texttt {Com}\circ \texttt {Lie}_{n}\) so that we can view \(\bigotimes _{i=1}^{r} \texttt {Lie}_{n}(U_{i})\) as a submodule of \(\texttt {e}_{n}(U)\)).
Finally, we easily check, using the identity \(\varepsilon _{A}(aa') = \sum _{(\varDelta _{A})} \pm \varepsilon _{A}(a \varDelta _{A}') \varepsilon _{A}(a' \varDelta _{A}'')\) (which in turns follows from the definition of \(\varDelta _{A}\)) that the pairing commutes with differentials (i.e. \(\langle d(),  \rangle = \pm \langle , d() \rangle \)). \(\square \)
Proof of Proposition 81
7 Outlook: The case of the 2sphere and oriented manifolds
Up to now, we were considering framed manifolds M in order to define the action of the (unframed) Fulton–MacPherson \(\texttt {FM}_{n}\) on \(\texttt {FM}_{M}\). When M is not framed, it is not possible to coherently define insertion of infinitesimal configurations from \(\texttt {FM}_{n}\) into the tangent space of M, because we lack a coherent identification of the tangent space at every point with \({\mathbb {R}}^{n}\). Instead, for an oriented (but not necessarily framed) manifold M, there exists an action of the framed Fulton–MacPherson operad obtained by considering infinitesimal configurations together with rotations of \(\mathrm {SO}(n)\) (see below for precise definitions).
In dimension 2, the formality of \(\texttt {FM}_{2}\) was extended to a proof of the formality of the framed version of \(\texttt {FM}_{2}\) in [20] (see also [45] for an alternative proof and [25] for a generalization for even n). We now provide a generalization of the previous work for the 2sphere, and we formulate a conjecture for higher dimensional closed manifolds that are not necessarily framed.
7.1 Framed little disks and framed configurations
Following Salvatore–Wahl [44, Definition 2.1], we describe the framed little disks operad as a semidirect product. If G is a topological group and \(\texttt {P}\) is an operad in Gspaces, the semidirect product \(\texttt {P}\rtimes G\) is the topological operad defined by \((\texttt {P}\rtimes G)(n) = \texttt {P}(n) \times G^{n}\) and explicit formulas for the composition. If H is a commutative Hopf algebra and \(\texttt {C}\) is a Hopf cooperad in Hcomodules, then the semidirect product \(\texttt {C}\rtimes H\) is defined by formally dual formulas.
The operad \(\texttt {FM}_{n}\) is an operad in \(\mathrm {SO}(n)\)spaces, the action rotating configurations. Thus we can form an operad \(\texttt {f}\texttt {FM}_{n} = \texttt {FM}_{n} \rtimes \mathrm {SO}(n)\), the framed Fulton–MacPherson operad, weakly equivalent to the standard framed little disks operad.
Given an oriented nmanifold M, there is a corresponding right module over \(\texttt {f}\texttt {FM}_{n}\), which we call \(\texttt {f}\texttt {FM}_{M}\) [52, Section 2]. The space \(\texttt {f}\texttt {FM}_{M}(U)\) is a principal \(\mathrm {SO}(n)^{\times U}\)bundle over \(\texttt {FM}_{M}(U)\). Since \(\mathrm {SO}(n)\) is an algebraic group, \(\texttt {f}\texttt {FM}_{n}\) and \(\texttt {f}\texttt {FM}_{M}(U)\) are respectively an operad and a module in semialgebraic spaces.
7.2 Cohomology of \(\texttt {f}\texttt {FM}_n\) and potential model
Definition 83
Proposition 84
The collection \(\{ \texttt {f}\texttt {G}_{A}(U) \}_{U}\) is a Hopf right \(\texttt {f}{\texttt {e}_{2}^{\vee }}\)comodule, with cocomposition given by the same formula as Eq. (24).
Proof
The proofs that the cocomposition is compatible with the cooperad structure of \(\texttt {f}{\texttt {e}_{2}^{\vee }}\), and that this is compatible with the quotient, is the same as in the proof of Proposition 16. It remains to check compatibility with differentials.
7.3 Connecting \(\texttt {f}\texttt {G}_{A}\) to \(\varOmega _{\mathrm {PA}}^*(\texttt {f}\texttt {FM}_{S^{2}})\)
Theorem 85
The Hopf right comodule \((\texttt {f}\texttt {G}_{A}, \texttt {f}{\texttt {e}_{2}^{\vee }})\), where \(A = H^{*}(S^{2}; {\mathbb {R}})\), is quasiisomorphic to the Hopf right comodule \((\varOmega _{\mathrm {PA}}^{*}(\texttt {f}\texttt {FM}_{S^2}), \varOmega _{\mathrm {PA}}^{*}(\texttt {f}\texttt {FM}_{2}))\).
Proof
The second map is given by the morphism of Proposition 41 on the \(\texttt {Graphs}^{\varepsilon }_{A}\) factor, composed with the pullback along the projection \(\texttt {f}\texttt {FM}_{S^2} \rightarrow \texttt {FM}_{S^2}\). The generator \(\alpha \in H\) is sent to a pullback of a global angular form \(\psi \) of the principal \(\mathrm {SO}(2)\)bundle \(\texttt {f}\texttt {FM}_{S^{2}}({\underline{1}}) \rightarrow \texttt {FM}_{S^{2}}({\underline{1}}) = S^2\) induced by the orientation of \(S^2\). This form satisfies \(d\psi = \chi (S^2) \mathrm {vol}_{S^2}\).
The proof of Giansiracusa–Salvatore [20] then adapts itself to prove that these two maps are maps of Hopf right comodules. The Künneth formula implies that the first map is a quasiisomorphism, and the second map induces an isomorphism on the \({\mathsf {E}}^{2}\)page of the Serre spectral sequence associated to the bundle \(\texttt {f}\texttt {FM}_{S^2} \rightarrow \texttt {FM}_{S^2}\) and hence is itself a quasiisomorphism. \(\square \)
Corollary 86
The CDGA \(\texttt {f}\texttt {G}_{H^{*}(S^2)}({\underline{k}})\) of Definition 83 is a real model for \(\mathrm {Conf}^{\mathrm {or}}_{k}(S^2)\), the \(\mathrm {SO}(2)^{\times k}\)principal bundle over \(\mathrm {Conf}_{k}(S^2)\) induced by the orientation of \(S^2\).
If M is an oriented nmanifold with \(n > 2\), Definition 83 readily adapts to define \(\texttt {f}\texttt {G}_{H^{*}(M)}\), by setting \(d\alpha \) to be the Euler class of M (when n is even), and \(d\beta _{i}\) to be the ith Pontryagin class of M. The proof of Proposition 84 adapts easily to this new setting, and \(\texttt {f}\texttt {G}_{H^{*}(M)}\) becomes a Hopf right \(\texttt {f}{\texttt {e}_{n}^{\vee }}\)comodule.
Conjecture 87
If M is a formal, simply connected, oriented closed nmanifold and if the framed little ndisks operad \(\texttt {fe}_{n}\) is formal, then the pair \((\texttt {f}\texttt {G}_{H^{*}(M)}, \texttt {f}{\texttt {e}_{n}^{\vee }})\) is quasiisomorphic to the pair \((\varOmega _{\mathrm {PA}}^{*}(\texttt {f}\texttt {FM}_{M}), \varOmega _{\mathrm {PA}}^{*}(\texttt {f}\texttt {FM}_{n}))\).
To directly adapt our proof for the conjecture, the difficulty would be the same as encountered by Giansiracusa–Salvatore [20], namely finding forms in \(\varOmega _{\mathrm {PA}}^{*}(\texttt {f}\texttt {FM}_{n})\) corresponding to the generators of \(H^{*}(\mathrm {SO}(n))\) and compatible with the Kontsevich integral. It was recently proved that the framed Fulton–MacPherson is formal for even n and not formal for odd \(n \ge 3\) [25, 38]. However, the proof that \(\texttt {f}\texttt {FM}_{n}\) is formal for even \(n \ge 4\), due to Khoroshkin and Willwacher [25], is much more involved than the proof of the formality of \(\texttt {fFM}_{2}\). In particular, the zigzag of maps is not completely explicit and relies on obstructiontheoretical arguments. It would be interesting to try and adapt the conjecture in this setting.
If M itself is not formal then it is also not clear how to define Pontryagin classes in some Poincaré duality model of M (the Euler class is canonically given by \(\chi (A) \mathrm {vol}_{A}\)). Nevertheless, for any oriented manifold M we get invariants of \(\texttt {fe}_{n}\)algebras by considering the functor \(\texttt {f}\texttt {G}_{H^{*}(M)}^{\vee } \circ ^{{\mathbb {L}}}_{\texttt {fe}_{n}} ()\). Despite not necessarily computing factorization homology, they could prove interesting.
Notes
Acknowledgements
I would like to thank several people: my (now former) advisor Benoit Fresse for giving me the opportunity to study this topic and for numerous helpful discussions regarding the content of this paper; Thomas Willwacher and Ricardo Campos for helpful discussions about their own model for configuration spaces ktheir explanation of propagators and partition functors, and for several helpful remarks; Ben Knudsen for explaining the relationship between the LS CDGAs and the Chevalley–Eilenberg complex; Pascal Lambrechts for several helpful discussions; Ivo Dell’Ambrogio, Julien Ducoulombier, Matteo Felder, and Antoine Touzé for their comments; and finally the anonymous referee, for a thorough and detailed report with numerous suggestions that greatly improved this paper. The author was supported by ERC StG 678156–GRAPHCPX during part of the completion of this manuscript.
Supplementary material
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