In this section we connect \(\texttt {G}_{A}\) to \(\varOmega _{\mathrm {PA}}^{*}(\texttt {FM}_{M})\) and we prove that the connecting morphisms are quasi-isomorphisms. We assume that M is a simply connected closed smooth manifold with \(\dim M \ge 4\) (see Proposition 45).
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 \(e_{12} e_{23} \dots e_{(k-1)k} e_{k1}\). 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 sub-CDGA 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 quasi-free 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 \(\mathrm {fGC}^{0}_{R} / (\mathrm {fLoop}_{R})\).
Let us note that the quotient map \(\mathrm {fGC}^{0}_{R} \rightarrow \mathrm {fGC}'_{R} = \mathrm {fGC}^{0}_{R} / I\) vanishes on \(\mathrm {fLoop}_{R}\), because \(\mathrm {fLoop}_{R}\) is included in R. Thus we have a diagram:
Let us prove that the morphism \(\mathrm {fGC}''_{R} \rightarrow \mathrm {fGC}'_{R}\) is a quasi-isomorphism. Define an increasing filtration on both algebras by letting \(F_{s}\mathrm {fGC}'_{R}\) (resp. \(F_{s}\mathrm {fGC}''_{R}\)) be the submodule spanned by graphs \(\varGamma \) such that \(\# \text {edges} - \# \text {vertices} \le s\). The splitting part of the differential strictly decreases the filtration, so only \(d_{R}\) and \(d_{\mathrm {contr}}\) remain on the first page of the associated spectral sequences.
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 sub-CDGA 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 quasi-isomorphic to \(\mathrm {fGC}_{R}'\) by Proposition 55. \(\square \)
The statement of the corollary is not concrete, as the “factorization” could go through a zigzag of maps. However, the CDGAs \(\mathrm {fGC}_{R}\) and \(\mathrm {fGC}'_{R}\) are both cofibrant (see Lemma 48 for \(\mathrm {fGC}_{R}\), whose proof can easily be adapted to \(\mathrm {fGC}_{R}'\)). Recall from Sect. 1.1 the following definition of homotopy. Let \(\pi : \mathrm {fGC}_{R} \rightarrow \mathrm {fGC}'_{R}\) be the quotient map. Recall that \(A_{\mathrm {PL}}^{*}(\varDelta ^{1}) = S(t,dt)\) is a path object for the CDGA \({\mathbb {R}}\), and \({{\mathrm{ev}}}_{0}, {{\mathrm{ev}}}_{1} : A_{\mathrm {PL}}^{*}(\varDelta ^{1}) \rightarrow {\mathbb {R}}\) are evaluation at \(t = 0\) and \(t = 1\). There exists some morphism \(Z_{\varphi }' : \mathrm {fGC}'_{R} \rightarrow {\mathbb {R}}\) and some homotopy \(h : \mathrm {fGC}_{R} \rightarrow A_{\mathrm {PL}}^{*}(\varDelta ^{1})\) such that the following diagram commutes:
Definition 57
Let \(A_{\mathrm {PL}}^{*}(\varDelta ^{1})_{h}\) be the \(\mathrm {fGC}_{R}\)-module induced by h, and let
$$\begin{aligned} \texttt {Graphs}'_{R}(U) = A_{\mathrm {PL}}^{*}(\varDelta ^{1})_{h} \otimes _{\mathrm {fGC}_{R}} {{\mathrm{Tw}}}\texttt {Gra}_{R}(U). \end{aligned}$$
Definition 58
Let \(Z_{\varepsilon }: \mathrm {fGC}_{R} \rightarrow {\mathbb {R}}\) be the algebra morphism that sends a graph \(\gamma \) with a single vertex labeled by \(x \in R\) to \(\varepsilon _{A}(\rho (x))\), and that vanishes on all the other connected graphs. Let \({\mathbb {R}}_{\varepsilon }\) be the one-dimensional \(\mathrm {fGC}_{R}\)-module induced by \(Z_{\varepsilon }\), and let
$$\begin{aligned} \texttt {Graphs}^{\varepsilon }_{R}(U) = {\mathbb {R}}_{\varepsilon } \otimes _{\mathrm {fGC}_{R}} {{\mathrm{Tw}}}\texttt {Gra}_{R}(U). \end{aligned}$$
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
For each finite set U, we have a zigzag of quasi-isomorphisms of CDGAs:
$$\begin{aligned} \texttt {Graphs}^{\varepsilon }_{R}(U) \xleftarrow {\sim } \texttt {Graphs}'_{R}(U) \xrightarrow {\sim } \texttt {Graphs}^{\varphi }_{R}(U). \end{aligned}$$
If \(\chi (M) = 0\), then \(\texttt {Graphs}'_{R}\) and \(\texttt {Graphs}^{\varepsilon }_{R}\) are right Hopf \(\texttt {Graphs}_{n}\)-comodules, and the zigzag defines a zigzag of Hopf right comodule morphisms.
Proof
We have a commutative diagram:
The \(\mathrm {fGC}_{R}\)-module \({{\mathrm{Tw}}}\texttt {Gra}_{R}(U)\) is cofibrant. Indeed, it is quasi-free, because \({{\mathrm{Tw}}}\texttt {Gra}_{R}(U)\) is freely generated as a graded \(\mathrm {fGC}_{R}\)-module by reduced graphs. Moreover, we can adapt the proof of Lemma 48 to filter the space of generators in an appropriate manner and show that \({{\mathrm{Tw}}}\texttt {Gra}_{R}(U)\) is cofibrant.
Therefore the functor \({{\mathrm{Tw}}}\texttt {Gra}_{R}(U) \otimes _{\mathrm {fGC}_{R}} (-)\) preserves quasi-isomorphisms. The two evaluation maps \({{\mathrm{ev}}}_{0}, {{\mathrm{ev}}}_{1} : A_{\mathrm {PL}}^{*}(\varDelta ^{1}) \rightarrow {\mathbb {R}}\) are quasi-isomorphisms. It follows that all the maps in the diagram are quasi-isomorphisms.
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.
The differential of R leaves the number of internal vertices constant, therefore if \(\varGamma \) already had an internal vertex, so do all the summands of \(d_{R}\varGamma \). The contracting part \(d_{\mathrm {contr}}\) of the differential decreases the number of internal vertices by exactly one, so let us assume that \(\varGamma \) has exactly one internal vertex. This vertex is at least univalent, as we consider reduced graphs. Then there are several cases to consider, depending of the valence of the internal vertex:
-
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 \(\texttt {Graphs}_{n} \rightarrow {\texttt {e}_{n}^{\vee }}\)).
Finally, the splitting part of the differential leaves the number of internal vertices constant, unless it splits off a whole connected component with only internal vertices, in which case the component is evaluated using the partition function \(Z_{\varepsilon }\). If that connected component consists of a single internal vertex, then we saw in the previous item that splitting the edge connecting this univalent vertex to the rest of the graph cancels with the contraction of that edge. Otherwise, if the graph has more than one vertex, then by definition \(Z_{\varepsilon }\) vanishes on that graph. \(\square \)
The morphisms are quasi-isomorphisms
In this section we prove that the morphisms constructed in Proposition 41 and Proposition 61 are quasi-isomorphisms, 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 semi-algebraic 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 quasi-isomorphisms 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 quasi-isomorphic sub-CDGA of trivial forms.
Recall that \(\varphi \in \varOmega _{\mathrm {PA}}^{n-1}(\texttt {FM}_{M}({\underline{2}}))\) 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
(Precise version of Theorem 3) Let M be a simply connected closed smooth manifold of dimension at least 4. Using the notation recalled above, the following zigzag, where the maps were constructed in Proposition 41, Proposition 60, and Proposition 61, is a zigzag of quasi-isomorphisms of \({\mathbb {Z}}\)-graded CDGAs for all finite sets U:
$$\begin{aligned} \texttt {G}_{A}(U) \xleftarrow {\sim } \texttt {Graphs}^{\varepsilon }_{R}(U) \xleftarrow {\sim } \texttt {Graphs}'_{R}(U) \xrightarrow {\sim } \texttt {Graphs}^{\varphi }_{R}(U) \xrightarrow {\sim } \varOmega _{\mathrm {PA}}^{*}(\texttt {FM}_{M}(U)). \end{aligned}$$
If \(\chi (M) = 0\), then the left-pointing maps form a quasi-isomorphism of Hopf right comodules:
$$\begin{aligned} (\texttt {G}_{A}, {\texttt {e}_{n}^{\vee }}) \xleftarrow \sim (\texttt {Graphs}^{\varepsilon }_{R}, \texttt {Graphs}_{n}) \xleftarrow \sim (\texttt {Graphs}'_{R}, \texttt {Graphs}_{n}). \end{aligned}$$
If moreover M is framed, then the right-pointing maps also form a quasi-isomorphism of Hopf right comodules:
$$\begin{aligned} (\texttt {Graphs}'_{R}, \texttt {Graphs}_{n}) \xrightarrow \sim (\texttt {Graphs}^{\varphi }_{R}, \texttt {Graphs}_{n}) \xrightarrow \sim (\varOmega _{\mathrm {PA}}^{*}(\texttt {FM}_{M}), \varOmega _{\mathrm {PA}}^{*}(\texttt {FM}_{n})). \end{aligned}$$
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 quasi-isomorphism 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 quasi-isomorphisms. 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 quasi-isomorphisms \(\texttt {Graphs}^{\varepsilon }_{R}(U) \rightarrow \texttt {Graphs}^{\varepsilon }_{A}(U)\), 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 quasi-isomorphism; 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 quasi-isomorphism. \(\square \)
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 quasi-isomorphism.
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 quasi-isomorphism 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 \(F_{-\#U-1} \texttt {Graphs}^{\varepsilon }_{A}(U) = 0\) 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. \(\square \)
The symmetric algebra \(S(\omega _{uv})_{u \ne v \in U}\) has a weight grading by the word-length on the generators \(\omega _{uv}\). This induces a weight grading on \({\texttt {e}_{n}^{\vee }}(U)\), because the ideal defining the relations is compatible with the weight grading. This grading in turn induces an increasing filtration \(F'_{s} \texttt {G}_{A}\) on \(\texttt {G}_{A}\) (the extra differential strictly decreases the weight). Define a shifted filtration on \(\texttt {G}_{A}\) by:
$$\begin{aligned} F_{s} \texttt {G}_{A}(U) {:}{=}F'_{s + \# U} \texttt {G}_{A}(U). \end{aligned}$$
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
For each partition \(\pi \) of U, \(\texttt {Graphs}^{\varepsilon }_{A}\langle \pi \rangle \) is a subcomplex of \({\mathsf {E}}^{0} \texttt {Graphs}^{\varepsilon }_{A}(U)\), and \({\mathsf {E}}^{0} \texttt {Graphs}^{\varepsilon }_{A}(U)\) splits as the sum over all partitions \(\pi \):
$$\begin{aligned} {\mathsf {E}}^{0} \texttt {Graphs}^{\varepsilon }_{A}(U) = \bigoplus _{\pi } \bigotimes _{V \in \pi } \texttt {Graphs}^{\varepsilon }_{A}\langle \{V\} \rangle . \end{aligned}$$
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 3-term relation). Finally, for a given monomial in \(\texttt {G}_{A}(U)\), the induced partition of U is still well-defined.
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
For each partition \(\pi \) of U, \(\texttt {G}_{A} \langle \pi \rangle \) is a subcomplex of \({\mathsf {E}}^{0} \texttt {G}_{A}(U)\), and \({\mathsf {E}}^{0} \texttt {G}_{A}(U)\) splits as the sum over all partitions \(\pi \) of U:
$$\begin{aligned} {\mathsf {E}}^{0} \texttt {G}_{A}(U) = \bigoplus _{\pi } \bigotimes _{V \in \pi } \texttt {G}_{A}\langle \{V\} \rangle . \end{aligned}$$
\(\square \)
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. \(\square \)
We have a commutative diagram of complexes:
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 quasi-isomorphism [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 quasi-isomorphism 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 chain-level 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 well-defined chain map, which is clearly a section of the morphism in the lemma, hence the morphism of the lemma is surjective on cohomology. \(\square \)
We now use a proof technique similar to the proof of [34, Lemma 8.3], working by induction. The dimension of \(H^{*}(\texttt {Graphs}'_{n}\langle \{U\} \rangle ) = {\texttt {e}_{n}^{\vee }}\langle \{U\} \rangle = \texttt {Lie}_{n}^{\vee }(U)\) is well-known:
$$\begin{aligned} \dim H^{i} ( \texttt {Graphs}'_{n} \langle \{U\} \rangle ) = {\left\{ \begin{array}{ll} (\#U - 1) !, &{} \text {if } i = (n-1)(\#U - 1); \\ 0, &{} \text {otherwise.} \end{array}\right. }\quad \end{aligned}$$
(33)
Lemma 73
For all sets U with \(\#U \ge 1\), the dimension of \(H^{i}(\texttt {Graphs}^{\varepsilon }_{A} \langle \{U\} \rangle )\) is the same as the dimension:
$$\begin{aligned} \dim H^{i}(A \otimes \texttt {Graphs}'_{n} \langle \{U\} \rangle ) = (\#U - 1)! \cdot \dim H^{i - (n-1)(\#U-1)}(A). \end{aligned}$$
The proof will be by induction on the cardinality of U. Before proving this lemma, we will need two additional sub-lemmas.
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.
There is an explicit homotopy h that shows that \({\mathcal {I}}\) is acyclic. Given a graph \(\varGamma \) with a single external vertex and at least one internal vertex, define \(h(\varGamma )\) to be the same graph with the external vertex replaced by an internal vertex, a new external vertex labeled by \(1_{A}\), and an edge connecting the external vertex to the new internal vertex:
The differential in \(\texttt {Graphs}^{\varepsilon }_{A} \langle {\underline{1}} \rangle \) only retains the internal differential of A and the contracting part of the differential. Contracting the new edge in \(h(\varGamma )\) gives \(\varGamma \) back, and it is now straightforward to check that \(d h(\varGamma ) = \varGamma \pm h(d \varGamma )\). \(\square \)
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.
We now get to the core of the proof of Lemma 73. The idea (adapted from [34, Lemma 8.3]) is to “push” the labels of positive degree away from the chosen vertex u through a homotopy. Roughly speaking, we use Fig. 2 to move labels around up to homotopy.
Lemma 75
The inclusion \(\texttt {Graphs}^{u}_{A} \langle \{U\} \rangle \subset \texttt {Graphs}^{\varepsilon }_{A} \langle \{U\} \rangle \) is a quasi-isomorphism.
Proof
Let \({\mathcal {Q}}\) be the quotient. We will prove that it is acyclic. The module \({\mathcal {Q}}\) further decomposes into a direct sum of modules (but the differential does not preserve the direct sum):
-
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 \(A^{> 0}\).
We now filter \({\mathcal {Q}}\) as follows. For \(s \in {\mathbb {Z}}\), let \(F_{s} {\mathcal {Q}}_{1}\) be the submodule of \({\mathcal {Q}}_{1}\) spanned by graphs with at most \(s+1\) edges, and let \(F_{s} {\mathcal {Q}}_{2}\) be the submodule spanned by graphs with at most s edges. This filtration is preserved by the differential of \({\mathcal {Q}}\).
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 \(k \ge 1\).
Let U be of cardinality \(k+1\). Choose some \(u \in U\) and define \(\texttt {Graphs}^{u}_{A} \langle \{U\} \rangle \) as before. By Lemma 75 we only need to show that this complex has the right cohomology. It splits as:
and therefore using the induction hypothesis:
$$\begin{aligned} \dim H^{i}(\texttt {Graphs}^{u}_{A} \langle \{U\} \rangle )&= k \cdot \dim H^{i - (n-1)} ( \texttt {Graphs}^{\varepsilon }_{A} \langle \{U {\setminus } \{u\}\} \rangle ) \\&= k! \cdot \dim H^{i - k (n-1)}(A). \end{aligned}$$
\(\square \)
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 quasi-isomorphism. 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 quasi-isomorphism.
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 quasi-isomorphism.
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 quasi-isomorphism, 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 quasi-isomorphism.
Now assume that \(U = \{ u \} \sqcup V\), where \(\# V \ge 1\), and assume that the claim is proven for sets of vertices of size at most \(\# V = \#U - 1\). By Eq. (23), we may represent any cohomology class of \(\texttt {FM}_{M}(U)\) by an element \(z \in \texttt {G}_{A}(U)\) satisfying \(dz = 0\). Using the relations defining \(\texttt {G}_{A}(U)\), we may write z as
$$\begin{aligned} z = z' + \sum _{v \in V} \omega _{uv} z_{v}, \end{aligned}$$
where \(z' \in A \otimes \texttt {G}_{A}(V)\) and \(z_{v} \in \texttt {G}_{A}(V)\). The relation \(dz = 0\) is equivalent to
$$\begin{aligned}&\displaystyle dz' + \sum _{v \in V} (p_{u} \times p_{v})^{*}(\varDelta _{A}) \cdot z_{v} = 0, \end{aligned}$$
(35)
$$\begin{aligned}&\displaystyle \text {and } dz_{v} = 0 \text { for all } v. \end{aligned}$$
(36)
By the induction hypothesis, for all \(v \in V\) there exists a cocycle \(\gamma _{v} \in \texttt {Graphs}'_{R}(V)\) such that \(\omega (\gamma _{v})\) represents the cohomology class of the cocycle \(z_{v}\) in \(H^{*}(\texttt {FM}_{M}(V))\), and such that \(\sigma _{*}(\gamma _{v})\) is equal to \(z_{v}\) up to a coboundary.
By Eq. (3536), the cocycle
$$\begin{aligned} {{\tilde{\gamma }}} = \sum _{v \in V} (p_{u} \times p_{v})^{*}(\varDelta _{R}) \cdot \gamma _{v} \in R \otimes \texttt {Graphs}'_{R}(V) \end{aligned}$$
is mapped to a coboundary in \(A \otimes \texttt {G}_{A}(V)\). The map \(\sigma _{*} : R \otimes \texttt {Graphs}'_{R}(V) \rightarrow A \otimes \texttt {G}_{A}(V)\) is a quasi-isomorphism, hence \({{\tilde{\gamma }}} = d {{\tilde{\gamma }}}_{1}\) is a coboundary too.
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 \(H^{*}(A \otimes \texttt {G}_{A}(V)) = H^{*}(M \times \texttt {FM}_{M}(V))\).
We now let \(\gamma ' = -{{\tilde{\gamma }}}_{1} + {{\tilde{\gamma }}}_{2}\), hence \(d\gamma ' = -{{\tilde{\gamma }}} + 0 = - {{\tilde{\gamma }}}\) and \(\sigma _{*}(\gamma ')\) is equal to \(z'\) up to a coboundary. By abuse of notation we still let \(\gamma '\) be the image of \(\gamma '\) under the obvious map \(R \otimes \texttt {Graphs}'_{R}(V) \rightarrow \texttt {Graphs}'_{R}(U)\), \(x \otimes \varGamma \mapsto \iota _{u}(x) \cdot \varGamma \). Then
$$\begin{aligned} \gamma = \gamma ' + \sum _{v \in V} e_{uv} \cdot \gamma _{v} \in \texttt {Graphs}'_{R}(U) \end{aligned}$$
is a cocycle, and \(\omega (\gamma )\) represents the cohomology class of z in \(\varOmega _{\mathrm {PA}}^{*}(\texttt {FM}_{M}(U))\). We have shown that the morphism \(\texttt {Graphs}'_{R}(U) \rightarrow \varOmega _{\mathrm {PA}}^{*}(\texttt {FM}_{M}(U))\) is surjective on cohomology, and hence it is a quasi-isomorphism. \(\square \)
Proof of Theorem 62
The zigzag of the theorem becomes, after factorizing the first map through \(\texttt {Graphs}^{\varepsilon }_{A}\):
$$\begin{aligned}&\texttt {G}_{A}(U) \leftarrow \texttt {Graphs}^{\varepsilon }_{A}(U) \leftarrow \texttt {Graphs}^{\varepsilon }_{R}(U) \leftarrow \\&\texttt {Graphs}'_{R}(U) \rightarrow \texttt {Graphs}^{\varphi }_{R}(U) \rightarrow \varOmega _{\mathrm {PA}}^{*}(\texttt {FM}_{M}(U)) \end{aligned}$$
All these maps are quasi-isomorphisms by Lemma 63, Proposition 60, Proposition 64, and Proposition76. Their compatibility with the comodule structures (under the relevant hypotheses) are due to Proposition 41, Proposition 60, and Proposition 61. \(\square \)
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 quasi-isomorphic 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 quasi-isomorphic, hence there exists some cofibrant S and quasi-isomorphisms \(f : S \xrightarrow {\sim } A\) 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 quasi-isomorphism \(f_{*} : \texttt {Graphs}_{S}^{\varepsilon , \varDelta } \rightarrow \texttt {G}_{A}\) 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 quasi-isomorphism \(\texttt {Graphs}_{S}^{\varepsilon ,\varDelta } \simeq \texttt {Graphs}_{S}^{\varepsilon ',\varDelta '}\) 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 \(n-1\). 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 dg-module S is cofibrant and \({\mathbb {R}}[-n]\) is fibrant (like all dg-modules). 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 quasi-isomorphisms \(\texttt {Graphs}_{S}^{\varepsilon ,\varDelta } \simeq \texttt {Graphs}_{S}^{\varepsilon ',\varDelta }\) (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] \(\mathrm {Conf}_{k-1}(M {\setminus } *) \hookrightarrow \mathrm {Conf}_{k}(M) \rightarrow M\) 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 \(\mathrm {Conf}_{k}(M)\). \(\square \)
Models for configurations on the 2- and 3-spheres
The degree-counting 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 2-sphere, \(S^{2}\). 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 2-sphere \(S^{2}\) is studied in greater detail in Sect. 6, where we study the action of the framed little 2-disks operad on a framed version of \(\texttt {FM}_{S^{2}}\).
In dimension 3, the only simply connected smooth closed manifold is the 3-sphere 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
The claim is clear for \(k = 0\). Since \(S^{3}\) is a Lie group, the Fadell–Neuwirth fibration is trivial [12, Theorem 4]:
$$\begin{aligned} \mathrm {Conf}_{k}({\mathbb {R}}^{3}) \hookrightarrow \mathrm {Conf}_{k+1}(S^{3}) \rightarrow S^{3} \end{aligned}$$
The space \(\mathrm {Conf}_{k+1}(S^{3})\) is thus identified with \(S^{3} \times \mathrm {Conf}_{k}({\mathbb {R}}^{3})\), which is rationally formal with cohomology \(H^{*}(S^{3}) \otimes {\texttt {e}_{3}^{\vee }}({\underline{k}})\). It thus suffices to build a quasi-isomorphism between \(\texttt {G}_{A}(\underline{k+1})\) and \(H^{*}(S_{3}) \otimes {\texttt {e}_{n}^{\vee }}(k)\).
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 \(f(1 \otimes \omega _{ij}) = \omega _{ij} + \omega _{0i} - \omega _{0j}\).
The Arnold relations show that this is a well-defined algebra morphism. Let us prove that \(d \circ f = 0\) on the generator \(\omega _{ij}\) (the vanishing on \(\upsilon \otimes 1\) is clear). We may assume that \(k = 2\) and \((i,j) = (1,2)\), and then apply \(\iota _{ij}\) to get the general case. Then we have:
$$\begin{aligned} (d \circ f)(\omega _{12})= & {} (1 \otimes 1 \otimes \upsilon - 1 \otimes \upsilon \otimes 1) + (1 \otimes \upsilon \otimes 1 - \upsilon \otimes 1 \otimes 1) \\&- (1 \otimes 1 \otimes \upsilon - \upsilon \otimes 1 \otimes 1) = 0 \end{aligned}$$
We know that both CDGAs have the same cohomology, so to check that f is a quasi-isomorphism it suffices to check that it is surjective in cohomology. The cohomology \(H^{*}(\texttt {G}_{A}({\underline{k}}_{+})) \cong H^{*}(S^{3}) \otimes {\texttt {e}_{3}^{\vee }}({\underline{k}})\) is generated in degrees 2 (by the \(\omega _{ij}\)’s) and 3 (by the \(\iota _{i}(\upsilon )\)’s), so it suffices to check surjectivity in these degrees.
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 \(d\omega \ne \iota _{0}(\upsilon )\).
It remains to prove that \(H^{2}(f)\) is surjective. We consider the quotient map \(p : \texttt {G}_{A}({\underline{k}}_{+}) \rightarrow {\texttt {e}_{3}^{\vee }}({\underline{k}})\) that maps \(\iota _{i}(\upsilon )\) and \(\omega _{0i}\) to zero for all \(1 \le i \le k\). We also consider the quotient map \(q : H^{*}(S^{3}) \otimes {\texttt {e}_{3}^{\vee }}({\underline{k}}) \rightarrow {\texttt {e}_{3}^{\vee }}({\underline{k}})\) sending \(\upsilon \otimes 1\) to zero. We get a morphism of short exact sequences:
We consider part of the long exact sequence in cohomology induced by these short exact sequences of complexes:
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. \(\square \)