Benoit Fresse
email: benoit.fresse@univ-lille.fr

Victor Turchin
email: turchin@ksu.edu

Thomas Willwacher
email: thomas.willwacher@math.ethz.ch

ON THE RATIONAL HOMOTOPY TYPE OF EMBEDDING SPACES OF MANIFOLDS IN R n

We study the spaces of embeddings of manifolds in a Euclidean space. More precisely we look at the homotopy fiber of the inclusion of these spaces to the spaces of immersions. As a main result we express the rational homotopy type of connected components of those embedding spaces through combinatorially defined L ∞ -algebras of diagrams.

1. Introduction 1.1. Background. Let L be a smooth compact submanifold possibly with boundary in R m . Let Emb(L, R n ) be the space of smooth embeddings L ֒→ R n , n-m ≥ 2. We allow L to be disconnected with components of possibly different dimensions. For example, one can choose L = r k=1 S m k ⊂ R m , m = max(m k ) + 1. The sets of isotopy classes of such links were studied by Haefliger in [START_REF]Knotted (4k -1)-spheres in 6k-space[END_REF][START_REF]Differentiable embeddings of S n in S n+q for q > 2[END_REF][START_REF]Enlacements de sphères en codimension supérieure à 2[END_REF].

Let i : L ֒→ R n be the trivial embedding, which we define by composing the canonical inclusion L ⊂ R m with a fixed linear embedding of R m into R n (for instance, we can choose to identify R m with the subspace spanned by the m first coordinate axes of R n ). Define the space of embeddings modulo immersions as the homotopy fiber over the trivial embedding i of the obvious map from the space of smooth embeddings to the space of immersions: Emb(L, R n ) := hofiber(Emb(L, R n ) → Imm(L, R n )).

(

) 1
Key words and phrases. Embedding calculus, graph-complexes, infinitesimal bimodules over an operad, rational homotopy theory, graph-complexes, isotopuy classes of embeddings, Maurer-Cartan elements.

B.F. acknowledges support from the Labex CEMPI (ANR-11-LABX-0007-01) and from the FNS-ANR project OCHoTop (ANR-18CE93-0002-01). V.T. has benefited from a visiting position of the Labex CEMPI (ANR-11-LABX-0007-01) at the Université de Lille and a visiting position at the Max Planck Institute for Mathematics in Bonn for the achievement of this work. He was also partially supported by the Simons Foundation travel grant, award ID: 519474. B.F. and V.T. are grateful to the ETH for hospitality. T.W. has been partially supported by the NCCR SwissMAP funded by the Swiss National Science Foundation, and the ERC starting grant GRAPHCPX (ERC StG 678156).

Roughly, Emb(L, R n ) can be thought of as a space of embeddings trivial as immersions. More precisely a point in such a space is a pair ( f, h), where f : L ֒→ R n is a smooth embedding and h : L × [0, 1] → R n is a regular homotopy from f to the inclusion i. The rational homology groups of these spaces Emb(L, R n ) have been extensively studied in [START_REF] Arone | Calculus of functors, operad formality, and rational homology of embedding spaces[END_REF][START_REF] Arone | On the rational homology of high-dimensional analogues of spaces of long knots[END_REF] under the codimension restriction n ≥ 2m + 1. (Note that the embedding spaces are connected in this codimension range.) In this paper we enlarge the range of codimension to n ≥ m + 2 and we describe the rational homotopy type of the components of such spaces. In addition we show that the results of rational homotopy theory allow one to obtain some information about the set of isotopy classes π 0 Emb(L, R n ).

For technical reasons we replace L by its open tubular neighborhood N(L) ⊂ R m . Note that the map L ֒→ N(L) induces a weak homotopy equivalence on our embedding spaces (see [58, Proof of Proposition 1.2]):

Emb(N(L), R n ) ≃ - → Emb(L, R n ).
(

) 2
We also want to study the case where some of the components of the manifold L extend to infinity. We assume for simplicity that any such unbounded component of L is closed and coincides with a Euclidean subspace near infinity. We then consider the space of embeddings modulo immersions Emb ∂ (L, R n ) in which embeddings and immersions coincide with the inclusion L ⊂ R m ⊂ R n outside some ball of a sufficiently big radius. For example, one can take L = r i=1 R m i ⊂ R m , m = max(m i ) + 1 (or just m = m 1 if r = 1), a disjoint union of r affine spaces R m i parallel to the coordinate plane spanned by the first m i basis vectors. These spaces Emb ∂ ( r i=1 R m i , R n ), called spaces of string links modulo immersions, have been studied in [START_REF] Arone | On the rational homology of high-dimensional analogues of spaces of long knots[END_REF][START_REF]Graph-complexes computing the rational homotopy of high dimensional analogues of spaces of long knots[END_REF][START_REF] Arnaud | The rational homology of spaces of long links[END_REF][START_REF] Arnaud | Rational homology and homotopy of high-dimensional string links[END_REF]. It was shown in [START_REF] Arnaud | Rational homology and homotopy of high-dimensional string links[END_REF]Corollary 2.6] that the rational homotopy groups of these spaces are computed by certain hairy graph complexes, provided that n ≥ 2max(m i ) + 2. Moreover, it was conjectured in [START_REF] Arnaud | Rational homology and homotopy of high-dimensional string links[END_REF]Conjecture 3.1] that the result holds in the lower range n ≥ max(m i ) + 3. This conjecture is proven by our Corollary 1.3. Note that the sets π 0 Emb( r i=1 S m i , R n ) = π 0 Emb ∂ ( r i=1 R m i , R n ) form finitely generated abelian groups in this range (see [START_REF]Knotted (4k -1)-spheres in 6k-space[END_REF][START_REF]Differentiable embeddings of S n in S n+q for q > 2[END_REF][START_REF]Enlacements de sphères en codimension supérieure à 2[END_REF], [58, Theorem 4.2, Lemma 4.7], Proposition 5.9). We study these spaces and we discuss the comparison of our results with Haefliger's in sections 5.5-5.7.

For technical reasons, we again replace such a "long manifold" L by N ∞ (L) := N(L) ∪ (R m \ D m R ), the tubular neighborhood of L in R m union the complement of a ball of a sufficiently big radius R. We still have a weak homotopy equivalence:

Emb ∂ (N ∞ (L), R n ) ≃ - → Emb ∂ (L, R n ). (3)
Note that the bounded framework (the case where L ⊂ R m is a compact submanifold) can be reduced to the unbounded one by taking N ∞ (L) = N(L) ∪ (R m \ D m R ), where D m R is a ball that contains N(L), so that:

Emb(N(L), R n ) ≃ Emb ∂ (N ∞ (L), R n ). (4)
1.2. Main results. From now on we assume that M ⊂ R m is the complement of a compact submanifold in R m (possibly with boundary). We also consider the space M * := M ∪ {∞}, topologized as a subset of S m = R m ∪ {∞}. The Goodwillie-Weiss Taylor tower of Emb ∂ (M, R n ) has been expressed by Arone and the second author through derived mapping spaces of infinitesimal bimodules (see [2, Proposition 6.9], [START_REF]Context-free manifold calculus and the Fulton-MacPherson operad[END_REF]Section 6]). The results obtained in these references, combined with the convergence result of Goodwillie, Klein and Weiss [START_REF] Goodwillie | Multiple disjunction for spaces of smooth embeddings[END_REF][START_REF] Thomas | Spaces of smooth embeddings, disjunction and surgery[END_REF][START_REF] Goodwillie | Embeddings from the point of view of immersion theory[END_REF] 1 , imply that, for nm ≥ 3 or nm ≥ 2 and M R m , one has a weak equivalence

Emb ∂ (M, R n ) ≃ IBimod h F m (IF M , F n ), (5)
where F k is the Fulton-MacPherson operad equivalent to the little discs operad E k (see [START_REF] Salvatore | Configuration spaces with summable labels[END_REF]), IF M is a sequence of locally compactified configuration spaces of points in M * , and IBimod h F m (. . . ) is the derived mapping space in the category of infinitesimal bimodules (see Section 2.7). In the case of a bounded submanifold, one has a similar formula in which instead of infinitesimal bimodules, the mapping space of right modules is used (see [START_REF] Arone | On the rational homology of high-dimensional analogues of spaces of long knots[END_REF]Proposition 6.9], [9, Proposition 6.1], [START_REF]Context-free manifold calculus and the Fulton-MacPherson operad[END_REF]Theorem 2.1]). The additional infinitesimal left action is necessary to encode the behavior of embeddings at infinity. We assume that the reader is familiar with this formula and with the underlying algebraic objects.

The main goal of this paper is to understand the right-hand side of (5) in the realm of the rational homotopy theory. We formulate our result in terms of graph complexes. We use the Sullivan rational homotopy theory and the Sullivan differential graded algebras of piece-wise linear differential forms Ω * (X) which can be associated to any space X (for instance, we are going to consider the case X = M * ). We say, by abusing classical conventions, that a differential graded commutative algebra R defines a Sullivan model of the space X when R is quasiisomorphic to the Sullivan differential graded algebra Ω * (X). 2 1 For the convergence one needs the ambient dimension n minus the handle dimension of M * to be at least three. 2 In the literature, a Sullivan model usually refers to an object equipped with a connected cell complex structure in the model category of differential graded commutative algebras. But we do Let A be some (possibly non-unital) differential graded commutative algebra. (In our result, we take for A the augmentation ideal of a Sullivan model of the pointed space M * .) We denote by HGC A,n the complex of Q-linear series of graphs with "hairs" (external legs), each of which decorated by an element of our algebra, as in the following example:

a 1 a 2 a 3 , a 1 , a 2 , a 3 ∈ A.
We assume that the graphs of this complex HGC A,n are connected and have internal vertices of valence greater than or equal to three. Multiple edges and tadpoles (edges connecting a vertex to itself) are allowed. We also assume that the decoration of the hairs is Q-multilinear in A.

We equip this decorated hairy graph complex HGC A,n with a homological grading (as opposed to the differential graded algebra A, which we equip with a cohomological grading, as usual in the Sullivan rational homotopy theory). To be explicit, we endow a graph Γ ∈ HGC A,n with #E edges, #V internal vertices, and with hairs labelled by homogeneous elements a 1 , . . . , a k ∈ A, with the homological degree

deg(Γ) = (n -1)#E -n#V - k i=1 |a i |,
where |a i | denotes the cohomological degree of the elements a i in A.

We endow the decorated hairy graph complex with a differential graded L ∞algebra structure, which we briefly sketch for the moment, referring to sections 3.2 and 3.3 for more details. The differential δ : HGC A,n → HGC A,n consists of three pieces δ = d A + δ split + δ join .

The piece d A is induced by the internal differential of the differential graded commutative algebra A. The piece δ split is given by the sum of the following splitting not need to require the existence of such a cell complex structure for our models in general. Therefore, we abusively use the phrase "Sullivan model" to refer to any differential graded commutative algebra quasi-isomorphic to Ω * (X). Besides we do not make any assumption on the spaces X to which we apply this concept. In particular, we do not necessarily assume that X is connected, so that the definition of a connected cell complex model of the differential graded algebra Ω * (X) does not make sense in our context in general.

operation on internal vertices:

→ , so that δ split Γ = v vertex ±Γ split v. ( 7
)
The piece δ join is defined by the sum of the operations that consists of joining a subset of the hairs of our graph into one hair and multiplying the corresponding decorations in A:

δ join Γ a 1 a 2 . . . a k = S ⊂hairs |S |≥2 ± Γ a 1 . . . j∈S a j . (8)
The higher L ∞ -operations are defined similarly to δ join . In brief, the rth L ∞operation ℓ r (Γ 1 , . . . , Γ r ), r ≥ 2, is the sum of the graphs that we obtain by joining hair subsets of the graphs Γ 1 , . . . , Γ r all together and by multiplying the corresponding decorations. For example, the Lie bracket has the following schematic description:

         Γ , Γ ′          = Γ Γ ′ , (9)
where we suppress the A decorations on hairs for simplicity. (They are multiplied whenever hairs are joined.) Note that, by convention, this Lie bracket has degree -1 (just as the Whitehead product in homotopy theory) and we adopt a similar convention for the higher L ∞ -operations. Thus, in comparison to the standard grading convention for L ∞ -algebra structures, we shift the degree of the L ∞operations by one (see Section 2.1). Our main result is the following.

Theorem 1.1. Let M ⊂ R m be a complement to a compact submanifold (possibly with boundary) and let R be a Sullivan model of the pointed space M * = M ∪ {∞} (with the base-point at infinity). We assume that R is equipped with an augmentation (corresponding to the base point). For nm ≥ 2, we have a weak homotopy equivalence

IBimod h F m (IF M , F Q n ) ≃ MC • (HGC R,n ),
where A = R denotes the augmentation ideal of our differential graded commutative algebra R and MC • (HGC R,n ) denotes the simplicial set of Maurer-Cartan forms with values in the complete L ∞ -algebra HGC R,n . 3To relate our computation to the right-hand side of (5) one may invoke the following result.

Theorem 1.2. Let M be a complement to a compact submanifold in R m . For n -m ≥ 3 or n -m ≥ 2 and M R m , the natural map IBimod h F m (IF M , F n ) → IBimod h F m (IF M , F Q n ) ( 10
)
defines a rational equivalence of nilpotent spaces componentwise and is finite-toone at the π 0 -level.

From Theorem 1.1 and equivalence [START_REF]Vassiliev homotopy string link invariants[END_REF], it follows that one can compute the rational homotopy type of the connected components Emb ∂ (M, R n ) ψ of the embedding spaces Emb ∂ (M, R n ) through our hairy graph complexes.

m : π 0 Emb ∂ (M, R n ) → MC(HGC R,n )/ ∼ . (11)
c. The model of the rational homotopy type of a connected component

Emb ∂ (M, R n ) ψ is given by the positive degree truncation of the twisted L ∞ -algebra HGC m(ψ) R,n , which we associate to the Maurer-Cartan element m(ψ) ∈ MC(HGC R,n ) corresponding to ψ ∈ Emb ∂ (M, R n ).

d. The Maurer-Cartan element that corresponds to the trivial embedding i

: M ֒→ R n is m(i) = 0.
Throughout the paper, the positive degree truncation of an L ∞ -algebra g refers to the L ∞ -subalgebra g >0 ⊂ g which agrees with g k in degree k ≥ 2, is given by the kernel ker(g 1 d -→ g 0 ) in degree k = 1, and vanishes in degree k ≤ 0. The third statement of the corollary (c) implies the identity

π Q k Emb ∂ (M, R n ) ψ H k (HGC m(ψ) R,n ),
for all k ≥ 1, where π Q k denotes the rationalization of the abelian group π k for k ≥ 2, and the Malcev completion of the nilpotent group π 1 for k = 1 (see [START_REF] Quillen | Rational homotopy theory[END_REF]Appendix A.3], see also [START_REF]Homotopy of Operads and Grothendieck-Teichmüller Groups. Part 1. The algebraic theory and its topological background[END_REF]Chapter I.8] for a general introduction to the Malcev completion of groups). The product on π

Q 1 = H 1 (HGC m(ψ) R,n
) is given by the Baker-Campbell-Hausdorff formula (see for example [START_REF] Berglund | Rational homotopy theory of mapping spaces via Lie theory for L ∞algebras[END_REF]Theorem 1.1]). The action of [START_REF] Buijs | Lie models in Topology[END_REF]Section 12.5.1]). The rational homology groups of Emb ∂ (M, R n ) ψ can also be computed as the Chevalley-Eilenberg homology of the L ∞ -algebra (HGC m(ψ) R,n ) >0 (see [START_REF] Berglund | Rational homotopy theory of mapping spaces via Lie theory for L ∞algebras[END_REF]Corollary 1.3], [START_REF] Getzler | Lie theory for nilpotent L ∞ -algebras[END_REF], Section 6). The last statement of the corollary (d) follows from the fact that the component of the trivial embedding under the composition of ( 5) and ( 10) is sent to the component of the map

π Q 1 on π Q k , k ≥ 2, is expressed as the exponent of the adjoint action of H 1 (HGC m(ψ) R,n ) on H k (HGC m(ψ) R,n ) (see
IF M → F m * - → F Q
n that factors through the commutative operad in sets Com (see Section 2.8). Recall that Com is given by the base-point in each arity Com(r) = * , so that we may write Com = * for this operad in what follows.

Theorem 1.2 follows from an application of Mienné's theory of Postnikov decompositions of operads and of bimodules over operads (see [START_REF] Mienné | Tours de Postnikov et invariants de Postnikov pour les opérades simpliciales[END_REF][START_REF]Postnikov decompositions of operads and of bimodules over operads[END_REF]). This theory enables one to adapt Haefliger's proof that the rationalization of the space of sections of a nilpotent fibration can be computed by a model (see [START_REF]Sur la cohomologie de l'algèbre de Lie des champs de vecteurs[END_REF][START_REF]Rational homotopy of the space of sections of a nilpotent bundle[END_REF]). We also refer to [START_REF] Sullivan | Infinitesimal computations in topology[END_REF] for the claim that the rationalization induces a finite-to-one correspondence on the sets of homotopy classes of maps with values in a nilpotent space. The proof of these counterparts of the claims of Theorem 1.2 in the context of spaces can be obtained by using a Postnikov decomposition of the target object of our mapping spaces. In the context of operads, we actually need to consider a decomposition by arity in addition to the decomposition by Postnikov sections. To ensure the convergence of this double decomposition, we need to prove that the source object of our mapping space is equivalent to a cell complex of free infinitesimal bimodules of bounded dimension arity-wise, and that the fibers of the arity decomposition of the target object have a connectivity n(r) that tends to ∞ faster than the dimension bounds of the cells of the source object. We basically check that these assumptions are fulfilled by the infinitesimal bimodules IF M and F n in order to get the conclusion of Theorem 1.2 for our mapping spaces. We explain this verification in detail in Section 4.2.

The results outlined in this introduction enable us to define a map

π 0 Emb ∂ (M, R n ) π 0 IBimod h F m (IF M , F n ) → π 0 MC • (HGC R,n ) = MC(HGC R,n
)/∼, and hence, an invariant of "M-knots". We know that this map is finite-to-one. However, we do not know yet how this map can be efficiently computed. It is likely that it can be expressed in terms of Bott-Taubes-Kontsevich type configuration space integrals. It is also possible that it can be expressed more simply in terms of the rational homotopy type of the complement of the embedding R n \ψ(M) and maybe in addition taking into account a chain version of the Alexander duality.

1.3. Range improvement. We reiterate that even though Theorem 1.1 and Corollary 1.3 are stated for a manifold M ⊂ R m , which is a complement to a compact submanifold, because of the equivalences ( 2) and ( 4), Corollary 1.3 applies to any compact manifold L (with components of possibly different dimensions) embeddable in R m . For R in HGC R,n , we take a Sullivan model of L. Similarly, because of equivalence (3), Corollary 1.3 can be applied to a closed submanifold L ⊂ R m , which near infinity looks like a finite disjoint union of affine subspaces. In the latter case for R we take an augmented Sullivan model of the one-point compactification L * = L ∪ {∞} of L, considered as pointed at ∞.

The range when Corollary 1.3 applies can be slightly improved. Namely, the manifold L does not need to be embeddable in R m , it is enough if it admits an immersion i : L R m . In case L is not compact, i is supposed to be proper, injective outside a compact subset, and to have as image near infinity a disjoint union of affine subspaces. We can then similarly define spaces Emb(L, R n ) and Emb ∂ (L, R n ). Define N(L) as the normal disc bundle over L, and

N ∞ (L) =: M as N(L) union R m \ D m
R the complement to a closed disc of some big radius R. The immersion i can be extended to an immersion N(L) R m , respectively N ∞ (L) R m . Then in this more general situation, the equivalences (2-4) still hold.

For the generalized Theorem 1.1 and Corollary 1.3, the manifold M can be taken as follows. First let |M * | be any compact pointed m-manifold with a basepoint * in its interior M * ⊂ |M * |. Let also

i : |M * | S m = R m ∪ {∞}, be an immersion such that i -1 (∞) = * . The manifold M is then defined as M := M * \ { * }. We consider the spaces Emb ∂ (M, R n ), Imm ∂ (M, R n ), Emb ∂ (M, R n )
of embeddings, immersions, and embeddings modulo immersions for which the

corresponding maps M → R n coincide with M i - → R m ⊂ R n near * . Corollary 1.3(a-c
) for a manifold M as above can be proved by exactly the same arguments, provided nm ≥ 3 or nm ≥ 2 and M * has no component S m . We need the codimension-two-requirement because in the proof we use the relative formality of the little discs operads (see Subection 2.8). Besides, we need to make sure that the ambient dimension n minus the handle dimension of M * is greater than or equal to three, because this assumption is necessary for the convergence of the Goodwillie-Weiss tower [START_REF] Goodwillie | Multiple disjunction for spaces of smooth embeddings[END_REF][START_REF] Thomas | Spaces of smooth embeddings, disjunction and surgery[END_REF][START_REF] Goodwillie | Embeddings from the point of view of immersion theory[END_REF], and for this reason, we do not allow the manifold M to have a closed component in the case where the codimension is two 4 . In Section 4.3 we explain how the main steps in the proof need to be adjusted for this more general situation.

1.4. Plan of the paper. The paper is organized as follows. In Section 2, we set our notation for later use and we briefly recall constructions of the literature that we use throughout the paper. More specifically, we review Pirashvili's Dold-Kan theory of Γ-modules, which we use in our study of our mapping spaces of infinitesimal bimodules over operads, and we review relative formality results for the little discs operads, which we use to reduce our mapping spaces to hairy graph complexes in the rational homotopy theory setting. We also briefly review the definition of the Fulton-MacPherson operad F n and of the infinitesimal bimodule IF M in this section. We introduce the hairy graph complexes HGC A,n in Section 3 and we complete the proof of our main theorems afterwards in Section 4. We eventually address some examples and applications in Section 5 where we also compare our results with previous work of the literature.

Acknowledgement. The authors thank G. Arone, U. Buijs, Y. Félix, P. Lambrechts, A. Skopenkov, D. Stanley and D. Tanré for communication.

Preliminaries

2.1. Generalities. We generally work over the ground field Q (i.e., all vector spaces, algebras, etc. will be defined over Q). The phrase differential graded will be abbreviated by dg. By a dg vector space we usually mean a cochain complex, which can also be considered as a chain complex by inversing the grading.

In general, we apply grading conventions inspired by topological applications. For instance, we apply cohomological conventions for dg commutative algebras as usual in rational homotopy theory (in particular, we assume that the differential of a dg commutative algebra increases degrees by 1), but we adopt homological conventions for dg Lie algebras and L ∞ -algebras (thus, we consider a differential that decreases degrees by 1 in the case of a dg Lie algebra). In addition, we assume that the bracket of a dg Lie algebra has degree -1 (like the Whitehead product in homotopy theory). We similarly assume that all higher brackets operations of an L ∞ -algebra have degree -1. With these grading conventions the Maurer-Cartan elements lie in degree 0.

We denote the category of non-negatively graded dg commutative algebras by dgCom. We equip it with the usual model structure, in which the weak equivalences are the quasi-isomorphisms and the fibrations are the maps that are surjective in all degrees.

In general, for a simplicial model category C, we use the name of the category to refer to the mapping space C(A, B), which we can associate to any objects A, B ∈ C. We also use the notation C(A, B) when C is not simplicial. (In this case, we assume that we have a canonical choice of simplicial frame or of cosimplicial frame.) The derived mapping space, denoted by

C h (A, B), is defined by taking C h (A, B) = C(Q A , R B ) for a cofibrant replacement Q A of A and a fibrant replacement R B of B.
We denote the morphism sets of a category by Mor C (. . . ) in general, but we will simplify this notation for certain categories of diagrams in dg vector spaces and in dg commutative algebras (see Section 3.1).

2.2.

Topological spaces and simplicial sets. By Top we denote the category of compactly generated (possibly non-Hausdorff) topological spaces. This category is cartesian closed, like the category of simplicial sets sset, and moreover we have a Quillen equivalence of model categories that preserves the cartesian closed structure (see [41, Section 2.4]5 ):

| . | : sset ⇄ Top : S • ,
where we consider the geometric realization functor | . | on the one hand and the singular complex functor S • on the other hand. By a "space" we understand either an object in Top or a simplicial set. Because of this equivalence, sometimes we will be sloppy and will not make a difference between these two categories.

2.3. On Γ-modules and Ω-modules. Let Γ be the category of finite pointed sets. We use the expression 'right Γ-module' for the category of contravariant functors T : Γ op → C with values in any category C, whereas the expression 'left Γmodule' refers to the category of covariant functors T : Γ → C. We also consider the category Ω of finite (non-pointed) sets with surjections as morphisms. We then use the expression 'right Ω-module' for the category of contravariant functors T : Ω op → C and 'left Ω-module' for the category of covariant functors T : Ω → C. We denote the category of left Γ-modules in C by C Γ , and the category of left Ω-modules by C Ω .

Suppose now that M is a left Γ-module in some abelian category C. For S * = { * } ⊔ S a pointed set and s ∈ S , we define π s,S * : S * → S * \ {s} to be the map that sends s to the base-point and all other elements to themselves. Then we have a left Ω-module cr M, the cross-effect, such that cr M(S ) ⊂ M(S * ) is the joint kernel of the maps M(π s,S * ). The following result is due to Pirashvili (see [START_REF] Pirashvili | Dold-Kan type theorem for Γ-groups[END_REF]).

Proposition 2.1 (Pirashvili). The cross-effect functor cr : C Γ → C Ω between left Γ-modules and left Ω-modules in an abelian category C is an isomorphism of categories. In particular, for a pair of left Γ-modules A, B ∈ C Γ , we have the identity:

Mor C Γ (A, B) = Mor C Ω (cr A, cr B).
In what follows, we apply this statement in the case where C is the category of dg vector spaces. We also consider left Γ-modules in the category of dg commutative algebras (which is non abelian). We use the terminology 'left Hopf Γmodules' for this category of left Γ-modules. In general, if C is a cofibrantly generated model category, then we can equip the category of left Γ-modules in C with the projective model structure. The weak equivalences in this model category C Γ are the objectwise weak equivalences and the fibrations are the objectwise fibrations.

2.4. Right Γ-modules in topological spaces and in simplicial sets. For our purpose, we also consider the injective model structure on the category of right Γmodules in simplicial sets sset Γ op . The weak equivalences of this model category are the objectwise weak equivalences again, while we take objectwise cofibrations as cofibrations.

The category sset Γ op can also be equipped with a Reedy model structure, defined in [START_REF]On an extension of the notion of Reedy category[END_REF], with the same class of weak-equivalences as the projective and injective model structures, but where the class of cofibrations and the fibrations depends on the definition of latching maps and of matching maps in the category of right Γ-modules. This definition of a Reedy model structure on sset Γ op is a generalization of the classical definition of the Reedy model category of diagrams on a Reedy indexing category in the case where the indexing category, like Γ op , contains non trivial automorphisms (see also [START_REF]On an extension of the notion of Reedy category[END_REF] for this subject).

The Reedy model structure on the category of right Γ-modules in simplicial sets is Quillen equivalent to both the injective model structure and the projective model structure (use that the identity functor carries the Reedy cofibrations to injective cofibrations, the Reedy fibrations to projective fibrations, and preserves all weak-equivalences). This observation implies that we can use any of these model structures to compute the (derived) mapping spaces of right Γ-modules

sset Γ op ,h (• • • ).
We can also adapt the definition of the Reedy model structure for the category of right Γ-modules in topological spaces. (We will explain in Section 2.6 that this Reedy model structure is a particular case of the Reedy model structures associated to the categories of operadic infinitesimal bimodules that we use in this paper.) The Quillen equivalence of Section 2.2, between topological spaces and simplicial sets, extends to a Quillen equivalence between the Reedy model category of right Γ-modules in topological spaces and the Reedy model category of right Γ-modules in simplicial sets:

| . | : sset Γ op ⇄ Top Γ op : S • ,
We can actually see that the functors | . | and S • preserve all weak-equivalences of right Γ-modules (not only the weak-equivalences between cofibrant or fibrant objects), because such an assertion holds in the model categories of simplicial sets and topological spaces. We can again use this equivalence to pass from results on the homotopy of right Γ-modules in topological spaces to results on the homotopy of right Γ-modules in simplicial sets.

2.5. Rational homotopy theory of Γ-modules. By standard results of rational homotopy theory (see [START_REF] Bousfield | On PL de Rham theory and rational homotopy type[END_REF]Section 8], see also [24, Section II.7.2]), the Sullivan functor of piece-wise linear differential forms Ω * and the adjoint Sullivan realization functor G define a Quillen adjunction

G : dgCom ⇄ sset op : Ω * .
(Recall that dgCom denotes the category of non-negatively graded dg commutative algebras.) The derived unit of this Quillen adjunction sends a space X to its rationalization. We explicitly have X Q := LG(Ω * (X)), for any X ∈ sset, where LG denotes the derived functor of G.

In what follows, we also apply the Sullivan functor of piece-wise linear differential forms to topological spaces. Then we set by an abuse of notation Ω * (X) = Ω * (S • (X)), where we consider the image of our space X under the singular complex functor S • (see Section 2.2). Note that the existence of the Quillen adjunction implies that we have an equivalence of mapping spaces sset h (X, LG(R)) ≃ dgCom h (R, Ω * (X)) when we consider the image of dg commutative algebra R under the derived functor of the Sullivan realization functor G. In particular, we have sset h (X, Y Q ) ≃ dgCom h (R, Ω * (X)) when we consider the rationalization of a space Y, for any choice of a dg commutative algebra R quasi-isomorphic to Ω * (Y). (Recall that we use the phrase "Sullivan model of the space Y" for any choice of such a dg commutative algebra R.)

By objectwise application of the functors Ω * and G, we obtain an adjunction:

G : dgCom Γ ⇄ (sset Γ op ) op : Ω * , (12)
between the category of right Γ-modules in simplicial sets sset Γ op and the category of left Γ-modules in dg commutative algebras dgCom Γ . (Recall that we use the phrase 'left Hopf Γ-module' to refer to this category of left Γ-modules dgCom Γ .) The above adjunction is clearly a Quillen adjunction for the projective model structure on dgCom Γ and the injective model structure on sset Γ op (the functor Ω * preserves the weak equivalences and carries the cofibrations of the injective model structure on sset Γ op to fibrations in the projective model category of diagrams dgCom Γ since it does so objectwise) 6 . Thus we can upgrade the classical Sullivan rational homotopy theory of spaces to right Γ-modules.

We can obviously compute the model of a right Γ-module Ω * (X) objectwise since the functor Ω * preserves all weak-equivalences of simplicial sets. In what follows, we also apply the Sullivan model functor to right Γ-modules in topological spaces and we still write Ω * (X) = Ω * (S • (X)) in this case, by the same abuse of notation as in the category of spaces. We easily check that a cofibrant object of the category of left Hopf Γ-modules dgCom Γ is cofibrant in the category of dg commutative algebras objectwise. We deduce from this observation that the rationalization of a right Γ-module X Q reduces to the rationalization of the spaces underlying our object objectwise. We still have the relation

sset Γ op ,h (X, Y Q ) ≃ dgCom Γ,h (R, Ω * (X))
at the mapping space level, for any choice of a left Hopf Γ-module R quasi-isomorphic to Ω * (Y). We more generally have the relation sset Γ op ,h (X, LG(R)) ≃ dgCom Γ,h (R, Ω * (X)) when we take the image of a left Hopf Γ-module R under the derived realization functor LG. In subsequent 6

The same observation holds if we equip dgCom Γ and sset Γ op with the Reedy model structure (using the general result of [START_REF] Berger | Axiomatic homotopy theory for operads[END_REF] in the case of the category dgCom Γ ), but we only use the case of the projective and injective model structures in this paper, because the left Γ-module in dg commutative algebras that we consider in our applications is naturally cofibrant with respect to the projective model structure.

arguments, we also use an extension of this relation for mapping spaces of right Γmodules in topological spaces by using that the Quillen equivalence between the categories of right Γ-modules in simplicial sets and in topological spaces gives an equivalence at the mapping space level.

2.6. Homotopy theory of operads and infinitesimal bimodules. We let Σ be the category of finite sets with bijections as morphisms and Λ be the category of finite sets with injective maps as morphisms. We again consider the category of right Σ-modules, which we define as the category of contravariant functors T : Σ op → C with values in any category C, the category of right Λ-modules, which consists of the contravariant functors T : Λ op → C, and the symmetrically defined categories of left Σ-modules and of left Λ-modules. In the literature, the expression 'symmetric sequence' or 'collection' is also used for our categories of right Σ-modules. In [START_REF]Homotopy of Operads and Grothendieck-Teichmüller Groups. Part 1. The algebraic theory and its topological background[END_REF][START_REF]Homotopy of Operads and Grothendieck-Teichmüller Groups. Part 2. The applications of (rational) homotopy theory methods[END_REF], the expression 'Λ-sequence' is used for the category of right Λ-modules while the expression 'covariant Λ-sequence' is used for the category of left Λ-modules.

In what follows, we notably consider the category of topological right Σmodules Top Σ op and the category of topological right Λ-modules Top Λ op . We equip Top Σ op with the projective model structure and Top Λ op with the Reedy model structure (see [24, Sections II.8.1 and II.8.3]).

The topological operads that we use in this paper are reduced in the sense that their arity zero component is reduced to a point. Recall that the underlying collection of a reduced operad inherits a right Λ-module structure. We use this observation to equip the category of reduced operads with the Reedy model structure transferred from the category of right Λ-modules (see [START_REF]Homotopy of Operads and Grothendieck-Teichmüller Groups. Part 2. The applications of (rational) homotopy theory methods[END_REF]Section II.8.4]). The Fulton-MacPherson operad F m , of which we briefly recall the definition in the next subsection, is Reedy cofibrant.

Recall that an infinitesimal bimodule over an operad P is a right Σ-module M, endowed with a right P-module structure, governed by right composition products

• i : M(k) ⊗ P(ℓ) → M(k + ℓ -1), k ≥ 1, ℓ ≥ 0, 1 ≤ i ≤ k,
together with a compatible infinitesimal left P-action, governed by composition products of the form

• i : P(k) ⊗ M(ℓ) → M(k + ℓ -1), k ≥ 1, ℓ ≥ 0, 1 ≤ i ≤ k.
For details, see [START_REF] Ducoulombier | Projective and Reedy model category structures for (infinitesimal) bimodules over an operad[END_REF]. The category of infinitesimal P-bimodules is denoted by IBimod P .

For any reduced well-pointed topological operad P, the category IBimod P can be equipped with the projective model structure, transferred by adjunction from the projective model structure on Top Σ op . The category IBimod P can also be equipped with the Reedy model structure, which is transferred by adjunction from the Reedy model structure on Top Λ op . (By [START_REF] Ducoulombier | Projective and Reedy model category structures for (infinitesimal) bimodules over an operad[END_REF]Theorem 5.1], this construction returns a valid model structure as soon as the operad P is well-pointed in the sense that the operadic unit defines a cofibration of topological spaces in arity one * → P(1).) Note that the derived mapping spaces IBimod h P (• • • ) coming from the projective and Reedy model structures are equivalent (see [START_REF] Ducoulombier | Projective and Reedy model category structures for (infinitesimal) bimodules over an operad[END_REF]Theorem 5.9], or use the general result of [START_REF] Dwyer | Function complexes in homotopical algebra[END_REF]Proposition 4.4]).

In the paper, we use that the structure of a right Γ-module is the same as the structure of an infinitesimal Com-bimodule, where Com is the set-theoretic operad of unital commutative algebras (the commutative operad), given by the one-point set Com(r) = * in every arity r ≥ 0 (see [START_REF] Victor Turchin | Hodge-type decomposition in the homology of long knots[END_REF]Lemma 4.3]). Thus, with our notation, we have a category identity Top Γ op = IBimod Com . The Reedy model structure of the category of infinitesimal bimodules IBimod Com , which we consider in this paragraph, is also identified with the Reedy model structure of the category of right Γ-modules in topological spaces which we considered in the previous paragraphs.

2.7.

Fulton-MacPherson compactified configuration space. The Fulton-MacPherson operad, denoted by F n , is a classical model of E n -operad. This operad was introduced by Getzler-Jones [START_REF] Getzler | Operads, homotopy algebra and iterated integrals for double loop spaces[END_REF], by using a real differential-geometric variant of the Fulton-MacPherson compactifications of the configuration spaces of points in R n . 7 Recall that the points of F n (S ), where S is any finite sets, are represented by rooted trees with leaves indexed by S , and where each internal vertex with k children is labelled by a configuration of k points in R n modulo scaling and translation. The internal vertices are required to have at least 2 children, as in the following picture, where we take S = {1, . . . , 6} as index set for the leaves of our tree: We consider a similarly defined infinitesimal F n -bimodule IF n [61, Section 6]8 . Points in IF n (S ) are rooted trees with leaves indexed by S , one distinguished vertex, the pearl, and with the following decorations. Non-pearl vertices with k children are labelled by a configuration of k points in R n modulo scaling and translation. The pearl (with say k children) is labelled by a configuration of k points in R n (but not modulo scaling or translation). The pearl is allowed to have any arity ≥ 0, while all the other internal vertices have an arity ≥ 2. If the pearl has arity zero, then this means that all the points escape to infinity in R n ∪ {∞} = S n . The following picture gives the representation of such a pearled tree. We need the following result.

Proposition 2.2. The infinitesimal F m -bimodule IF M is cofibrant in IBimod F m equipped with the Reedy model structure.

Proof. Implicitly this result appeared in the proof of [START_REF]Context-free manifold calculus and the Fulton-MacPherson operad[END_REF]Theorem 6.5]. Recall that, by [START_REF] Ducoulombier | Projective and Reedy model category structures for (infinitesimal) bimodules over an operad[END_REF]Theorem 5.4], an infinitesimal bimodule over a reduced operad P is cofibrant in the Reedy model structure if and only it is cofibrant as an infinitesimal bimodule over P >0 in the projective model structure, where P >0 is the operad such that P >0 (0) = ∅ and P >0 (k

) = P(k), for k > 0. The fact that IF M is cofibrant in IBimod (F m ) >0 is analogous to [61, Lemma 2.2] 9 .
2.8. Relative rational formality of the little disks operads: Recollections from [START_REF] Fresse | The intrinsic formality of E n -operads[END_REF][START_REF] Turchin | Relative (non-)formality of the little cubes operads and the algebraic Cerf Lemma[END_REF]. It is shown in [START_REF] Fresse | The intrinsic formality of E n -operads[END_REF] that the natural maps F m → F n , induced by the inclusion R m → R n , are rationally formal for nm ≥ 2. Concretely, reformulating slightly [27, Theorems C and D], one has a commutative diagram of reduced operads Let us note that by generalities on model categories we may assume that the upper left vertical arrow of our diagram is a fibration 10 . Then we may use that object considered in [START_REF]Context-free manifold calculus and the Fulton-MacPherson operad[END_REF] and denoted by F m is Reedy cofibrant and invert the arrow. Furthermore, note that, trivially, all objects on the right act on themselves considered as operadic infinitesimal bimodules. For later reference we shall hence record the following diagram of operads (three left-hand columns) and infinitesimal bimodules (right-hand column).

F m F Q n R m R Q n * H n ∼ ∼ ∼ relating the map F m → F Q n to
C * [•, M * ] is more complicated -it is obtained from C * [•, M * ] =: IF M by adding non-labelled hairs of length ≤ 1.
F m F m F Q n F Q n R m R Q n R Q n * H n H n = ∼ ∼ ∼ ∼ ∼ (13)
We will make use of the restriction functors associated to the operad morphism of the right-hand side for our infinitesimal bimodule structures. We can for instance regard the object H n as an infinitesimal Com-bimodule by restriction through the operad morphism Com = * → H n . Recall that this implies that H n inherits a natural right Γ-module structure in the category of topological spaces. We go back to this subject in Section 2.10.

2.9. Graph complexes and graph (co)operads. We shall use the graph cooperads Graphs n defined by Kontsevich [START_REF] Kontsevich | Operads and motives in deformation quantization[END_REF]. Let us briefly recall the definition. An admissible graph with r external and k internal vertices is an undirected graph such that the following holds.

• The external vertices are numbered 1, . . . , r.

• There is at least one external vertex in any connected component.

• All internal vertices have at least valence 3.

Tadpoles and multiple edges are allowed. Here is an example of an admissible graph 1 2 3 4 5 .

A graph with #E edges and #V internal vertices is assigned degree (n-1)#E -n#V.

An n-orientation on an admissible graph is the following data:

• For n even it is an ordering of the set of edges up to even permutations.

• For n odd it is an ordering of the set of half-edges and vertices up to even permutations.

We call an admissible graph with orientation data an oriented graph. We now define the space Graphs n (r) as the Q-linear combinations of isomorphism classes of (n-)oriented admissible graphs with r external vertices, together with an orientation. We identify an oriented graph with minus the same graph with opposite orientation.

The spaces Graphs n (r) assemble into a dg Hopf cooperad. The differential is obtained by edge contraction.

δ i = i δ =
The dg commutative algebra structure is given by gluing graphs along external vertices.

1 2 3 4 ∧ 1 2 3 4 = 1 2 3 4
Finally the cooperadic cocompositions are defined by subgraph contraction. Note that each dg commutative algebra Graphs n (r) is quasi-free, generated by the internally connected graphs IG n (r) ⊂ Graphs(r), so that, as graded commutative algebra Graphs n (r) = S (IG n (r)). There is a natural map of dg Hopf cooperads

Graphs n → H * (F n )
given on the algebra generators IG n by sending any graph with internal vertices to zero, and by sending the edge between vertices i and j to the generator ω i j ∈ H * (F n ). We then have the following important statement.

Theorem 2.3 (Kontsevich [START_REF] Kontsevich | Operads and motives in deformation quantization[END_REF], Lambrechts-Volić [START_REF] Lambrechts | Formality of the little N-disks operad[END_REF]). The map of dg Hopf cooperads

Graphs n → H * (F n ) is a quasi-isomorphism for every n ≥ 2.
2.10. The left Γ-module structure of the graph cooperad. We recalled in Section 2.8 that the graded Hopf cooperad

H * (F n ) is equipped with a morphism H * (F n ) → Com c
, where Com c is the dual cooperad of the operad that governs the category of commutative algebras in vector spaces. We explicitly have Com c (r) = Q. We have an immediate description of the restriction of this morphism to the dg Hopf cooperad of graphs as the morphism Graphs n → Com c such that γ → 0 if the graph γ has at least one edge. We mention in Section 2.8 that the operad H n = LG(H * (F n )), which we define by taking the image of the cohomology cooperad H * (F n ) under the left derived functor of the Sullivan realization functor LG, inherits the structure of an infinitesimal bimodule over the commutative operad, and as a consequence, forms a right Γ-module in the category of topological spaces. We can dually see that the cohomology cooperad H * (F n ) inherits the structure of an infinitesimal bicomodule over the commutative cooperad. We also have an isomorphism between the category of infinitesimal Com c -bicomodules in a category and the category of left Γ-modules (just as we observed that the category of infinitesimal Com-bimodules in topological spaces is isomorphic to the category of right Γ-modules). Hence, providing H * (F n ) with such an infinitesimal bicomodule structure amounts to providing this object with a left Hopf Γ-module structure. The right Γ-module structure of the object H n = LG(H * (F n )) actually corresponds to this left Hopf Γ-module structure of the cohomology H * (F n ) when we take the objectwise realization functor of ( 12) for Γ-modules.

By restriction, we get that the cooperad of graphs Graphs n inherits an infinitesimal Com c -bicomodule structure and hence, a left Hopf Γ-module structure like the cohomology cooperad H * (F n ). We then consider the collection Graphs n (S ) associated to arbitrary finite sets S , which we define by an obvious generalization of the definitions of Section 2.9. We take this object Graphs n (S ) to define the image of a pointed set S * = { * } ⊔ S under our functor on the category Γ, while we determine the morphism f * : Graphs n (S ) → Graphs n (T ) associated to a map f : S * → T * of pointed sets S * = { * } ⊔ S and T * = { * } ⊔ T as follows. Let γ ∈ Graphs(S ). We consider the external vertices v s indexed by the elements such that s ∈ f -1 ( * ). We take f * γ = 0 if some of these external vertices v s have an incident edge in γ. We discard these external vertices otherwise and we merge the external vertices v s associated to the elements s ∈ f -1 (t) in the fiber of an index t ∈ T to obtain the graph f * γ ∈ Graphs n (T ) (with the merged external vertex indexed by the corresponding element t ∈ T ).

The collection IG n is preserved by the left Γ-module structure of the cooperad Graphs n and hence, inherits the structure of a left Γ-module in the category of graded vector spaces. The cross effect of this collections cr IG n is identified with the collection of internally connected graphs with no isolated external vertices. We easily see that this left Ω-module cr IG n is free in the sense that every morphism of left Ω-modules on cr IG n is uniquely determined by a morphism of left Σ-modules on a left Σ-module of generators pIG n ⊂ cr IG n , which consists of graphs all of whose external vertices have valence exactly one. We will call such internally connected graphs primitive. For example: We endow the category dgCom Σ with a model structure by declaring the fibrations and weak equivalences to be objectwise fibrations and weak equivalences, respectively. We endow the category dgCom Γ with the projective model structure. Note that, in particular, every object in dgCom Γ is fibrant since every object in dgCom Σ is.

1 2 3
∈
We will need the following statement.

Lemma 3.1. For any object M ∈ dgCom Γ the simplicial left Hopf Γ-module

M ∆ • := M ⊗ Ω * (∆ • ) is a simplicial frame for M.
Proof. This result follows from the classical properties of the Sullivan dg algebra Ω * (∆ • ), namely that we have H * (Ω * (∆ k )) = Q, for any simplicial dimension k ≥ 0, and that the morphism Ω * (∆ • ) → Ω * (∂∆ • ) is surjective degree-wise (see for instance [24, Theorem II.7.1.5] for an account of the applications of the Sullivan dg algebra for the definition of simplicial frames in the category of dg commutative algebras). The first acyclicity claim implies that the morphism M = M ∆ 0 → M ∆ k is a quasi-isomorphism, while the latter claim implies that the matching morphism

M ∆ k → M ∂∆ k is a fibration, because the kth matching object M k (M ∆ • ) of the simplicial left Hopf Γ-module M ∆ • is identified with the tensor product M ∂∆ k = M ⊗ Ω * (∂∆ • ) (see loc. cit.
for the proof of an analogous identity in the category of dg commutative algebras).

3.2. Hairy graph complex.

Lemma 3.2. Suppose that M is a left Hopf Γ-module. Then there is a bijection

Mor HΓ (Graphs o n , M o ) Mor Σ (pIG o n , cr M o ),
where the superscript (-) o means that we consider objects equipped with a zero differential, cr is the cross-effect functor (see Section 2.3) and pIG n is the Σmodule of the primitive internally connected graphs as in Section 2.9.

Proof. Since Graphs o n is free as a left Hopf Σ-module, generated by the internally connected graphs IG n , we have that

Mor HΓ (Graphs o n , M o ) = Mor Γ (IG o n , M o ). ( 14
)
By Proposition 2.1 the cross effect functor induces an equivalence of categories between left Γ-modules and left Ω-modules in any abelian category. We hence find Mor

Γ (IG o n , M o ) = Mor Ω (cr IG o n , cr M o ). ( 15
)
Now cr IG o n is a free left Ω-module, generated by the Σ-module pIG n , so that

Mor

Ω (cr IG o n , cr M o ) = Mor Σ (pIG o n , cr M o ). (16)
We introduce the notation

HGC M,n := Hom Σ (pIG o n , cr M o )
and call HGC M,n the space of M-decorated hairy graphs. Here Hom Σ (-, -) denotes the graded vector space of homogeneous homomorphisms of left Σ-modules.

Proposition 3.3. The space HGC M,n is equipped with an L ∞ -structure {ℓ k } k≥1 (functorially in M), with differential δ = ℓ 1 and higher L ∞ -operations ℓ k , k ≥ 2, together with a descending complete filtration

HGC M,n = F 1 HGC M,n ⊃ F 2 HGC M,n ⊃ • • • that is compatible with the L ∞ -structure in the sense that ℓ k (F p 1 HGC M,n , . . . , F p k HGC M,n ) ⊂ F p 1 +•••+p k HGC M,n . ( 17
)
The Maurer-Cartan elements of HGC M,n are in one-to-one correspondence with the morphisms of left Hopf Γ-modules Graphs n → M.

We recall the explicit definition of a Maurer-Cartan element in a complete L ∞ -algebra later on (in Section 5.1), when we tackle examples of applications of our result. For the moment, we only need a conceptual definition of L ∞ -structures and of Maurer-Cartan elements in terms of zeros of vector fields on graded affine schemes that we shall quickly recall, see for instance [START_REF]Mapping Spaces for DG Hopf Cooperads and Homotopy Automorphisms of the Rationalization of E n -operads[END_REF]Section 4]. Consider an L ∞ -algebra g with differential δ = ℓ 1 and higher L ∞ -operations ℓ k , k ≥ 2, that is equipped with a complete descending filtration

g = F 1 g ⊃ F 2 g ⊃ • • •
compatible with the L ∞ -operations as in [START_REF] Ducoulombier | Delooping of high-dimensional spaces of string links[END_REF]. Then for any graded commutative algebra R we may consider the completed tensor product R ⊗ g, which is again a filtered complete L ∞ -algebra by R-linear extension of the operations. We may then consider the functions

U R : (R ⊗ g) 0 → (R ⊗ g) -1 x → k≥1 1 k! ℓ k (x, . . . , x). ( 18
)
We can recover the operations ℓ k , k ≥ 1, by graded polarization from these functions, for a varying R (see [START_REF]Mapping Spaces for DG Hopf Cooperads and Homotopy Automorphisms of the Rationalization of E n -operads[END_REF]Section 4]). We may conversely define a sequence of operations ℓ k , k ≥ 1, by providing the function U R , for each graded commutative algebra R, as long as we can verify that this function has a power series expansion as above, with terms ℓ k being R-linear extensions of multilinear functions defined over Q. We moreover get that the structure relations of L ∞ -algebras for the operations ℓ k are then equivalent to the relation

U R[ǫ] (x + ǫU R (x)) = U R[ǫ] (x), (19)
for the power series U R , for any graded commutative algebra R, any element x ∈ (R ⊗ g) 0 , and where ǫ is a formal variable of degree +1. Let us note that with, this approach, we define the differential of our L ∞ -algebra as the linear term δ = ℓ 1 of our power series U R , and the above relation [START_REF] Dwyer | Function complexes in homotopical algebra[END_REF] integrates the relation of differential δ 2 = 0 when we focus on the linear term. This representation of the structure of L ∞ -algebras has the advantage of avoiding signs. Geometrically, and as it is used in previous literature, one interprets g as a pro-algebraic graded variety with functor of points R → (R ⊗ g) 0 . Then the power series U R encodes a vector field Q on this graded variety of degree -1, and the L ∞ -relations [START_REF] Dwyer | Function complexes in homotopical algebra[END_REF] state that Q 2 = 0, i.e., that Q is a homological vector field. That said, we will never use this latter algebro-geometric interpretation, and the reader may safely ignore it.

Proof of Proposition 3.3. We equip HGC M,n = Hom Σ (pIG o n , cr M o ) with the descending complete filtration inherited from the filtration on pIG n by the number of edges in graphs. That is, F p HGC M,n consists of those homomorphisms that vanish on all graphs that have fewer than p edges. Now consider a graded algebra R and note that

R ⊗ HGC M,n Hom Σ/R (R ⊗ pIG o n , R ⊗ cr M o )
, where we take R-linear homomorphisms, i.e., we extend our ground ring to R. For this isomorphism we also use that there are only finitely many graphs with a given number of edges.

Lemma 3.2 naturally extends to R-coefficients to give us an isomorphism

(R ⊗HGC M,n ) 0 Mor Σ/R (R⊗pIG o n , R⊗cr M o ) Φ -→ Mor HΓ/R (R⊗Graphs o n , R⊗M o ).
Recall also that the inverse map is given by restriction to generators and projection to cogenerators. Explicitly, the inverse sends a morphism F on the right-hand side to π • F • ι, where ι : pIG n → Graphs n is the natural inclusion and π : M → cr M is the projection, hiding R-linear extension from the notation.

We then define the function

U R : (R ⊗ HGC M,n ) 0 → (R ⊗ HGC M,n ) -1 to be U R (x) := π • (d M • Φ(x) -Φ(x) • d Graphs n ) • ι. (20)
For this proof we shall also abbreviate the commutator with the differentials appearing in this formula to

[d, (-)] := d M • (-) -(-) • d Graphs n .
Let us first verify the L ∞ -relations in the form [START_REF] Dwyer | Function complexes in homotopical algebra[END_REF]. To this end we first note that for any morphism

F ∈ Mor HΓ/R (R ⊗ Graphs o n , R ⊗ M o ) the combination F + ǫ[d, F] : R[ǫ] ⊗ Graphs o n → R[ǫ] ⊗ M o is a morphism of Hopf Γ-modules over R[ǫ],
where ǫ is again a formal variable of degree +1. Note also that for any morphism F :

R[ǫ] ⊗ Graphs o n → R[ǫ] ⊗ M o of Hopf Γ-modules we have Φ(π • F • ι) = F and hence U R[ǫ] (π • F • ι) = π • [d, F] • ι.
We use this to verify [START_REF] Dwyer | Function complexes in homotopical algebra[END_REF] as follows:

U R[ǫ] (x + ǫU R (x)) = U R[ǫ] (π • Φ(x) • ι + ǫπ • [d, Φ(x)] • ι) = U R[ǫ] (π • (Φ(x) + ǫ[d, Φ(x)]) • ι) = π •          [d, Φ(x)] + [d, ǫ[d, Φ(x)]] =0          • ι = U R[ǫ] (x).
For the final simplification we used that the differentials square to zero. Furthermore, note that U R (x) is obtained by composing copies of x with Rlinear extensions of the structure maps (product, Γ structure and differential) of M and Graphs n . Hence it is clear that U R (x) is indeed of the form [START_REF] Ducoulombier | Projective and Reedy model category structures for (infinitesimal) bimodules over an operad[END_REF], obtained using the R-linear extensions of k-linear maps ℓ k on HGC M,n , and hence we obtain indeed an L ∞ -structure on HGC M,n . Furthermore, all structure maps used in the definition of ℓ k either leave the number of edges of graphs in Graphs n constant, or decrease the number of edges. Hence the compatibility [START_REF] Ducoulombier | Delooping of high-dimensional spaces of string links[END_REF] follows.

Finally, by very definition the Maurer-Cartan elements in R ⊗ HGC M,n are those satisfying U R (x) = 0, and hence precisely those elements that correspond under the isomorphism Φ to left Hopf Γ-module maps R ⊗ Graphs n → R ⊗ M.

As a corollary we find the following result.

Proposition 3.4. The left Hopf Γ-module Graphs n (see Section 2.9) is cofibrant in the projective model structure.

Proof. Recall that Com c denotes the dual cooperad of the operad of commutative algebras in vector spaces (see Section 2.10). This cooperad Com c is identified with the initial object of the category of left Hopf Γ-modules. Let the following lifting problem be given.

Com c A Graphs n B ∼ .
By the preceding proposition this lifting problem is equivalent to the problem of lifting some Maurer-Cartan element m ∈ HGC B,n to a Maurer-Cartan element m ′ ∈ HGC A,n along the morphism of L ∞ -algebras HGC A,n → HGC B,n .

However, this morphism is compatible with the descending complete filtrations on both sides, and the associated graded morphism is a surjection and a quasiisomorphism. Hence the lifting is unobstructed. (The obstructions lie in the homology of the kernels of the surjective quasi-isomorphisms gr p HGC A,n ∼ -→ gr p HGC B,n .) 3.3. Explicit combinatorial formulas for the L ∞ -structure on the hairy graph complex. The definition of the L ∞ -structure on HGC M,n in the proof of Proposition 3.3 is concise and in particular takes care of all prefactors and signs. However, it is not very explicit. In this section we work out the explicit combinatorial form of the operations ℓ k , modulo signs and prefactors, in the special case M = R ⊗• , where R is an augmented dg commutative algebra. The left Hopf Γ-module structure of this object M = R ⊗• is determined by the commutative algebra structure of R and we have cr M = R⊗• . For this M we denote the graph complex alternatively by HGC R,n := HGC M,n .

It will be advantageous to identify

HGC R,n = Hom Σ (pIG n , R• ) r≥1 pIG n (r) * ⊗Σ r R⊗r =: pIG * n ⊗Σ R• ,
where we complete the tensor product with respect to the filtration by number of edges. We shall think of elements of the dual space pIG * n of pIG n also as graphs. We can hence depict elements of HGC R,n as series of graphs with external legs which are decorated by elements of R.

a 1 a 2 a 3 , a 1 , a 2 , a 3 ∈ R.
Using this picture, let us describe combinatorially the L ∞ -structure on the graph complex HGC R,n , by tracing the construction of the proof of Proposition 3.3. Concretely, let us compute the L ∞ -operation

ℓ k (Γ 1 , . . . , Γ k ),
for Γ 1 , . . . , Γ k ∈ HGC R,n graphs with R-decorated legs. To this end we have to compute the ǫ 1 • • • ǫ k coefficient in the series (see [START_REF] Benoit Fresse | Operadic cobar constructions, cylinder objects and homotopy morphisms of algebras over operads[END_REF])

U Q[ǫ 1 ,...,ǫ k ] (ǫ 1 Γ 1 + • • • + ǫ k Γ k =:Γ ),
where ǫ j is a variable of degree negative to the degree of Γ j . Let us also write R := Q[ǫ 1 , . . . , ǫ n ]. Recall also that the linear term of our series U R , which corresponds to the case k = 1 of this computation, collects the terms of the differential of our L ∞ -algebra.

We first have to extend our Γ to a map

Γ ′ ∈ Mor Ω/R (R ⊗ cr IG o n , R ⊗ cr M o ) ⊂ Mor Σ/R (R ⊗ cr IG o n , R ⊗ cr M o ) R ⊗ cr IG * n ⊗Σ R•
according to [START_REF] Crowley | A classification of smooth embeddings of fourmanifolds in seven-space, II[END_REF]. Note that elements of the right-hand side can again be considered as graphs, but now with multiple external vertices that are R-decorated, and that can have valency ≥ 1. Tracing the construction one can then see that the desired element Γ ′ is combinatorially obtained by summing over all ways of fusing arbitrary subsets of hairs to external vertices, multiplying the decorations, as indicated in the following picture.

Γ j a 1 a 2 . . . a k → ± Γ j a 1 a 2 . . . . . .
Next we have to take Γ ′ and extend it further to a map

Γ ′′ ∈ Mor Γ/R (R ⊗ IG o n , R ⊗ M o ) R ⊗ IG * n ⊗Σ R •
according to [START_REF] Zhi | Transferring homotopy commutative algebraic structures[END_REF]. Graphically, this just amounts to summing over all ways of inserting zero-valent external vertices in our graphs, decorated by the unit element

1 ∈ R. Γ j a 1 a 2 . . . . . . → Γ j 1 a 1 a 2 . . . 1 . . . 1 1
Finally we have to work out the identification ( 14) to obtain an element

F ∈ Mor HΓ/R (R ⊗ Graphs o n , R ⊗ M o ) ⊂ Mor Σ (R ⊗ Graphs o n , R ⊗ M o ) ⊂ R ⊗ Graphs * n ⊗R ⊗• . ( 21
)
Given an element γ = γ 1 • • • γ r ∈ Graphs n that decomposes into r internally connected components, our desired map F acts as F(γ) = Γ ′′ (γ 1 ) • • • Γ ′′ (γ r ). Graphically this means that F, as an element of the very right-hand side of ( 21), is obtained by fusing several copies of Γ ′′ at the external vertices, thus taking formally an "exponential" of Γ ′′ . Here is a schematic picture of the graphs produced.

Γ 1 Γ 2 Γ 3 1 a 1 a 2 . . . 1 . . . a 1 a a
To the element F we then need to apply the differential, given by the differentials on source and target. Afterwards we restrict the map again to the generators pIG and project to R⊗• . Note in particular that the restriction to pIG is the same as the projection Graphs * n → pIG * that discards all graphs that are not internally connected or have external vertices of valency 1. Furthermore, recall that we only need to keep the coefficient of ǫ 1 • • • ǫ k , or equivalently only those terms in which each Γ j appears exactly once. Terms in F that contribute non-trivially to the end result ℓ k (Γ 1 , . . . , Γ k ) thus can easily be seen to have either of two forms. a. All external vertices have valence exactly one, except for one, which has higher valence. Then the differential on Graphs * n splits off all edges on the unique vertex to make it into a valence one vertex as well.

Γ 1 Γ 2 a a aaa a a → Γ 1 Γ 2 a a aaa a a
The pieces thus produced contribute to both the differential and higher L ∞operations as depicted in (8), [START_REF] Boavida | Manifold calculus and homotopy sheaves[END_REF].

b. The graph is internally connected (in particular k = 1) and all vertices are already of valence one. Then there are contributing pieces of the differential on Graphs * n splitting internal vertices, producing [START_REF]On an extension of the notion of Reedy category[END_REF], and also another piece by the differential on R.

In the case of the computation of ℓ 1 , we retrieve the terms of the differential of HGC R,n depicted in the introduction of the paper. Namely, in (a), we retrieve the term δ join of this differential δ = d R + δ split + δ join , whereas (b) gives the term δ split and the term d R . For the higher operations ℓ k , for k ≥ 2, we retrieve the picture given in the introduction (9) in the particular case k = 2. Indeed, from (a), we obtain that the operations ℓ k , k ≥ 2, take k graphs and glue a non-empty subset of hairs of each graph to a newly created vertex with a hair, which we decorate by the product of the decorations of the fused hairs.

In the sequel we use the following notation. where A * = Q1 ⊕ A is obtained by adjoining a unit to A, so that A = Ā * .

3.4. Hairy graph complex for homotopy commutative algebras. The L ∞algebra HGC R,n of the previous subsection depends on the choice of a model R for M * , which can be relatively big in general. In this section, we improve the situation by extending the construction HGC R,n to homotopy commutative algebras. This allows us to take for R the cohomology H * (M * ) of the space M * , which we equip with a homotopy commutative algebra structure that we can transfer from the Sullivan algebra Ω * (M * ) using perturbation techniques. We refer to Kadeishvili's memoir [START_REF] Kadeishvili | The A(∞)-algebra structure in cohomology, and rational homotopy type[END_REF] for the application of this idea in rational homotopy theory (see also [START_REF]Cohomology C ∞ -algebra and rational homotopy type[END_REF] for a survey). We actually modify Kadeishvili's definition of this model of the rational homotopy of a space on the cohomology. Namely, Kadeishvili uses C ∞ -algebras (called commutative A ∞ -algebras in the first cited reference), which are identified with algebras over the operadic cobar construction of the (operadic suspension of the) Lie cooperad ΩLie c (see [START_REF] Getzler | Operads, homotopy algebra and iterated integrals for double loop spaces[END_REF], see also [START_REF] Loday | Algebraic Operads, Grundlehren der mathematischen Wissenschaften[END_REF]Section 13.1] for an account of this correspondence). But the transfer argument used by Kadeishvili to provide the cohomology H * (M * ) with a homotopy commutative algebra structure can also be applied to the cobar construction ΩL c ∞ , where, instead of the Lie cooperad Lie c , we consider the cooperad of L ∞ -coalgebras L c

∞ . In what follows, we use the notation Lie for the operadic desuspension of the usual Lie operad in the category of graded vector spaces, so that the structure of an algebra over this operad Lie is governed by a Lie bracket of degree 1, as we assume in our grading conventions (see Section 2.1). Recall also that the operad that governs the category of L ∞ -algebras is identified with the operadic cobar construction L ∞ = ΩCom c + of the cooperad of cocommutative non-counital coalgebras Com c + (when we adopt our grading conventions for L ∞ -algebra structures again).

This operad L ∞ = ΩCom c + defines a resolution of the Lie operad Lie, with a quasi-isomorphism L ∞ = ΩCom c + ∼ -→ Lie deduced from the Koszul duality of operads (see again [START_REF] Getzler | Operads, homotopy algebra and iterated integrals for double loop spaces[END_REF] or [START_REF] Loday | Algebraic Operads, Grundlehren der mathematischen Wissenschaften[END_REF]Section 13.2]). The cooperads Lie c and L c ∞ , which we considered in the previous paragraph, are dual to the operads Lie and L ∞ = ΩCom c + in the category of (differential) graded vector spaces. We also have L c ∞ = BCom + , where we take the operadic bar construction B of the operad of non-unital commutative algebras Com + . We can dualize the quasi-isomorphism

L ∞ ∼ - → Lie to get a quasi-isomorphism of dg cooperads Lie c ∼ - → L c
∞ . This quasiisomorphism induces a quasi-isomorphism of dg operads when we pass to the cobar construction ΩLie c ∼ -→ ΩL c ∞ and we also have a quasi-isomorphism of dg operads ΩL c ∞ ∼ -→ Com + by the operadic bar duality (see loc. cit.). Hence, both dg operads ΩLie c and ΩL c ∞ are identified with resolutions of the operad of non-unital commutative algebras Com + .

We consider the category of algebras over the cobar construction of the cooperad of L ∞ -coalgebras ΩL c ∞ as a model for the category of homotopy commutative algebras rather than the category of algebras over the dg operad C ∞ = ΩLie c because we have an explicit formula for the L ∞ -structure of the hairy graph complex in this context, whereas we only have a theoretical result asserting the existence of such a structure in the case of algebras over the dg operad C ∞ = ΩLie c . We just give brief explanations on this theoretical result at the end of the section.

For the moment, we can assume that C is any dg cooperad among Lie c and L c ∞ . Briefly recall that the structure of an algebra A over the cobar construction of a dg cooperad ΩC is equivalent to the structure defined by a twisting coderivation D A on the cofree C-coalgebra C(A), where we call twisting coderivation a map D A : C(A) → C(A), of (cohomological) degree 1, which is a coderivation with respect to the operations of the C-coalgebra structure on C(A), which is trivial on the dg vector space A ⊂ C(A), and which satisfies the relation δD A + D A δ + D 2 A = 0 with respect to the natural differential δ induced by the internal differential of the cooperad C and of the dg vector space A on the cofree C-coalgebra C(A). This twisting coderivation can also be determined by a map ρ A : C(A) → A, of degree 1, and which is also trivial on the dg vector space A ⊂ C(A). Furthermore, we can express the equation of twisting coderivations δD A + D A δ + D 2 A = 0 in terms of this map (see [START_REF] Benoit Fresse | Operadic cobar constructions, cylinder objects and homotopy morphisms of algebras over operads[END_REF] for a detailed account of this correspondence). (The assumption that ρ A vanishes on the dg vector space A ⊂ C(A) reflects our convention that we integrate the unary operation of a homotopy commutative algebra structure in the differential d A of the dg vector space that underlies our object A.)

The vector spaces L c ∞ (r) = BCom + (r) are spanned by rooted trees T with r leaves indexed by 1, . . . , r. The degree of a tree T with p internal vertices is given by -p (in cohomological conventions). The map ρ A : C(A) → A, which determines the structure of an ΩL c ∞ -algebra on A, can therefore be defined by giving a collection of maps ρ T A : A ⊗r → A, associated to such trees T , and such that ρ T A (a σ(1) , . . . , a σ(r) ) = ρ σT A (a 1 , . . . , a r ) for every permutation σ, where σT is defined by applying the permutation σ to the indices of the leaves of our tree. We just take ρ | A = 0 for a trivial tree without vertices T = |, because this operation ρ | A represents the value of the map ρ A : C(A) → A on the summand A ⊂ C(A) (which is trivial under our conventions). We assume that ρ T A has degree 1p (for a tree with p vertices) and the relations of an ΩL c ∞ -algebra structure, equivalent to the equation of twisting coderivations for the coderivation D A corresponding to ρ A , are equivalent to the relations

d A ρ T A + ±ρ T A d A + e ρ T/e A + T =S ′ • ie S ′′ ρ S ′ A • i e ρ S ′′ A = 0,
for the operations associated to the trees T , where d A denotes the internal differential of the dg vector space A (which acts by derivation in the case of the tensor product A ⊗r ), the first sum runs over edge contraction operations T → T/e, and the second sum runs over the operadic decompositions T = S ′ • i e S ′′ in the category of trees. The sign ± corresponds to the permutation of the map d A of degree 1 and of the operation ρ T A of degree 1p, and is determined by the general conventions of differential graded algebra. Recall that the operadic composite S ′ • i e S ′′ of trees S ′ and S ′′ is obtained by plugging the tree S ′′ into the leave of the tree S ′ indexed by i e . In our sum, we actually consider decompositions T = S ′ • i e S ′′ where we can perform a shuffle of the indices of the "free" leaves of S ′ and S ′′ inside the tree T . The index i e is a dummy variable, because we actually assume that our sum runs over isomorphism classes of such decompositions T = S ′ • i e S ′′ . The operation ρ S ′ A • i e ρ S ′′ A is given by ρ S ′ A • i e ρ S ′′ A (a 1 , . . . , a r ) = ±ρ S ′ A (a i 1 , . . . , ρ S ′′ A (a j 1 , . . . , a j ℓ ), . . . , a i k ), for all a 1 , . . . , a r ∈ A, where (i 1 , . . . , i e , . . . , i k ) and ( j 1 , . . . , j ℓ ) reflects the shuffle of the indices of the "free" leaves of the trees S ′ and S ′′ inside T . (The sign ± corresponds to the permutation of variables involved in this formula and is again determined by the general conventions of differential graded algebra.)

Every (non-unital) dg commutative algebra is an ΩL c ∞ -algebra by restriction of structure through the operad morphism ΩL c ∞ → Com + . The ΩL c ∞ -algebra structure of a dg commutative algebra can also be determined by the following formula:

ρ T A (a 1 , . . . , a r ) =        0, if T has more than 1 vertex, a 1 • • • a r , otherwise.
By general results of the homotopy theory of algebras over operads (for which we refer to [START_REF] Hinich | Homological algebra of homotopy algebras[END_REF]), the category of ΩL c ∞ -algebras is endowed with a natural model structure (like the category of non-unital dg commutative algebras), which is transferred from the category of dg vector spaces. Besides, we obtain that the functors of extension of structure and of restriction of structure associated to the quasi-isomorphism ΩL c ∞ → Com + define a Quillen equivalence. In particular, every ΩL c ∞ -algebra is quasi-isomorphic to non-unital dg commutative algebra. We now describe our definition of an L ∞ -algebra structure on HGC A,n when A is an ΩL c ∞ -algebra. We modify the definition of the component δ join of the differential δ = d A + δ split + δ join by taking:

δ join Γ a 1 a 2 . . . a k = T S ⊂hairs |S |≥2 ± Γ a 1 . . . T ρ T A ({a i } i∈S )
.

The variable T runs over the elements of a tree basis of L c ∞ = BCom + . The variable S runs over the subsets of hairs of the graph Γ. The hairs of S are plugged in the leaves of the tree T and the root of that tree becomes a new hair decorated by ρ T A applied to the respective decorations of the hairs in S . In this process, we just discard the tree T = | (and we therefore assume that our hair set S has at least two elements) since we have ρ | A = 0. For graphs Γ 1 , . . . , Γ r ∈ HGC A,n with k 1 , . . . , k r hairs, decorated by elements a i 1 , . . . , a i k 1 ∈ A, i = 1, . . . , r, we define the higher r-ary L ∞ -operations ℓ r , r ≥ 2, by the following similar formulas:

ℓ r (Γ 1 , . . . , Γ r ) = T S ± Γ 1 Γ 2 • • • Γ r T ρ T A ({a i } i∈S )
.

The variable T runs again over the elements of a tree basis of L c ∞ = BCom + . The variable S runs over the subsets of hairs of the graphs Γ 1 , . . . , Γ r such that each graph contributes by at least one hair.

The validity of this construction is easy to check from our explicit description of the structure of an ΩL c ∞ -algebra. We also immediately see that this construction is functorial with respect to morphisms of ΩL c ∞ -algebras and that a quasiisomorphism of ΩL c ∞ -algebras induces a quasi-isomorphism on the hairy graph complex by a standard spectral sequence argument. Hence our construction provides a coherent generalization of the decorated graph complex of commutative algebras. Note simply that we deal with non-unital algebras when we consider the category of ΩL c ∞ -algebras, which are therefore equivalent to the augmentation ideal R of augmented unital dg commutative algebras R rather than to the (augmented) unital dg commutative algebras that we considered so far.

In the context of homotopy commutative algebras, we also have a class of homotopy morphisms (also called ∞-morphisms in the literature), which are equivalent to morphisms of the homotopy category. In general, a homotopy morphism of algebras over the cobar construction ΩC of a dg cooperad C, can be defined by giving a morphism of C-coalgebras F : (C(A), D A ) → (C(B), D B ), where we consider the twisted cofree C-coalgebras (C(A), D A ) and (C(B), D B ) associated to the ΩC-algebra structures on A and B (see [START_REF] Benoit Fresse | Operadic cobar constructions, cylinder objects and homotopy morphisms of algebras over operads[END_REF] for an account of the correspondence between this class of homotopy morphisms and the morphisms of the homotopy category of ΩC-algebras). Such a morphism can also be determined by a map ψ : C(A) → B, of degree 0, and which satisfies coherence constraints which are equivalent to the preservation of differentials when we pass to the morphism of C-coalgebras F : (C(A), D A ) → (C(B), D B ) (see again [START_REF] Benoit Fresse | Operadic cobar constructions, cylinder objects and homotopy morphisms of algebras over operads[END_REF] for an account of this correspondence). Note that we do not take the convention that the map ψ : C(A) → B vanishes on the summand A ⊂ C(A) in our definition of homotopy morphisms. This linear component of our map can actually be identified with a morphism of dg vector spaces f : A → B which underlies our homotopy morphism of ΩC-algebras.

In the case C = L c ∞ , the map ψ : C(A) → B, which determines a homotopy morphism of ΩL c ∞ -algebras, can be defined by giving a collection of maps

ψ T : A ⊗r → B,
associated to the trees T , and such that we again have the equivariance relation ψ T (a σ(1) , . . . , a σ(r) ) = ψ σT (a 1 , . . . , a r ) for every permutation σ. Note that the linear part of ψ, thus the morphism of dg vector spaces f : A → B underlying ψ, is encoded by the map ψ | : A → B associated to the trivial tree T = |. We assume that ψ T has degree -p (for a tree with p internal vertices) and the coherence constraints of homotopy morphisms is equivalent to the relations

d B ψ T -±ψ T d A + T =S ′ (S ′′ 1 ,...,S ′′ m ) ρ S ′ B (ψ S ′′ 1 , . . . , ψ S ′′ m ) = e ψ T/e + T =S ′ • ie S ′′ ψ S ′ • i e ρ S ′′

A

where d A (respectively, d B ) denotes the internal differential of the dg vector space A (respectively, B), and ρ S A (respectively, ρ S B ) denotes the operations that determine the ΩL c ∞ -algebra structure of A (respectively, B). The first sum runs over the operadic tree decompositions T = S ′ (S ′′ 1 , . . . , S ′′ m ), where S ′ is a tree with m leaves in which we plug the trees S ′′ 1 , . . . , S ′′ m . We also assume that we can perform a shuffle of the indices of the leaves of the trees S ′′ 1 , . whose r-ary component acts on graphs schematically as follows

Ψ r (Γ 1 , . . . , Γ r ) = ± Γ 1 Γ 2 • • • Γ r T k • • • T 1 ψ T 1 (. . . ) ψ T k (. . . )
Here we sum over all ways of connecting the hairs to a forest. Each hair of every graph Γ i must go into some tree T j (which can be the trivial tree |). The new hair at the root of the tree T j in the forest is decorated by ψ T j applied to the decorations of the hairs the tree connects to. Note that each map Ψ r has degree zero.

In the case of the cohomology H * (X) of a space X, the ΩL c ∞ -algebra that makes H * (X) a model of the rational homotopy of the space X is obtained by transfer, after picking a quasi-isomorphism of dg vector spaces H * (X)

∼ - → Ω * (X)
where we regard H * (X) as a dg vector space equipped with a trivial differential. The transferred structure can be defined by applying perturbation methods to the coalgebras (as in [START_REF] Kadeishvili | The A(∞)-algebra structure in cohomology, and rational homotopy type[END_REF] and in [START_REF] Zhi | Transferring homotopy commutative algebraic structures[END_REF]) or by model category arguments (see [START_REF] Berger | Axiomatic homotopy theory for operads[END_REF] and [START_REF]Props in model categories and homotopy invariance of structures[END_REF]). In all cases, the obtained object H * (X) is connected to Ω * (X) by a zigzag of quasi-isomorphisms of ΩL c ∞ -algebras, and therefore, we can take this model in our applications to graph complexes. Note simply that we take the augmentation ideal of the cohomology algebra for the pointed space X = M * rather than the cohomology algebra itself in these applications.

We mentioned at the beginning of this subsection that we can also lift the definition of the decorated hairy graph complex to the classical category of C ∞algebras (at least theoretically). We can obtain such a result by observing that we can define a morphism ΩL c ∞ → ΩLie c by lifting the quasi-isomorphism ΩLie c ∼ -→ Com + , using that ΩL c ∞ forms a cofibrant object in the category of dg operads. Indeed, this observation implies that every C ∞ -algebra A inherits an ΩL c ∞ -algebra structure by restriction of structure, and as a result, we can use the construction of this paragraph to associate a decorated hairy graph complex HGC A,n to A.

We finally note that to compute the set of equivalence classes of MC elements, it is enough to know a C ∞ structure of A. In other words, one does not need to produce its lift to an ΩL c ∞ structure, see Remark 5.4.

4. Proof of the main Theorems 4.1. Proof of Theorem 1.1. Our goal is to compute the homotopy type of the mapping space

IBimod h F m (IF M , F Q n ) = IBimod h F m (IF M , (Res F Q n F m F Q n ))
Here we consider the category of infinitesimal bimodules as equipped with the Reedy model structure. (Recall that the homotopy type of the derived mapping space between two objects is the same for the projective and Reedy model structures.) Let us transform the mapping space as follows, using diagram ( 13)

IBimod h F m (IF M , Res F Q n F m F Q n ) ≃ IBimod h F m (IF M , Res R Q n F m R Q n ) ≃ IBimod h F m (IF M , Res H n F m H n ) ≃ IBimod h F m (IF M , Res Com F m Res H n Com H n ).
Here we used that weak equivalences of infinitesimal bimodules induce weak equivalences on the derived mapping space. For the final equality we used that the lower composition in (13) factorizes through the commutative operad Com = * . We now use the adjunction between induction and restriction (Proposition 2.4) to obtain the following extra simplification:

IBimod h F m (IF M , Res Com F m Res H n Com H n ) IBimod h Com (LInd Com F m IF M , Res H n Com H n ).
We finally use Proposition 2.5 to find that

IBimod h F m (IF M , (Res F n F m F n ) Q ) ≃ IBimod h Com (M ו * , Res H n Com H n ).
Now, infinitesimal Com-bimodules are the same as topological right Γ-modules.

We can hence write the result of this relation as

IBimod h Com (M ו * , Res H n Com H n ) = Top Γ op ,h (M ו * , H n ).
Recall that the operad H n is given by H n = LG(H * (F n )), where we consider the derived functor of the Sullivan realization functor on dg commutative algebras G, and that this identity H n = LG(H * (F n )) also holds in the category of right Γmodules. From the observations of Section 2.5, we can simplify our result further to

Top Γ op ,h (M ו * , LG(H * (F n ))) ≃ dgCom Γ,h (H * (F n ), Ω * (M ו * )) ≃ dgCom Γ,h (H * (F n ), R ⊗• ),
where in the last relation we pick an (augmented) Sullivan model R of the space M * .

To compute the derived mapping space we use the graph complex model

Graphs n ≃ H * (F n )
from Section 2.9. By Proposition 3.4 we have that Graphs n is cofibrant as a left Hopf Γ-module (with respect to the projective model structure). Furthermore R ⊗• is fibrant since so is any object in the projective model structure. We can hence write

dgCom Γ,h (H * (F n ), R ⊗• ) ≃ dgCom Γ (Graphs n , R ⊗• )
to compute the mapping space we use the simplicial frame from Lemma 3.1. By combining this expression with the result of Proposition 3.3, we obtain:

dgCom Γ (Graphs n , R ⊗• ) = Mor HΓ (Graphs n , R ⊗• ⊗ Ω * (∆ • )) = MC(Def(Graphs n , R ⊗• ⊗ Ω * (∆ • ))) = MC(Def(Graphs n , R ⊗• ) ⊗Ω * (∆ • )) = MC • (HGC R,n ).
4.2. Proof of Theorem 1.2. Our next goal is to prove that the mapping spaces IBimod h F m (IF M , F n ) are nilpotent under the assumptions of Theorem 1.2, that the rationalization of these mapping spaces are equivalent to the mapping spaces

IBimod h F m (IF M , F Q n )
with values in the rationalization of the infinitesimal bimodule F n , and that the rationalization map is finite-to-one on homotopy classes.

We rely on Mienné's theory of Postnikov decompositions of operads and of infinitesimal bimodules over operads, as we briefly explained in the introduction of the paper. We refer to [START_REF] Mienné | Tours de Postnikov et invariants de Postnikov pour les opérades simpliciales[END_REF] for a detailed account of this theory in the context of operads empty in arity zero (the category of non-unitary operads in the terminology of loc. cit.). The extension of the theory to reduced operads (called unitary operads in loc. cit.) and to infinitesimal bimodules is the subject of the memoir [START_REF]Postnikov decompositions of operads and of bimodules over operads[END_REF] in preparation. The theory is very similar in this case, using concepts introduced in [START_REF]Homotopy of Operads and Grothendieck-Teichmüller Groups. Part 2. The applications of (rational) homotopy theory methods[END_REF] for the study of the homotopy of reduced operads and in [START_REF] Ducoulombier | Projective and Reedy model category structures for (infinitesimal) bimodules over an operad[END_REF] for the study of the homotopy of infinitesimal bimodules. Notably, we use that every reduced operad P has a decomposition P = lim s cosk Λ s P such that Mor Op (-, cosk ≤s P) = Mor Op ≤s (-, P),

where Op ≤s denotes the category of s-truncated (reduced) operads, the category of operads that are defined up to arity ≤ s, and we consider the image of the object P under the obvious forgetful functor (-) ≤s : Op → Op ≤s . The coskeletal decomposition P = lim s cosk Λ s P was introduced in [24, Section II.8.4] in the context of reduced operads and was used in [START_REF] Ducoulombier | Projective and Reedy model category structures for (infinitesimal) bimodules over an operad[END_REF] in the context of infinitesimal bimodules. In both settings cosk Λ s P is determined, as a symmetric sequence, by the Λ-structure of P:

cosk Λ s P(n) = lim u∈Mor Λ (r,n) r≤s P(r),
In what follows, we mainly use that we have cosk Λ s P(s) = P(s) and that cosk Λ s-1 P(s) is identified with M(P)(s), the sth matching object of the reduced operad P.

If P is Reedy fibrant, then the morphisms cosk Λ s P → cosk Λ s-1 P that define this coskeletal decomposition of the operad are Reedy fibrations, and in particular, form a fibration of simplicial sets cosk Λ s P(r) → cosk Λ s-1 P(r) in each arity r. The Postnikov decompositions that we consider are decompositions of these morphisms into towers of morphisms:

cosk Λ s P = lim t P t (cosk Λ s P/cosk Λ s-1 P) → • • • • • • → P t (cosk Λ s P/cosk Λ s-1 P) → P t-1 (cosk Λ s P/cosk Λ s-1 P) → • • • • • • → P 0 (cosk Λ s P/cosk Λ s-1 P) = cosk Λ s-1 P,
which we obtain by applying the classical construction of Postnikov decompositions to the morphisms cosk Λ s P(r) → cosk Λ s-1 P(r) in the category of simplicial sets (we just check that the operad structure passes to the Postnikov sections).

The morphisms P t (cosk Λ s P/cosk Λ s-1 P) → P t-1 (cosk Λ s P/cosk Λ s-1 P) are still Reedy fibrations.

In the simplifying case where P consists of simply connected spaces, the main result of Mienné's theory asserts that such morphisms fit in Postnikov extension diagrams of the following form in the category of s-truncated operads:

P t (cosk Λ s P/cosk Λ s-1 P) ≤s L(Nπ t P(s), t + 1) P t-1 (cosk Λ s P/cosk Λ s-1 P) ≤s K(Nπ t P(s), t + 1)
where Nπ t P(s) is the homotopy group π t of the fiber of the matching map P(s) → MP(s) regarded as an additive operad concentrated in arity s, we consider the associated Eilenberg-MacLane space K(Nπ t P(s), t + 1) and the corresponding path space fibration sequence K(Nπ t P(s), t) → L(Nπ t P(s), t+1) → K(Nπ t P(s), t+ 1) with L(Nπ t P(s), t + 1) = PK(Nπ t P(s), t + 1) ∼ * .

We have an analogous statement in the context of infinitesimal bimodules. In the case of an operad P (which we regard as an infinitesimal bimodule over itself), we can also use the Postnikov decompositions and the above Postnikov extension diagrams in the category of operads to get the Postnikov decomposition of our object P in the category of infinitesimal bimodules.

We use these decompositions to study mapping spaces of infinitesimal bimodules IBimod h R (M, P), where R is any operad endowed with an operad map R → P. We get, at the mapping space level, the tower decomposition

IBimod h R (M, P) = lim s IBimod h R,≤s (M, P) → • • • • • • → IBimod h R,≤s (M, P) ( * ) --→ IBimod h R,≤s-1 (M, P) → • • • • • • → IBimod h R,≤0 (M, P) = * ,
which we can associate to the coskeletal decomposition of the target object since IBimod h R,≤s (M, P) = IBimod h R (M, cosk Λ s P), and we decompose the morphisms of this tower further, as

IBimod h R,≤s (M, P) = lim t IBimod h R,≤s (M, P t (cosk Λ s P/cosk Λ s-1 P)) → • • • → IBimod h R,≤s (M, P t (cosk Λ s P/cosk Λ s-1 P)) ( * * ) --→ IBimod h R,≤s (M, P t-1 (cosk Λ s P/cosk Λ s-1 P)) → • • • → IBimod h R,≤s (M, P 0 (cosk Λ s P/cosk Λ s-1 P)) = IBimod h R,≤s-1 (M, P),
by using the Postnikov decomposition on the target. We get that the morphisms (**), in the latter tower of mapping spaces, are identified with the principal fibrations of simplicial sets with the mapping spaces G s, t = IBimod h R,≤s (M, K(Nπ t P(s), t)) as fibers. We have in general:

π i IBimod h R,≤s (M, K(π(s), t)) = Ht-i Σ s (LIndecM(s), π(s))
, for any Σ s -module π(s) (regarded as an infinitesimal bimodule concentrated in arity s), where LIndec denotes the (derived) functor of indecomposables from the category of infinitesimal bimodules to the category of right Λ-modules in pointed spaces, and we consider the reduced Σ s -equivariant cohomology of the based Σ s -space LIndecM(s) with coefficients in π(s). The indecomposable quotient IndecM of an infinitesimal bimodule M is defined by moding out the spaces M(r) by the Σ r -subspaces spanned by the image of the composition operations

• i : R(k) → M(r -k + 1) → M(r) and • i : M(r -k + 1) → R(k) → M(r)
such that k ≥ 2 (when we assume that the operad R is reduced). This object IndecM naturally inherits the structure of a right Λ-module in pointed simplicial sets. Note that, in the above formula, we consider the reduced homology relative to the natural base point of the indecomposable quotient.

This refined tower decomposition of mapping spaces of infinitesimal bimodules is used to establish the claims of the following statement. We proceed by induction, by using the expression of the fibers of our decomposition, and we use an analysis of the connectivity of these fibers to pass to the limit of the tower whenever necessary. Theorem 4.1 ([48]). Let P be a Reedy fibrant object in the category of infinitesimal bimodules over an operad R. For simplicity, we are still going to assume that P consists of simply connected spaces. Let M be any infinitesimal R-bimodule.

a. If we have a finite dimension bound m(s), in each arity

s ≥ 1, such that H i Σ s (LIndecM(s), π(s)) = 0 for i ≤ m(s)
, for any choice of Σ s -module of coefficients π(s), then the mapping spaces IBimod h R,≤s (M, P) are nilpotent, for all s ≥ 1, and satisfy the relation

IBimod h R,≤s (M, P) Q φ ∼ IBimod h R,≤s (M, P Q ) φ,
for any choice of base point φ ∈ IBimod h R,≤s (M, P), where φ denotes the composite of φ : M ≤s → P ≤s with the rationalization map P ≤s → P Q ≤s .

b. If we assume further that the matching map P(s) → MP(s) is n(s)connected for a connectivity bound n(s) such that n(s)-m(s) → ∞, then the connectivity of the maps IBimod h R,≤s (M, P) → IBimod h R,≤s-1 (M, P), which connect the truncated mapping spaces of the previous assertion, tends to infinity when s → ∞, and we can also pass to the limit s → ∞ in our statement. Thus, we get that the "total" mapping space IBimod h R (M, P) is also nilpotent in this case, and we still have the relation

IBimod h R (M, P) Q φ ∼ IBimod h R (M, P Q ) φ
at this level, for any choice of base point φ ∈ IBimod h R (M, P), where we again denote by φ the composite of this morphism φ : M → P with the rationalization map P → P Q . c. In the previous assertions, if we assume in addition that the homotopy groups Nπ t P(s) associated to the infinitesimal bimodule P are finitely generated in each degree t and in each arity s, and that the cohomology groups H i Σ s (LIndecM(s), π(s)) are finitely generated abelian groups in each degree i, for any choice of finitely generated Σ s -module of coefficients π(s), then the rationalization map P → P Q also induces a finite-to-one map on the connected components of our mapping spaces.

We checked in [START_REF] Fresse | The rational homotopy of mapping spaces of E n operads[END_REF]Lemma 10.7] that the matching map P(s) → MP(s) is (n -2)(s -1)-connected in the case of the operad P = F n . On the other hand, one has that IF M is cofibrant and therefore LIndec(IF M ) = Indec(IF M ). In each arity, IF M (s) is a manifold with corners and

Indec(IF M )(s) = IF M (s)/∂IF M (s) = M ∧s * /∆ s M * .
The right-most space is the quotient of the s-th smash power of M * by the fat diagonal ∆ s M * . The space M ∧s * /∆ s M * is a Σ s -cofibrant pointed space of dimension ms. In case the codimension nm ≥ 3, we can take m(s) = ms and one has that

n(s) -m(s) = (n -2)(s -1) -ms → ∞.
In case the codimension nm = 2, one has that M R m and therefore the handle dimension of M * is ≤ m -1. The assignment M * → M ∧s * /∆ s M * is homotopy invariant and would give an equivalent pointed Σ s -space if M * is replaced by a homotopy equivalent pointed CW-complex X * of dimension ≤ m -1. The dimension of X ∧s * /∆ s X * is m(s) = (m -1)s and therefore the convergency requirement n(s)m(s) → ∞ is again satisfied. Finiteness property follows from the fact that any homotopy group of the configuration space C(s, R n ) ≃ F n (s), n ≥ 3, is finitely generated (see again [25, Lemma 10.7] 11 ) and that M (and therefore M ∧s * /∆ s M * as well) is homotopy equivalent to a finite CW complex.

IF i(U)
as a colimit in the category of infinitesimal F m -bimodules. Note that as a space IF M (k) is the Fulton-MacPherson-Axelrod-Singer local compactification [START_REF] Sinha | Manifold-theoretic compactications of conguration spaces[END_REF] of the configuration space of k + 1 distinct labeled points in M * , with the first being fixed to be * .

The equivalence ( 5) is proved in exactly the same way as [61, Theorem 2.1 or Theorem 6.3]. Propositions 2.2, 2.5 and Theorems 1.1, 1.2 are also proved by exactly the same arguments.

Examples and applications

In this section we provide some applications. In general the cohomology of the graph complex HGC A,n is not known. However, one can perform low degree computations and make certain qualitative statements.

5.1.

Maurer-Cartan elements. The goal of this section is to understand the type of hairy diagrams that appear in degree 0 and that can contribute to Maurer-Cartan (MC) elements. Recall that a MC element of a complete L ∞ -algebra g is, under our grading convention, a degree zero element m ∈ g that satisfies the MC equation:

dm + 1 2 [m, m] + 1 3! [m, m, m] + . . . = ∞ i=1 1 i! ℓ i (m, . . . , m i ) = 0.
11 An expression of π * F n (s) in terms of the homotopy groups of spheres is given in the proof of this reference.

The m-twisted L ∞ algebra g m has the differential

d m = d + [m, -] + 1 2 [m, m, -] + . . . = ∞ i=0 1 i! ℓ i+1 (m, . . . , m i , -)
and (higher) brackets

ℓ m k (-, . . . , - k ) = ∞ i=0 1 i! ℓ i+k (m, . . . , m i , -, . . . , - k ).
Also recall that MC • (g) is the simplicial set MC(g ⊗Ω * (∆ • )) of MC elements. In particular the gauge relations are 1-simplices in this set, or, in other words, MC elements of g ⊗Ω * (∆ 1 ).

For a dg commutative algebra or homotopy commutative algebra A we denote by dim(A), the maximal degree where A is non-zero. However, we do not know in general whether a space M * of the form considered in our main theorem admits a Sullivan model of finite dimension. On the other hand, under the assumptions of Theorem 1.2, one can always choose a homotopy commutative algebra model A of M * of algebraic codimension ndim(A) ≥ 3. For example, we can take the augmentation ideal of the cohomology algebra H * (M * ) equipped with the transferred homotopy commutative algebra structure that makes this object a model of the pointed space M * in the category of homotopy commutative algebras (see the account of Section 3.4). In many relevant applications natural dg commutative algebra models of finite type and of that codimension exist also, and hence, we can take them instead, avoiding the use of homotopy commutative algebras. Proposition 5.1. Let A be a dg commutative algebra or homotopy commutative algebra of finite type with ∆ := ndim(A) -3 ≥ 0, then the complex HGC A,n is bounded below and is finite-dimensional in every degree. Moreover, its part of non-positive degree HGC A,n ≤0 is spanned by trees with ≤ n-3 1+∆ leaves. Any such tree with N leaves has all its leaves labeled by elements of A of degree ≥ (N -1)(1 + ∆) + 1.

Proof. The first statement and the fact that all hairy graphs of genus ≥ 1 appear in strictly positive degrees are easy to check, see [START_REF] Arnaud | Rational homology and homotopy of high-dimensional string links[END_REF]Lemma 4.3] for the proof of a similar result. For the second statement we note that the trees with N leaves have the smallest degree when they are unitrivalent. Such trees have 2N -3 edges and N -2 internal vertices. The smallest possible degree of such a tree is (2N -3)(n -1) -(N -2)n -N(n -3 -∆). Assuming that this degree is ≤ 0 implies N ≤ n-3 1+∆ .

If a unitrivalent tree of degree ≤ 0 has one leaf labeled by an element of degree I and all the other N -1 leaves labeled by elements of the maximal possible degree n -3 -∆, one has (2N -3)(n -1) -(N -2)n -(N -1)(n -3 -∆) -I ≤ 0. The latter inequality implies I ≥ (N -1)(1 + ∆) + 1. By A >0 we understand the naive truncation (A

>0 ) i =        A i , i > 0, 0, i = 0.
Proof. By Proposition 5.1, only linear combinations of trees can produce a MC element of HGC A,n . Moreover all its A labels must be of degree ≥ (N -1)(1

+ ∆) + 1 ≥ (2 -1)(1 + 0) + 1 = 2.
Similarly only trees with A labels of degree ≥ 1 can contribute to the gauge relations. For H 0 the argument is the same.

MC elements in terms of unitrivalent trees.

In this subsection we explain how MC elements can be encoded in terms of unitrivalent trees modulo IHX relations. We hope that this description will be easier to relate to the known higher dimensional knot invariants, see sections 5.5-5.9.

For any dg commutative algebra A and an integer n ≥ 3, consider the space UTT A,n defined as follows. It is spanned by the unitrivalent trees of HGC A,n (which means trees, whose all internal vertices are trivalent). In addition to the orientation relation as in HGC A,n , we quotient this space by IHX relations -images of δ split of trees whose all internal vertices are trivalent except one, which is four-valent. Note that UTT A,n is a quotient-complex of HGC A,n , moreover UTT A,n inherits from HGC A,n a dg Lie algebra structure by means of this quotient map. Proof. Denote by TGC A,n the L ∞ -algebra spanned by all trees of HGC A,n . It is naturally a quotient L ∞ -algebra of HGC A,n . It follows from Proposition 5.1, that TGC A,n has the same set of Maurer-Cartan elements and gauge relations as HGC A,n . On the other hand, both TGC A,n and UTT A,n = TGC A,n /IHX admit a complete filtration by the number of hairs minus one. The quotient map TGC A,n → HGC A,n induces surjective quasi-isomorphisms of associated graded quotients gr i TGC A,n → gr i UTT A,n , i ≥ 1. We use here the fact that the cyclic L ∞ operad is quasi-isomorphic to the cyclic Lie operad. By the same argument as in the proof of Proposition 3.4, we get an equivalence of simplicial sets MC • (TGC A,n ) ≃ MC • (UTT A,n ). In particular, we get a bijection on the sets of connected components.

Remark 5.4. The discussion and result apply mutatis mutandis to the case of a homotopy commutative algebra A, except that in this case UTT A,n is generally not a dg Lie algebra, but inherits an L ∞ -structure from HGC A,n . It is easy to see that the induced L ∞ -algebra structure of UTT A,n is determined by the C ∞ -algebra structure of A restricted along the operad inclusion ∞ → ΩL c ∞ . 5.3. Finiteness. One important question is when the set of isotopy classes of embeddings is finite. Thanks to Corollary 1.3.b we are able to conclude that the set π 0 Emb ∂ (M, R n ) is finite if the corresponding set MC/∼ of gauge equivalence classes of Maurer-Cartan elements is finite.

Example 5.5. π 0 Emb(S 3 × S 3 , R 11 ) is finite. In particular there are finitely many isotopy classes of embeddings S 3 × S 3 ֒→ R 11 which are trivial as immersions. 12 Indeed, first we notice that S 3 × S 3 is formal. On the other hand, according to Proposition 5.1, degree zero diagrams in HGC H * (S 3 )⊗H * (S 3 ),11 are trees with ≤

∂ (M, R n ) i ≃ Q Emb ∂ (M ′ , R n ) i ′ .
Example 5.7. It is easy to check that Emb(RP 2 , R 6 ) is connected. 13 Since RP 2 is embeddable in R 4 and In fact, by the Haefliger-Zeeman unknotting theorem (see [START_REF] Haefliger | Plongements différentiables de variétés dans variétés[END_REF][START_REF] Zeeman | Isotopies and knots in manifolds, Topology of 3-manifolds and related topics[END_REF], [54, Theorem 2.6.b]), the space Emb(S 3 × S 3 , R 11 ) is connected. [START_REF] Budney | On the homology of the space of knots[END_REF] This is because Emb(RP 2 , R 6 ) is connected and Imm(RP 2 , R 6 ) is simply-connected.

Ω * (RP 2 ) ≃ Q, one gets Emb(RP 2 , R 6 ) ≃ Q * . More generally, Emb( r RP 2 , R 6 ) ≃ Q C(r, R 6 ) ≃ F 6 (r).
Proof. Let m be a Maurer-Cartan element of HGC R,n corresponding to ψ. In the complex HGC m R,n consider the filtration by the genus of the graphs plus the number of hairs. The differential d 0 in the associated spectral sequence is the non-deformed differential of HGC R,n . Thus the E 1 term in positive degrees is isomorphic to the positive degree homology of HGC R,n which exactly consists of the rational homotopy groups of Emb ∂ (M, R n ) i .

In Section 5.6.1 we show that the component of the trivial embedding and that of the Hopf link of Emb(S m 1 S m 2 , R m 1 +m 2 +1 ) have different rational homotopy groups.

String links. In this subsection we apply our results to the spaces Emb

∂ r i=1 R m i , R n , n -max(m i ) ≥ 3,
of string links modulo immersions which were defined in the introduction. In this case the corresponding manifold M * = M ∪ {∞} deformation retracts to a wedge of spheres ∨ r i=1 S m i . We can hence take as a Sullivan model

R = Q[ω 1 , . . . , ω r ]/(ω i ω j ) i, j=1,...,r ,
where ω i is a generator of degree m i , and all products of generators vanish. In particular R is an r-dimensional graded vector space spanned by ω 1 , . . . , ω r . It follows that HGC R,n is a complex of hairy graphs with hairs coming in r colors, corresonding to the possible decorations of the hairs by the ω i . This graph complex was denoted by HGC m 1 ...m r ;n in [START_REF] Arnaud | Rational homology and homotopy of high-dimensional string links[END_REF].

In this case the L ∞ -structure is fairly simple: The differential is just δ = δ split (cf. ( 6)), since d R = 0 and δ join = 0 by the vanishing of all products in R. By the same reason the L ∞ -structure is trivial. This reflects the fact that the space of string links is in fact a loop space [START_REF] Ducoulombier | Delooping of high-dimensional spaces of string links[END_REF].

By our computations we can then conclude that for nmax(m i ) ≥ 3 and

ψ ∈ Emb ∂ r i=1 R m i , R n we have that Emb ∂ ⊔ r i=1 R m i , R n ψ ≃ Q MC • (HGC R,n ) α >0 ,
where α is a Maurer-Cartan element corresponding to (the connected component of) ψ. In particular this means that the rational homotopy groups may be computed as

π k (Emb ∂ (M, R n ), ψ) ⊗ Q H k (HGC α R,n ) = H k (HGC R,n ) for k ≥ 1.
For the last identity we used that the L ∞ -structure is trivial and hence the twist does not alter our complex. (Note also that all components of the space of string links have the same homotopy type as the space in question is a loop space.) The statement is true for k = 1 since by the loop space structure, π 1 is abelian. This result has been proved for n ≥ 2 max(m i ) + 2 in [START_REF] Arnaud | Rational homology and homotopy of high-dimensional string links[END_REF] and was conjectured to hold in codimension greater than or equal to three [START_REF] Arnaud | Rational homology and homotopy of high-dimensional string links[END_REF]Conjecture 3.1]. Similarly, we obtain that

H * (Emb ∂ (M, R n ) ψ , Q) S H >0 (HGC R,n ) .
In other words, the homology is just the symmetric product of the homotopy, which again could be deduced by the fact the space of string links is a loop space. Since the L ∞ structure is abelian, the Maurer-Cartan equation and gauge action are abelian as well and we get

MC(HGC R,n )/∼ = H 0 (HGC R,n ). ( 22
)
On the other hand, using Haefliger's work on the isotopy classes of spherical links [START_REF]Knotted (4k -1)-spheres in 6k-space[END_REF][START_REF]Enlacements de sphères en codimension supérieure à 2[END_REF] together with [START_REF] Fresse | The rational homotopy of mapping spaces of E n operads[END_REF]Corollary 20] of the authors' work on the rational homotopy type of the delooping of Emb ∂ (R m , R n ), nm ≥ 3, Songhafouo-Tsopméné and the second author proved [58, Theorem 4.2] that π 0 Emb ∂ r i=1 R m i , R n with respect to the concatenation of links is a finitely generated abelian group isomorphic to H 0 (HGC R,n ) when tensored with Q:

Q ⊗ π 0 Emb ∂ ⊔ r i=1 R m i , R n ≃ H 0 (HGC R,n ). ( 23
)
Note also that by Proposition 5.1 only trees can contribute to the non-positive degrees. Using this and the fact that the cyclic L ∞ operad is quasi-isomorphic to the cyclic Lie operad, the space H 0 (HGC R,n ) can be described as the space spanned by unitrivalent trees of degree zero with leaves labeled by 1, . . . , r, and quotiented out by IHX relations. This fact is quite remarkable as it relates the study of spaces of higher dimensional links to the theory of Vassiliev invariants of classical knots and links in R 3 , where the spaces of unitrivalent diagrams modulo IHX relations naturally appear [START_REF] Bar-Natan | On the Vassiliev knot invariants[END_REF][START_REF]Vassiliev homotopy string link invariants[END_REF]. For more general embedding spaces, the MC element as a knot invariant can also be formulated purely in terms of unitrivalent trees modulo IHX, see Proposition 5.3.

5.6. Spherical links. A. Haefliger proved that π 0 Emb r i=1 S m i , R n , n > max(m i ) + 2, is always a finitely generated abelian group [START_REF]Knotted (4k -1)-spheres in 6k-space[END_REF][START_REF]Differentiable embeddings of S n in S n+q for q > 2[END_REF][START_REF]Enlacements de sphères en codimension supérieure à 2[END_REF]. The product on this set is defined as follows. Given two links f and g, we place them in two disjoint balls. We then connect each f (S m i ) with g(S m i ) by a thin tube along any path. Note that because of the codimension condition the complement is simply connected and the choice of the paths does not matter for the resulting isotopy class of the link. The product is obviously commutative and associative. There are inverses because in codimension > 2 the isotopy classes of links coincide with concordance classes.

This product can be easily extended to π 0 Emb r i=1 S m i , R n , n > max(m i )+2. Proposition 5.9. For n > max(m i ) + 2,

π 0 Emb ⊔ r i=1 S m i , R n = π 0 Emb ∂ ⊔ r i=1 R m i , R n .
Proof. The surjective projection

π 0 Emb ⊔ r i=1 S m i , R n → r i=1 π 0 Emb (S m i , R n )
splits. Denote its kernel by π U 0 Emb r i=1 S m i , R n . The upper script U stands for "unknots" -every component of such links is unknotted. Similarly define the groups π

U 0 Emb r i=1 S m i , R n , π U 0 Emb ∂ r i=1 R m i , R n , π U 0 Emb ∂ r i=1 R m i , R n .
One can easily see that all these four groups are isomorphic:

π U 0 Emb ⊔ r i=1 S m i , R n = π U 0 Emb ⊔ r i=1 S m i , R n = π U 0 Emb ∂ ⊔ r i=1 R m i , R n = π U 0 Emb ∂ ⊔ r i=1 R m i ,
π 0 Emb (S m i , R n ) = π 0 Emb ∂ (R m i , R n )
is proved in [26, Theorem 1.1] (see also Section 5.7).

In fact one also has a bijection on the corresponding sets MC/∼ of Maurer-Cartan elements. In this case M * is homotopy equivalent to

r i=1 S m i { * }. Hence, R = H * ⊔ r i=1 S m i = ⊕ r i=1 H * (S m i ). ( 24
)
Let 1 i ∈ H 0 (S m i ) and ω i ∈ H m i (S m i ) denote the generating cohomology classes of the summand S m i . The corresponding hairy graph complex is spanned by graphs with hairs of 2r possible labels: 1 1 , ω 1 , . . ., 1 r , ω r . The relations are ω i ω j = 0, 1 i 1 j = δ i j 1 j , 1 i ω j = δ i j ω j , 1 ≤ i, j ≤ r.

It follows from Corollary 5.2, Proposition 5.9 and equations ( 22) and [START_REF]Homotopy of Operads and Grothendieck-Teichmüller Groups. Part 1. The algebraic theory and its topological background[END_REF], that in the range n > max(m i ) + 2,

Q ⊗ π 0 Emb ⊔ r i=1 S m i , R n = Q ⊗ π 0 Emb ∂ ⊔ r i=1 R m i , R n = H 0 HGC H * (⊔ i S m i ),n = H 0 HGC H * (∨ i S m i ),n = MC HGC H * (⊔ i S m i ),n ∼ = MC HGC H * (∨ i S m i ),n ∼.

Example of components of different homotopy type.

Unlike in the case of string links, the rational homotopy type of different components of Emb ⊔ r i=1 S m i , R n can be different. For example, we consider the space Emb S m 1 ⊔ S m 2 , R m 1 +m 2 +1 . In the corresponding non-deformed hairy graph complex, the graph m = ω 1 ω 2 is a cycle of degree zero. We expect that, in the Maurer-Cartan element assigned to a link i :

S m 1 ⊔ S m 2 ֒→ R m 1 +m 2 +1
, the coefficient of this graph m is given by the linking number of the components of the link. In particular, the Maurer-Cartan element corresponding to the Hopf link is m. Now if we look at the graph

S = 1 1 1 2 ,
it is a non-trivial cycle (of degree m 1 + m 2 ) in the non-deformed complex. But it is no more a cycle for the m-twisted graph complex:

d m (S ) = [m, S ] = ω 1 ω 2 1 1 + ω 1 ω 2 1 2 .
Topologically this corresponds to the fact that the fibration

Emb S m 1 ⊔ S m 2 , R m 1 +m 2 +1 → S m 1 +m 2 (25)
that assigns to a link the direction of the vector between the images of the basepoints of the spheres, has a section for the component of the trivial link, and does not admit a section for the Hopf link. In the former case the homotopy groups of S m 1 +m 2 appear as direct summands of those of the link space, while for the latter case this is not true even rationally.

) → S n-m-1 → B Emb ∂ (R m , R n ), (26)
nm > 2, where B denotes the classifying space functor.

On the level of graph-complexes this fiber sequence can be seen as follows. The graph complexes corresponding to Emb ∂ (R m , R n ) and Emb(S m , R n ) are HGC R,n and HGC R,n , respectively, where

R = H * (S m ) = Q[ω]/ω 2 = Q1 ⊕ Qω.
Explicitly, elements of HGC R,n are hairy graphs with hairs being either decorated by 1 or ω, while elements of HGC R,n are hairy graphs all of whose hairs are decorated by ω. The graphs

L = 1 ω T = 1 ω ω
are the ones that correspond to the rational homotopy of S n-m-1 . The graph L is always non-zero and T is nonzero if and only if nm is odd. Let U be a subspace in HGC R,n spanned by L and T . One has that the (graded vector space) direct sum U ⊕ HGC R,n is a subcomplex in HGC R,n . The authors proved (see [26, Theorem 2.1]) that the inclusion

U ⊕ HGC R,n ⊂ HGC R,n
is a quasi-isomorphism. The fiber sequence (26) implies that π 0 Emb(S m , R n ) = π 0 Emb ∂ (R m , R n ) and that all components of Emb(S m , R n ) have the same homotopy type. The latter fact can also be seen from the observation that Emb ∂ (R m , R n ) acts on Emb(S m , R n ) for which π 0 Emb(S m , R n ) is a torsor of π 0 Emb ∂ (R m , R n ).

The group π 0 Emb(S m , R n ) = π 0 Emb ∂ (R m , R n ), nm > 2, is a finitely generated abelian group usually torsion except two cases (a) n = 4k -1, m = 2k -1, k ≥ 2;

(b) n = 6k, m = 4k -1, k ≥ 1.

In the latter two cases this group is infinite and has rank one [START_REF] Fresse | The rational homotopy of mapping spaces of E n operads[END_REF]Corollary 20]. 14 The corresponding Maurer-Cartan elements (generators of the groups H 0 (HGC R,n )) are, respectively, L ω = ω ω and T ω = ω ω ω .

Geometrically, the class L ω appears as image, under inclusion

Ω 2k V 2k-1 (R 4k-1 ) → Emb ∂ (R 2k-1 , R 4k-1 ),
of the SO(2k) Euler class in the rational homotopy of the Stiefel manifold V 2k-1 (R 4k-1 ) = SO(4k-1)/SO(2k). In case (b), the MC element T ω corresponds to the Haefliger trefoil S 4k-1 ֒→ R 6k [START_REF]Knotted (4k -1)-spheres in 6k-space[END_REF].

The twisting by L ω changes neither the differential, nor any (higher) bracket of HGC H (S 2k-1 ),4k-1 . This is because for even codimension nm, any graph with two ω-hairs attached to an internal vertex is zero. The twisting by T ω changes both the differential and the L ∞ structure of HGC H * (S 4k-1 ),6k . One can show that HGC T ω H * (S 4k-1 ),6k is L ∞ isomorphic to the non-deformed one HGC H * (S 4k-1 ),6k . In [26], we also determine the rational homotopy type of Emb(S m , R n ), n-m > 2 (see [26, Corollary 1.3]). We prove that each component of Emb(S m , R n ) is rationally a product of K(Q, j)'s in all cases except when n is odd and m is even. In the latter case Emb(S m , R n ) ≃ Q Emb ∂ (R m , R n ) × S n-m-1 and the failure of not being rationally abelian is only in the factor S n-m-1 . 5.8. Non-linear Maurer-Cartan equation. In all the examples that we considered so far the Maurer-Cartan equation for a hairy graph complex HGC A,n was reduced to a linear equation and the set MC(HGC A,n )/∼ is identified with H 0 (HGC A,n ). However, it is not generally the case. Consider, for example, Emb(S 2 ×S 2 , R 7 ). Since S 2 ×S 2 is embeddable in R 5 , our approach can be applied. One has

R = Q[ω 1 , ω 2 ]/ ω 2 1 =ω 2 2 =0
, where |ω 1 | = |ω 2 | = 2. In degree zero HGC R,n has two graphs

L 1 = ω 1 ω 1 ∧ ω 2 ; L 2 = ω 2 ω 1 ∧ ω 2 .
(In fact there is one more graph of degree zero: the H-shaped one with all its four hairs labelled by ω 1 ∧ ω 2 , but it can be killed by gauge transformations as it is the 14 This fact can also be easily deduced from Haefliger's [36, Corollary 6.7 and Remark 6.8].

boundary of the X-shaped graph again with all its four hairs labelled by ω 1 ∧ ω 2 . Compare also with Proposition 5.3.) If m = λ 1 L 1 + λ 2 L 2 , λ 1 , λ 2 ∈ Q, then the Maurer-Cartan equation becomes

0 = 1 2 [m, m] = λ 1 λ 2 × ω 1 ∧ ω 2 ω 1 ∧ ω 2 ω 1 ∧ ω 2 .
Thus, the set

MC/∼ = {λ 1 L 1 + λ 2 L 2 | λ 1 = 0 or λ 2 = 0}.
We conjecture that in this case, the MC element is related to the Boéchat-Haefliger invariant [START_REF] Boéchat | Plongements de variétés différentiables orientées de dimension 4k dans R 6k+1[END_REF][START_REF] Boéchat | Plongements différentiables des variétés orientées de dimension 4 dans R 7 , Essays on Topology and Related Topics[END_REF] BH : π 0 Emb(S 2 × S 2 , R 7 ) → H 2 (S 2 × S 2 , Z) = Z 2 , which was shown in [START_REF] Boéchat | Plongements différentiables des variétés orientées de dimension 4 dans R 7 , Essays on Topology and Related Topics[END_REF] to have as image 2Z × 0 0 × 2Z. Up to an action of the torsion π 0 Emb(S 4 , R 7 ) = Z 12 (that preserves BH), this invariant determines the isotopy class of embedding [START_REF] Crowley | A classification of smooth embeddings of fourmanifolds in seven-space, II[END_REF][START_REF]A classification of smooth embeddings of 4-manifolds in 7-space, I[END_REF]. We conjecture that the coefficient λ in the MC element corresponds to the Whitney invariant [55, Section 1] W : π 0 Emb(S 1 × S 2 , R 6 ) → H 1 (S 1 × S 2 , Z) ≃ H 2 (S 1 × S 2 , Z) ≃ Z, while the coefficient µ corresponds to the action of π 0 Emb(S 3 , R 6 ) = Z ≃ Q π 0 Emb(S 3 , R 6 ) (the generators are Haefliger's trefoil in π 0 Emb(S 3 , R 6 ) = Z [START_REF]Knotted (4k -1)-spheres in 6k-space[END_REF], and the tripod T ω ∈ H 0 (HGC H * (S 3 ),6 ) = Q, see Section 5.7). It was shown by A. Skopenkov in [START_REF]A classification of smooth embeddings of 3-manifolds in 6-space[END_REF] that

• for any integer i there exists an embedding S 1 × S 2 ֒→ R 6 of Whitney invariant i;

• two isotopy classes of such embeddings have the same Whitney number if and only if they are in the same orbit of the π 0 Emb(S 3 , R 6 )-action;

• all π 0 Emb(S 3 , R 6 )-orbits in π 0 Emb(S 1 × S 2 , R 6 ) are finite except the one of Whitney invariant 0.

In fact in [START_REF]A classification of smooth embeddings of 3-manifolds in 6-space[END_REF] A. Skopenkov completely classifies isotopy classes of smooth embeddings of any closed connected orientable 3-manifold in R 6 in terms of the Whitney invariant and the π 0 Emb(S 3 , R 6 )-action. On the other hand any compact orientable 3-manifold is parallelizable [START_REF] John | Characteristic classes[END_REF] and therefore is immersible in R 4 . According to Section 1.3 our approach can be applied. In fact, the above computation stays exactly the same, except that for a compact connected orientable M the class L α above gets replaced by classes L α i labelled by a basis {α i } i of H 1 (M). This can be compared to and is precisely in agreement with Skopenkov's results. Note that in the general case we can still take the cohomology R = H * (M) as a model for M, but now we must account for higher homotopy commutative operations. These higher operations however all produce graphs with ≥ 2 vertices and hence do not enter the above computation.

Sullivan models

For completeness, let us remark that Theorem 1.1 also allows us to write down a Sullivan model for the components of the embedding space, using standard results in the literature. Recall that by HGC H * (M * ),n we understand the L ∞ algebra induced by the homotopy commutative structure of H * (M * ), see Section 3.4. The statement about the Chevalley-Eilenberg complex is obtained by applying [8, Corollary 1.3]. Remark 6.2. One can also build a model (essentially) out of the Chevalley-Eilenberg complex of (HGC m R,n ) >0 for a general dg commutative algebra model R of M * of finite type. One just has to interpret the Chevalley-Eilenberg complex appropriately, as follows. One first notes that HGC R,n is in fact the dual of an L ∞ -coalgebra HG R,n . Concretely, while elements of HGC R,n are formal (possibly) infinite series of graphs, elements of HG R,n are finite linear combinations of (the dual) graphs. Then the homological Chevalley complex of the positive degree truncation (HG m R,n ) >0 of the twist of HG R,n , is a dg commutative algebra model for the corresponding connected component of MC • (HGC R,n ). To see this one first notes that we have a map of L ∞ -coalgebras HG R,n → HG H * (M * ),n and accordingly a map of the (homological) Chevalley-Eilenberg complexes

C CE * (HG m R,n ) >0 → C CE * (HG m ′ H * (M * ),n ) >0 ,
where m is the image of the Maurer-Cartan element m ′ ∈ MC(HGC H * (M * ),n ). The map above is again a quasi-isomorphism of dg commutative algebras, as one can see from the spectral sequence associated to the bounded below exhaustive filtration on the number of edges in graphs on both sides.

Corollary 1 . 3 .

 13 Under the assumptions of Theorem 1.2, we have the following statements:a. Every component of Emb ∂ (M, R n ) is nilpotent.b. We have a naturally defined finite-to-one map from the set of connected components of the space Emb ∂ (M, R n ) to the set of Maurer-Cartan elements of the L ∞ -algebra HGC R,n modulo gauge equivalence:

7

  This type of compactification is sometimes called Fulton-MacPherson-Axelrod-Singer compactification.

6

  Now let M ⊂ R m be an open subset containing a neighborhood of ∞. Recall that we write M * := M ∪ {∞} for the space obtained by adding to M the point at infinity in S m = R m ∪ {∞}. We consider the infinitesimal F m -bimodule IF M ⊂ IF m composed of those configurations that lie in the pre-image of M ×r * under the map IF m (r) → (S m ) r . As a space, IF M (k) is the Fulton-MacPherson-Axelrod-Singer local compactification of the configuration space C * (k, M * ) of k + 1 points in M * , one of which is fixed to be ∞.

  one factorizing through the commutative operad Com = * . Here R m and R n are intermediate operads whose precise form is not relevant at this point, whereas H n = LG(H * (F n )) is defined by taking the image of the cohomology of the topological operad F n under the derived functor of the Sullivan realization functor LG. (We then regard the object H * (F n ) as a dg Hopf cooperad equipped with a trivial differential.) Recall that, for n ≥ 2, the cohomology of the Fulton-MacPherson operad H * (F n ) is identified with the n-Poisson cooperad Pois c n , the dual of the operad Pois n that governs the category of graded commutative algebras equipped with a Poisson bracket of degree n -1. Let Com c denote the dual cooperad of the operad of commutative algebras in vector spaces. We have a morphism of graded Hopf cooperads H * (F n ) = Pois c n → Com c , dual to the morphism of graded operads Com → Pois n , which reflects the obvious restriction of structure functor from the category of graded Poisson algebras governed by the operad Pois n to the category of graded commutative algebras. We have LG(Com c ) = G(Com c ) = * and the morphism * → H n in the above diagram is identified with the image of the morphism of graded Hopf cooperads H * (F n ) → Com c under the derived functor of the Sullivan realization functor LG.

10 Let

 10 A ← B → C be a zigag in a model category. We may factorize the second map into B ∼ ֒-→ B ′ ։ C and then lift to obtain a zigzag A ← B ′ ։ C. In our case we apply this construction to the arrow category equipped with the projective model structure.

1 2 3 4 5

 5 → 1 2 * ⊗ 3 4 5 + 1 2 * ⊗ 3 4 5

Notation 3 . 5 .

 35 For a non-unital dg commutative algebra A, we define HGC A,n := HGC Ā * ,n ,

4. 3 .

 3 Range improvement. In this subsection we explain why Theorem 1.1 and Corollary 1.3.a-c hold for more general manifolds M m as described in Section 1.3. One has M ⊂ M * = M ∪ { * } and the immersion i : M * S m = R m ∪ {∞} satisfies i -1 (∞) = * . The corresponding infinitesimal F m -bimodule IF M is defined as follows. Let iO(M) denote the category of open subsets U ⊂ M, such that U ∪ { * } is open in M * and i| U is injective. Define IF M := colim U∈iO(M)

Corollary 5 . 2 .

 52 Under the assumptions of Proposition 5.1, we have the relations MC HGC A,n ∼ = MC HGC A >0 ,n ∼ and H 0 HGC A,n = H 0 HGC A >0 ,n .

Proposition 5 . 3 .

 53 Assuming ndim(A) ≥ 3, the quotient map HGC A,n → UTT A,n induces a bijection on the sets of gauge equivalence classes of MC elements.

11 - 3 1+2 = 2 Corollary 5 . 6 .

 113256 + 2 3 leaves. But any such tree α β has never degree zero. Thus the set MC HGC H * (S 3 )⊗H * (S 3 ),11 has only one element 0. 5.4. M-unknots. The hairy graph complex HGC R,n depends only on the Sullivan model R of M * . The following is an immediate consequence of Corollary 1.3.c-d. Let i : M ⊂ R m ⊂ R n and i ′ : M ′ ⊂ R m ′ ⊂ R n satisfy the assumptions of Theorem 1.2. If the models Ω * (M * ) ≃ Ω * (M ′ * ) are quasi-isomorphic as augmented dg commutative algebras, then the components of the trivial embedding are rationally equivalent: Emb

Proposition 5 . 8 .

 58 Under the assumptions of Theorem 1.2, the rational homotopy groups of any component Emb ∂ (M, R n ) ψ are subquotients of those of the component of the trivial embedding Emb ∂ (M, R n ) i .

12

 12

5. 9 .

 9 Non-linear gauge relation. Even if the MC equation happens to be linear and the MC elements are just zero-cycles, the gauge action can still be non-linear. As an example consider Emb(S 1 × S 2 , R 6 ). Since the manifold S 1 × S 2 is formal, we takeR = H * (S 1 × S 2 ) = Q[α, β] β 2 ,where |α| = 1, |β| = 2. Consider the following diagrams in HGC R,[START_REF] Berger | Axiomatic homotopy theory for operads[END_REF] :L α = α α ∧ β L β = β α ∧ β T α∧β = α ∧ β α ∧ β α ∧ βBy Proposition 5.1, L β and T α∧β are the only graphs in HGC R,6 of degree zero. Any MC element has the form λL β + µT α∧β , λ, µ ∈ Q. We claim that the gauge relation isλL β + µT α∧β ∼ λL β if λ 0. (27)Indeed, the followingm = λL β + µtT α∧β -µ λ dtL α ∈ MC HGC R,6 ⊗ Ω * (∆ 1 )is a MC element (note that [L α , L β ] = T α∧β ) and m| t=0 = λL β , m| t=1 = λL β + µT α∧β . It is easy to see that all the other gauge equivalences are generated by this one. We conclude MC/∼ = {λL β + µT α∧β | µ = 0 or λ = 0}.

Proposition 6 . 1 .

 61 In the setting of Theorem 1.1 and for n ≥ 4 and H * (M * ) concentrated in degrees ≤ n -3 we have that every connected component of MC • (HGC H * (M * ),n ) is nilpotent and of finite (homological and homotopical) type. Let m ∈ MC(HGC H * (M * ),n ) be a Maurer-Cartan element, let HGC m H * (M * ),n be the twisted L ∞ -algebra and let (HGC m H * (M * ),n ) >0 be its positive degree truncation. Then the (cohomological) Chevalley-Eilenberg complex C * CE (HGC m H * (M * ),n ) >0 is a Sullivan model for the connected component MC • (HGC H * (M * ),n ) m of m. In particular, the cohomology of this component is the Chevalley-Eilenberg cohomologyH k (MC • (HGC H * (M * ),n ) m ) H k CE (HGC m H * (M * ),n ) >0 .Proof. By [8, Theorem 5.5] the homotopy groups of the connected component corresponding to the MC element m are computed by H * (HGC m H * (M * ),n ), which is clearly degree-wise nilpotent and of finite type as the underlying L ∞ -algebra is. Hence the Maurer-Cartan spaces are nilpotent.

  = M ∪ {∞} denotes the pointed m-manifold defined by adding a point at infinity to our open subset M ⊂ R m ⊂ R m ∪ {∞} = S m . Below, by M ו * , we denote the right Γ-module (equivalently the infinitesimal Com-bimodule) that assigns the space of pointed maps S * → M * to any finite pointed set S * . The object IF M is already Reedy cofibrant by Proposition 2.2. Hence LIndCom F m IF M ≃ Ind Com F m IF M .The latter object is obtained by contracting all boundary components to points, and hence we obtain just M ו Model structure. Recall that we use the expression 'left Hopf Γ-module' for the category of left Γ-modules in dg commutative algebras. In what follows, we similarly call 'left Hopf Σ-modules' the objects of the category of left Σ-modules in dg commutative algebras. To avoid clumsy notation, we will abbreviate the notation of the morphism sets Mor dgCom Σ (. . . ) of the category of left Hopf Σ-modules dgCom Σ by Mor HΣ (. . . ) and we will similarly abbreviate the notation of the morphism sets Mor dgCom Γ (. . . ) of the category of left Hopf Γ-modules dgCom Γ by Mor HΓ (. . . ). We will also use the short notation Mor Σ , Mor Γ , Mor Ω for the morphisms sets of the categories of left Σ-modules, of left Γ-modules and of left Ωmodules in dg vector spaces.

	We denote the derived functor associated to Ind Q P by LInd Q P .
	Recall that M Proposition 2.5. We have LInd Com F m IF M ≃ M ו * .
	Proof.

pIG n is a primitive graph. 2.11. Induction and restriction of operadic bimodules. Proposition 2.4. Let P → Q be a morphism of well-pointed reduced operads (see Section 2.6). Then the induction and restriction functors Ind Q P : IBimod P ⇄ IBimod Q : Res Q P form a Quillen adjunction, where we equip the categories of infinitesimal bimodules with the Reedy model structure. Proof. This statement is the counterpart for infinitesimal bimodules and the Reedy model structure of the claim of [21, Theorem 15.B] for right modules and the projective model structure. The argument of this reference applies without change in our setting. Namely, we just use that fibrations and weak equivalences of infinitesimal bimodules over operads are created in the category of right Λ-modules and that Res Q P is the identity at the Λ-module level to conclude that the restriction functor Res Q P preserves fibrations and weak equivalences. The result follows. * * . 3. Hopf Γ-modules and hairy graph complexes 3.1.

  . . , S ′′ m inside T . The map ρ S ′ B (ψ S ′′ 1 , . . . , ψ S ′′ m ) is given by ρ S ′ B (ψ S ′′ 1 , . . . , ψ S ′′ m )(a 1 , . . . , a r ) = ) reflects again the shuffle of the indices of the leaves of the trees S ′′ 1 , . . . , S ′′ m inside T . To any such homotopy morphism, we associate the L ∞ -morphism Ψ : HGC A,n → HGC B,n

	±ρ S ′ B (ψ S ′′ 1 (a i 1 1 , . . . , a i 1 k 1 (i 1 1 , . . . , i 1 k 1 , . . . , i m 1 , . . . , i m ), . . . , ψ S ′′ 1 (a i m 1 , . . . , a i m km )), for all a 1 , . . . , a r ∈ A, where k m

  5.7.Spherical and long embeddings. The approach with graph complexes that we develop in this paper can also be used to compare the homotopy type of different embedding spaces. The first non-trivial question is how the homotopy type of Emb(S m , R n ) is related to that of Emb ∂ (R m , R n ). In[26] we study this problem. Note that[START_REF] Budney | On the homology of the space of knots[END_REF] Section 4] gives a connection between the embedding spaces Emb(S m , R n ) and the long embedding spaces Emb ∂ (R m , R n ). It turns out though that working modulo immersions and assuming nm > 2 makes this connection more straightforward. By the authors' [26, Theorem 1.1], one has a fiber sequence Emb(S m , R n

We recall the explicit definition of a Maurer-Cartan element in a complete L ∞ -algebra and the definition of this simplicial set of Maurer-Cartan forms in Section 5.1.

Such a component must be S m , as an immersion of a closed m-manifold in an m-manifold is always a covering map.

This category is denoted by K in this reference.

In[START_REF]Context-free manifold calculus and the Fulton-MacPherson operad[END_REF], this infinitesimal bimodule is denoted by C * [•, S n ].

In[START_REF]Context-free manifold calculus and the Fulton-MacPherson operad[END_REF], the second author considers the projective model structure on IBimod F m , in which for the cofibrancy of an object, the action of F m (0) should also be free. Because of this the cofibrant