Benoit Fresse

Little discs operads, graph complexes and

Grothendieck-Teichmüller groups

Introduction

The operads of little discs (and the equivalent operads of little cubes) were introduced in topology, in the works of Boardman-Vogt [START_REF] Boardman | Homotopy-everything H-spaces[END_REF][START_REF] Boardman | Homotopy invariant algebraic structures on topological spaces[END_REF] and May [START_REF] May | The geometry of iterated loop spaces[END_REF], for the recognition of iterated loop spaces. We refer to the paper [START_REF] Carlsson | Stable homotopy and iterated loop spaces[END_REF], in the handbook of algebraic topology, for an account of these applications. We also refer to the literature for the general definition of an operad in a category and for the precise definition of the little discs operads, which we denote by D n throughout this chapter.

The aim of this chapter is to survey new applications of the little discs operads which were motivated by the works of Kontsevich [START_REF] Kontsevich | Formality conjecture[END_REF][START_REF] Kontsevich | Operads and motives in deformation quantization[END_REF] and Tamarkin [START_REF] Dmitry | Another proof of M. Kontsevich formality theorem[END_REF] on the deformation-quantization of Poisson manifolds and by the Goodwillie-Weiss embedding calculus in topology [START_REF] Goodwillie | Embeddings from the point of view of immersion theory[END_REF][START_REF] Weiss | Embeddings from the point of view of immersion theory[END_REF]. For our purpose, we also consider the general class of E n -operads, which consists of the operads that are weakly-equivalent to the operad of little n-discs (equivalently, to the operad of little n-cubes). Besides, we deal with E n -operads in the category of differential graded modules, which we similarly define as the class of operads that are weakly-equivalent (quasi-isomorphic) to the operad of singular chains on the little n-discs operad (the chain little n-discs operad) C * (D n ). In Kontsevich's approach, the proof of the existence of deformationquantizations of Poisson manifolds reduces to the construction of a comparison map of differential graded Lie algebras between on the one hand, the Hochschild cochain complex, which governs the deformations of an associative algebra structure, and on the other hand, the algebra of polyvector fields, equipped with the Schouten-Nijenhuis bracket of polyvectors, which can be used to govern the deformations of a Poisson structure on a manifold.

Kontsevich used an explicit definition of such a comparison map in his first proof of the existence of deformation-quantizations. The theory of E 2operads actually occurs in a second generation of proofs of this theorem. The idea is that the differential graded Lie algebra structure of both the Hochschild cochain complex and the algebra of polyvector fields can be integrated into an action of a differential graded E 2 -operad and the algebra of polyvector fields 1 Little discs operads, graph complexes and Grothendieck-Teichmüller groups is rigid (has a unique realization up to quasi-isomorphism) as an algebra over an E 2 -operad.

The action of an E 2 -operad on the complex of Hochschild cochains was initially conjectured by Deligne. The proof of the latter statement, now established in a wide context (which includes the topological counterpart of the Hochschild cohomology theory in the stable homotopy theory framework), can be interpreted as a measure of the degree of commutativity of the Hochschild cochain complex, regarded as a derived version of the center of associative algebras.

The algebra of polyvector fields comes actually equipped with the structure of a 2-Poisson algebra, where in general an n-Poisson algebra refers to a form of graded Poisson algebra such that the Poisson bracket is an operation of degree n -1 (which is actually the case of the Schouten-Nijenhuis bracket for n = 2). The operad that governs this category of graded Poisson algebras, the n-Poisson operad Pois n , represents the homology of the operad of little n-discs H * (D n ). Therefore, the proof that the algebra of polyvector fields inherits an action of an E 2 -operad, and actually, the crux of the operadic proof of the existence of deformation-quantizations, is equivalent to an operadic formality claim, which asserts that the chain operads of little 2-discs C * (D 2 ) is quasiisomorphic to the 2-Poisson operad Pois 2 . In fact, such a statement holds for all n ≥ 2:

C * (D n ) ∼ Pois n ,
and one the main objectives of this chapter will be to explain this result in details. For the moment, simply mention that the case n = 2 of this formality claim was established by Tamarkin by using the theory of Drinfeld's associators. This operadic approach gives deep insights on structures carried by the set of solutions of the deformation-quantization problem, when we consider the set of all deformation-quantization functors as a whole. Indeed, from Tamarkin's arguments, we can deduce the more precise result that a formality quasi-isomorphism for the chain operad of little 2-discs (and as a consequence, a deformation-quantization functor for Poisson manifolds) is associated to any Drinfeld associator. This observation hints that the rational version of the Grothendieck-Teichmüller group GT (Q) acts on the moduli space of deformation-quantizations just because the set of Drinfeld's associators Ass(Q) defines a torsor under an action of this group. We explain shortly that this connection reflects a finer identity between the Grothendieck-Teichmüller group and the group of homotopy automorphisms of E 2 -operads. Recall simply for the moment that the Grothendieck-Teichmüller group models the relations that can be gained from actions of the absolute Galois group on curves. In deformation-quantization theory, we just consider a pro-algebraic version of this group.

To complete this overview, let us mention that higher dimensional generalizations of the deformation-quantization problem, which involve structures governed by any class of n-Poisson algebras, have been studied by Little discs operads, graph complexes and Grothendieck-Teichmüller groups 3 Calaque-Pantev-Toën-Vaquié-Vezzosi in the realm of derived algebraic geometry (see [START_REF] Calaque | Shifted Poisson structures and deformation quantization[END_REF]).

The link between the operads of little discs and the embedding calculus comes from certain descriptions of the Goodwillie-Weiss towers, which are towers of "polynomial" approximations of the embedding spaces Emb(M, N ), where (M, N ) is any pair of smooth manifolds (see [START_REF] Goodwillie | Embeddings from the point of view of immersion theory[END_REF][START_REF] Weiss | Calculus of embeddings[END_REF]). We refer to Arone-Ching's paper, in this handbook volume, for a comprehensive introduction to the embedding calculus.

In what follows, we focus on the case of Euclidean spaces M = R m , N = R n , and we consider a space of embeddings with compact support Emb c (R m , R n ), whose elements are the embeddings f : R m → R n such that there exists a compact domain K ⊂ R m with f R m \K = i, where i : R m → R n denotes the standard embedding i : (x 1 , . . . , x m ) → (x 1 , . . . , x m , 0, . . . , 0). Then we consider an analogously defined space of immersions with compact support Imm c (R m , R n ) and we take the homotopy fiber of the obvious forgeftul map Emb c (R m , R n ) → Imm c (R m , R n ). We use the notation Emb c (R m , R n ) for this space.

In general, one can prove that the Goodwillie-Weiss approximations are weakly-equivalent to mapping spaces of truncated (bi)modules over the little discs operads, where the notion of a truncated operadic (bi)module refers to a a (bi)module which is defined up to some arity only. This result was established by Arone-Turchin in [START_REF] Arone | On the rational homology of highdimensional analogues of spaces of long knots[END_REF], after a pioneering work of Dev Sinha [START_REF] Dev | Operads and knot spaces[END_REF] on the particular case of the spaces of long knots Emb c (R, R n ). In the case of the space of embeddings with compact support modulo immersions Emb c (R m , R n ), one can prove further that the Goodwillie-Weiss approximations are weaklyequivalent to m + 1-fold loop spaces of mapping spaces of truncated operads with the little m-discs operad as source object and the little m-discs operad as target object. This finer result has been established in full generality by Boavida-Weiss in [START_REF] Boavida | Spaces of smooth embeddings and configuration categories[END_REF], by an improvement of the methods used in the study of the Goodwillie-Weiss calculus of embedding spaces, while other authors have obtained general results on mapping spaces of (truncated) operadic bimodules which permit one to recover this delooping relation from the results obtained by Sinha and Arone-Turchin (see the articles of Dwyer-Hess [START_REF] Dwyer | Long knots and maps between operads[END_REF] and Turchin [START_REF] Turchin | On the other side of the bialgebra of chord diagrams[END_REF] for the case m = 1, and the article of Ducoulombier-Turchin [START_REF] Ducoulombier | Delooping the functor calculus tower[END_REF] for the case of general m ≥ 1).

In the case nm ≥ 3, we can use convergence statements to deduce an equivalence of total spaces from the operadic interpretation of the Goodwillie-Weiss tower, so that we have a weak homotopy equivalence:

Emb c (R m , R n ) ∼ Ω m+1 Map h Op (D m , D n ),
where Map h Op (-, -) denotes a derived mapping space bifunctor on the category of operads in topological spaces.

The formality of the little discs operads over the rationals can be used to determine the rational homotopy type of the operadic derived mapping

Handbook of Homotopy Theory spaces Map h

Op (D m , D n ) which occur in this description of the embedding spaces Emb c (R m , R n ). For this purpose, we use the fact that we have a rational homotopy equivalence

Map h Op (D m , D n ) ∼ Q Map h Op (D m , D Q n )
as soon as nm ≥ 3, where D Q n denotes a rationalization of the topological operad of little n-discs D n which is given by an operadic extension of the Sullivan rational homotopy theory of spaces (we explain this construction with more details later on). In fact, we use an improved version of the formality which implies that this rational operad D Q n has a model H * (D n ) which is determined by the rational cohomology of the operad of little n-discs H * (D n ) = H * (D n , Q) (equivalently, by the dual object of the n-Poisson operad Pois n ). This result gives an effective approach to compute the rational homotopy of the operadic mapping spaces Map h

Op (D m , D n ), and hence, to compute the rational homotopy of the embedding spaces Emb c (R m , R n ) by the Goodwillie-Weiss theory of embedding calculus.

Besides the homotopy of the mapping spaces Map h Op (D m , D Q n ), we can compute the rational homotopy type of the spaces of homotopy automorphisms Aut h

Op (D Q n ) in the category of operads. We will actually see that

Aut h Op (D Q 2 ) is weakly-equivalent to a semi-direct product GT (Q) SO(2) Q , where GT (Q)
is the rational Grothendieck-Teichmüller group, and this statement gives a theoretical explanation for the occurrence of the rational Grothendieck-Teichmüller group in deformation-quantization. We have an analogue of this result in the realm of profinite homotopy theory. We explain both statements in this chapter.

In fact, the main objective of this survey is to explain the result of these computations of mapping spaces and of homotopy automorphism spaces of operads. We mainly address this subject. We organize our account as follows.

We devote the first section of our survey to the particular case n = 2 of the homotopy theory of E n -operads. We will explain that the little 2-discs operad has a model given by an operad shaped on braid groupoids. We use this model to obtain our weak equivalence between the Grothendieck-Teichmüller group and the homotopy automorphism space Aut h Op (D Q 2 ). We explain the formality of E n -operads in the second section and we tackle the applications to the computation of the rational homotopy of mapping spaces of E n -operads in the third section. We use graph complexes in the proof of the formality of E n -operads. We therefore retrieve graph complexes, the graph complexes alluded to in the title of this paper, in our expression of the rational homotopy of the mapping spaces of E n -operads. The ultimate goal of this survey is precisely to explain this graph complex description of the rational homotopy type of the mapping spaces of E n -operads.

In general, in this chapter, we use the term 'differential graded module' and the language of differential graded algebra, rather than the language of chain complexes. In fact, we only use the expression '(co)chain complex' for specific constructions of differential graded modules, like the singular complex of a topological space, the Hochschild cochain complex, . . . For short, we also use Little discs operads, graph complexes and Grothendieck-Teichmüller groups 5 the prefix 'dg' for any category of structured objects that we may form within a base category of differential graded modules (like dg-algebras, dg-operads, . .

. ).

In what follows, we generally define a differential graded module (thus, a dg-module for short) as the structure, equivalent to a (possibly unbounded) chain complex, which consists of a module M equipped with a Z-graded decomposition M = n∈Z M n and with a differential δ : M → M such that δ(M * ) ⊂ M * -1 . If necessary, we use the phrase 'lower graded dg-modules' to refer to the objects of this category of dg-modules.

In some cases, we also deal with 'upper graded dg-modules', which are modules M equipped with a Z-graded decomposition of the form M = n∈Z M n and with a differential δ : M → M such that δ(M * -1 ) ⊂ M * . In general, we can use the standard correspondence M * = M - * to convert an upper graded dg-module structure into a lower graded dg-module structure, but we prefer to keep upper graded dg-module structures when this representation is the usual convention in the literature (for instance, in rational homotopy theory).

We equip the category of dg-modules with its standard tensor product so that this category inherits a symmetric monoidal structure, with a symmetry operator defined by using the usual sign rule of homological algebra,

In our study, we freely use the language and the results of the theory of model categories. In particular, in what follows, we rather use the generic term 'weak equivalence' for the class of quasi-isomorphisms, because the quasiisomorphisms represent the class of weak equivalences of the usual model categories of dg-objects (dg-modules, dg-algebras, dg-operads, . . . ).

Braids and the homotopy theory of E 2 -operads

We devote this section to the study of the homotopy of E 2 -operads.

In general, we have a homotopy equivalence of spaces D n (r) ∼ -→ F ( Dn , r), for each r ∈ N, where we consider the underlying spaces of the operad of little n-discs D n (r), and F ( Dn , r) denotes the configuration space of r points in Dn . In the case n = 2, this result implies that D 2 (r) forms an Eilenberg-MacLane space K(P r , 1), where P r is the pure braid group on r strands in D2 .

The first purpose of this section is to explain that we can elaborate on this result in order to get a model of the class of E 2 -operads in the category of operads in groupoids. In short, we check that we have a relation D 2 ∼ B(CoB), where we consider the classifying spaces of a certain operad in groupoids, the operad of colored braids CoB. We will see that this operad CoB governs the category of strict braided monoidal categories as a category of algebras. We use a variant of this operad, the operad of parenthesized braids, which we associate to the category of general braided monoidal categories, in order to define the Grothendieck-Teichmüller group as a group of automorphisms of an operad in the category of groupoids. We will explain that, when we pass to topological spaces, this identity gives an equivalence between the space of homotopy automorphisms of the little 2-discs operad and a semi-direct product of the Grothendieck-Teichmüller group with the group of rotations. The statement of this result is the second and main objective of this section. To complete this survey, we also explain the definition of the notion of a Drinfeld associator from the viewpoint of the theory of operads.

In our constructions, we deal with versions of the Grothendieck-Teichmüller which are associated to various completions of operads in groupoids, and as a consequence, we actually consider various completions of the little 2-discs operad (namely, the profinite completion and the rationalization) when we examine the relationship between the Grothendieck-Teichmüller group and the homotopy of E 2 -operads. In this section, we use a simple definition of these completion operations which we form at the level of the groupoid models of our operads. In the next section, we revisit the definition of the particular case of the rationalization of operads by using the Sullivan rational homotopy theory of spaces.

The operad of colored braids

Briefly recall that a braid on r-strands is an isotopy class of paths α : [START_REF] Alekseev | Kontsevich deformation quantization and flat connections[END_REF], and where we assume that α(0) = (α 1 (0), . . . , α r (0)) (respectively, α(1) = (α 1 (1), . . . , α r (1))) is a permutation of fixed contact points ((z 0 1 , 0), . . . , (z 0 r , 0)) (respectively, ((z 0 1 , 1), . . . , (z 0 r , 1))) on the equatorial line y = 0 of the disc D 2 ×{0} (respectively, D 2 ×{1}). Thus, we have z 0 i = (x 0 i , 0) for i = 1, . . . , r, and by convention we can also assume that the contact points are ordered so that

[0, 1] → F ( D2 × [0, 1], r) with α(t) = (α 1 (t), . . . , α r (t)) ∈ F ( D2 × [0, 1], r) such that α i (t) = (z i (t), t) for each t ∈ [0,
x 0 1 < • • • < x 0 r .
In what follows, we use the usual representation of the isotopy class of a braid in terms of a diagram which is given by a projection onto the plane y = 0 in the space D2 × [0, 1].

The assumption α(t) ∈ F ( D2 × [0, 1], r) is equivalent to the requirement that we have z i (t) = z j (t) for all pairs i = j. In this definition, we assume that the strands of a braid are indexed by the set {1, . . . , r}. This assumption is not standard in the definition of a braid, but we use this convention in our definition of colored braids. Intuitively, the indices i ∈ {1, . . . , r} are colors which we assign to the strands of our braids, as in the following picture:

α = 1 2 3 4 1 2 3 4 3 4 1 2
.

(1.1)

Formally, the operad of colored braids is an operad in the category of groupoids CoB ∈ Grd Op whose components CoB(r) are groupoids with the Little discs operads, graph complexes and Grothendieck-Teichmüller groups 7 permutations on r letters as objects and the isotopy classes of colored braids as morphisms. The source (respectively, the target) of a morphism is the permutation of the set {1, . . . , r} that corresponds to the permutation of the contact points ((z 0 1 , ), . . . , (z 0 r , )) in the sequence α(0) = (α 1 (0), . . . , α r (0)) (respectively, α(1) = (α 1 (1), . . . , α r (1))). For instance, the above example of colored braid depicts a morphism with the permutation s = (1 3 4) as source object and the permutation t = (1 4) as target object. The composition of braids is given by the standard concatenation operation on paths. Note simply that this operation preserves the indexing when we consider a pair of composable morphisms in our groupoid. Note also that our convention is to orient braids from the top to the bottom and we compose braids accordingly.

The action of the symmetric group Σ r on CoB(r) is given by the obvious re-indexing operation of the strands of our braids. The operadic composition operations • i : CoB(k) × CoB(l) → CoB(k + l -1) are functors which are defined on morphisms by a cabling operation on the strands of our braids. In brief, to define a composite α • i β, where α ∈ CoB(k) and β ∈ CoB(l), we insert the braid β on the ith strand of the braid α, as in the example given in the following picture:

1 2 1 2 1 2 • 1 1 2 1 2 1 2 = 1 2 3 2 3 1 1 2 3 . (1.2)
The operadic unit 1 ∈ CoB(1) is the trivial braid with one strand. By convention, we also assume that the component of arity zero of the colored braid operad is identified with the one-point set CoB(0) = * .

In the introduction of this section, we mentioned that this operad CoB governs the category of strict braided monoidal categories. We give more explanations on this interpretation of the colored braid operad later on, when we explain a similar interpretation of an operad that governs the category of general braided monoidal categories (see Theorem 1.1.5).

The following theorem gives the connection between the operad of little 2-discs and the operad of colored braids: [START_REF] Benoit Fresse | Homotopy of operads and Grothendieck-Teichmüller groups[END_REF]Theorem I.5.3.4]). We have an equivalence in the category of operads in groupoids π D 2 ∼ CoB, where π D 2 is the operad in groupoids defined by the fundamental groupoids π D 2 (r) of the spaces of little 2-discs D 2 (r), r ∈ N.

1.1.2 Theorem (see
In the category of operads in groupoids Grd Op, we say that a morphism is an equivalence φ : P ∼ -→ Q when this morphism defines an equivalence of categories arity-wise φ : P(r) ∼ -→ Q(r), for each r ∈ N. Then we say that operads in groupoids P, Q ∈ Grd Op are equivalent when these operads can Handbook of Homotopy Theory be connected by a zigzag of equivalences

P ∼ ← -• ∼ -→ • . . . • ∼ -→ Q in the category of operads in groupoids.
We just use that the fundamental groupoid functor is strongly symmetric monoidal in order to equip the collection of fundamental groupoids π D 2 = {π D 2 (r), r ∈ N} with an operad structure. We refer to the cited reference [START_REF] Benoit Fresse | Homotopy of operads and Grothendieck-Teichmüller groups[END_REF]Theorem I.5.3.4] for the explicit definition of a zigzag of equivalences of operads in groupoids between this object π D 2 and the colored braid operad CoB.

We can apply the classifying space functor B(-) to go back from the category of groupoids towards the category of spaces (or towards the category of simplicial sets). This functor B : Grd → T op is also strongly symmetric monoidal, and hence, preserves operad structures. For our purpose, we consider the operad B(CoB) defined by the collection of the classifying spaces of the colored braid groupoids B(CoB(r)), r ∈ N. We have the following result:

1.1.3 Theorem (Z. Fiedorowicz [START_REF] Fiedorowicz | The symmetric bar construction[END_REF], see also [26, §I.5.2]). We have a weak equivalence D 2 ∼ B(CoB) in the category of topological operads.

This theorem is established in [START_REF] Fiedorowicz | The symmetric bar construction[END_REF] by arguments of covering theory. In [26, §I.5.3], it is explained that we can also deduce this weak equivalence relation D 2 ∼ B(CoB) from the result of the previous theorem. In brief, the observation that each space D 2 (r) is an Eilenberg-MacLane space implies that we have a weak equivalence of spaces D 2 (r) ∼ B(π D 2 (r)), in each arity r ∈ N. We can elaborate on the proof of this relation to establish that we actually have a weak equivalence of operads D 2 ∼ B(π D 2 ) between the operad of little 2-discs D 2 and the classifying space of the fundamental groupoid operad π D 2 . Then we just use that the equivalence of operads in groupoids of Theorem 1.1.2 induces a zigzag of weak equivalences of operads in topological spaces B(π D 2 ) ∼ B(CoB) when we pass to classifying spaces.

The operad of parenthesized braids

The operads of colored braids are not sufficient for our purpose. To define the Grothendieck-Teichmüller group, we need a variant of this operad, which we call the parenthesized braid operad PaB.

The objects of the colored braid operad form an operad in sets Ob CoB, which is identified with the permutation operad , the operad Π defined by the collection of the symmetric groups Π = {Σ r , r ∈ N}. The permutation operad also represents a set-theoretic version of the operad of unital associative algebras, or in other words, the operad in sets that governs the structure of a monoid.

To define the operad of parenthesized braids, we just take a pullback of the operad of coloured braids PaB = ω * CoB along a morphism ω : Ω → Π, where Ω is the operad in sets that governs the category of non-commutative magmas with a fixed unit element (in another terminology, the category of non-commutative non-associative monoids). This operad Ω, the magma operad, has one free generator in arity two µ ∈ Ω(2), equipped with a free action of the symmetric group Σ 2 , and an extra arity-zero element * ∈ Ω(0) such that µ • 1 * = 1 = µ • 2 * , where 1 ∈ Ω(1) denotes the operadic unit. In positive arity, the elements of this operad π ∈ Ω(r) are formal operadic composites of the operations µ ∈ Ω(2) and (1 2) ∈ Ω(2). The result of these operadic composition operations can also be represented as planar binary rooted trees with r leaves indexed by the values of a permutation on r letters σ = (σ(1), . . . , σ(r)), as in the following examples:

σ • µ = σ(1) σ(2) , σ • µ • 1 µ = σ(1) σ(2) σ(3) , σ • µ • 2 µ = σ(1) σ(2) σ(3)
, . . . .

In arity zero, we just take Ω(0) = * .

The morphism ω : Ω → Π, which we consider in the our pullback operation PaB = ω * CoB, carries µ ∈ Ω(2) to the identity permutation on 2 letters id 2 ∈ Σ 2 . The groupoids PaB(r) underlying this operad PaB = ω * CoB are defined by taking Ob PaB(r) := Ω(r) and Mor PaB(r) (p, q) := Mor CoB(r) (ω(p), ω(q)) for the morphism sets, for all p, q ∈ Ω(r). The operadic composition operations are defined by taking an obvious lifting of the composition operations of the operad of colored braids. In [26, §I.6.2], we represent a parenthesized braid by a braid whose contact points form the centers of a dyadic decomposition of the axis y = 0 in the disc D2 (a decomposition obtained by dividing intervals into equal pieces), because one can observe that such decompositions are in bijection with the elements of the magma operad. For instance, the following braid

β = 1 2 3 4 1 2 3 4
.

(1.3) represents a morphism of the groupoid PaB with the object p = (1 2 4)

• µ • 2 (µ • 1 µ) as source and the object q = (1 4 2 3) • µ • 1 (µ • 1 µ) as target.
In the operad PaB, we consider the morphisms

τ = 1 2 1 2 and α = 1 2 3 1 2 3 , (1.4)
which we call the braiding and the associator respectively. We aim to give an interpretation of the parenthesized braid operad in classical algebraic language. The object µ ∈ Ω(2) can be regarded as an abstract operation on 2 variables µ = µ(x 1 , x 2 ). We use the notation of a tensor product for this operation µ(x 1 , x 2 ) = x 1 ⊗x 2 , because we are going to see that µ represents a universal tensor product operation within the operad PaB. The element (1 2)µ = µ(x 2 , x 1 ) represents an operation µ(x 2 , x 1 ) = x 2 ⊗ x 1 , where the variables (x 1 , x 2 ) are transposed when we use this variable interpretation of our operation. We also get that µ • 1 µ = µ(µ(x 1 , x 2 ), x 3 ) represents the result of the substitution of the variable x 1 by the operation µ = µ(x 1 , x 2 ) in µ = µ(x 1 , x 2 ), while µ • 2 µ = µ(x 1 , µ(x 2 , x 3 )) represents the result of the substitution of the second variable x 2 by the same operation µ = µ(x 1 , x 2 ) with an index shift of the variables. We equivalently have µ(µ(x 1 , x 2 ), x 3 ) = (x 1 ⊗ x 2 ) ⊗ x 3 and µ(x 1 , µ(x 2 , x 3 )) = x 1 ⊗ (x 2 ⊗ x 3 ). We accordingly get that the braiding τ = τ (x 1 , x 2 ) represents an isomorphism such that

τ (x 1 , x 2 ) : x 1 ⊗ x 2 → x 2 ⊗ x 1 (1.5)
in the morphism set Mor PaB(2) (µ, (1 2)µ) of our operad in groupoids PaB, while the associator α = α(x 1 , x 2 , x 3 ) represents an isomorphism such that

α(x 1 , x 2 , x 3 ) : (x 1 ⊗ x 2 ) ⊗ x 3 → x 1 ⊗ (x 2 ⊗ x 3 ) (1.6) in Mor PaB(3) (µ • 1 µ, µ • 2 µ). The operadic composition formulas µ • 1 * = 1 = µ • 2 * are equivalent to the relations x 1 ⊗ * = x 1 = * ⊗ x 1 , (1.7)
so that the arity zero object * ∈ Ω(0) = Ob PaB(0) can be interpreted as a unit object with respect to this tensor product operation µ(

x 1 , x 2 ) = x 1 ⊗ x 2 .
We easily see that the braiding and the associator satisfy the following coherence relations with respect to this unit object:

α( * , x 1 , x 2 ) = α(x 1 , * , x 2 ) = α(x 1 , x 2 , * ) = id and τ (x 1 , * ) = id = τ ( * , x 1 ).
(1.8) We easily check, moreover, that the associator satisfies the pentagon relation

x 1 ⊗ α(x 2 , x 3 , x 4 ) • α(x 1 , x 2 ⊗ x 3 , x 4 ) • α(x 1 , x 2 , x 3 ) ⊗ x 4 = α(x 1 , x 2 , x 3 ⊗ x 4 ) • α(x 1 ⊗ x 2 , x 3 , x 4 ) (1.9)
in PaB, as well as the hexagon relations

x 2 ⊗ τ (x 1 , x 3 ) • α(x 2 , x 1 , x 3 ) • τ (x 1 , x 2 ) ⊗ x 3 = α(x 2 , x 3 , x 1 ) • τ (x 1 , x 2 ⊗ x 3 ) • α(x 1 , x 2 , x 3 ), (1.10) τ (x 1 , x 3 ) ⊗ x 2 • α(x 1 , x 3 , x 2 ) -1 • x 1 ⊗ τ (x 2 , x 3 ) = α(x 3 , x 1 , x 2 ) -1 • τ (x 1 ⊗ x 2 , x 3 ) • α(x 1 , x 2 , x 3 ) -1 . (1.11)
Note however that the isomorphism τ (x 1 , x 2 ) is not involutive in the sense that τ (x 2 , x 1 ) • τ (x 1 , x 2 ) = id , because the braid which represents this isomorphism in the braid group is not involutive either.

From this examination, we conclude that the object µ(x 1 , x 2 ) = x 1 ⊗ x 2 ∈ Ob PaB(2) can be interpreted as an abstract tensor product operation that Little discs operads, graph complexes and Grothendieck-Teichmüller groups 11 can be used to govern the structure of a braided monoidal category with a strict unit, which is given by the arity zero element of our operad * ∈ Ω(0) = Ob PaB(0), but where we have a general associativity isomorphism α = α(x 1 , x 2 , x 3 ), which is depicted in Eqn. 1.4 together with the braiding isomorphism τ = τ (x 1 , x 2 ). The operad PaB, equipped with these generating elements, is actually the universal operad that governs such structures, as shown in the following statement:

1.1.5 Theorem (see [START_REF] Benoit Fresse | Homotopy of operads and Grothendieck-Teichmüller groups[END_REF]Theorem I.6.2.4]). Let M ∈ Cat Op be an operad in the category of small categories C = Cat. Fixing an operad morphism φ : This theorem is established in the cited reference. The morphism associated to the quadruple (m, e, a, c) given in the theorem is obviously determined by the formulas φ(µ) = m, φ( * ) = e, φ(α) = a and φ(τ ) = c. The claim is that this assignments determines a well-defined morphism on PaB. The proof of this result follows from a combination of an operadic interpretation of the MacLane coherence theorem and of the classical presentation of the braid group by generators and relations. We have an analogous statement for the operad of colored braids CoB. In this case, we just require a = id in our statement, because we represent the tensor product operation by the identity permutation on 2-letters id 2 ∈ Σ 2 in the object-sets of this operad CoB and this operation satisfies a strict associativity relation.

PaB → M amounts to fixing a unit object e ∈ Ob M(0), a product object m ∈ Ob M(2), an associativity isomorphism a ∈ Mor M(3) (m • 1 m, m • 2 m),
Theorem 1.1.5 implies that the category of algebras governed by the operad PaB in the category of categories is identified with a category of braided monoidal categories with a strict unit but general associativity isomorphisms. The operad of colored braids CoB has a similar interpretation (already mentioned in the introduction), but we then consider the category of braided monoidal categories with strict associativity identities instead of associativity isomorphisms. We refer to [26, §I.6.2] for more detailed explanations on these topics.

The Grothendieck-Teichmüller group

The Grothendieck-Teichmüller group is defined as a group of automorphisms of the parenthesized braid operad. To be more precise, we have to consider completions of this operad in applications. These completions operations are performed at the groupoid level. In what follows, we mainly consider the case of the Malcev completion, which we denote by G Q for any groupoid G ∈ Grd , and the profinite completion, which we denote by G (yet another natural example of completion operation is the p-profinite completion, but we do not consider this variant of the profinite completion in this survey). In all cases, the considered completion operation does not change the object sets of our groupoids and is a natural generalization, for groupoids, of the corresponding classical completion operation on groups. Recall simply that the Malcev completion of groups is an extension of the classical rationalization of abelian groups which combines a pro-nilpotent completion with a rationalization operation. In the case of a free group for instance, we can identify the elements of the Malcev completion with infinite products of iterated commutators with rational exponents. (We refer to [26, §I.8] for a detailed survey of this subject.)

To define the rationalization of our operad PaB Q , we just perform the aritywise completion operation PaB Q (r) = PaB(r) Q . Then we define the rational Grothendieck-Teichmüller group GT (Q) as the group of automorphisms of the operad PaB Q which reduce to the identity mapping on the object sets of our operad. In principle, we regard the object PaB Q as an operad in a category of Malcev complete groupoids, where the morphisms satisfy a continuity constraint, and we assume that our automorphisms satisfy such a condition in the definition of the Grothendieck-Teichmüller group. But all morphisms are automatically continuous in the case of the Malcev completion of the operad PaB (see [START_REF] Benoit Fresse | Homotopy of operads and Grothendieck-Teichmüller groups[END_REF]Proposition I.10.1.5]), and therefore, we can neglect this issue in what follows.

We use a similar construction to define the profinite completion of our operad PaB and the profinite Grothendieck-Teichmüller group GT . (We just need to take care of the continuity constraints in the definition of morphisms in this case.) We examine the definition of automorphisms on these completions of the parenthesized braid operad to get more insights into the definition of these Grothendieck-Teichmüller groups. We explain our constructions in full details in the case of the rational Grothendieck-Teichmüller group only, because the profinite analogues of these constructions is obvious.

Any morphism φ : PaB → PaB Q admits a unique extension to the completed operad φ : PaB Q → PaB Q . By Theorem 1.1.5, such a morphism φ : PaB → PaB Q is fully determined by giving a triple (m, a, c) such that m = φ(µ), a = φ(α) and c = φ(τ ). Note that we automatically have φ( * ) = * since PaB(0) = * ⇒ PaB(0) Q = * . For our purpose, we also set m = φ(µ) = µ since we only consider morphisms that are given by the identity mapping on objects in the definition of the Grothendieck-Teichmüller group.

We necessarily have φ(τ ) = τ • τ 2ν for some parameter ν ∈ Q, where we identify τ 2 ∈ Mor PaB(2) (µ, µ) with an element of the pure braid group on 2 strands P 2 and we use the expression τ 2ν with ν ∈ Q to represent an element in the Malcev completion of this group (P 2 ) Q . We similarly have φ

(α) = α • f for some morphism f ∈ Mor PaB(3)Q (µ • 1 µ, µ • 1 µ),
which is represented by an element of the Malcev completion of the pure braid group on three strand (P 3 ) Q . We have P 3 = K × x 12 , x 23 , where K denotes a central element in P 3 , which is defined by the expression:

K = , (1.12)
Little discs operads, graph complexes and Grothendieck-Teichmüller groups 13 while x 12 and x 23 denote the pure braids such that:

x 12 = , x 23 = . (1.13)
The notation -, which we use in this expression P 3 = K × x 12 , x 23 , refers to the free group generated by a collection of elements.

We can easily deduce from the unit relation a • 2 * = id that the associator φ(α) = α • f has no factor in K Q . Hence, our morphism φ is determined by an assignment of the form:

φ(µ) := µ, φ(τ ) := τ • τ 2ν = τ λ , φ(α) := α • f (x 12 , x 23 ), (1.14)
where we set λ = 1+2ν for ν ∈ Q and we assume

f = f (x 12 , x 23 ) ∈ x 12 , x 23 Q .
In what follows, we use the notation of a formal series on two abstract variables f = f (x, y) to represent this element f in the Malcev completion of the free group F = x 12 , x 23 .

One can prove that the unit relations a • 1 e = id = a • 2 e in the coherence constraints of Theorem 1.1.5 are equivalent to the identities:

f (x, 1) = x = f (1, x), (1.15)
while the hexagon relations are equivalent to the following system of equations

f (x, y) • f (y, x) = 1, (1.16) f (x, y) • x ν • f (z, x) • z ν • f (y, z) • y ν = 1, (1.17)
for a triple of variables (x, y, z) such that zyx = 1. The pentagon equation is equivalent to the following relation in the Malcev completion of the pure braid group on 4 strands (P 4 ) Q (respectively, in the profinite completion P 4 ):

f (x 23 , x 34 )f (x 13 x 12 , x 34 x 24 )f (x 12 , x 23 ) = f (x 12 , x 24 x 23 )f (x 23 x 13 , x 34 ), (1.18
) where in general, we use the notation x ij for the pure braid group elements such that:

x ij = i j • • • • • • • • • • • • • • • • • • • • • • • • • • • . (1.19)
(We refer to [START_REF] Drinfel | On quasitriangular quasi-Hopf algebras and on a group that is closely connected with Gal(Q/Q)[END_REF] and to [26, Proof of Proposition I.11.1.4] for a more detailed analysis of these equations.)

The composition of morphisms corresponds to the following operation on this set of pairs (λ, f (x, y)):

(λ, f (x, y)) * (µ, g(x, y)) = (λµ, f (x, y) • g(x λ , f (x, y) -1 • y λ • f (x, y))). (1.20)

Handbook of Homotopy Theory

Thus, an element of the Grothendieck-Teichmüller group γ ∈ GT (Q), which corresponds to a morphism φ : PaB → PaB Q that induce an isomorphism on the Malcev completion φ : PaB Q -→ PaB Q , can be uniquely determined by giving a pair (λ, f ) ∈ Q × x 12 , x 23 Q , which satisfies the constraints of Eqn. 1.15-1.18 and which is invertible with respect to this composition operation. A necessary and sufficient condition for this invertibility condition is given by λ ∈ Q × (see [START_REF] Drinfel | On quasitriangular quasi-Hopf algebras and on a group that is closely connected with Gal(Q/Q)[END_REF] and [START_REF] Benoit Fresse | Homotopy of operads and Grothendieck-Teichmüller groups[END_REF]Proposition I.11.1.5]).

The elements of the profinite Grothendieck-Teichmüller group GT have a similar representation as pairs (λ, f ) ∈ Z × x 12 , x 23 , where we now consider the profinite completion of the integers Z for the parameter λ and the profinite completion of the free group x 12 , x 23 for the formal series f = f (x 12 , x 23 ). (We just lack a simple characterization of the invertibility of morphisms in the profinite setting.)

This representation of the elements of the Grothendieck-Teichmüller group in terms of pairs (λ, f ) and the equations of Eqn. 1.15-1.18 are actually Drinfeld's original definition of the Grothendieck-Teichmüller group in [START_REF] Drinfel | On quasitriangular quasi-Hopf algebras and on a group that is closely connected with Gal(Q/Q)[END_REF]. The correspondence between this definition and the operadic definition which we summarize in this paragraph is established with full details in the book [26, §I.11.1], but the ideas underlying this operadic interpretation were already implicitly present in Drinfeld's work [START_REF] Drinfel | On quasitriangular quasi-Hopf algebras and on a group that is closely connected with Gal(Q/Q)[END_REF]. We also refer to [START_REF] Bar-Natan | On associators and the Grothendieck-Teichmuller group[END_REF] for another formalization of this interpretation, which uses ideas close to the language of universal algebra. In the introduction of this chapter, we mentioned that the Grothendieck-Teichmüller group was defined by using ideas of the Grothendieck program in Galois theory. In fact, we have an embedding Gal ( Q/ Q) → GT which is defined by using an action of the absolute Galois group on genus zero curves with marked points (see [START_REF] Drinfel | On quasitriangular quasi-Hopf algebras and on a group that is closely connected with Gal(Q/Q)[END_REF]). For the rational Grothendieck-Teichmüller group, a result of F. Brown's (see [START_REF] Brown | Mixed Tate motives over Z[END_REF]) implies that we have an analogous embedding Gal MT (Z) → GT (Q), where Gal MT (Z) now denotes the motivic Galois group of a category of integral mixed Tate motives (see also [START_REF] Deligne | Groupes fondamentaux motiviques de Tate mixte[END_REF][START_REF] Terasoma | Mixed Tate motives and multiple zeta values[END_REF] for the definition of this group and for the definition of this mapping).

We go back to the definition of the Grothendieck-Teichmüller group GT (Q) in terms of operad isomorphisms φ γ : PaB Q -→ PaB Q . We can regard the classifying space operad E Q 2 = B(PaB Q ) as a model for the rationalization of the E 2 -operad E 2 = B(PaB). We deduce from the functoriality of the classifying space construction that any element γ ∈ GT (Q) induces an automorphism φ γ :

E Q 2 ∼ -→ E Q
2 at the topological operad level, and hence, defines an element in the homotopy automorphism space Aut h

Op (E Q 2 ). We claim that this correspondence γ → φ γ induces a bijection when we pass to the group of homotopy classes of homotopy automorphisms. We can deduce this statement from the following more precise statement:

1.1.7 Theorem (B. Fresse [26, Theorem III.5.2.5]). We have Aut h Op (E Q 2 ) ∼ GT (Q) B(Q).

Little discs operads, graph complexes and Grothendieck-Teichmüller groups 15

The factor B(Q) in the expression of this theorem corresponds to a rationalization of the group of rotations SO(2) ∼ B(Z) which acts on the little 2-discs model of the class of E 2 -operads E 2 = D 2 by rotating the configurations of little 2-discs in this operad D 2 . We equip this factor B(Q) with the obvious additive group structure.

We have Mor PaB(2)Q (µ, µ) = (P 2 ) Q , and as a consequence, any element of the Grothendieck-Teichmüller group γ ∈ GT (Q) determines an automorphism of the Malcev completion of the pure braid group (P 2 ) Q Q through its action on the automorphism group of the object µ ∈ Ob PaB [START_REF] Arnol'd | The cohomology ring of the group of dyed braids[END_REF]. We use this observation to determine the action of the Grothendieck-Teichmüller group on the group B(Q) that we consider in the definition of the semi-product GT (Q) B(Q). (We refer to [26, §III.5.2] for details.)

Let us insist that we consider derived homotopy automorphism spaces in the statement of this theorem. In the model category approach, these homotopy automorphism spaces are defined by taking the actual spaces of homotopy automorphism spaces associated to a cofibrant-fibrant replacement

R Q 2 of our operad E Q 2 .
To associate an element of this derived homotopy automorphism space to an element of the Grothendieck-Teichmüller group γ ∈ GT (Q), we use the fact that an automorphism φ γ :

E Q 2 → E Q 2 automatically admits a lifting to this cofibrant-fibrant replacement R Q 2 .
The claim is that all homotopy automorphisms of this cofibrant-fibrant model R Q 2 are homotopic to such morphisms, and that this correspondence gives all the homotopy of the space Aut h

Op (E Q 2 ) up to the factor B(Q). The book [26, § §III. [START_REF] Alekseev | Kontsevich deformation quantization and flat connections[END_REF][START_REF] Arnol'd | The cohomology ring of the group of dyed braids[END_REF][START_REF] Arone | On the rational homology of highdimensional analogues of spaces of long knots[END_REF][START_REF] Arone | Graph-complexes computing the rational homotopy of high dimensional analogues of spaces of long knots[END_REF][START_REF] Bar-Natan | On associators and the Grothendieck-Teichmuller group[END_REF] gives a proof of this result by using spectral sequence methods and an operadic cohomology theory which provides approximations of our homotopy automorphism spaces. This method is close to the methods which are used in the next sections, when we study the homotopy automorphism spaces of E n -operads for any value of the dimension parameter n ≥ 2.

We now consider the profinite version of the Grothendieck-Teichmüller group GT . We take the classifying space operad E 2 = B(PaB ) as a model for the profinite completion of the E 2 -operad E 2 = B(PaB) and we use the same construction as in the rational setting to define a mapping from the profinite Grothendieck-Teichmüller group towards the homotopy automorphism space of this object Aut h Op (D 2 ). Then we have the following analogue of the result of Theorem 1.1.8:

1.1.8 Theorem (G. Horel [START_REF] Horel | Profinite completion of operads and the Grothendieck-Teichmüller group[END_REF]). We have

Aut h Op (D 2 ) ∼ GT B(Z ).
The article [START_REF] Horel | Profinite completion of operads and the Grothendieck-Teichmüller group[END_REF] gives a proof of the result of this theorem by using the correspondence with groupoids. In short, the idea of this paper is to observe that the operad PaB represents a cofibrant object with respect to some model structure on the category of operads in groupoids. Then we can use model category arguments (combined with higher category methods) to prove that we can transport the computation of the homotopy automorphism space Aut h

Op (E 2 )

Handbook of Homotopy Theory

to the computation of the homotopy automorphism space associated to this object PaB in the category of operads in groupoids.

1.1.9 The Drinfeld-Kohno Lie algebra operad

Besides the colored braid and the parenthesized braid operads, which are defined by using the structures of the braid groups, we consider operads in Lie algebras which are associated to infinitesimal versions of the pure braid groups. To be explicit, in these infinitesimal versions, we consider the Drinfeld-Kohno Lie algebras (also called the Lie algebras of infinitesimal braids), which are defined by a presentation of the form:

p(r) = L(t ij , {i, j} ⊂ {1, . . . , r})/ < [t ij , t kl ], [t ij , t ik + t jk ] >, (1.21)
for r ∈ N, where L(-) denotes the free Lie algebra functor, we associate a generator t ij such that t ij = t ji to each pair {i = j} ⊂ {1, . . . , r}, and we take the ideal generated by the commutation relations

[t ij , t kl ] ≡ 0, (1.22)
for all quadruples {i, j, k, l} ⊂ {1, . . . , r} such that {i, j, k, l} = 4, together with the Yang-Baxter relations

[t ij , t ik + t jk ] ≡ 0, (1.23)
for all triples {i, j, k} ⊂ {1, . . . , r} such that {i, j, k} = 3. This definition makes sense over any ground ring k, but from the next paragraph on, we will assume that the ground ring is a field of characteristic zero. Note that this Lie algebra p(r) inherits a weight grading from the free Lie algebra since this ideal is generated by homogeneous relations. If we use the notation L m = L m (-), for the homogeneous component of weight m of the free Lie algebra L = L(t ij , {i, j} ⊂ {1, . . . , r}), then we have the decomposition p(r) = m≥1 p(r) m , where we set p(r

) m = L m /L m ∩ < [t ij , t kl ], [t ij , t ik + t jk ] >, for m ≥ 1.
In fact, we have the identity p(r) * = gr Γ * P r , where on the right-hand side we consider the graded Lie algebra of the sub-quotients of the central series filtration of the pure braid group gr Γ * P r (see [26, Theorem I.10.0.4] for a detailed proof of this statement).

The collection p = {p(r), r ∈ N} inherits the structure of an operad in the category of Lie algebras, where we take the direct sum of Lie algebras to define our symmetric monoidal structure. The action of the symmetric group Σ r on the Lie algebra p(r) is defined, on generators, by the obvious re-indexing operation σ • t ij = t σ(i)σ(j) , for all σ ∈ Σ r . The composition products are given by Lie algebra morphisms of the form

• i : p(k) ⊕ p(l) → p(k + l -1), (1.24)
defined for all k, l ∈ N and i ∈ {1, . . . , k}, and which satisfy the equivariance, Little discs operads, graph complexes and Grothendieck-Teichmüller groups 17

unit and associativity relations of operads in the category of Lie algebras. For generators t ab ∈ p(k) and t cd ∈ p(l), we explicitly set:

t ab • i 0 =                t a+l-1b+l-1 , if i < a < b, t ab+l-1 + • • • + t a+l-1b+l-1 , if i = a < b, t ab+l-1 , if a < i < b, t ab + • • • + t aj+l-1 , if a < i = b, t ab , if a < b < i, (1.25)
and

0 • i t cd = t c+i-1d+i-1 for all i. (1.26)
The operadic unit is just given by the zero morphism 0 : 0 → p(1) with values the zero object p(1) = 0. In fact, these operations reflect the composition structure of the operad of colored braids, in the sense that we can identify the components of homogeneous weight of this operad p(-) m , m ≥ 1, with the fibers of a tower of operads CoB / Γ m CoB, m ≥ 1, which we deduce from the central series filtration of the pure braid group. (We refer to [26, §I.10.1] for more explanations on this correspondence.)

We call this operad p the Drinfeld-Kohno Lie algebra operad. We consider generalizations of this operad when we study the Sullivan model of E noperads. This subsequent study is our main motivation for the recollections of this paragraph, but the Drinfeld-Kohno Lie algebra operad also occurs in the theory of Drinfeld's associators and in the definition of a graded version of the Grothendieck-Teichmüller group. We just give a brief overview of this subject to complete the account of this section.

The operad of chord diagrams and associators

To define the set of Drinfeld's associators, we consider an operad in groupoids, the chord diagram operad CD k , defined over any characteristic zero field k, and such that we have the relation CD(r) k = exp p(r) for each r ∈ N, where we consider the exponential group associated to a completion of the Drinfeld Kohno Lie algebra p(r).

To be more precise, we explained in the previous paragraph that the Drinfeld Kohno Lie algebra admits a weight decomposition p(r) = m≥1 p(r) m . To form the completed Lie algebra p(r), we just replace the direct sum of this decomposition by a product. Thus we have p(r) = m≥1 p(r) m so that the elements of this completed Lie algebra p(r) are represented by infinite series of Lie polynomials (modulo the ideal generated by the commutation and the Yang-Baxter relations). The exponential group CD(r) k = exp p(r) consists of formal exponential elements e ξ , ξ ∈ p(r), together with the group operation given by the Campbell-Hausdorff formula at the level of the completed Lie algebra Handbook of Homotopy Theory p(r). We identify this group CD(r) k = exp p(r) with a groupoid with a single object when we regard the chord diagram operad CD k = {exp p(r), r ∈ N} as an operad in the category of groupoids. The structure operations of this operad CD k are induced by the structure operations of the Drinfeld-Kohno Lie algebra operad on p.

We also have an identity CD(r) = G( Û(p(r))), where we consider the set of group-like elements G(-) in the complete enveloping algebra of the Drinfeld-Kohno Lie algebra Û(p(r)). Indeed, the group-like elements are identified with actual exponential series of Lie algebra elements ξ ∈ p(r) within the complete enveloping algebras Û(p(r)). The name "chord diagram" comes from the theory of Vassiliev invariants, where a monomial t i1j1 • • • t imjm ∈ Û(p(r)) is associated to a diagram with r vertical strands numbered from 1 to r, and l chords corresponding to the factors t i k j k , as in the following picture:

t 12 t 12 t 36 t 24 = 1 2 3 4 5 6 • • • • • • • • . (1.27)
The composition products of the chord diagram operad have a simple description in terms of chord diagram insertions too.

In the previous paragraph, we explained that the components of homogeneous weight of the Drinfeld-Kohno Lie algebra operad represent the fibers of a tower decomposition of the parenthesized braid operad (and of the colored braid operad equivalently). In fact, a stronger result holds when we work over a field of characteristic zero. To be more explicit, we consider the Malcev completion of the operad PaB, and a natural extension of this construction for ground fields such that Q ⊂ k. Then we may wonder about the existence of equivalences of operads in groupoids φ α : PaB k ∼ -→ CD k , which would be equivalent to a splitting of this tower decomposition over k. By Theorem 1.1.5, the morphism of operads in groupoids φ a : PaB ∼ -→ CD k which would induce such an equivalence on the completion is determined by the choice of a braiding c ∈ exp p(2) and of an associativity isomorphism a ∈ exp p(3). The braiding has the form c = exp(κt 12 /2), for some parameter κ ∈ k × , since p(2) = k t 12 , and one can prove that the associator is necessarily of the form a = exp f (t 12 , t 23 ), for some Lie power series f (t 12 , t 23 ) ∈ L(t 12 , t 23 ). Thus, the existence of an equivalence of operads in groupoids φ α : PaB k ∼ -→ CD k reduces to the existence of such a Lie power series f (t 12 , t 23 ) ∈ L(t 12 , t 23 ) such that a = exp f (t 12 , t 23 ) satisfies the unit, pentagon and hexagon constraints of Theorem 1.1.5, for some given parameter κ ∈ k × .

The set of Drinfeld's associators precisely refers to this particular set of associators a = exp f (t 12 , t 23 ) which we associate to the chord diagram operad CD k . This notion was introduced by Drinfeld in the paper [START_REF] Drinfel | On quasitriangular quasi-Hopf algebras and on a group that is closely connected with Gal(Q/Q)[END_REF], to which we also refer for an explicit expression of the unit, pentagon and hexagon constraints (see also the survey of [26, §I.10.2]). Further reductions occur in the pentagon and hexagon constraints in the definition of Drinfeld's associators. In fact, a result of Furusho (see [START_REF] Furusho | Pentagon and hexagon equations[END_REF]) implies that the hexagon constraints are satisfied as soon as we have a power series that fulfills the unit and the pentagon constraints.

We have the following main result:

1.1.11 Theorem (V.I. Drinfeld [START_REF] Drinfel | On quasitriangular quasi-Hopf algebras and on a group that is closely connected with Gal(Q/Q)[END_REF]). The set of Drinfeld's associators is not empty, for any choice of field of characteristic zero as ground field k (including k = Q), so that we do have an operad morphism φ α : PaB → CD Q which induces an equivalence when we pass to the Malcev completion φ α :

PaB Q ∼ -→ CD Q .
In [START_REF] Drinfel | On quasitriangular quasi-Hopf algebras and on a group that is closely connected with Gal(Q/Q)[END_REF], Drinfeld gives an explicit construction of a complex associator by using the monodromy of the Knizhnik-Zamolodchikov connection. This associator, which is usually called the Knizhnik-Zamolodchikov associator in the literature, can also be identified with a generating series of polyzeta values. Descent arguments can be used to establish the existence of a rational associator from the existence of this complex associator (see again [START_REF] Drinfel | On quasitriangular quasi-Hopf algebras and on a group that is closely connected with Gal(Q/Q)[END_REF] and [START_REF] Bar-Natan | On associators and the Grothendieck-Teichmuller group[END_REF] for different proofs of this descent statement), so that the result of this theorem holds over k = Q, and not only over k = C. Another explicit definition of an associator, defined over the reals, is given by Alekseev-Torossian in [START_REF] Alekseev | Kontsevich deformation quantization and flat connections[END_REF], by using constructions introduced by Kontsevich in his proof of the formality of the operads of little discs (see [START_REF] Kontsevich | Operads and motives in deformation quantization[END_REF]).

1.1.12 The operad of parenthesized chord diagrams, the graded Grothendieck-Teichmüller group, and other related objects

The existence of associators can be used to get insights into the structure of the rational Grothendieck-Teichmüller group GT (Q). Indeed, the definition implies that the set of associators inherits a free and transitive action of the rational Grothendieck-Teichmüller group. To go further into the applications of associators, one introduces a parenthesized version of the chord diagram operad PaCD Q (by using the same pullback construction as in the case of the parenthesized braid operad PaB) and a group of automorphisms, denoted by GRT (Q), which we associate to this object PaCD Q . One can easily check that every equivalence of operads in groupoids φ a : PaB Q ∼ -→ CD Q lifts to an isomorphism φ a : PaB Q -→ PaCD Q so that the existence of rational associators implies the existence of a group isomorphism GT (Q) GRT (Q) by passing to automorphism groups. This group GRT (Q) is usually called the graded Grothendieck-Teichmüller group in the literature, because this group is identified with the pro-algebraic group associated to a graded Lie algebra such that grt = m≥0 grt m . We moreover have grt m F m GT (Q)/ F m+1 GT (Q), for all m ≥ 1, for some natural filtration of the Grothendieck-Teichmüller group

GT (Q) = F 0 GT (Q) ⊃ F 1 GT (Q) ⊃ • • • ⊃ F m GT (Q) ⊃ • • • .
We again refer to [START_REF] Drinfel | On quasitriangular quasi-Hopf algebras and on a group that is closely connected with Gal(Q/Q)[END_REF] for a proof of these results (see also the survey of [26, §I.11.4]).

The groups GT (Q), GRT (Q), and the Lie algebra grt are also related to other objects of arithmetic geometry and group theory. In §1.1.6, we al-ready recalled that, by a result of F. Brown (see [START_REF] Brown | Mixed Tate motives over Z[END_REF]), the Grothendieck-Teichmüller group GT (Q) contains a realization of the motivic Galois group of a category of integral mixed Tate motives Gal MT (Z) . In fact, one conjectures that these groups are isomorphic (Deligne-Ihara). Furthermore, one can prove that the Galois group Gal MT (Z) reduces to the semi-direct product of the multiplicative group with the prounipotent completion of a free group on a sequence of generators s 3 , s 5 , . . . , s 2n+1 , . . . (see [START_REF] Deligne | Groupes fondamentaux motiviques de Tate mixte[END_REF]). This result implies that the embedding Gal MT (Z) → GT (Q) is equivalent to an embedding of the form k ⊕ L(s 3 , s 5 , . . . , s 2n+1 , . . . ) → grt when we pass to the category of Lie algebras. In this context, one can re-express the Deligne-Ihara conjecture as the conjecture that this embedding of Lie algebras is an isomorphism.

In our comments on Theorem 1.1.11, we also explained that the Knizhnik-Zamolodchikov associator represents the generating series of polyzeta values. The polyzeta values satisfy certain equations, called the regularized double shuffle relations, which can be expressed in terms of the Knizhnik-Zamolodchikov associator, and one conjectures that all relations between polyzetas follow from the double shuffle relations and from the fact that the Knizhnik-Zamolodchikov associator defines a group-like power series. By a result of Furusho [START_REF] Furusho | Double shuffle relation for associators[END_REF], the pentagon condition for associators implies the regularized double shuffle relations. This result implies that the Grothendieck-Teichmüller group embeds in a group defined by solutions of regularized double shuffle relations with a degeneration condition, and one conjectures again that this embedding is an isomorphism.

The theory of associators is also used by Alekseev-Torossian in the study of the solutions of the Kashiwara-Vergne conjecture, a problem about the Campbell-Hausdorff formula motivated by questions of harmonic analysis. These authors notably proved in [START_REF] Alekseev | Kontsevich deformation quantization and flat connections[END_REF] that the set of Drinfeld's associators embeds into the set of solutions of the Kashiwara-Vergne conjecture. In particular, one can deduce the existence of such solutions from the existence of associators. In addition, one can prove that the action of the Grothendieck-Teichmüller group on Drinfeld's associators lifts to the set of solutions of the Kashiwara-Vergne conjecture. This action is still free so that we get that the Grothendieck-Teichmüller group embeds into a group of automorphisms associated to this set of solutions. The conjecture is that this group embedding is an isomorphism, yet again.

The rational homotopy of E n -operads and formality theorems

The goal of this section is to explain the definition of rational models of E n -operads and a characterization of the class of E n -operads up to rational homotopy equivalence. In what follows, we just focus on the case n ≥ 2, Little discs operads, graph complexes and Grothendieck-Teichmüller groups 21 because we can put the case n = 1 apart. Indeed, we have D 1 ∼ Π, where we regard the permutation operad Π = {Σ r , r ∈ N} (see §1.1.4) as a discrete operad in topological spaces. Hence, the class of E 1 -operads is also identified with the class of operads that are weakly-equivalent to this discrete operad Π, and such a class of objects is fixed by the rationalization.

Recall that a map of simply connected topological spaces is a rational homotopy equivalence f :

X ∼ Q --→ Y if this map induces a bijection on homotopy groups f * : π * (X) ⊗ Z Q -→ π * (Y ) ⊗ Z Q.
In what follows, we consider a generalization of this notion in the context of spaces which, like the underlying spaces of the little 2-disc operad, are (connected but) not necessarily simply connected. In this case, we assume that a rational homotopy equivalence also induces an isomorphism on the Malcev completion of the fundamental group

f * : π 1 (X, x) Q -→ π 1 (Y, f (x)) Q .
In the context of operads, we just consider operad morphisms φ : P ∼ Q --→ Q which define a rational homotopy equivalence of spaces arity-wise φ : P(r)

∼ Q --→ Q(r)
. and we write P ∼ Q Q when our objects P and Q can be connected by a zigzag of such rational homotopy equivalences. We aim to determine the class of operads such that R ∼ Q D n .

We develop a rational homotopy theory of operads to address this problem. We rely on the Sullivan rational homotopy of spaces, which we briefly review in the next paragraph. We explain the construction of an operadic extension of the Sullivan model afterwards. We eventually check that the n-Poisson cooperad, the dual structure of the n-Poisson operad, defines a Sullivan model of the little n-discs operad D n , and as such determines a model for the class of E n -operads up to rational homotopy. We need a cofibrant resolution of the n-Poisson cooperad to perform computations with this model. We will explain that such a cofibrant resolution is given by the Chevalley-Eilenberg cochain complex of a graded version of the Drinfeld-Kohno Lie algebra operad of the previous section. We actually consider another resolution in our construction, namely a cooperad of graphs, and we also explain the definition of this object. We use the latter model in the next section, when we explain a graph complex description of the rational homotopy type of mapping spaces of E n -operads.

In order to apply the methods of rational homotopy theory, we take k = Q as a ground ring for our categories of modules from now on, and we also consider the cohomology with coefficients in this field H * (-) = H * (-, Q). We similarly take H * (-) = H * (-, Q) for the homology.

Recollections on the Sullivan rational homotopy theory of spaces

Recall that we call 'upper graded dg-module' the structure formed by a module M equipped with a decomposition such that M = n∈Z M n and with a differential δ : M → M such that δ(M * -1 ) ⊂ M * . We say that such a dg-module M is non-negatively graded when we have M n = 0 for n < 0. Let dg * Com be the category of commutative algebras in upper non-negatively graded dg-modules (the category of commutative cochain dg-algebras for short). The Sullivan model for the rational homotopy of a space takes values in this category dg * Com and is obtained by applying the Sullivan functor of PL differential forms, a version of the de Rham cochain complex which is defined over Q (instead of R) and on the category of simplicial complexes (or simplicial sets), instead of the category of smooth manifolds (see [START_REF] Sullivan | Infinitesimal computations in topology[END_REF]). For our purpose, we consider the simplicial set variant of this functor:

Ω * : sSet op → dg * Com . (1.28)
In the particular case of a simplex

∆ n = {0 ≤ x 1 ≤ • • • ≤ x n ≤ 1}, we explicitly have: Ω * (∆ n ) = Q[x 1 , . . . , x n , dx 1 , . . . , dx n ], (1.29)
where dx 1 , . . . , dx n represents the differential of the variables x 1 , . . . , x n in this commutative cochain dg-algebra. The Sullivan functor Ω * : sSet op → dg * Com has a left adjoint

G • : dg * Com → sSet op , (1.30)
which is given by the formula G • (A) = Mor dg * Com (A, Ω * (∆ • )), for any commutative cochain dg-algebra A ∈ dg * Com, and this pair of adjoint functors (G • , Ω) defines a Quillen adjunction. (We refer to [START_REF] Bousfield | On PL de Rham theory and rational homotopy type[END_REF] for this application of the formalism of model categories to Sullivan's constructions.) Then we set:

A := derived functor of G • (A) = Mor dg * Com (R A , Ω * (∆ • )), (1.31)
where R A ∼ -→ A is any cofibrant resolution of A in dg * Com. If X satisfies reasonable finiteness and nilpotence assumptions, then the space

X Q := Ω * (X) (1.32)
defines a rationalization of the space X in the sense that we have the identities

π * (X Q ) := π * (X) ⊗ Z Q, for * ≥ 2, π 1 (X) Q , for * = 1, (1.33)
where we again use the notation (-) Q for the Malcev completion functor on groups. Besides, one can prove that the unit of the derived adjunction relation between the functors G • and Ω * defines a map η : X → X Q which corresponds to the usual rationalization map at the level of these homotopy groups.

The category of Hopf cochain dg-cooperads

To extend the Sullivan model to operads, the idea is to consider cooperads in the category of commutative cochain dg-algebras, where the cooperad is a structure which is dual to an operad in the sense of the theory of categories.

In general, a cooperad in a symmetric category C consists of a collection Little discs operads, graph complexes and Grothendieck-Teichmüller groups 23 of objects C = {C (r), r ∈ N}, together with an action of the symmetric group Σ r on C (r), for each r ∈ N, and composition coproducts

• * i : C (k + l -1) → C (k) ⊗ C (l), (1.34)
defined for all k, l ∈ N, i ∈ {1, . . . , k}, and which satisfy equivariance, unit and coassociativity relations dual to the equivariance, unit and coassociativity axioms of operads. To handle difficulties, we consider a subcategory of cooperads such that C (0) = C (1) = 1 where 1 is the unit object of our base category, and we use the notation Op c * 1 for this category of cooperads. This restriction enables us to simplify some constructions, because the composition coproducts are automatically conilpotent when we put the component of arity zero apart and we assume C (1) = 1. In some cases, we consider a category of cooperads Op c * N such that we still have C (0) = 1, but where C (1) may not reduce to the unit object. More care is necessary in this case, and we actually assume an extra conilpotence condition for the composition coproducts that involve the component of arity one. (We refer to [START_REF] Benoit Fresse | The extended rational homotopy theory of operads[END_REF] for the precise expression of this conilpotence condition.)

In [START_REF] Benoit Fresse | Homotopy of operads and Grothendieck-Teichmüller groups[END_REF]§II.12] the author considers a category of Λ-cooperads, whose objects have no term in arity zero, but a diagram structure over the category of finite ordinals and injective maps which extends the action of the symmetric groups on ordinary operads. This category of Λ-cooperads is isomorphic to the category of cooperads which we consider in this paragraph Op c * 1 , so that the results of this reference [START_REF] Benoit Fresse | Homotopy of operads and Grothendieck-Teichmüller groups[END_REF] can immediately be transposed to our setting. (The structure of a Λ-cooperad is used to overcome technical difficulties that occurs in the construction of the Sullivan model of operads, but we can neglect these issues in this overview.)

We use the name 'Hopf cochain dg-cooperad' for the category of cooperads in the category of commutative cochain dg-algebras C = dg * Com and we also adopt the notation dg * Hopf Op c * 1 for this category of cooperads. We also consider a category of operads in simplicial sets satisfying P(0) = P(1) = * in order to deal with the restrictions imposed by the definition of our category of cooperads in our model. We use the notation sSet Op * 1 for this category of operads. We have the following statement: -For a cofibrant operad P ∈ sSet Op * 1 such that H * (P(r)) forms a finite dimensional Q-module in each arity r ∈ N and in each degree * ∈ N, we have a weak equivalence

Ω * (P)(r) ∼ -→ Ω * (P(r))
between the component of arity r of the Hopf cochain dg-cooperad Ω * (P) ∈ dg * Hopf Op c * 1 and the image of the space P(r) under the Sullivan functor Ω * (-), for any r ∈ N.

The first claim of this theorem follows from the observation that the functor G • (-) is strongly symmetric monoidal. The functor Ω * (-), on the other hand, is only weakly monoidal. To be more precise, in the case of this functor, we have a Künneth morphism ∇ : Ω * (X) ⊗ Ω * (Y ) → Ω * (X × Y ) which is a quasiisomorphism but not an isomorphism. Hence, for an operad in simplicial sets P, we only get that the composition product • i : P(k) × P(l) → P(k + l -1) induces a morphism which fits in a zigzag of morphisms of commutative cochain dg-algebras Ω * (P(k+l-1))

• * i -→ Ω * (P(k)×P(l)) ∼ ← -Ω * (P(k))⊗Ω * (P(l)).
The idea is to use the adjoint lifting theorem (see for instance [11, §4.5]) to produce the functor of the second claim of the theorem Ω * : sSet Op op * 1 → dg * Hopf Op c * 1 and to fix this problem. Then the crux lies in the verification of the third claim, for which we refer to the cited reference.

For an operad in simplicial sets P ∈ sSet Op * 1 , we now set: For any operad P ∈ sSet Op * 1 such that H * (P(r)) = H * (P(r), Q) forms a finite dimensional Q-module in each arity r ∈ N and in each degree * ∈ N, we have:

P Q := R Ω * (P) ,
P Q (r) ∼ P(r) Q ,
where we consider the component of arity r of the operad P Q on the left-hand side and the Sullivan rationalization of the space P(r) on the right-hand side.

Little discs operads, graph complexes and Grothendieck-Teichmüller groups 25

For the operad of little n-discs D n , we now set R Ω * (D n ) = Ω * (E n ), where E n is any cofibrant model of E n -operad in simplicial sets such that E n (0) = E n (1) = * , and we still write

D Q n = R Ω * (D n ) .
To apply the rational homotopy theory to the class of E n -operads, we aim to determine the model of these objects R Ω * (D n ).

Recall that we have a homotopy equivalence D n (r) ∼ F (R n , r) between the underlying spaces of the operad of little n-discs D n (r) and the configuration spaces of the Euclidean space F (R n , r). (In §1.1.1, we use an equivalent homotopy equivalence D n (r) ∼ F ( Dn , r), where we take the open disc Dn ∼ = R n rather than the Euclidean spaces R n .) In a first step, we recall the following result about the cohomology algebras of these spaces:

1.2.5 Theorem (V.I. Arnold [START_REF] Arnol'd | The cohomology ring of the group of dyed braids[END_REF], F. Cohen [START_REF] Frederick | The homology of C n+1 -spaces, n ≥ 0. In The homology of iterated loop spaces[END_REF]). Let n ≥ 2. For each r ∈ N, the graded commutative algebra H * (D n (r)) H * (F (R n , r)) has a presentation of the form:

H * (F (R n , r)) = (ω ij , 1 ≤ i < j ≤ r) (ω 2 ij , ω ij ω ik -ω ij ω jk + ω ik ω jk )
where the elements ω ij correspond to cohomology classes of degree n -1.

In the expression of this theorem, the notation (-) represents the free graded commutative algebra generated by the variables ω ij . The result established by V.I. Arnold in [START_REF] Arnol'd | The cohomology ring of the group of dyed braids[END_REF] concerns the case n = 2 of this statement. The already cited work of F. Cohen [START_REF] Frederick | The homology of C n+1 -spaces, n ≥ 0. In The homology of iterated loop spaces[END_REF] gives the general case n ≥ 2. We also refer to Sinha's survey [START_REF] Dev | The (non-equivariant) homology of the little disks operad[END_REF] for a gentle introduction to the computation of this theorem. The classes ω ij ∈ H * (F (R n , r)) represent the pullbacks of the fundamental class of the n -1-sphere ω ∈ H * (S n-1 ) under the maps π ij : F (R n , r) → S n-1 such that π ij (a 1 , . . . , a r ) = (a j -a i )/||a j -a i ||. We can also consider unordered pairs {i, j} in this definition. We just have ω ij = (-1) n ω ji in this case, since ω ij corresponds to the image of ω ji under the action of the antipodal map on the sphere. In what follows, we refer to these cohomology classes ω ij as the Arnold classes, we refer to the identity ω ij ω ikω ij ω jk + ω ik ω jk = 0 as the Arnold relation, and we refer to the presentation of the above theorem as the Arnold presentation.

The homology of the operad D n now inherits the structure of an operad in graded modules. The cohomology H * (D n ) inherits a dual cooperad structure, because the homology of the spaces D n (r) has a finite dimension as a Q-module in each arity and in each degree, so that we have the arity-wise duality relation H * (D n (r)) = Hom gr Mod (H * (D n (r)), Q) in the category of graded modules. Note that we have

D n (0) = * ⇒ H * (D n (0)) = Q and D n (1) ∼ * ⇒ H * (D n (1)) = Q, so that the collection H * (D n ) = {H * (D n (
r), r ∈ N} satisfies our connectedness condition in the definition of a cooperad. One can easily check that this cooperad structure is compatible with the graded commutative algebra structure of the cohomology, so that the object H * (D n ) actually forms a Hopf cooperad in the category of graded modules.

We aim to determine this cooperad structure. We use the following identity, already mentioned in the introduction of this chapter:

1.2.6 Theorem (F. Cohen [START_REF] Frederick | The homology of C n+1 -spaces, n ≥ 0. In The homology of iterated loop spaces[END_REF]). For n ≥ 2, we have an isomorphism of operads in graded modules H * (D n )

Pois n , where Pois n is the operad that governs the category of n-Poisson algebras.

Recall that the structure of an n-Poisson algebra refers to a graded version of Poisson structure where we have a commutative product µ(x 1 , x 2 ) = x 1 x 2 of degree 0 and a Poisson bracket λ(x 1 , x 2 ) = [x 1 , x 2 ] of degree n -1. This Poisson bracket satisfies the symmetry relation λ(x 1 , x 2 ) = (-1) n λ(x 2 , x 1 ), a graded version of the Jacobi identity and of the Poisson distribution relation. The n-Poisson operad Pois n is defined by the corresponding presentation by generators and relations in the category of operads. Equivalently, we can represent an element of the graded module Pois n (r) as a Poisson polynomial π = π(x 1 , . . . , x r ) of degree one in each variable x i .

For our purpose, we actually consider a unitary version of the n-Poisson operad, where we have an extra arity zero operation e ∈ Pois(0) such that µ • 1 e = 1 = µ • 2 e and λ • 1 e = 0 = λ • 2 e. This operation corresponds to a unit in the structure of an n-Poisson algebra and reflects the identity D n (0) = * at the level of the topological operad D n .

We get the following result when we pass to the cohomology: Let -, -: Pois n (r) ⊗ H * (D(r)) → Q denote the duality pairing which we obtain by using this relation H * (D n ) ∼ = Pois n . For a Poisson monomial π(x 1 , . . . , x r ) ∈ Pois n (r), we have the formula:

ω ij , π(x 1 , . . . , x r ) = 1, if π(x 1 , . . . , x r ) = x 1 . . . [x i , x j ] • • • x j • • • x r , 0, otherwise,
where we consider the generating classes We can now regard the object Pois c n H * (D n ) as a Hopf cochain dgcooperad equipped with a trivial differential. We have to make explicit a cofibrant resolution of this object for the applications of our methods of the rational homotopy theory of operads. In the next paragraphs, we explain a first definition of such a resolution by using graded analogues of the Drinfeld-Kohno Lie algebra operad of the previous section.

ω ij ∈ H * (F (R n , r)) = H * (D(

The graded Drinfeld-Kohno Lie algebra operads and the associated Chevalley-Eilenberg cochain complexes

The graded analogues of the Drinfeld-Kohno Lie algebra operad, which we define for every value of the parameter n ≥ 2, are denoted by p n . The ungraded Drinfeld-Kohno Lie algebra operad of §1.1.9 corresponds to the case n = 2. Thus, we have p = p 2 with the notation of §1.1.9.

To define the Lie algebras p n (r), we use the same presentation as in Eqn. 1.21:

p n (r) = L(t ij , {i, j} ⊂ {1, . . . , r})/ < [t ij , t kl ], [t ij , t ik + t jk ] >, (1.35)
but we now take deg(t ij ) = n-2 and we assume the graded symmetry relation t ji = (-1) n t ij , for every pair {i, j} ⊂ {1, . . . , r}. Then we take the same construction as in §1.1.9 to provide these Lie algebras with an action of the symmetric groups and with additive composition products

• i : p n (k)⊕p n (l) → p n (k + l -1)
, so that the collection p n = {p n (r), r ∈ N} inherits the structure of an operad in the category of graded Lie algebras. Note that the graded Lie algebras p n (r) still inherit a weight grading from the free Lie algebra, and hence, form weight graded objects in the category of graded modules. Besides, we can form a completed version of the operads pn , as in the case n = 2 in §1.1.9, but for n ≥ 3, we trivially have pn = p n because the components of homogeneous weight m ≥ 1 of the Lie algebras p n (r) are concentrated in a single degree * = m(n -2) and we have m(n -2) → ∞ when n ≥ 3.

We consider the Chevalley-Eilenberg cochain complexes C * CE (ĝ) associated to the complete Lie algebras ĝ = pn (r). The cofibrant objects of the category of commutative cochain dg-algebras are retracts of dg-algebras of the form R = (S(V ), ∂), where S(V ) is the symmetric algebra on an upper graded dgmodule V equipped with a filtration The commutative cochain dg-algebras C * CE (p n (r)) inherit an action of the symmetric groups by functoriality of the Chevalley-Eilenberg cochain complex, as well as composition coproducts

F 1 V ⊂ F 2 V ⊂ • • • ⊃ F m V ⊂ • • • ⊂ V and where we have a differential ∂ such that ∂(F m V ) ⊂ S(F m-1 V ).
• * i : C * CE (p n (k + l -1)) → C * CE (p n (k)) ⊗ C * CE (p n (l))
, which are given by the composites of the morphisms

• * i : C * CE (p n (k + l -1)) → C * CE (p n (k) ⊕ pn (l)
) induced by the composition products of the operad p n with the Künneth isomorphisms

C * CE (p n (k) ⊕ pn (l)) C * CE (p n (k)) ⊗ C * CE (p n (l))
. Hence, we get that the collection C * CE (p n ) = {C * CE (p n (r)), r ∈ N} inherits the structure of Hopf cochain dg-cooperad. In addition, one can prove that this Hopf cochain dg-cooperad is cofibrant (see [START_REF] Benoit Fresse | Homotopy of operads and Grothendieck-Teichmüller groups[END_REF]Theorem II.14.1.7]).

Then we have the following statement:

1.2.9 Theorem (T. Kohno [START_REF] Kohno | Série de Poincaré-Koszul associée aux groupes de tresses pures[END_REF]). We have a quasi-isomorphism of commutative cochain dg-algebras

C * CE (p n (r)) ∼ -→ H * (F ( Dn , r))
such that t ∨ ij → ω ij for each pair {i, j} ⊂ {1, . . . , r} and p ∨ → 0 when p ∨ is the dual basis element of a homogeneous Lie polynomial p ∈ p n (r) m of weight m > 1.

The cited reference gives the case n = 2 of this statement. The general result can be deduced from the observation that the cohomology algebra H * (F ( Dn , r) forms a Koszul algebra with the enveloping algebra of the Lie algebra p n (r) as dual associative algebra. We refer to [START_REF] Benoit Fresse | Homotopy of operads and Grothendieck-Teichmüller groups[END_REF]Theorem II.14.1.14] for further explanations on this approach. Now we can easily check that the quasi-isomorphisms of this theorem preserve cooperad structures. Hence, we get the following statement:

1.2.10 Proposition. The quasi-isomorphisms of Theorem 1.2.9 define a weak equivalence of Hopf cochain dg-cooperads

C * CE (p n ) ∼ -→ H * (D n ) = Pois c n ,
where we regard the cohomology of the little n-discs operad H * (D n ) as a Hopf cochain dg-cooperad equipped with a trivial differential.

We deduce from this proposition that the object C * CE (p n ) defines a cofibrant resolution of the object Pois c n = H * (D n ) in the category of Hopf cochain dg-cooperads. In our constructions, we actually consider a second cofibrant resolution, which is given by a Hopf cochain dg-cooperad of graphs Graphs c n , and we explain the definition of this object in the next paragraph.

Little discs operads, graph complexes and Grothendieck-Teichmüller groups 29

The graph cooperad

The Hopf cochain dg-cooperad Graphs c n precisely consists of graphs γ ∈ Graphs c n (r) with unnumbered internal vertices • and external vertices indexed by 1, . . . , r, as in the following picture:

γ = • 1 2 3 . (1.36)
The degree of such a graph is determined by assuming that each internal vertex • contributes to the degree by deg(•) = n and that each edge contributes to the degree by deg(-) = 1n (in the lower grading convention). Thus, we have deg(γ) = (1n)v + ne (in the lower grading convention again), where v denotes the number of internal vertices and e denotes the number of edges in the graph γ ∈ Graphs c n (r). In fact, we can regard our graphs as tensor products of symbolic elements given by the internal vertices and by the edges of our objects. In particular, we assume that graphs equipped with odd symmetries vanish in Graphs c n (r). We also assume that each edge is oriented and that a reversal of orientation is equivalent to the multiplication by a sign (-1) n in Graphs c n (r). For our purpose, we allow graphs with double edges, but not loops (edges with the same origin and endpoint) and we assume that each internal vertex is at least trivalent though the latter conditions are not essential. Besides, we assume that each connected component of our graph contains at least one external vertex.

The differential of graphs is defined by contracting edges in order to merge internal vertices together or in order to merge internal vertices with external vertices, as shown schematically in the following picture:

δ • • • • • • • • = • • • • • • • and δ • • • • • • • • • • i = • • • i . (1.37)
For instance, we have the formula:

δ • 1 2 3 = 1 2 3 ± 1 2 3 ± 1 2 3 (1.38)
in Graphs c n (3). The product is given by the amalgamated sum of graphs along external vertices. For instance, we have the formula:

1 2 3 = 1 2 3 • 1 2 3 . (1.39)
The cooperad coproduct • * i : Graphs c n (k+l-1) → Graphs c n (k)⊗Graphs c n (l), where we fix k, l ∈ N, i ∈ {1, . . . , k}, has the form • * i (γ) = α⊂γ γ/α ⊗ α, where the sum runs over all the subgraphs α ⊂ γ that contain the external vertices indexed by i, . . . , i + l -1, and γ/α denotes the graph obtained by collapsing this subgraph to a single external vertex (which we index by i in Handbook of Homotopy Theory the result of the operation, while we shift the index of the vertices such that j > i by j → jl + 1). Note that we have Graphs c n (1) = Q in general, so that our object Graphs c n belongs to the extended category of Hopf cochain dgcooperads dg * Hopf Op c * N but not to the category of connected Hopf cochain dg-cooperads dg * Hopf Op c * 1 . We easily see that the commutative cochain dg-algebras of graphs defined in this paragraph Graphs c n (r) have a structure of the form Graphs c n (r) = (S(Q[-1]⊗ICGraphs c n (r)), ∂) (like the Chevalley-Eilenberg cochain dg-algebras of the previous paragraph), where ICGraphs c n (r) is a complex of graphs which are connected when we remove the external vertices inside Graphs c n (r). (In what follows, we refer to such graphs as internally connected graphs.) We just perform an extra degree shift in the definition of this complex of internally connected graphs in order to get a Q[-1] factor on the generating dg-module of our symmetric algebra (as in the definition of the Chevalley-Eilenberg cochain complex of a Lie dg-algebra). We can actually use this expression to identify the object ICGraphs c n (r) with the dual of an L ∞ -algebra (a strongly homotopy Lie algebra). We can use this symmetric algebra structure Graphs c n (r) = (S(Q[-1]⊗ICGraphs c n (r)), ∂) to prove that Graphs c n forms a cofibrant object in the category dg * Hopf Op c * N , and we also have the following proposition:

1.2.12 Proposition (M. Kontsevich [START_REF] Kontsevich | Operads and motives in deformation quantization[END_REF]). We have a quasi-isomorphism of Hopf dg-cooperads

Graphs c n ∼ -→ H * (D n ) = Pois c n
which carries the graph γ ij ∈ Graphs c n (r) with a single edge i j to the Arnold class ω ij and which cancel the internally connected graphs with a non-empty set of internal vertices.

The assignment of this proposition determines the map Graphs c n (r)

∼

-→ H * (D n (r)) as a morphism of graded commutative algebras since the internally connected graphs generate the object Graphs c n (r) as a graded commutative algebra. (Note that the graphs γ ij of the proposition represent the internally connected graphs with an empty set of internal vertices.) We just check that this map preserves differentials (and hence, gives a well-defined morphism of commutative cochain dg-algebras in each arity r ∈ N), as well as the cooperad structures, so that our collection of maps define a morphism of Hopf dgcooperads. We refer to the cited reference [START_REF] Kontsevich | Operads and motives in deformation quantization[END_REF] and to [START_REF] Lambrechts | Formality of the little N -disks operad[END_REF] for a proof that this morphism defines a quasi-isomorphism. Observe simply that the differential identity of Eqn. 1.38 is carried to the Arnold relation in H * (D n (3)).

Recall that we set R Ω * (D n ) = Ω * (E n ) for the topological operad of little n-discs D n , where E n is any cofibrant model of E n -operad in simplicial sets such that E n (0) = E n (1) = * . We have the following result: This theorem asserts that the operad of little n-discs is formal in the sense of our operadic counterpart of the Sullivan rational homotopy theory of spaces. The cited reference [29, Theorem A'] proves an intrinsic formality theorem which implies this operadic formality result in the case n ≥ 3. (In the next statement, we will explain that the case n = 2 of this theorem follows from the existence of Drinfeld's associators.)

The result of this theorem can also be deduced from Kontsevich's proof of the formality of E n -operads when we pass to real coefficients (see [START_REF] Kontsevich | Operads and motives in deformation quantization[END_REF]). Indeed, the construction of Kontsevich can be used to define a collection of quasi-isomorphisms Graphs c n (r)

∼ -→ Ω * sa (FM n (r))
, where FM n is a model of E n -operad given a real oriented analogue of the Fulton-MacPherson compactification of the configuration spaces (see [START_REF] Getzler | Operads, homotopy algebra and iterated integrals for double loop spaces[END_REF]) and Ω * sa (-) denotes a cochain dg-algebra functor of semi-algebraic forms (see [START_REF] Hardt | Real homotopy theory of semi-algebraic sets[END_REF][START_REF] Lambrechts | Formality of the little N -disks operad[END_REF]). One can observe that these morphisms can be associated to a strict morphism of Hopf cochain dg-cooperads Graphs c n ∼ -→ Ω * (FM n ) (by using a general coherence statement of [START_REF] Benoit Fresse | Homotopy of operads and Grothendieck-Teichmüller groups[END_REF]Proposition II.12.1.3]).

The approach of the cited reference [START_REF] Fresse | The intrinsic formality of E n -operads[END_REF] does not use this constructions and gives a formality quasi-isomorphism which is defined over the rationals by using obstruction theory methods. The claim of this reference is that the E n -operads are intrinsically rationally formal for n ≥ 3 in the sense that every Hopf cochain dg-cooperad A n which satisfies H * (A n ) Pois c n and is equipped with an extra-involution operad J : A n → A n such that J(λ) = -λ in the case 4|n satisfies A n ∼ Pois c n . We apply this claim to the Hopf cochain dg-cooperad A n = R Ω * (D n ) to get the statement of the theorem. We have an extension of this formality result for the morphisms D m → D n which link the operads of little discs when nm ≥ 2 (see [START_REF] Fresse | The intrinsic formality of E n -operads[END_REF]Theorem C]).

Recall that we set D Q n := R Ω * (D n ) to define a model for the rationalization of the little n-discs operad in topological spaces. The result of the previous theorem has the following corollary:

Handbook of Homotopy Theory

Hence, the results of this section gives a simple algebraic model of the rational homotopy type of E n -operads. In the case n = 2, we have an identity C * CE (p) ∼ B(CD Q ), where we consider the chord diagram operad of §1.1.10 (recall also that we use the notation p = p2 for the ungraded Drinfeld-Kohno Lie algebra operad which occurs in this case n = 2). Thus, since we have on the other hand We now examine a counterpart of the formality result of Theorem 1.2.13 in the category of dg-modules dg * Mod . We use the notation C * (-) for both the singular complex functor from the category of topological spaces to the category of dg-modules and for the standard normalized chain complex functor on simplicial sets. These functors are lax symmetric monoidal and therefore carry operads in topological spaces (respectively, in simplicial sets) to operads in dg-modules. Furthermore, in the case of a cofibrant operad in simplicial sets R, we have the duality relation Ω * (R) ∨ ∼ C * (R) in the category of dgoperads dg Op * when we consider the dual in dg-modules of the Hopf cochain dg-cooperad Ω * (R) of Theorem 1.2.13. Therefore, the result of Theorem 1.2.13 implies the following statement, which was also obtained by the authors cited in this statement by other method:

D Q 2 = B(PaB Q ) (see §1.
1.2.15 Theorem (D. Tamarkin [START_REF] Dmitry | Formality of chain operad of little discs[END_REF], M. Kontsevich [START_REF] Kontsevich | Operads and motives in deformation quantization[END_REF]). We have the relation

Pois n ∼ C * (D n ),
in the category of dg-operads.

This result is exactly the formality theorem mentioned in the introduction of this chapter for the class of E n -operads in dg-modules. Tamarkin's proof of this theorem, which works in the case n = 2, relies on the correspondence between formality equivalences and associators, whereas Kontsevich's proof, which works for every n ≥ 2 but requires to pass to real coefficients, relies on the definition of semi-algebraic forms associated to graphs (as we explain in our survey of Theorem 1.2.13). In [START_REF] Boavida | On the formality of the little disks operad in positive characteristic[END_REF], Boavida and Horel have given a new proof of the formality result of this theorem by using a generalization of classical formality criterion of mixed Hodge theory in the context of operads (see also [START_REF] Petersen | Minimal models, GT-action and formality of the little disk operad[END_REF] for an application of this approach in the case n = 2).

The rational homotopy of mapping spaces on the operads of little discs

We now tackle the main objective of this chapter, namely the computation of the homotopy of the mapping spaces Map

(D m , D Q n ) ∼ Map h T op Op (D m , D n ) Q when n-m ≥ 3
, so that the results explained in this section gives a full computation of the rational homotopy of the mapping spaces Map h

T op Op (D m , D n ) that occur in the operadic description of the introduction for the embedding spaces Emb c (R m , R n ).

To carry out these computations, we use the graph complex model Graphs c n of the rational homotopy of the operad D n . Hence, we naturally obtain, as a main outcome, a graph complex description of the homotopy type of the spaces

Map h T op Op (D m , D Q n ) and Aut h Op (D Q n ).
In the case of the mapping spaces Map h T op Op (D m , D Q n ), we express the result as the Maurer-Cartan space MC • (HGC mn ) associated to a Lie dg-algebra of hairy graphs HGC mn . We explain the definition of this object first and we explain our computation afterwards.

In the case of the automorphism spaces Aut h Op (D Q n ), we get that our object is homotopy equivalent to a cartesian product of Eilenberg-MacLane spaces (like any H-group in rational homotopy theory). Thus, we can focus on the computation of the homotopy groups in this case. We give a description of these groups in terms of the homology of a non-hairy graph complex GC n of which we also explain the definition beforehand. This graph complex GC n is a graded version of a complex introduced by Kontsevich in [START_REF] Kontsevich | Formal (non)commutative symplectic geometry[END_REF], and therefore, this complex is usually called the Kontsevich graph complex in the literature.

The hairy graph complex

The hairy graph complex HGC mn explicitly consists of formal series of connected graphs with internal vertices •, internal edges •-•, which link internal vertices together, and external edges • -(the hairs), which are open at one extremity, as in the following examples:

, , . (1.40)
This complex HGC mn is equipped with a lower grading. The degree of a graph γ ∈ HGC mn is determined by assuming that each vertex contributes by deg(

δ • • • • • • • = • • • • • • • • . (1.41)
We equip the hairy graph complex with the Lie bracket such that:

γ 1 • • • , γ 2 • • • = ± γ 1 • • • γ 2 • • • - ± γ 2 • • • γ 1 • • • , (1.42)
where the first sum runs over the re-connections of a hair of the graph γ 1 to a vertex of the graph γ 2 , and similarly in the second sum, with the role of the graphs γ 1 and γ 2 exchanged. In the case m = 1, we have to consider a deformation of this Lie dg-algebra structure which we call the Shoikhet L ∞ -structure (a strongly homotopy Lie algebra). We just refer to [START_REF] Willwacher | Deformation quantization and the gerstenhaber structure on the homology of knot spaces[END_REF] for the explicit definition of this structure.

In the next theorem, we consider the Maurer-Cartan space MC • (L) associated to the Lie dg-algebra L = HGC m,n . This simplicial set MC • (L) is defined by the sets of flat L-valued PL connections on the simplices ∆ n , n ∈ N. To be more precise, in the definition of this object MC • (L), we generally assume that L forms a complete Lie dg-algebra with respect to a filtration

L = F 1 L ⊃ F 2 L ⊃ • • • ⊃ F k L ⊃ • • • such that [F k L, F l L] ⊂ F k+l L.
In the case L = HGC mn , we assume that F k L = F k HGC mn consists of power series of graphs γ ∈ H mn such that ev ≥ k, where e denotes the number of edges and v denotes the number of internal vertices in γ. Then we explicitly set:

MC n (L) = ω ∈ (L ⊗ Ω * (∆ n )) 1 | δ(ω) + 1 2 [ω, ω] = 0} , (1.43)
for every simplicial dimension n ∈ N, where (L ⊗ Ω * (∆ n )) 1 denotes the component of upper degree 1 in the completed tensor product of the Lie dg-algebra L with the Sullivan cochain dg-algebra of PL forms Ω * (∆ n ). The face and degeneracy operators of this simplicial set are inherited from the simplices. This construction has a natural extension for L ∞ -algebras (see for instance [START_REF] Getzler | Lie theory for nilpotent L ∞ -algebras[END_REF]). We now have the following main result: For any n ≥ m ≥ 2, we have the relation:

Map h T op Op (D m , D Q n ) ∼ MC • (HGC mn )
, where HGC mn is the hairy graph complex. This relation extends to the case n > m = 1 when we equip HGC 1n equipped with the Shoikhet L ∞ -structure.

The results of the previous section imply that we have the following weakequivalences:

Map h T op Op (D m , D Q n ) ∼ Map h dg * Hopf Op c * 1 (R Ω * (D n ), R Ω * (D m )) (1.44) ∼ Map h dg * Hopf Op c * 1 (Pois c n , Pois c m ), (1.45)
where Map h dg * Hopf Op c * 1 (-, -) denote a derived mapping space bifunctor in the category of Hopf cochain dg-cooperads, we use the Quillen adjunction between the functors G • (-) and Ω * (-) in the first equivalence (1.44) and the formality result of Theorem 1.2.13 in the second equivalence (1.45). These equivalences reduce the proof of Theorem 1.3.2 to a problem of algebra.

To compute the derived mapping space of Hopf cochain dg-cooperads Map h dg * Hopf Op c * 1 (Pois c n , Pois c m ), we need to pick a cofibrant resolution of the object Pois c n on the source and a fibrant resolution of the object Pois c m on the target. For this purpose, we take the cofibrant Hopf cochain dg-cooperad in the category of L ∞ -algebras (see [28, §8]).

R n = C * CE (p n ) (see §1.
The result of this theorem has the following corollary:

1.3.3 Corollary. For any n ≥ m ≥ 2 (and for n > m = 1), we have the identity: π * (Map h T op Op (D m , D Q n ), ω) = H * -1 (HGC ω mn ), for any ω ∈ MC 0 (HGC mn ), where HGC ω mn is the complex HGC mn equipped with the twisted differential δ ω = δ + [ω, -] + (extra terms in the L ∞ -case).

The identity of this statement follows from the result of Theorem 1.3.2 and from a general result about the homotopy groups of Maurer-Cartan spaces MC • (L) for which we refer to [START_REF] Berglund | Rational homotopy theory of mapping spaces via Lie theory for L ∞ -algebras[END_REF].

A computation of the rational homotopy groups of the embedding spaces Emb c (R m , R n ), analogous to the result established in this corollary, is given in [START_REF] Arone | Graph-complexes computing the rational homotopy of high dimensional analogues of spaces of long knots[END_REF] (see also [START_REF] Lambrechts | Homotopy graph-complex for configuration and knot spaces[END_REF] for the case m = 1 of these computations). These previous computations are based on the interpretation in terms of mapping spaces of operadic bimodules of the Goodwillie-Weiss tower of the embedding spaces Emb c (R m , R n ) (or of the equivalent interpretation of the Goodwillie-Weiss tower in terms of Sinha's cosimplicial model in the case m = 1). In [START_REF] Lambrechts | The rational homology of spaces of long knots in codimension > 2[END_REF], the formality of E n -operads in chain complexes is also used to get a description of the homology of the embedding spaces Emb c (R 1 , R n ) in terms of a Hochschild Handbook of Homotopy Theory cohomology theory for operads (we apply this Hochschild cohomology theory to the n-Poisson operad). The graph operad model of the n-Poisson can also be used to deduce a graph complex model of the homology of the embedding space Emb c (R 1 , R n ) from this algebraic approach.

In fact, we can use the result of the above corollary and the equivalence between the embedding space Emb c (R m , R n ) and the m + 1-fold iterated loop space of the operadic mapping space Map h T op Op (D m , D n ) given in the introduction to get applications of the result of Theorem 1.3.2 in the theory embedding spaces. For this purpose, we also use the following theorem: We refer to the cited reference [28, §10] for the detailed proof of this statement, which relies on an analogous result for spaces, established by Haefliger in [START_REF] Haefliger | Rational homotopy of the space of sections of a nilpotent bundle[END_REF].

We examine the rational homotopy of the spaces of homotopy automorphisms to complete the result of this section. We first explain the definition of the Kontsevich graph complexes GC n which occur in this computation.

The Kontsevich graph complex

The definition of the complex GC n is the same as the definition of the hairy graph complex HGC mn , except that we now consider graphs without hairs, as in the following examples:

, . (1.47)
We determine the degree of a graph in GC n by assuming that each vertex contributes by deg(•) = n and each edge contributes by deg(•-•) = 1n as in the case of hairy graphs. We still assume that every vertex of a graph in GC n is at least trivalent and we do not allow loops (edges with the same origin and endpoint). The differential is defined by the blow-up of vertices again. The space of homotopy automorphisms Aut For each λ ∈ Q × , we have the identity:

π * (h -1 (λ)) = H * (GC n ) ⊕ Q, if * ≡ -n -1(4), 0, otherwise,
where GC n denotes the Kontsevich graph complex.

We deduce this statement from the result of Theorem 1.3.2, by using that the identity morphism is represented by the Maurer-Cartan element such that ω = | in the hairy graph complex HGC nn . We just consider versions of the graph complexes GC 2 n and HGC 2 mn where bivalent vertices are allowed. We have HGC 2 mn ∼ HGC mn for any n ≥ m ≥ 2, whereas for the graph complex GC 2 n , we have:

H * (GC 2 n ) = H * (GC n ) ⊕ Q L * , if * ≡ -n -1(4), 0, otherwise, (1.48)
where L * denotes the homology classes of graphs of the form:

L * = • • • (1.49)
We easily see that the operation [ω, -] in the differential δ ω = δ + [ω, -] of the twisted complex (HGC 2 nn ) ω associated to the Maurer-Cartan element ω = | is given by the addition of a hair | to any graph γ ∈ HGC 2 nn . We can then use a spectral sequence to check that we have a quasi-isomorphism

Q | ⊕ Q[-1] ⊗ GC 2 n ∼ -→ HGC 2
nn where we consider the mapping γ → γ-which associates a graph with one hair γ-∈ HGC 2 nn to any graph γ ∈ GC n . We refer to [29, Proposition 2.2.9] for the detailed line of arguments.

In the case n = 2, we have H 0 (GC n ) = grt by a result of T. Willwacher, where grt is the graded Grothendieck-Teichmüller Lie algebra (see [START_REF] Thomas Willwacher | Kontsevich's graph complex and the Grothendieck-Teichmüller Lie algebra[END_REF]). Therefore, in this case, the result of Theorem 1.3.6 reflects the relation that we obtained in Theorem 1.1.7.

Outlook

Throughout this survey, we have focused on the study of the homotopy of E noperads themselves, but one can use variants of the definition of an E n -operad to associate operadic right module structures to any n-manifold M .

Handbook of Homotopy Theory

For this purpose, we can use again the Fulton-MacPherson operad FM n , the model of E n -operad, given by a real oriented version of the Fulton-MacPherson compactification of the configuration spaces, which was considered by Kontsevich in his proof of the formality of E n -operads (see §1.2). These Fulton-MacPherson compactifications have a natural generalization for the configuration spaces of manifolds F (M, r), and when M is a framed manifold, this construction returns a collection of spaces FM M = {FM M (r), r ∈ N} which inherits the structure of a right module (in the operadic sense) over the Fulton-MacPherson operad FM n .

This object is equivalent to constructions used by Ayala-Francis in the definition of the factorization homology of manifolds (see Ayala-Francis's chapter, in this handbook volume, for a survey of this subject). In particular, one can use a relative composition product of the object FM M over the operad FM n to compute the factorization homology of any framed manifold M . The methods used by Kontsevich to prove the formality of E n -operads have been used by several authors to define models of the rational homotopy type of this right module FM M , and hence, to tackle the rational homotopy computations in factorization homology theory. To cite a few works on this subject, let us mention that a graph complex model of the object FM M , which extends the graph cooperad of §1.2.11 when M is a simply connected compact manifold without boundary, is defined by Campos-Willwacher in [START_REF] Campos | A model for configuration spaces of points[END_REF], while an extension of Arnold's presentation is used by Idrissi in [START_REF] Idrissi | The lambrechts-stanley model of configuration spaces[END_REF] to get a small model of the object FM M . Idrissi's result provides a generalization of Knudsen's description of the factorization homology for higher enveloping algebras of Lie algebras [START_REF] Knudsen | Higher enveloping algebras[END_REF]. The paper [START_REF] Campos | Configuration spaces of manifolds with boundary[END_REF] provides an extension of the constructions of [START_REF] Campos | A model for configuration spaces of points[END_REF][START_REF] Idrissi | The lambrechts-stanley model of configuration spaces[END_REF] for manifolds with boundary, while the paper [START_REF] Campos | A model for framed configuration spaces of points[END_REF] addresses an extension of the definition of these operadic module structures by using a framed version of the operads of little discs.

In § §1.2-1.3, we entirely focus on the rational homotopy theory framework, but we may wonder which information we may still retrieve by our methods in positive characteristic. For instance, partial formality results have been obtained by Cirici-Horel in [START_REF] Cirici | étale cohomology, purity and formality with torsion coefficients[END_REF] when we take an arity-wise truncation of operads below the characteristic of the coefficients (see also [START_REF] Boavida | On the formality of the little disks operad in positive characteristic[END_REF] for an improvement of these partial formality results). In fact, the E n -operads are not formal as symmetric operads in chain complexes in positive characteristic, because their components are not formal as representations of the symmetric groups. Nevertheless, we may wonder whether E n -operads are formal as non-symmetric operads, which is enough for the study of mapping spaces over an E 1 -operad. The case n > 2 of this question is still open, but Salvatore has proved in [START_REF] Salvatore | Planar non-formality of the little discs operad in characteristic two[END_REF] that E 2 -operads are not formal as non-symmetric operads over F 2 .

  and a braiding isomorphism c ∈ Mor M(2) (m, (1 2)m) which satisfy the strict unit relations m • 1 e = 1 = e • 1 m together with the coherence constraints of the unit, pentagon and hexagon relations of Eqn. 1.7-1.11 in the operad M.

1. 2 . 3

 23 Theorem (B. Fresse [26, §II.10, §II.12]). -The letf adjoint of the Sullivan functor G • : dg * Com → sSet op induces a functor G • : dg * Hopf Op c * 1 → sSet Op op * 1 from the category of Hopf cochain dg-cooperads dg * Hopf Op c * 1 to the category of operads in simplicial sets sSet Op op * 1 . For an object A ∈ dg * Hopf Op c * 1 , we set G • (A)(r) = G • (A(r)) and we use the fact that G • (-) is strongly symmetric monoidal to equip the collection of these simplicial sets G • (A) = {G • (A(r)), r ∈ N} with the structure of an operad. Handbook of Homotopy Theory -This functor G • : dg * Hopf Op c * 1 → sSet Op op * 1 admits a right adjoint Ω * : sSet Op op * 1 → dg * Hopf Op c * 1 and the pair of functors (G • , Ω * ) defines a Quillen adjunction.

  where we use the notation R Ω * (-) for the right derived functor of the functor of the previous theorem Ω * : sSet Op op * 1 → dg * Hopf Op c * 1 , and we again use the notationfor the left derived functor of the Sullivan realization on operads G • : dg * Hopf Op c * 1 → sSet Op op * 1 . The equivalence Ω * (P)(r) ∼ Ω * (P(r)) implies that we have the following result at the level of this realization: 1.2.4 Theorem (B. Fresse [26, Theorem II.10.2.1 and Theorem II.12.2.1]).

1. 2 . 7

 27 Proposition. The cohomology algebras H * (D n (r)), r ∈ N, form a Hopf cooperad in graded modules such that H * (D n ) ∼ = Pois c n , where Pois c n denotes the cooperad dual to Pois n in graded modules. The n-Poisson cooperad Pois c n is explicitly defined by taking the dual graded modules of the components of the n-Poisson operad Pois c n (r) = Hom gr Mod (Pois n (r), Q). We take the adjoint morphisms of the composition products of the n-Poisson operad to provided this collection of graded modules Pois c n (r) with a cooperad structure. Therefore, the relation of this proposition H * (D n ) ∼ = Pois c n follows from the result of the previous theorem H * (D n ) ∼ = Pois n and the duality between the homology and the cohomology H * (D n (r)) = Hom gr Mod (H * (D n (r)), Q).

  r)) of the Arnold presentation of Theorem 1.2.5. This duality relation is immediate in arity 2, because the Poisson bracket operation λ = λ(x 1 , x 2 ) corresponds to the fundamental class of the n -1-sphere in the homology of D n (2), where we use the relationD n (2) ∼ F (R n , r) ∼ S n-1 .The general formula followsLittle discs operads, graph complexes and Grothendieck-Teichmüller groups 27 from the fact that the maps π ij : F (R n , r) → F (R n , 2) in the definition of the classes ω ij correspond to composites with the zero-ary operation * ∈ D n (0) which represents our algebra unit e ∈ Pois n (0) when we pass to the n-Poisson operad Pois n . We refer to the paper[START_REF] Dev | The (non-equivariant) homology of the little disks operad[END_REF] for a more thorough study of this duality relation between the n-Poisson polynomials and the elements of the cohomology algebras H * (D n (r)) = H * (F (R n , r)) in the Arnold presentation.

  The Chevalley-Eilenberg cochain complex is precisely defined by an expression of this form C * CE (g) = (S(Q[-1] ⊗ ĝ∨ ), ∂), where Q[-1] = Q e denotes the graded Handbook of Homotopy Theory module generated by a single element e in lower degree -1 (equivalently, in upper degree one) and ĝ∨ denotes the (continuous) dual of the completed Lie algebra ĝ = pn (r). The differential ∂ is induced by the dual map of the Lie bracket [-, -] on ĝ.

1. 2 .

 2 13 Theorem (B. Fresse and T. Willwacher [29, Theorem A']). We have the relation Pois c n ∼ R Ω * (D n ), Little discs operads, graph complexes and Grothendieck-Teichmüller groups 31 in the category of Hopf cochain dg-cooperads.

1. 2 .

 2 14 Corollary. We have D Q n = Pois c n , for any n ≥ 2, where we consider the image of the dual cooperad of the n-Poisson operad Pois c n under the operadic upgrading of the Sullivan realization functor -.We just use the implication Poisc n ∼ R Ω * (D n ) ⇒ Pois c n ∼ R Ω * (D n )to get the result of this corollary. This result, together with the observations of Proposition 1.2.10 and Proposition 1.2.12, implies that we can take eitherPois c n = G • (C * CE (p n )) or Pois c n = G • (Graphs c n ) to get a model of the rationalization D Q n . We have an identity G • (C * CE (p n (r))) = MC • (p n (r)), for each r ∈ N, where we consider a Maurer-Cartan space associated to the complete Lie algebra pn (r) (we review the definition of this construction in the next sections).

2 ∼

 2 1), we can deduce the existence of a weak equivalence D Q Pois c n from the operadic interpretation of Drinfeld's associators given in §1.1.10 (see [26, §II.14.2]).

  •) = n, that each internal edge contributes by deg(•-•) = 1n, that each hair contributes by deg(• -) = mn + 1, and by adding a global degree shift by -m. Thus, we have deg(γ) = nv + (1n)e + (mn + 1)hm, where v denotes the number of internal vertices, the letter e denotes the number of internal edges and h denotes the number of hairs of the graph γ ∈ HGC mn . The differential of the hairy graph complex is defined by the Handbook of Homotopy Theory blow-up of internal vertices:

1. 3 . 2

 32 Theorem (B. Fresse, V. Turchin, and T. Willwacher [28, Theorem 1]).

2 . 8 )

 28 and we adapt the classical Boardman-Vogt Wconstruction of operads to define a natural fibrant resolution functor W c (-) on the category of Hopf cochain dg-cooperad (see[28, §5] for a detailed definition of this functor). By analyzing the definition of maps on these Hopf cochain dg-cooperads, one sees that the mapping space Map h dg * Hopf Op c * 1 (Pois c n , Pois c m ) is weakly equivalent to the Maurer-Cartan space associated to an L ∞ -algebra of biderivations BiDer(C * CE (p n ), W c (Pois c m )) (see [28, §6]). The object C * CE (p n ) in this complex of biderivations can be replaced by the graph cooperad model of the n-Poisson cooperad Graphs c n ∼ Pois c n . The connection of the derived mapping space Map h dg * Hopf Op c * 1 (Pois c n , Pois c m ) with the hairy graph complex of the theorem comes from an ultimate reduction of this L ∞ -algebra of biderivations, which yields a relation of the form HGC mn ∼ BiDer(Graphs c n , W c (Pois c m )) (1.46)

1. 3 . 4

 34 Theorem (B. Fresse, V. Turchin, and T. Willwacher[START_REF] Benoit Fresse | The rational homotopy of mapping spaces of E n -operads[END_REF] Theorem 15]).In the case nm ≥ 3, the space Map h T op Op (D m , D n ) is nm -1connected, and we moreover have the relation:Map h T op Op (D m , D n ) Q ∼ Map T op Op (D m , D Q n )in the homotopy category of spaces.

  h T op Op (D Q n ) is the sum of the connected components of the mapping spaces Map h T op Op (D n , D Q n ) associated to the morphisms φ which are invertible in the homotopy category of operads. Let h : Aut h T op Op (D Q n ) → Aut Hopf Op (H * (D n , Q)) be the natural map which carries any such morphism to the associated homology morphism. For n ≥ 2, we have a bijection Aut Hopf Op (H * (D n , Q)) = Q × which is determined by taking the action of an automorphism φ ∈ Aut Hopf Op (H * (D n , Q)) on the representative of the Poisson bracket operation λ ∈ Pois n in the operad Pois n = H * (D n ). We get the following result: Little discs operads, graph complexes and Grothendieck-Teichmüller groups 37 1.3.6 Theorem (B. Fresse, V. Turchin, and T. Willwacher [28, Corollary 5]).

  Little discs operads, graph complexes and Grothendieck-Teichmüller groups 33 topy automorphism spaces Aut h Op (D Q n ), for all n ≥ 2. Thus, we aim to generalize the computation carried out in §1.1 in the case of the automorphism space Aut h Op (D Q 2 ). In the case of the mapping spaces Map h T op Op (D m , D Q n ), we are also going to check that we have the relation Map hT op Op

h T op Op (D m , D Q n )

and of the homo-