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The minimal model for the Batalin-Vilkovisky operad

The purpose of this paper is to explain and to generalize, in a homotopical way, the result of Barannikov-Kontsevich and Manin which states that the underlying homology groups of some Batalin-Vilkovisky algebras carry a Frobenius manifold structure. To this extent, we first make the minimal model for the operad encoding BV-algebras explicit. Then we prove a homotopy transfer theorem for the associated notion of homotopy BV-algebra. The final result provides an extension of the action of the homology of the Deligne-Mumford-Knudsen moduli space of genus 0 curves on the homology of some BV-algebras to an action via higher homotopical operations organized by the cohomology of the open moduli space of genus zero curves. Applications in Poisson geometry and Lie algebra cohomology and to the Mirror Symmetry conjecture are given.

Nearly all the examples of BV-algebras appearing in the aforementioned fields actually have some homology groups as underlying spaces. Therefore they are some shadow of a higher structure: that of a homotopy BV-algebra.
Algebra and homotopy theories do not mix well together a priori. The study of the homotopy properties of algebraic structures often introduces infinitely many new higher operations of higher arity. So one uses the operadic calculus to encode them.
The study of the homotopy properties of algebraic structures often introduces infinitely many new higher operations of higher arity. This is the case for Batalin-Vilkovisky algebras, which do not have homotopy invariance properties, like the transfer of structure under homotopy equivalences, see [LV10,Section 10.3]. To solve this, we have defined, in [GCTV09], a notion of homotopy Batalin-Vilkovisky algebra with the required homotopy properties. To do so, we have constructed a quasi-free, thus cofibrant, resolution of the operad BV encoding Batalin-Vilkovisky algebras, using the inhomogeneous Koszul duality theory.
While quite "small", this resolution carries a non-trivial internal differential; so it is not minimal in the sense of D. Sullivan [Sul77]. The purpose of the present paper is to go even further and to produce the minimal model of the operad BV, that is, a resolution as a quasi-free operad with a decomposable differential and a certain grading on the space of generators.
The first main result of this paper is the following computation of the homology groups of the bar construction for the operad BV as a deformation retract.
Theorem (2.1). The various maps defined in Section 2 form the following deformation retract in the category of differential graded S-modules This result provides the space of generators for the minimal model of the operad BV. But, on the opposite to the Koszul duality theory, this graded S-module is not endowed with a cooperad structure but with a homotopy cooperad structure. This means that there are higher decomposition maps which split elements, not only into 2 but also into 3, 4, etc. Finally, the differential of the minimal model is made up of these decomposition maps.
Let us recall that the problem of making minimal models explicit in algebraic topology is related to the following notions. Sullivan models [Sul77] are dg commutative algebras generated by the (dual of the) rational homotopy groups π • X ⊗ Q of a topological space X, where the differential is given by the Whitehead products. Quillen models [Qui69] are dg Lie algebras generated by the (dual of the) rational homology groups H • (X, Q) of a topological space X, where the differential is given by the Massey products. The Steenrod algebra is an inhomogenous Koszul algebra, whose Koszul dual dg algebra is the Λ algebra, see [Pri70]. The minimal model of the Steenrod algebra is generated by the underlying homology groups of the Λ algebra and the differential is related to the Adams spectral sequence [BCK + 66, Wan67]. Recall that the Λ algebra is the first page of the Adams spectral sequence, which computes the homotopy groups of spheres. The operad BV behaves exactly in the same way as the Steenrod algebra with respect to the inhomogeneous Koszul duality theory, see [GCTV09]. But, in contrast to the Steenrod algebra, we are able to compute, in this paper, the underlying homology groups of its Koszul dual (co)operad together with its algebraic structure. We also provide a topological explanation for our result. It was shown by E. Getzler [Get94a] that the operad BV is the homology of the framed little discs operad. Its minimal model is generated by a homotopy cooperad extension of the cooperad H • (M 0,n+1 ) by a free resolution T c (δ) of the circle S 1 .
We call the algebras over the minimal model of the operad BV skeletal homotopy Batalin-Vilkovisky algebras, since they involve fewer generating operations than the notion of a homotopy BV-algebra given in [GCTV09]. We provide new formulae for the homotopy transfer theorem for algebras over a quasi-free operad on a homotopy cooperad. To prove them, we have to introduce a new operadic method, based on a refined bar-cobar adjunction, since the classical methods of [LV10,10.2] (classical bar-cobar adjunction), and of A. Berglund [Ber09] (homological perturbation lemma) failed to apply. This gives the homotopy transfer theorem for skeletal homotopy BV-algebras.
The d∆-condition, also called the dd-lemma or dd c -lemma in [DGMS75], is a particular condition, coming from Kähler geometry, between the two unary operators: the underlying differential d and the BV-operator ∆. Under this condition, S. Barannikov and M. Kontsevich [BK98], and Y.I. Manin [Man99] proved that the underlying homology groups of a dg BV-algebra carry a Frobenius manifold structure. Such a structure is encoded by the homology operad H • (M 0,n+1 ) of the Deligne-Mumford-Knudsen moduli space of stable genus 0 curves. Its is also called an hypercommutative algebra (with a a compatible non-degenerate pairing). Tree formulae for such a structure have been given by A. Losev and S. Shadrin in [LS07].
We show that these results are actually a consequence of the aforementioned homotopy transfer theorem. This allows us to prove them under a weaker and optimal condition, called the non-commutative Hodgeto-de Rham condition. We recover the Losev-Shadrin formulae and thereby explain their particular form. Moreover our approach gives higher non-trivial operations, which are necessary to recover the homotopy type of the original dg BV-algebra. Theorem (7.8). Let (A, d, •, ∆, , ) be a dg BV-algebra with non-commutative Hodge-to-de Rham degeneration data.
The underlying homology groups H(A, d) carry a homotopy hypercommutative algebra structure, which extends the hypercommutative algebras of M. Kontsevich and S. Barannikov [BK98], Y.I. Manin [Man99], A. Losev and S. Shadrin [LS07], and J.-S. Park [Par07], and such that the rectified dg BV-algebra Rec(H(A)) is homotopy equivalent to A in the category of dg BV-algebras.
In geometrical terms, this lifts the action of the operad of moduli space of genus 0 stable curves (cohomological field theory) into a certain action of the cooperad of the open moduli space of genus 0 curves (extended cohomological field theory): [BK−M−LS−P ] 7 7 n n n n n n n n n n n n We conclude this paper with applications to the Poisson geometry, Lie algebra cohomology and the Mirror Symmetry conjecture. To conclude, this paper develops the homotopy theory for dg BV-algebras (homotopy skeletal BV-algebras, ∞-quasi-isomorphisms) necessary to study the Mirror Symmetry conjecture, in the same way as the homotopy theory of dg Lie algebras was used to prove the deformation-quantization of Poisson manifolds by M. Kontsevich in [Kon03].
Some of the results of the present paper were announced in [DCV09]. While we were typing it, V. Dotsenko and A. Khoroshkin computed in [DK09] the homology of the bar construction of the operad BV without the action of the symmetric groups. They used the independent method of Gröbner basis for shuffle operads developed in [DK10].
The paper is organized as follows. We begin by recalling the Koszul resolution of the operad BV given in [GCTV09]. In the second section, we compute the homology of the Koszul dual dg cooperad BV ¡ and we write it as a deformation retract of BV ¡ . In the third section, we recall the notion of homotopy cooperad with its homotopy properties: the homotopy transfer theorem for homotopy cooperads. With these tools in hand, we produce the minimal model of the operad BV at the end of Section 3. In a fourth section, we describe the associated notion of algebra, called skeletal homotopy BV-algebras. Section 5 deals with a generalization of the bar-cobar adjunction between operads and homotopy cooperads. The last section contains the homotopy transfer theorem for skeletal homotopy BV-algebras and the extension of the result of Barannikov-Kontsevich and Manin. The reader is supposed to be familiar with the notion of an operad and operadic homological algebra, for which we refer to the book [LV10]. In the present paper, we use the same notations as used in this reference.
We work over a field K of characteristic 0 and all the S-modules M = {M (n)} n∈N are reduced, that is, M (0) = 0.

RECOLLECTION ON HOMOTOPY BV-ALGEBRAS
In this section, we recall the main results of [GCTV09] needed in the rest of the text. In loc.cit., we made explicit a resolution of the operad BV using the Koszul duality theory. It is given by a quasi-free operad on a dg cooperad, which is smaller than the bar construction of BV.
1.1. BV-algebras. Definition 1.1 (Batalin-Vilkovisky algebras). A differential graded Batalin-Vilkovisky algebra, or dg BValgebra for short, is a differential graded vector space (A, d A ) endowed with ⊲ a symmetric binary product • of degree 0, ⊲ a symmetric bracket , of degree +1, ⊲ a unary operator ∆ of degree +1, such that d A is a derivation with respect to each of them and such that £ the product • is associative, £ the bracket satisfies the Jacobi identity £ the unary operator ∆ satisfies ∆ 2 = 0, £ the bracket is the obstruction to ∆ being a derivation with respect to the product • £ the operator ∆ is a graded derivation with respect to the bracket The operad encoding BV-algebras is the operad defined by generators and relations where T (V ) denotes the free operad on the S-module with K 2 being the trivial representation of the symmetric group S 2 . The space of relations R is the sub-Smodule of T (V ) generated by the relations '£' given above. The basis elements •, , , ∆ are of degree 0, 1, and 1. Since the relations are homogeneous, the operad BV is graded by this degree, termed the homological degree. We denote by Com the operad generated by the symmetric product • and the associativity relation. We denote by Lie 1 the operad generated by the symmetric bracket , and the Jacobi relation; it is the operad encoding Lie algebra structures on the suspension of a space. The operad G governing Gerstenhaber algebras is defined similarly. Its underlying S-module is isomorphic to Com • Lie 1 , on which the operad structure is given by means of distributive laws, see [LV10, Section 8.6].
1.2. Quadratic analogue. We consider the homogeneous quadratic analogue qBV of the operad BV. This operad is defined by the same spaces of generators V and relations except for the inhomogeneous relation which is changed into the homogenous relation: We denote this homogenous quadratic space of relations by qR. This operad qBV = T (V )/(qR) is also given by means of distributive laws on the S-module 1.3. Koszul dual cooperad of the operad qBV. We denote by s the homological suspension, which shifts the homological degree by +1. Recall that the Koszul dual cooperad of a quadratic operad T (V )/(qR) is defined as the sub-cooperad C(sV, s 2 qR) ⊂ T c (sV ) cogenerated by the suspension sV of V with corelators in the double suspension s 2 qR of qR, see [LV10,Chapter 7]. Namely, it is the "smallest" sub-cooperad of the cofree cooperad on sV which contains the corelators s 2 qR.
We denote by S c := End c Ks −1 = {Hom((Ks −1 ) ⊗n , Ks −1 )} n∈N the suspension cooperad of endomorphisms of the one dimensional vector space s −1 K concentrated in degree −1. The desuspension S c C of a cooperad C is the cooperad defined by the aritywise tensor product, called the Hadamard tensor product, 1.4. Koszul dual dg cooperad of the operad BV. We consider the map ϕ : qR → V defined by and 0 on the other relations of qR, so that the graph of ϕ is equal to the space of relations R. The induced map qBV ¡ → sV extends to a square-zero coderivation d ϕ on the cooperad qBV ¡ , see [GCTV09, Lemma 5]. The dg cooperad is called the Koszul dual dg cooperad of the inhomogeneous quadratic operad BV. We use the notation ⊙ for the 'symmetric' tensor product, that is, the quotient of the tensor product under the permutation of terms. In particular, we denote by δ m ⊗ L 1 ⊙ · · · ⊙ L t a generic element of T c (δ) • S c Com c 1 • S c Lie c with L i ∈ S c Lie c , for i = 1, . . . , t; the elements of S c Com c 1 being implicit. Under these notations, the coderivation d ϕ is explicitly given by where L ′ i ⊙ L ′′ i is Sweedler-type notation for the image of L i under the binary part of the decomposition map of the cooperad S c Lie c . The sign, given by the Koszul rule, is equal to ε i = (|L 1 | + · · · + |L i−1 |). The image of d ϕ is equal to 0 when m = 0 or when L i ∈ S c Lie c (1) = K I for all i.
1.5. Koszul resolution of the operad BV. We denote by BV ∞ the quasi-free operad given by the cobar construction on BV ¡ : where d 1 is the unique derivation which extends the internal differential d ϕ and where d 2 is the unique derivation which extends the infinitesimal (or partial) coproduct of the cooperad qBV ¡ , see [LV10, Section 6.5]. The total derivation d = d 1 + d 2 squares to zero and faithfully encodes the algebraic structure of the dg cooperad on BV ¡ . The space of generators of this quasi-free operad is isomorphic to It is called the Koszul resolution of BV. Notice that it is much smaller than the bar-cobar resolution ΩB BV ∼ − → BV. The Koszul resolution and the bar-cobar resolution are both quadratic. But they are not minimal resolutions: they are both quasi-free operads with a differential which is the sum of a quadratic term (d 2 ) and a non-trivial linear term (d 1 ).
Algebras over the operad BV ∞ are called homotopy BV-algebras. For an explicit description of this algebraic notion together with its homotopy properties, we refer the reader to [GCTV09].
1.6. Homotopy transfer theorem for homotopy BV-algebras. We consider the data of two chain complexes, where i and p are chain maps and where h has degree 1. It is called a homotopy retract when id A − ip = d A h + hd A and when, equivalently, i or p is a quasi-isomorphism. If, moreover, the composite pi is equal to id H , then it is called a deformation retract.
Theorem 33] Any homotopy BV-algebra structure on A transfers to H through a homotopy retract such that i extends to an ∞-quasi-isomorphism.

THE HOMOLOGY OF BV ¡ AS A DEFORMATION RETRACT
The purpose of this section is to construct an explicit contracting homotopy for the chain complex BV ¡ := (qBV ¡ , d ϕ ). This is a necessary ingredient for the construction of the minimal model of the operad BV given in the next section. As a byproduct, this computes the homology of the bar construction of the operad BV in terms of the homology of the moduli space M 0,n+1 of genus 0 curves. The main result of this section is the following theorem.
Theorem 2.1. The various maps defined in this section form the following deformation retract: A reduced rooted tree is a rooted tree whose vertices have at least one input. We consider the category of reduced rooted trees with leaves labeled bijectively from 1 to n, denoted by Tree. We define a total order on the vertices of a tree by reading its planar representation from leaf 1 to the root by following the internal edges without crossing them. See Figure 1 for an example.
2.2. Free operad and cofree cooperad. The underlying S-module of the free operad T (V ) on an Smodule V is given by the direct sum t∈Tree t(V ), where t(V ) is the treewise tensor module obtained by labeling every vertex of the tree t with an element of V according to the arity and the action of the symmetric groups. The operadic composition map is given the the grafting of trees. Dually, the underlying S-module of the conilpotent cofree cooperad T c (V ) is equal to the same direct sum over trees and its decomposition map is given by cutting the trees horizontally; see [LV10, Chapter 5] for more details.
The subcategory of trees with n vertices is denoted by Tree (n) . The number of vertices endows the free operad T (V ) ∼ = n∈N T (V ) (n) and the conilpotent cofree cooperad T c (V ) ∼ = n∈N T c (V ) (n) with a weight grading. We represent a labeled tree by t(v 1 , . . . , v n ), using the aforementioned total order on vertices. This is a classical generalization of the characterization of coderivation for cofree coalgebras. Here are two simple but useful examples, for more details see [LV10, Section 6.3.14].
(1) If η factors through the projection T c (M ) ։ T c (M ) (1) = M , then d η is given on a decorated tree as a signed sum over the vertices of the tree. The summand corresponding to a vertex v is the same tree with η applied to the decoration of v and all other decorations the same. The sign is the Koszul sign.
(2) If η factors through the projection T c (M ) ։ T c (M ) (2) , then d η is given on a decorated tree as a signed sum over the internal edges of the tree. The summand corresponding to an edge e has the edge contraction along e of the original tree as its underlying tree. The decorations away from the contraction vertex are the same; the decoration on the contraction vertex is given by applying η to the two decorated vertices involved in the contraction, viewed as a two-vertex decorated tree in T c (M ) (2) . The sign is the Koszul sign.
2.4. A contracting homotopy for a cofree cooperad. Let M be the S-module which is the linear span of elements µ and β, both of arity two, in degrees 1 and 2 respectively, both with trivial symmetric group action.
Let ψ denote the degree one morphism of graded S-modules ψ : T c (M ) → M which first projects T c (M ) to the cogenerators M and then takes µ to β and β to zero. ψ can be extended uniquely to a degree one coderivation d ψ of T c (M ) by Lemma 2.2. We will construct a degree −1 chain homotopy H of graded S-modules on T c (M ) so that d ψ H + Hd ψ is the identity outside arity 1 and the zero map on arity 1 (which is one dimensional, spanned by a representative of the coimage of the counit map).
To do this, we will need a combinatorial factor.
Definition 2.3. Let T be a binary tree. The vertex v has some number of leaves m v above one of its incoming edges, and another number n v above the other (we need not concern ourselves which is which). Let the weight ω(v) be their product m v n v . For an illustration, see Figure 2. Lemma 2.4. [Lod04] The sum of the weights of all the vertices of a binary tree with n vertices is n+1 2 . Definition 2.5. Let h : M → M be the degree −1 morphism of graded S-modules given by taking β to µ and µ to 0. We will use h to define the contracting homotopy H.
Let the homotopy H be defined on a decorated tree with n vertices in T c (M ) as a sum over the vertices. For the vertex v, the contribution to the sum is ω(v) ( n+1 2 ) times the decorated tree obtained by applying h to v (including the Koszul sign). So it has a similar flavor to extending h as a coderivation, but also includes combinatorial factors.
Lemma 2.6. The map d ψ H + Hd ψ is zero in arity one and the identity in all other arities.
Proof. First, applied to the coidentity subspace of T c (M ), the degree zero part of T c (M )(1), this sum is clearly zero. Next, consider a tree in T c (M ) with at least one vertex. The map H acts on it by taking the signed and weighted sum of replacing each β with a µ; the coderivation d ψ acts by taking the signed sum over all the µ and replacing it with a β. To act first with one and then with the other means that either (1) The maps H and d ψ act on two distinct vertices of the tree, or (2) they act on the same vertex, changing it first from β to µ or vice versa and then back, ending up with the same tree, with a combinatorial factor.
The first type come in pairs, one from d ψ H and one from Hd ψ , with the same combinatorial factors. They have the opposite sign, because the sign conventions for H and d ψ are the same, and in one of the cases, there is one more or fewer µ than the other in a position that induces a sign. This means all of these terms cancel.
For the second type, note first of all that the induced signs from H and d ψ will be the same sign acting on whichever vertex we have chosen, and every vertex will be acted on by precisely one of Hd ψ and d ψ H nontrivially, depending on whether it begins decorated by β or µ, so the final result of acting in this way on every vertex will be the sum over all vertices of the underlying tree T : Applying d 2 to a decorated tree in T c (M ) gives a sum of trees, each of which has one special 4valent vertex decorated by an element of sG (2) obtained by contraction of one edge and composition of the associated two operations. The rest of the vertices are trivalent and decorated with an element of sG (1) = M . Because M is one dimensional in each degree, we can specify that each trivalent vertex is decorated by either µ or β, with an overall coefficient on the decoration of the special vertex. Then in order that two separate terms be in the same summand of T c (sG) so that they might cancel, the underlying trees must be the same and the decorations on each trivalent vertex must be the same.
Definition 2.7. A contraction tree is a tree with one undecorated 4-valent vertex and all other vertices trivalent and decorated by either µ or β.
Note that we can induce a fixed order on the leaves of the special 4-valent vertex of a contraction tree by using the order on the leaves of the whole tree; order the leaves of the special vertex by the smallest number of a tree leaf above each one, see [LV10,Section 8

.2]
A sum of decorated trees c T T ∈ T c (M ) is in the kernel of d 2 if and only if once we sum over all possible edge contractions, any summands that have the same underlying contraction tree cancel with each other.
Therefore it is important to know which decorated binary trees T can have the same contraction tree. There are precisely three underlying trees that can give rise to a given contraction trees by an edge contraction, corresponding to the three distinct binary trees with two vertices: Definition 2.8. There are twelve distinct two-vertex binary trees with vertices decorated by µ and/or β, which form a K-linear basis for the twelve dimensional space T c (M ) (2) . We will refer to these basis trees with the notation t i (a, b) where a and b are each one of the symbols µ and β and i is one of 1, 2, and 3. The numbers correspond, respectively, to the trees pictured above, while a and b are decorations of the two vertices, following the vertex order convention established in subsection 2.1. If S is a contraction tree, then S(t i (a, b)) is the decorated binary tree obtained by replacing the 4-valent vertex with t i (a, b) and Up to scale, there are precisely twelve decorated trees that can contract to yield the contraction tree S if each vertex is given either µ or β as a decoration; these are the trees S(t i (a, b)). These twelve are clearly in correspondence with the set {t i (a, b)}. We have shown: Lemma 2.9. The following are equivalent: (1) The element c T T is in G ¡ .
(2) For each contraction tree S, c S(ti(a,b)) S[t i (a, b)] = 0 where the sum runs over the twelve decorated trees S(t i (a, b)) that can yield S as a contraction.
(3) For each contraction tree S, c S(ti(a,b)) d 2 (t i (a, b)) = 0 where the sum runs over the twelve decorated trees S(t i (a, b)) that can yield S as a contraction.
In words, a sum of decorated trees can only be in G ¡ if there is local cancellation for every possible contraction tree. Global cancellation is sufficient but local cancellation is necessary (this means that they are equivalent, but it will be easier to use local cancellation to check global cancellation in the sequel).
Proof. Let c T T be a sum of decorated trees in G ¡ . Then for every contraction tree S, the sum over the twelve basis elements c S(ti(a,b)) d 2 (t i (a, b)) is zero.
Let us consider applying H to c T T . By definition, this is a sum over every vertex of the decorated tree T . To show that the resultant sum is in G ¡ , we then apply d 2 and demonstrate that we get zero. Applying d 2 involves applying the desuspension of the infinitesimal composition map γ (1) on each set of two adjacent vertices, summing over all such pairs. We will confuse such subsets with internal edges, with which they are in bijection, as described in Section 2.3.
In total, to apply H and then d 2 to a decorated tree T involves summing over all choices of a vertex and edge of T ; each individual summand is the application of first a weighted multiple of h to the chosen vertex, and then infinitesimal composition γ (1) to the chosen edge.
This sum splits into those pairs of vertex and edge which are distinct, and those pairs where the chosen edge is incident on the chosen vertex. We will show that each of these two constituent sums is zero individually.
If the vertex and edge are distinct, then, up to sign, the application of h on the vertex and infinitesimal composition on the edge commute. For a given contraction tree and choice of trivalent vertex on the contraction tree, the overall sign of commuting the shifted infinitesimal composition on the edge of one of the twelve decorated trees which yields the given contraction tree and h on the corresponding vertex will be independent of the particular choice of decorated tree within the twelve. Let S v [t i (a, b)] be obtained from S[t i (a, b)] by applying the appropriate weighted multiple of h to the vertex v. Since the sum c S(ti(a,b)) S[t i (a, b)] over the twelve corresponding decorated trees without any application of h is zero, The other case to consider is when the edge involved in the contraction is incident on the vertex where h is applied. Let us fix a contraction tree S; because the original sum is in G ¡ , it is true that )) is zero. We will replace d 2 (t i (a, b)) with the weighted sum of applying h to the top and bottom vertex of t i (a, b), followed by d 2 , and show that the result is still zero.
The two-vertex component of the Koszul dual cooperad to a quadratic operad is isomorphic to the space of relations, up to a degree shift. In this case, we have: Lemma 2.11. The kernel of d 2 on the linear span of t i (a, b) is six dimensional, spanned by the shifted Gerstenhaber relations: (1) the two dimensional space of associativity relations (2) the three dimensional space of Leibniz relations spanned by

(note that the signs are different than in the usual Leibniz relation because of the shift, and that the presentation is not symmetric in our basis), and
(3) the one-dimensional space of the Jacobi relation t 1 (β, β) + t 2 (β, β) + t 3 (β, β).
For the weighting of h, it is necessary to look at the shape and decorations of the contraction tree S. Choose a representative so that the two vertices involved in the contraction edge are adjacent in the total ordering of vertices. Let the number of leaves above the edge i of the contraction vertex be n i .
Then up to an overall sign and overall factor of 1 ( n+1 2 ) , the weighted sum of applying h to both vertices of each of the basis elements is given by: . Applying these formulae to the kernel described above gives: ( (3) and: So the kernel of d 2 in this twelve dimensional space is stable under the weighted application of h, no matter the particular trees that define the weights. This means that for each contraction tree S, the sum obtained from c S(ti(a,b)) d 2 (t i (a, b)) by replacing d 2 (t i (a, b)) with the weighted sum of applying h to the top and bottom vertex of t i (a, b), followed by d 2 , is still zero, as desired.
is zero, then d restricts to be a coderivation of O ¡ .
Proof. Since the Koszul dual cooperead O ¡ = C(sN, s 2 R) is a quadratic cooperad, this proof is dual to the proof that a derivation of the free operad T (N ) passes to the quotient Corollary 2.13. The coderivation d ψ defined on T c (M ) restricts to G ¡ . We will refer to the restriction with the same notation.
Proof. In order to check this, we need check only that elements of G ¡ which d ψ takes into T c (M ) (2) land in G ¡(2) . For degree reasons, such elements must belong to G ¡(2) , which is described by Lemma 2.11. A direct calculation verifies that d ψ takes an associativity relation to a difference of two Leibniz relations, takes each Leibniz relation to the Jacobi relation, and takes the Jacobi relation to zero.
Remark. One can easily check that the dual of the chain complex (G ¡ , d ψ ) is isomorphic to both the Koszul complex Lie ¡ • κ Lie (see [LV10,Section 7.4]) and the Chevalley-Eilenberg complex of the free Lie algebra. This isomorphism along with the preceding proposition implies as a corollary the well-known facts that the Lie and commutative operads are Koszul, and that, equivalently, the Chevalley-Eilenberg homology of the free Lie algebra is trivial.
Definition 2.15. We define a map of S-modules as follows. We will describe the image of δ m ⊗ x where x has underlying tree T . Let λ range over assignments of a non-negative integer to each edge of T so that the sum of all the integers is m.
Then the image of δ m ⊗ x has underlying tree T ′ which is obtained from T by inserting λ(e) bivalent vertices on each edge e, labeled by δ.
Proof. First, let x ∈ G ¡ . We will verify that θ(δ n ⊗x) is in qBV ¡ by checking that d 2 θ is zero on T c (δ)⊗G ¡ (here d 2 is the differential induced by composition in qBV).
The map θ inserts vertices decorated by δ, and d 2 composes pairs of adjacent vertices. The sum involved in applying d 2 includes compositions involving 0, 1, and 2 vertices decorated by δ. Each of these vanishes for a different reason.
(1) The insertion of a vertex decorated with δ commutes up to sign with compositions that do not involve it, so inserting m vertices decorated with δ and then contracting an edge whose vertices are decorated by µ or β is the same as contracting the edge first and then inserting vertices decorated with δ. But since d G ¡ 2 coincides with d qBV ¡ 2 on the δ 0 component of qBV ¡ , and we are starting in the kernel of d G ¡ 2 to begin with, this summand is zero.
(2) Contracting an edge whose vertices are both decorated by δ gives a bivalent vertex whose decoration is s(∆ • ∆), which is zero since ∆ • ∆ = 0 in qBV.
(3) Finally, consider contracting an edge between a vertex v decorated by a µ or β and an adjacent vertex decorated by a δ. Let λ ′ be a map from the edges of T to the natural numbers so that the sum of the images adds to m − 1. There are precisely three choices of λ with a δ adjacent to v which can be forgotten to yield an element whose underlying tree is T with vertices inserted according to λ ′ . The sum of the three contractions with v associated to λ ′ together make up a relation of qBV.
Now consider ρθ(δ m ⊗x). Because ρ first decomposes and then projects, it is zero on any tree decorated by β, δ, and µ unless all of the vertices decorated by δ are below all of the other vertices. There is precisely one summand in the sum defining θ which satisfies this condition. That is the summand corresponding to the partition λ with λ of the outgoing edge of the root equal to m and λ of every other edge equal to zero. The map ρ splits this into two levels and then projects; the only way for the projection to be nonzero is for it to split with δ m as the bottom level; then ρθ(δ m ⊗ x) = (δ m ⊗ x). Because ρ is an isomorphism, a one-sided inverse is an inverse.
Lemma 2.17. Under the above isomorphism θ, the differential δ −1 ⊗ d ψ is sent to d ϕ : Proof. It is enough to prove it on the level of the cofree cooperads. We show that the following diagram is commutative whered ϕ is the unique coderivation of the cofree cooperad T c (M ⊕ Kδ) which extends the map ϕ. Since δ −1 ⊗ d ψ is a coderivation, it is enough to prove it by projecting onto the space of cogenerators M ⊕ Kδ. We conclude by showing that the only non-trivial component is Proof. Write the chain complex as The homology is then one dimensional by Proposition 2.14 everywhere except at 1 ⊗ G ¡ , where everything is in the kernel of the differential so the homology is just the quotient by the image of d ψ .
Proof of Theorem 2.1. We prove that the following data form a deformation retract, where the projection map p is the sum of the projection onto T c (δ) and the projection onto G ¡ composed with Hd ψ . Assume that x is in the coaugmentation coideal G ¡ . Since H is a contracting homotopy for d ψ , Since G ¡ is contractible and x is in the coaugmentation coideal, this means that x − Hd ψ x is in the image of d ψ , therefore in the image of d ϕ . This shows that Hd ψ x is in the same homology class as x. It is independent of choice of representative because it gives zero on all of Im d ψ . A quick calculation verifies that d ϕ (δ ⊗ H) − (δ ⊗ H)d ϕ gives the projection onto δ m ⊗ G ¡ except on the rightmost factor, where it gives id −Hd ψ . This concludes the proof of the theorem, with the exception of the rightmost identification with the dual to the Gravity operad given in the next section.
2.8. The homology of BV ¡ in terms of the moduli space of curves and the Gravity operad. Let us recall from E. Getzler's papers [Get94b,Get95] the definition of the quadratic operad Grav encoding gravity algebras. It is generated by skew-symmetric operations [x 1 , . . . , x n ] of degree 2 − n for any n ≥ 2, which satisfy the following relations: The sign is the Koszul sign coming from the permutation of the elements. We consider the moduli space M 0,n+1 of genus 0 curves with n + 1 marked points. The gluing along two points and the Poincaré residue map induce an operad structure on the suspension sH • (M 0,n+1 ) of its homology, see [Get95, Section 3.4]. Let S −1 denote both the desuspension operad and cooperad structure on End Ks −1 .
Proposition 2.19 ([Get94b]). The gravity operad is related to the homology of the moduli space of genus 0 curves by the following isomorphism of operads: Proposition 2.20. The quotient G ¡ /Im d ψ is a cooperad isomorphic to S −1 Grav * .
Proof. This is the cooperadic dual of Theorem 4.5 of [Get94a]. The aritywise linear dual of the differential graded quadratic cooperad (G ¡ , d ψ ), with degree 1 coderivation, is a differential graded quadratic operad, with degree 1 derivation. (We consider the opposite homological degree on the linear dual). By [GJ94, Theorem 3.1], the underlying operad is isomorphic to (G ¡ ) * ∼ = S 2 G := End Ks 2 ⊗ H G, which admits the same quadratic presentation as the operad G except for the −2 degree shift of the generators s −2 • and s −2 , . By the universal property of quadratic operads, the derivation t d ψ is characterized by the images of these generators, that is s −2 • → s −2 , and s −2 , → 0. Therefore, up to the degree shift, the derivation t d ψ is equal to the derivation ∆ on G defined in [Get94a]. Theorem 4.5 of loc. cit. states that This concludes the proof of Theorem 2.1.

Theorem 2.21. There exist isomorphisms of graded S-modules
Proof. The first isomorphism is a general fact about Koszul operads. In the case of an inhomogenous Koszul operad P, it is proved as follows. The degree −1 map qP ¡ ։ sV → V P is a twisting morphism κ : P ¡ = (qP ¡ , d ϕ ) → P ∈ Tw(P ¡ , P), see [GCTV09, Appendix A] or [LV10, Section 7.8]. By the general properties of the bar-cobar adjunction [LV10, Section 6.5], it induces a morphism of dg cooperads f κ : P ¡ → B P, which is equal to the following composite: On the right-hand side, the operad P comes equipped with a filtration; we consider the induced filtration on the bar construction. On the left-hand side, we consider the filtration given by the weight grading on the cooperad qP ¡ . The coderivation d ϕ lowers this filtration by 1 and the morphism f κ preserves the respective filtrations. By the Poincaré-Birkhoff-Witt theorem [GCTV09, Theorem 39], gr P ∼ = qP, the first page (E 0 , d 0 ) of the right hand-side spectral sequence is isomorphic to B qP. So the map f κ induces the map fκ : (qP ¡ , 0) → B qP, on the level of the first pages of the spectral sequences, whereκ is the twisting morphism associated to the homogeneous quadratic operad qP. Since it is Koszul, the morphism fκ is a quasi-isomorphism and we conclude by the convergence theorem of spectral sequences associated to bounded below and exhaustive filtrations [ML95, Chapter 11]. The second isomorphism follows from Theorem 2.1.

Remarks.
⋄ While we were writing this paper, V. Dotsenko and A. Khoroshkin in [DK09] proved, with another method (Gröbner bases for shuffle operads), the second isomorphism on the level of graded Nmodules, i.e. without the action of the symmetric groups. ⋄ The cooperad G ¡ with the action of d ψ is the Koszul dual cooperad of the operad G with the action of ∆ is the sense of Koszul duality theory of operads over Hopf algebras, see the Ph.D. Thesis of O. Bellier [Bel11] for more details.

THE MINIMAL MODEL OF THE OPERAD BV
In this section, we recall the notion of a homotopy cooperad, and we develop a transfer theorem for such structures across homotopy equivalences. We apply this result to the deformation retract given in the previous section. This allows us to make the minimal model of the operad BV explicit.
Definition 3.1 (Homotopy cooperad). A homotopy cooperad structure on a graded S-module C is the datum of a square-zero degree −1 derivation d on the free operad T (s −1 C) which respects the augmentation map. An ∞-morphism C D of homotopy cooperads is a morphism of augmented dg operads between the associated quasi-free operads (T (s −1 C), d) → (T (s −1 D), d ′ ). We denote this category by ∞-coop ∞ .
Since the map d is a derivation on a free operad, it is completely characterized by its image on generators ∆ : C → T (C), under the above isomorphism. The substitution of a tree t at the i th vertex by a tree t ′ is denoted by t • i t ′ , see [LV10, Section 5.5] for more details.
where the sum runs over the elements t(c 1 , . . . , c l ) and A homotopy cooperad structure on a graded S-module C with vanishing maps ∆ t = 0 for trees t ∈ Tree (≥3) with more than 3 vertices is equivalent to a coaugmented dg cooperad structure on C := C ⊕ I. In this case, the definition in terms of a square-zero derivation on the free operad is equivalent to the differential of the cobar construction Ω C.
In the same way, the datum of an ∞-morphism F : , that is, a family of morphisms {f t : C → t(D)} t∈Tree , satisfying some relations. An interpretation in terms of Maurer-Cartan elements is given in [MV09a,Section 4.7].
The projection C → T (C) ։ C of d on the graded S-module C endows it with a differential denoted by d C , which is equal to the sum d C = ∆ t over the corollas t. The images on corollas of any ∞-morphism define a morphism of dg S-modules (C, d C ) → (D, d D ). When this latter map is a quasi-isomorphism, the ∞-morphism is called an ∞-quasi-isomorphism.

Homotopy transfer theorem for homotopy cooperads.
Theorem 3.3. Let (C, {∆ t }) be a homotopy cooperad. Let (H, d H ) be a dg S-module, which is a homotopy retract of the dg S-module (C, d C ): There is a homotopy cooperad structure on the dg S-module (H, d H ), which extends the transferred composition maps t(p) • ∆ t • i and such that the map p extends to an ∞-quasi-isomorphism.
Proof. For any corolla t, the transferred structure map ∆ t on H is given by the differential d H . For any tree t ∈ Tree with at least 2 vertices, we consider all the possible ways of writting it by successive substitutions of trees with at least 2 vertices: The transferred structure map ∆ t : H → t(H) is then given by where the notation (∆ t ′ h) • j ∆ t means here the composite of ∆ t with ∆ t ′ h at the j th vertex of the tree t.
The extension of the map p : C → H into an ∞-morphism p ∞ : C → T (H) is given by the same kind of formula. On corollas, it is given by the map p, and for any tree t ∈ Tree (≥2) with at least 2 vertices, it is given by When C is a dg cooperad, these formulae are the exact duals to the ones given by [Gra07] for dg (pr)operads. The rest of the proof is straightforward, following the ideas of loc. cit.

The homotopy cooperad structure on H(B BV). Let us denote the graded S-module
Theorem 2.1 provides us with the following deformation retract in the category of dg S-modules: Proof. This is a direct application of the Homotopy Transfer Theorem 3.3 for homotopy cooperads.
3.4. The minimal model of the operad BV.
Definition 3.5. A minimal operad is a quasi-free dg operad (T (X), d) ⋄ with a decomposable differential, that is d : X → T (≥2) (X), and ⋄ such that the generating degree graded S-module admits a decomposition into X = k≥1 X (k) ). A minimal model of a dg operad P is the data of a minimal operad (T (X), d) together with a quasiisomorphism of dg operads (T (X), d) ∼ / / / / P , which is an epimorphism. (This last condition is always satisfied when the differential of P is trivial).
The generalization of the notion of a minimal model from dg commutative algebras [DGMS75, Sul77] to dg operads was initiated by M. Markl in [Mar96], see also [MSS02, Section II.3.10]. Notice however that the aforementioned definition is strictly more general than loc. cit. and includes the crucial case of dg associative algebras, since we do not require that X(1) = 0 here. (A minimal operad in the sense of Markl is minimal in the present sense: the extra grading is given by the arity grading X (k) := X(k + 1)). The present definition faithfully follows Sullivan's ideas: the increasing filtration F k := k i=1 X (i) is the Sullivan triangulation assumption. The extra grading X (k) is called the syzygy degree. Notice that any non-negatively graded quasi-free operad with decomposable differential is minimal; one only has to consider X (k) := X k−1 .
The following lemma compares the two approaches of Quillen (cofibrant) and Sullivan (minimal) of homotopical algebra. [Hin97].

Proof. This is a particular case of [MV09b, Corollary 40].
Since the definition is different, one needs a more general proof for the uniqueness of minimal models.
Proposition 3.7. Let P be a dg operad. When it exists, the minimal model of the operad P is unique up to isomorphism.
Proof. We work with the model category structure on dg operads defined by V. Hinich in [Hin97]. Let M and M ′ be two minimal models of the graded operad P. Since the differentials are decomposable, we get d X = 0 and d X ′ = 0. So the aforementioned quasi-isomorphism is actually an isomorphism of graded S-modules X ∼ = X ′ . Therefore, the map f is an isomorphism of dg operads.
Proof. First, the quasi-free operad T (s −1 (T c (δ) ⊕ S −1 Grav * )), d is minimal since it is non-negatively graded with the decomposable differential coming from the transferred homotopy cooperad structure on H.
Then, the ∞-quasi-isomorphism p ∞ : qBV ¡ H of Corollary 3.4 induces a morphism of dg operads P : Ω BV ¡ → (T (s −1 H), d). It is a quasi-isomorphism by the following argument. We consider the filtration F • on Ω BV ¡ , and respectively F ′ • on (T (s −1 H), d), given by the number of vertices of the underlying tree: The first terms of the respective associated spectral sequences are and 1 H), 0). The morphism of dg operads P preserves the aforementioned filtrations. Moreover, it satisfies E 0 (P ) = T (s −1 p). So it is a quasi-isomorphism by the Künneth formula. The two filtrations are obviously exhaustive. At fixed arity, they are bounded below: for a fixed degree, the number of vertices is limited since the generator of arity one have degree greater or equal to 1. We conclude the argument by means of the classical convergence theorem for spectral sequences [ML95, Chapter 11].
Finally, we define a morphism of operads F : T (s −1 H) → BV by and the rest being sent to 0. We now check the commutativity of the differentials on the generators. It is straightforward on s −1 δ m . The only elements of Im Hd ψ whose image under d are trees with vertices labeled only by µ and δ are in Im Hd ψ (3). Indeed, let t be an element of Im Hd ψ (n), which is the sum of trees with k vertices labeled by µ and with n − 1 − k vertices labeled by β. To get trees labeled only by µ and δ, one has to apply h = δ ⊗ H a total of n − 1 − k times. This introduces the n − 1 − k power of δ and applies the coproduct of the cooperad T c (δ) ⊗ G ¡ a total of n − k times. In the end, we get trees labeled by n − 1 copies of µ and n − 1 − k copies of δ split n − k times. To get totally split trees, we should have n − k = 2n − 3 − k, which implies n = 3.
The one-dimensional space Lie It remains to show that the following diagram is commutative to conclude that F is a quasi-isomorphism. It is enough to check it on the generators, which is equivalent to the commutativity of the following diagram It is easily checked on δ, µ, and β. Both maps vanish on the rest of qBV ¡ by the same arguments as above: the only element which produces a non-trivial element in T (Ks −1 δ ⊕ Ks −1 µ) under p ∞ is β, which concludes the proof.
We denote by d n : s −1 H → T (s −1 H) (n) the part of the differential d, which splits elements into n pieces. The component d 2 coincides with the decomposition map on the cooperad S −1 Hyc ¡ * . Proposition 3.9. The differential of the minimal model of the operad BV has the following shape: Up to the desuspension s −1 , the image of an element of degree k of S −1 Grav k * under the map d n is a sum of trees with n vertices labeled by elements of S −1 Grav * and of T c (δ), such that the total degree of the elements from S −1 Grav * is equal to k − n + 2 and such that the total weight, i.e. the total power, of the elements coming from T c (δ) is equal to n − 2. For instance, this induces d n (S −1 Grav k * ) = 0 for n > k + 2 .
Proof. By direct inspection of the various formulae.
We denote the minimal model of the operad BV by

Remarks.
⋄ The results about TCFT, two-fold loop spaces, and the cyclic Deligne conjecture, obtained in [GCTV09] using the cofibrance property of the Koszul resolution of the operad BV hold as well with this minimal model. The proof of the Lian-Zuckerman conjecture with this minimal model requires further work and will be the subject of another paper. ⋄ The same method can be applied to [HL11] to make explicit the minimal model of the inhomogeneous quadratic operad H 0 (SC) , where the operad SC is Kontsevich Swiss-cheese operad.

SKELETAL HOMOTOPY BV-ALGEBRAS
We call algebras over the minimal model of the operad BV skeletal homotopy BV-algebras. We make this notion explicit and we give a description in terms of Maurer-Cartan elements in a homotopy Lie algebra.

Second definition of homotopy BV-algebras.
Definition 4.1. A skeletal homotopy Batalin-Vilkovisky algebra is an algebra over the minimal operad BV ∞ .
Recall that a skeletal homotopy BV-algebra structure on a dg module (A, d A ) is the datum of a morphism of dg operads BV ∞ → End A . The differential ∂ A of the operad End A is equal to We denote µ the image of an element µ of BV ∞ into End A .

Proposition 4.2. A skeletal homotopy BV-algebra is a chain complex (A, d A ) endowed with operations
and µ : A ⊗n → A, of degree |µ| + n − 2, for any µ ∈ Grav * (n), where the first sum runs over the decomposition product of the cooperad structure on Grav * , ∆ Grav * (µ) = µ 1 • i µ 2 , and where the second sum corresponds to composites of at least three operations with at least one ∆ m .

Proposition 4.3. A skeletal homotopy BV-algebra with vanishing operators ∆ m , for m ≥ 1, is a homotopy hypercommutative algebra.
Proof. This is a direct corollary of Proposition 4.2 together with the fact that the operad HyperCom is Koszul, that is Ω(S −1 Grav * ) ∼ − → HyperCom, see [Get95].
In operadic terms, this means that is a short exact sequence of dg operads, where s −1 T c (δ) is the ideal of BV ∞ generated by s −1 T c (δ).
Equivalently, the short sequence of homotopy cooperads Proof. Let D ∞ denote T (s −1 T c (δ)). This is the minimal resolution of the algebra of dual numbers D.
The pushout of I ← D ∞ BV ∞ gives a representative of the homotopy quotient of BV by ∆ since D ∞ BV ∞ is a cofibration and since all the operads in the diagram are cofibrant, see [Hir03,Chapter 15].
A map from this diagram to an operad is the same thing as a map of the generators of HyperCom ∞ that respects the differentials; since the augmentation ideal of D ∞ vanishes in any map from this diagram, the differentials coincide with those of HyperCom ∞ . So the image of HyperCom in the homotopy category of dg operads gives the homotopy quotient of BV by ∆.
We refer the reader to [Mar09,DK09,KMS11] further studies on this topic. This result on the level of homology allows us to conjecture that the homotopy quotient of the framed little disk by the circle is the compactified moduli space of genus zero stable curves M 0,n+1 . This will be the subject of another paper.
Remark. Since the generators of HyperCom ∞ form a cooperad, one can define the notion of ∞-morphism of homotopy hypercommutative algebras using [LV10,10.2]. In the case of the operad BV ∞ , to define the notion of ∞-morphism of skeletal homotopy BV-algebras, one has to refine the arguments, using the homotopy pullback of endomorphism operads for instance. With these definitions, Proposition 4.3 shows that the category of homotopy hypercommutative algebras with ∞-morphisms is a subcategory of the category of skeletal homotopy BV-algebras with ∞-morphisms, but not a full subcategory.

Maurer-Cartan interpretation. Recall from [MV09a, Theorem 28] that the module of morphisms of S-modules
Hom S (H, End A ) := n≥1 Hom Sn (H(n), End A (n)) carries an L ∞ -algebra structure, {ℓ n } n≥1 , given in terms of the homotopy cooperad structure on H by the formula: where γ End A is the composition map of operations of End A and where the sign is the Koszul sign due to the permutation of the graded elements {f i }.

Proposition 4.5. There is a natural bijection
Proof. This follows from Theorem 54 of [MV09a].
This result gives an interpretation of skeletal homotopy BV-algebra structures in term of Maurer-Cartan elements in an L ∞ -algebra.
We denote by (Im Hd ψ ) [k] the subspace of Im Hd ψ spanned by the tree monomials with k vertices labelled by µ.
Lemma 4.6. The isomorphism of Theorem 2.21 preserves the respective gradings:

Proof. By direct inspection.
This result allows us to organize the operations of a skeletal homotopy BV-algebra into strata. The first stratum is described as follows. Since S −1 Grav * (1) ∼ = HLie ¡ 1 , in weight 1, is equal to the trivial representation of S n , then we get Hom S (Grav * (1) , End A ) ∼ = Hom(S c(≥2) (A), A), up to suspension.
∆ t lands entirely in weight one only when t has precisely two vertices. So a twisting element α vanishing outside the weight 1 part of Grav * actually satisfies the truncated Maurer-Cartan equation ∂α + 1 2 ℓ 2 (α, α) = 0. This corresponds to the definition of a Frobenuis manifold in terms of a hypercommutative algebra structure, see Y.I. Manin [Man99].

HOMOTOPY BAR-COBAR ADJUNCTION
In this section, we introduce a new bar-cobar adjunction between the category of augmented dg operads and the category of homotopy cooperads. This bar construction relies on the notion of a cofree homotopy cooperad, which we make explicit in terms of nested trees.

Cofree homotopy cooperad.
We consider now the category of homotopy cooperads with (strict) morphisms.
of homotopy cooperads is a morphism of graded S-modules C → D which commutes with the structure maps.
A morphism of homotopy cooperads is an ∞-morphism with vanishing components C → T (D) (n) for n ≥ 2. The associated category is denoted by coop ∞ . There is a forgetful functor which retains only the underlying dg S-module structure of a homotopy cooperad.

Definition 5.2.
A nested tree is a tree t ∈ Tree\{|} equipped with a set of subsets of vertices {T i } i , called nests, such that: ⋄ each nest T i corresponds to a subtree of the tree t, ⋄ each nest T i has at least two elements, ⋄ if T i ∩ T j = ∅, then T i ⊂ T j or T j ⊂ T i , and ⋄ the full subset corresponding to the tree t is a nest as long as t has more than one vertex.
The associated category is denoted by NestedTree. See Figure 4 for an example. We consider the following total order on nests. The innermost nests are the largest ones. We compare them using their minimal element. Then we forget about these nests and proceed in the same way until reaching the full nest, which is the minimal nest. In the example of Figure 4, it gives To any dg S-module (V, d V ), we associate the S-module spanned by nested trees with vertices labeled by elements of V . It is denoted by Using the order on vertices given in Section 2.1 and the above order on nests, we write a simple element of Its homological degree is equal to n k=1 |v k | + N − n + 1. So the degree of a labeled corolla t(v) is equal to |v|.
where the notation T i means that we forget the nest T i . The sign is given as usual by the Koszul rule as follows. To every nest T i , we associate the tree t i obtained from the subtree of t defined by T i after contracting all its proper subnests. Each vertex thereby obtained is labeled by the least element of the contracted nest. The degree of a nest T i is equal to |T i | := 2 − #t i , where #t i stands for the number of vertices of the tree t i . (In the example of Figure 4, one has |T 1 | = −2.) If T j T i , then i < j. So we first permute T j with the nests T j−1 , . . . , T i+1 to bring it next to T i . Then we apply the differential to the pair (T i , T j ), that is we forget about the nest T j . This comes with a sign equal to (−1) to the power #t i + #t j + k + des(t j , t i ), where k is the number of vertices of t i smaller than the smallest vertex of t j and where des(t j , t i ) is the number of descents, that is the number of pairs (a, b) of vertices of t j and t i respectively such that a > b. But the differential has to "jump over" the nests T 1 , . . . , T i−1 . In the end, it produces the sign (−1) ε , with We consider the differential on N T (V ) given by the sum over all the vertices of the image of the labeling element of V under d V . By a slight abuse of notation, it is still denoted d V : We consider maps {∆ t : N T (V ) → t(N T (V ))} t∈Tree (≥2) defined as follows. Let τ be a simple element of N T (V ). We consider the aforementioned tree t 1 associated to the full nest T 1 , which is obtained by contracting all the subtrees corresponding to the interior nests. If t = t 1 , then ∆ t (τ ) := 0. Otherwise, if t = t 1 , the image of τ under ∆ t is equal to the tree t 1 with vertices labeled by the nested trees obtained from τ by forgetting its full nest.

Proposition 5.3. For any dg
) form a homotopy cooperad. This defines a functor N T : dg-S-Mod → coop ∞ which is right adjoint to the forgetful functor U : coop ∞ → dg-S-Mod.
Proof. The three first points of the equivalent definition of a homotopy cooperad given in Proposition 3.2 are trivially satisfied by N T (V ). The last point is straightforward to check.
Let C be a homotopy cooperad. We consider the morphism of S-modules ∆ iter : C → N T (C) defined as follows. For any tree t, the extra data given by the nests t(T 1 , . . . , T N ) is equivalent to the decomposition of t into successive substitutions where the trees {t i } are associated to the nests {T i } as defined above. The image of the map ∆ iter on a nested tree t(T 1 , . . . , T N ) is defined by ) . Let V be a dg S-module. To any morphism of dg S-modules f : U(C) → V , we associate a morphism F : C → N T (V ) defined by the composite The map F is a morphism of homotopy cooperads which satisfies the following universal property which concludes the proof.
Hence the homotopy cooperad N T (V ) is called the cofree homotopy cooperad on V .

Remarks.
⋄ The endofunctor U • N T in dg-S-Mod can be endowed with a comonad structure: decompose a nested tree into all the possible ways of seeing it as a nested tree of nested subtrees. Proposition 5.3 and its proof are equivalent to saying that the category of homotopy cooperads is the category of coalgebras over the comonad U • N T . ⋄ Recall that the notion of an A ∞ -algebra can be encoded geometrically by the Stasheff polytopes, also called the associahedra. In the same way, the notion of a homotopy cooperad can be encoded by a family of polytopes, defined by by means of graph associahedra labelled by nested trees as introduced by M.P. Carr and S.L. Devadoss in [CD06,DF08]. Notice that this notion generalizes the nested sets of C. De Concini and C. Procesi [DCP95]. For instance, the chain subcomplex of nested trees with fixed underlying tree t should be isomorphic to the cochain complex This surely deserves further study, which we leave to a future work or to the interested reader.
Proof. Checking this is a straightforward calculation.
In plain words, the pair of functors Ω π and B π are adjoint and this adjunction is represented by the twisting morphism bifunctor.
Proof. The first natural bijection is given by [MV09a,Theorem 54]. The second one is described as follows. Proposition 5.3 already provides us with a natural bijection Under this bijection, a morphism of S-modules f : C → sP induces a morphism of homotopy cooperads F : C → B π P if and only if the following diagram commutes Remark. The universal operadic twisting morphism π : B(SAs) → SAs induces a pair of adjoint functors B π and Ω π between the category of dg associative algebras and the category of homotopy coalgebras by [GJ94], see also [LV10,Chapter 11]. One can prove that it coincides with the restriction of the above bar and cobar constructions B π and Ω π to S-modules concentrated in arity one, which explains the notation.

HOMOTOPY TRANSFER THEOREM
In this section, we prove the homotopy transfer theorem and the rectification theorem for skeletal homotopy BV-algebras. We consider the map G(End A ) : B End A → B π End A defined, for any t ∈ Tree, by t(sf n , . . . , sf 1 ) ∈ T c (sEnd A ) → ± t(T 1 , . . . , T n−1 ; sf n , . . . , sf 1 ) ∈ N T (sEnd A ) , where the sum runs over all the maximal nestings, that is the ones with a maximal number of nests. Since the bar construction B End A is a cooperad, it carries a homotopy cooperad structure; the map G(End A ) is a quasi-isomorphism of homotopy cooperads.

Proposition 6.1. Let (H, d H ) be a homotopy retract of a chain complex (A, d A ). There exists a quasiisomorphism of homotopy cooperads
such that the following diagram, made up of quasi-isomorphisms of homotopy cooperads, is commutative, Proof. Let us first give the proof in arity 1; so here End A = Hom(A, A). We consider the quasiisomorphism of cooperads G : As c ∼ − → B(SAs). The map G(End A ) is equal to where κ ′ := Sκ : As c = SAs ¡ → SAs is the Koszul morphism coming from the Koszul duality of the operad As. By the Comparison Lemma [LV10, Lemma 6.4.13], the quasi-isomorphism G induces a quasi-isomorphism of quasi-free left SAs-modules (or equivalently of quasi-free anti-associative algebras in the category of S-modules). By the left lifting property, it admits a homotopy inverse quasi-isomorphism Under the bar-cobar adjunction, the quasi-isomorphism of cooperads Ψ is equivalent to the quasi-isomorphism of operads Ψ : ΩB End A ∼ − → End H . Finally, we define the morphism of homotopy cooperads Φ : B π End A ∼ − → B π End H to be the map corresponding to the quasi-isomorphism of operads under the homotopy bar-cobar adjunction.
One extends these arguments to higher arity by using the colored Koszul operad of [VdL03], which encodes operads, instead of the Koszul (non-symmetric) operad As which encodes associative algebras.
By definition, the following diagram is commutative which concludes the proof.
The morphism of homotopy cooperads Φ : B π End A ∼ − → B π End H is completely characterized by its projection onto the space of cogenerators, which we denote by φ : N T (sEnd A ) → sEnd H .
There is a skeletal homotopy BV-algebra on (H, d H ), which extends the transferred operations p µi ⊗n , for any µ ∈ H. If we denote by α ∈ Tw ∞ (H, End A ) the skeletal homotopy BV-algebra structure on A, such a transferred skeletal homotopy BV-algebra structure on H is given by Proof. We apply the bar-cobar adjunction of Theorem 5.7 to So a skeletal homotopy BV-algebra structure α : H → End A on A is equivalently given by a morphism of homotopy cooperads F α : H → B π End A . The transferred skeletal homotopy BV-algebra on H is then obtained by pushing along the morphism Φ: which is equivalent to the following twisting morphism Remark. We proved the homotopy transfer theorem for homotopy BV-algebras, i.e. for the Koszul model BV ∞ of the operad BV in [GCTV09,Theorem 33]. Since the S-module of generators of the minimal model BV ∞ forms a homotopy cooperad and not a cooperad, we cannot apply the arguments of [GCTV09, Appendix B.3] and of [LV10, Section 10.3] based on the classical bar-cobar adjunction. Neither can we use the homological perturbation lemma of [Ber09]. Notice that the existence of the homotopy transferred structure follows from model category arguments by [Rez96,BM03]. But we need here an explicit formula for the application to Frobenius manifolds in the next section.
Needless to say that the Homotopy Transfer Theorem 6.2 holds for any algebras over a quasi-free operad generated by a homotopy cooperad. In the case of a quasi-free operad generated by a dg cooperad, Koszul models or bar-cobar resolutions for instance, we recover the formulae of [GCTV09] and of [LV10, Chapter 10] as follows. Proposition 6.3. Let P be a Koszul operad, eventually inhomogeneous. Let A be a homotopy P-algebra and let (H, d H ) be a homotopy retract of the chain complex (A, d A ).
The transferred homotopy P-algebra structure on H given by [GCTV09,Theorem 47] and by [LV10, Theorem 10.3.6] is equal to the transferred homotopy P-algebra structure on H given by Theorem 6.2.
Proof. The proof relies on the following diagram being commutative: The two homotopy transfer theorems for homotopy BV-algebras and skeletal homotopy BV-algebras commute under the functor P * : skeletal homotopy BV-algebras → homotopy BV-algebras as follows.
Proposition 6.4. Let (H, d H ) be a homotopy retract of a chain complex (A, d A ). Consider a skeletal homotopy BV-algebra structure on A. The associated homotopy BV-algebra structure P * (A) on A transfers to a homotopy BV-algebra to H by Theorem 33 of [GCTV09]. This homotopy BV-algebra structure on H is equal to the homotopy BV-algebra associated, under P , to the transferred skeletal BV-algebra given by Theorem 6.2.
Proof. The proof relies on the commutativity of the following diagram: Rectification Theorem for homotopy BV-algebras provides us with an ∞-quasi-isomorphism P * (H) ∼ Ω κ B ι P * (H). Finally, the two dg BV-algebras are linked by a zig-zag of quasi-isomorphism of dg BV-algebras by Theorem 11.4.14 of [LV10].
This theorem gives homotopy control of the transferred structure. It plays a key role in the interpretation of the main result in the next section.

FROM BV-ALGEBRAS TO HOMOTOPY FROBENIUS MANIFOLDS
We apply the Homotopy Transfer theorem to endow the underlying homology of a dg BV-algebra with Massey products. When the induced action of ∆ is trivial, we recover and extend up to homotopy the Barannikov-Kontsevich-Manin Frobenius manifold structure. Applications of this general result are given in Poisson geometry and Lie algebra cohomology and to the Mirror Symmetry conjecture. 7.1. Massey products. Working over a field K, one can always write the underlying homology (H • (A, d A ), 0) of a dg BV-algebra A as a deformation retract of (A, d A ).
Definition 7.1. We call Massey-Batalin-Vilkovisky products the operations composing the transferred skeletal homotopy BV-algebra structure on the homology H(A) of a dg BV-algebra given by the Homotopy Transfer Theorem 6.2.
Recall that the homology of any dg commutative (associative) algebra carries partial Massey products, see [Mas58]. For instance, the partial Massey triple-product x, y, z is defined for three homology classes x, y, z ∈ H(A) such that xy = 0 = yz as follows. Letx,ȳ,z ∈ A be cycles which represent x, y, and z respectively and let a, b ∈ A such thatxȳ = da,ȳz = db. For dg Lie algebras, partial Massey products were defined by V.S. Retakh in [Ret93]. The present Massey-Batalin-Vilkovisky products generalize both the partial commutative and Lie Massey products.
Theorem 6.5 shows that the data of the Massey products allow one to reconstruct the homotopy type of the initial dg BV-algebra. A mixed chain complex is a graded vector space A equipped with two anti-commuting square-zero operators d and ∆ of respective degree −1 and 1.
where d Hn = 0, ∆ Hn = 0, and where the maps of the following commutative diagram are isomorphisms A dg BV-algebra, which satisfies this condition, is called a Hodge dg BV-algebra by A. Losev and S. Shadrin in [LS07]. (In this case, the obvious homotopy h, which contracts A to its homology H, is such that [h, ∆] = h∆ + ∆h = 0.) Definition 7.6. [Par07] A mixed chain complex is called semi-classical if every homology class has a representative in the kernel of ∆. Proof. The first assertion is given by Lemma 7.5. To prove the second one, it is enough to write the homology H(A) as a deformation retract of A, with representatives in Ker ∆. In this case, ∆i = 0, which concludes the proof.
The existence of NC Hodge-to-de Rham degeneration data is therefore the most general condition that naturally supports this notion of the trivialization of the action of ∆ on the homology of a dg BV-algebra

Examples.
⋄ Let M be a compact Kähler manifold, with complex structure denoted by J. The space of differential forms (Ω • (M), d DR , ∆ := Jd DR J) forms a dg BV-algebra which satisfies the d∆-condition, see P. Deligne, P. Griffiths, J. Morgan and D. Sullivan [DGMS75]. (Notice that here the operator ∆ has order less than 1). ⋄ Let M be a Calabi-Yau manifold. The Dolbeault complex of anti-holomorphic differential forms with coefficients into holomorphic polyvector fields (Γ(M, ∧ •T * M ⊗ ∧ • T M ), d :=∂, ∧, ∆ := div, , S ) is a dg BV-algebra satisfying the d∆-condition, see S. Barannikov and M. Kontsevich [BK98]. This is an extension, from vector fields to polyvector fields, of the Kodaira-Spencer dg Lie algebra [KS58,KS60], which encodes the complex structures of a manifold. ⋄ Let (M, w) be a Poisson manifold. The space of differential forms (Ω • (M), d DR , ∧, ∆ := [i w , d DR ]) form a dg BV-algebra, see [Kos85,Bry88]. When (M, ω) is a compact symplectic manifold of dimension n, O. Mathieu proved in [Mat95] that M satisfies the hard Lefschetz condition, i.e. the cup product [ω k ] : H n−k (M ) → H n+k (M ) is an isomorphism, for k ≤ n/2, if and only if this dg BV-algebra is semi-classical. S. Merkulov further proved that this is equivalent to the d∆-condition in [Mer98]. This is the case when M is a Kähler manifold, see [Bry88]. ⋄ Let V be finite dimensional vector space with basis {v i } 1≤i≤n . We consider the free commutative algebra A := S(V ⊕ s −1 V * ) of functions on the cotangent bundle of V * , equipped with the order 2 and degree 1 operator ∆ := . These data define the prototypical example of BValgebras, see [BV81]. Any element w of degree −2 such that ∆(w) = w, w = 0 gives rise to a dg BV-algebra (A, d w := w, − , •, ∆, , ). One can find dg BV-algebras of this type equipped with NC Hodge to de Rham degeneration data but which does not satisfy the d∆-condition, see [Par07, Example 9] and [Ter08, Section 3.2].
The underlying homology groups H(A, d) carry a homotopy hypercommutative algebra structure, which extends the hypercommutative algebras of M. Kontsevich and S. Barannikov [BK98], Y.I. Manin [Man99], A. Losev and S. Shadrin [LS07], and J.-S. Park [Par07], and such that the rectified dg BV-algebra Rec(H(A)) is homotopy equivalent to A in the category of dg BV-algebras.
Proof. The transferred skeletal homotopy BV-algebra structure on homology given by Theorem 6.2 forms a homotopy hypercommutative algebra by Proposition 7.2.
We make explicit the various constructions of [BK98] as follows. When a dg BV-algebra satisfies the d∆-condition, there is a zig-zag of quasi-isomorphisms of dg Lie algebras (smooth formality) Tracing through the aforementioned constructions, one can see that the associated potential is given by the same kind of sums of labelled trees but with a normalizing coefficient given by the number of automorphisms of the trees. We recover the explicit formula of [LS07]. Manin [Man99] and Park [Par07] use obstruction theory, for which choices can be made to produce the above structure.
The first stratum of operations composing the transferred homotopy hypercommutative algebra is equal to the tree formulae of Losev-Shadrin as follows. Lemma 4.6 shows that the weight 1 part of Grav * is isomorphic to HLie ¡ 1 . For any n ≥ 2, the space Lie ¡ 1 (n) is one dimension and generated by the element, which in T c (β) is the sum of all binary tree with vertices labeled by β. The image of such trees under the formula of Theorem 6.2 is made up of binary trees with each vertex labelled by •, one leaf labelled by ∆, and with edges labelled by h. (One can see that the image of a maximal nesting under the map Φ is given by labeling all interior edges by h.) Under the d∆-condition, the relations p∆ = ∆i = h∆ + ∆h = ∆ 2 = 0 make many trees cancel and this produces the aforementioned Losev-Shadrin formulae.
The last assertion is a direct corollary of Theorem 6.5.

Remarks.
⋄ First, this theorem conceptually explains the result of Barannikov-Kontsevich, Manin, Losev-Shadrin, and Park in terms of the homotopy transfer theorem, thereby answering a question asked by the referee of [Par07, Section 5]. ⋄ Since there is no differential on homology, the first stratum of operations of this homotopy hypercommutative algebra satisfies the relations of an hypercommutative algebra. So Theorem 7.8 proves the existence of such a structure under a weaker condition (NC Hodge-to-de Rham degeneration data) than in [BK98,Man99,Par07] (d∆-condition, semiclassical). ⋄ Unlike the framework of Frobenius manifolds, we do not work here with cyclic unital BV-algebras.
First, a cyclic BV-algebra is equipped with a non-degenerate bilinear form which forces its dimension to be finite. The present method works in the infinite dimensional case. Then, the operad which encodes BV-algebras with unit is not augmented, so it does not admit a minimal model. To make a cofibrant replacement explicit, one would need to use the more general Koszul duality theory developed by J. Hirsh and J. Millès in [HM10]. ⋄ Finally, Theorem 7.8 provides higher structure on homology, which is shown to be necessary to recover the homotopy type of the original dg BV-algebra and not to lose any homotopy data when passing to homology, see also Example 7.4 below.
In geometrical terms, we have lifted the action of the Deligne-Mumford-Knudsen moduli space of genus 0 curves to an action of the open moduli space of genus 0 curves as follows. The map f is the morphism of operads given by [BK98,Man99,LS07,Par07]. The map κ is the twisting Koszul morphism from the cooperad H •+1 (M 0,n+1 ) given in [Get95]. It sends the cohomological class corresponding to H 0 (M 0,n+1 ) to the fundamental class of M 0,n+1 . The construction given in Theorem 7.8 corresponds to the map α, which is a twisting morphism from the cooperad H •+1 (M 0,n+1 ). The map κ vanishes outside the top dimensional classes and the restriction of the map α to these top dimensional classes is equal to the composite f • κ. Such a morphism of operads f defines the genus zero part of what Kontsevich-Manin call a Cohomological Field Theory in [KM94].
(The product by u and by v is equal to zero.) The differential map is defined on the generators by dz := xy, dv := u , and by 0 otherwise.
The algebra A is finite dimensional and spanned by the 9 elements: x, y, xy, z, u, v, xz, yz, xyz. Its underlying homology H • (A, d) is five dimensional and spanned by the classes of: x, y, xz, yz, xyz.
We define the degree +1 operator ∆ on the aforementioned elements by Proof. It is straightforward to see that ∆ commutes with d, that it has order less than 2 (but not less than 1) and that it squares to 0. A decomposition such as the one of Lemma 7.5 is given by Therefore, this dg BV-algebra satisfies the d∆-condition.
The first Massey product in the second stratum of the transferred homotopy hypercommutative algebra structure is the first homotopy in the associated C ∞ -algebra structure, since S −1 Grav * (2) (3) ∼ = Com ¡ (3). In the present example, this product is not trivial since it is equal to −yz on the elements x, y, y. So this provides an example of a dg BV-algebra, which satisfies the d∆-condition, the strongest condition, and for which the Barannikov-Kontsevich-Manin structure of a Frobenius manifold on homology is not enough to recover the original homotopy type of the dg BV-algebra. 7.5. Application to Poisson geometry and Lie algebra cohomology. Let M be an n-dimensional manifold. We consider the Gerstenhaber algebra of polyvector fields A := Γ(M, Λ • T M ) on M, equipped with the Schouten-Nijenhuis bracket , SN . Recall from J.-L. Koszul [Kos85,Proposition (2.3)] that any torsion-free connection ∇ on T M which induces a flat connection on Λ n T M gives rise to a square-zero order 2 operator D ∇ making (A, ∧, D ∇ , , SN ) into a BV-algebra. For instance, this is the case when M is orientable with volume form Ω or when M is a Riemannian manifold with the Levi-Civita connection.
For instance, this is the case when M is orientable with unimodular Poisson stucture, i.e. D Ω (w) = 0. The homology groups associated to the differential d w form the Poisson cohomology of the manifold M, see [Lic77]. (For similar constructions in non-commutative geometry, we refer the reader to [GS10]). Recall that the homology groups associated to the differential ∆ on the left-hand side form the Poisson homology of the manifold M. The Poisson homology and cohomology are proved to be isomorphic under the weaker condition that the Poisson manifold is orientable and unimodular, see P. Xu in [Xu99].
Theorem 7.12. The de Rham cohomology of a Poisson manifold M carries a skeletal homotopy BValgebra, whose rectified dg BV-algebra is homotopy equivalent to the dg BV-algebra (Ω • (M), d DR , ∧, ∆). The Poisson cohomology of an orientable Poisson manifold M carries a skeletal homotopy BV-algebra, whose rectified dg BV-algebra is homotopy equivalent to the dg BV-algebra (Γ(M, Λ • T M ), d w , ∧, ∆, , SN ).
The de Rham cohomology and the Poisson cohomology of a symplectic manifold are isomorphic skeletal homotopy BV-algebras. When the manifold M is compact and satisfies the hard Lefsechtz condition, this isomorphism reduces to an isomorphism of homotopy hypercommutative algebras.
Proof. This is a direct corollary of Theorem 7.8 and Proposition 7.11.
Let us now describe the linear case. Under the same notations as in the last example of Section 7.2, when V = g * is the linear dual of a finite dimensional Lie algebra, the transpose of the bracket produces a degree −2 element w in g ⊗ Λ 2 g * satisfying w, w = 0, by the Jacobi relation. In this case, the twisted differential d w is equal to the Chevalley-Eilenberg differential on A ∼ = S(g) ⊗ Λ(g * ) ⊂ C ∞ (g * ) ⊗ Λ(g * ), which computes the cohomology of g with coefficients in S(g) and the adjoint action. If the Lie algebra g is unimodular, that is Tr( x, − ) = 0, for any x ∈ g, then ∆(w) = 0 and the Chevalley-Eilenberg complex (S(g) ⊗ Λ(g * ), d w , •, ∆, , ) is a dg BV-algebra.
Remark. It would be now interesting to study the relationship with the Duflo isomorphism, the analogue of the space of differential forms, and the symplectic and the hard Lefschetz condition, in this linear case. 7.6. Application to Mirror Symmetry.
Theorem 7.14. The Dolbeault cohomology of a Calabi-Yau manifold carries a homotopy hypercommutative algebra structure, which extends the hypercommutative algebra structure of [BK98] and whose rectified dg BV-algebra is homotopy equivalent to the Dolbeault complex (Γ(M, ∧ •T * M ⊗∧ • T M ),∂, ∧, div, , S ). The moduli space M of Maurer-Cartan elements associated to the Dolbeault complex is an extension of the moduli space M classical associated to the Kodaira-Spencer dg Lie subalgebra, which encodes deformations of complex structures. The notion of generalized complex geometry was introduced by N. Hitchin in [Hit03] and then developed by his students M. Gualtieri [Gua04] and G.R. Cavalcanti [Cav05] as a framework which encompasses both complex and symplectic geometries. In this sense, the moduli space M was shown by Gualtieri to correspond to deformations of generalized complex structures. Several versions of the d∆-condition were shown to hold in this setting, see [AG07,Cav07]. Finally the dg BV algebra structure of [Li05] allows us to apply the same argument which produces a version of Theorem 7.14 in the context of generalized complex geometry.
S. Barannikov generalized in [Bar02] the notions of periods and variations of Hodge structure from M classical to M . He showed, for instance, that the image of these generalized periods on H • (M, C) coincide with the Gromov-Witten invariants. This is based on the fact that the Dolbeault cohomology admits not one but a family of Frobenius manifold structures. This remark coincides with the present approach: there are many choices in the Homotopy Transfer theorem. Moreover, the various transferred structures are related by the group of ∞-isomorphisms, see [LV10,Theorem 10.3.15]. In the case of homotopy BV-algebras, this group should be related to the Givental group [Giv01a,Giv01b].
The Mirror Symmetry conjecture [Kon95] claims that the Fukaya . The results of the present paper show that it is actually enough to prove the existence of an ∞-quasiisomorphism of dg BV-algebras to get the aforementioned isomorphism on the cohomology level and to relate the two associated deformation functors.