aa r X i v : . [ m a t h . A T ] J un HOMOTOPY DERIVATIONS OF THE FRAMED LITTLE DISCS OPERADS
SIMON BRUN
Abstract.
We study the homotopy derivations of the framed little discs operads, which correspondto the homotopy derivations of the BV n operads. By extending a result by Willwacher about thehomotopy derivations of the e n operads we show that the homotopy derivations of the BV n operadsmay be described through the cohomology of a suitable graph complex. We will present an explicitquasi-isomorphic map. Contents
1. Introduction 12. Preliminaries and basic notation 22.1. General notation 22.2. BV operad and its Koszul dual operad 32.3. Convolution dg Lie algebra and deformation complex 42.4. Graph complexes 43. Homotopy derivations of the BV n operads 63.1. Homotopy derivations as deformation complex 63.2. A map of coalgebras 83.3. A quasi-isomporhism 103.4. Cohomology of the homotopy derivations of the BV n operads 133.5. Zeroth cohomology isomorphic to grt Introduction
The little n-discs operad D n is an operad in the category of topological spaces. Its arity k operations D n ( k ) correspond to the space of embeddings of k copies of the unit n-disc to itself, Q k D n → D n , insuch a way that the embedding maps are a composition of translations and dilations. The framed littlen-discs operad f D n allows additionally for rotations in the embeddings. Precisely, the n-discs operad D n has a left action of SO ( n ) by rotating the little n-discs around their center. Then, the framed littlen-discs operad f D n is a topological operad defined as semidirect product f D n = D n ⋊ SO ( n )with the operations in arity k given by f D n ( k ) = D n ( k ) ⋊ SO ( n ) k as described in Definition 2.1 of [SW03]. Its operadic composition γ fD n : ( D n ⋊ SO ( n ))( k ) × (( D n ⋊ SO ( n ))( n ) × · · · × ( D n ⋊ SO ( n ))( n k )) → ( D n ⋊ SO ( n ))( n + · · · + n k )is defined as γ fD n (( a, g ) , (( b , h ) , . . . , ( b k , h k ))) := ( γ D n ( a, ( g b , . . . , g k b k )) , g h , . . . , g k h k )where a, b , . . . , b k ∈ D n and g , h , . . . , h k ∈ SO ( n ).While the homology of the little n-discs operad corresponds to the Gerstenhaber operad e n H ( D n ) ∼ = e n , Mathematics Subject Classification. he homology of the framed little 2n-discs operad is described by the BV n operad, precisely, H ( f D n ) ∼ = BV n ⋊ H ( SO (2 n − , as stated in theorem 5.4 of [SW03].Batalin-Vilkovisky algebras, i.e. algebras over the BV operad, have their origin in Physics and weremathematically introduced by Getzler in [Get94].Kontsevich [Kon99] as well as Lambrechts and Volić [LV14] showed that the little n-discs operad D n isformal. Ševera [Š10] as well as Giansiracusa and Salvatore [GS10] extended this proof to the framed little2-discs operad f D , i.e. there exists a zig-zag of quasi-isopmorhisms of of dg operads between the operadof rational singular chains of the framed little 2-discs operad and its homology, which is isomorphic tothe BV operad: C • ( f D ) ˜ ← . ˜ → H • ( f D ) ∼ = BV , [GCTV12, Theorem 10].Gálvez-Carrillo, Tonks and Vallette [GCTV12] extended the theory about Koszul dual operads ofquadratic operads to inhomogeneous quadratic operads, as for example the BV n operad. We willdescribe the Koszul dual operad BV !2 n in section 2.2. The cobar construction of the Koszul dual cooperad BV n, ∞ := Ω BV ¡2 n provides quasi-free, but not minimal, resolution of the BV n operad as stated intheorem 6 of [GCTV12]: BV n, ∞ := Ω BV ¡2 n ˜ → BV n . Furthermore there is a quasi-isomorphism of dg operadsΩ BV ¡2 ˜ → C • ( f D )which lifts the resolution BV , ∞ ˜ → BV , [GCTV12, Theorem 11 and 13].Hence, we can deduce important applications for homotopy BV algebras, i.e. algebras over BV , ∞ ,as pointed out in corollary 12 and 14 of [GCTV12]: Any topological conformal field theory carries ahomotopy BV algebra structure. The same is true for the singular chain complex of the double loopspace of a topological space endowed with an action of the circle.In [Wil15] Willwacher discribes the homotopy derivations of the e n operads governing n-algebrasthrough the cohomology of Kontsevich’s graph complex GC n . Precisely, theorem 1.3 in [Wil15] statesthat H ( Der ( e n, ∞ )) ∼ = S + (cid:0) H ( GC ≥ n,conn )[ − n − ⊕ R [ − n − (cid:1) [ n + 1] . where GC ≥ n,conn denotes the connected graphs of Kontsevich’s graph complex with at least bivalentvertices.Expanding this result to the BV n operads, we show that the homotopy derivations of the BV n operads are quasi-isomorphic to the homology of a suitable graph complex. Precisely, in theorem 2 andequation 29 we show, that H ( Der ( BV n, ∞ )) ∼ = S + R [[ u ]] (cid:16)(cid:16) H (cid:16) GC ≥ n,conn (cid:17) [ − n − ⊕ R [ − n − (cid:17) [[ u ]] (cid:17) [2 n + 1] , where u is an even variable emerging in the Koszul dual operad BV !2 n , since BV n is an inhomogeneousquadratic operad. The differential in the considered graph complex has a first order contribution in u additional to the vertex splitting differential inherent to Kontsevich’s graph complex.Furthermore, we will present an explicit combinatorial map and prove that it is a quasi-isomorphism.In [Wil15] Willwacher proved that the zeroth cohomology of the homotopy derivations of the e operadis isomorphic to the Grothendieck-Teichmüller Lie algebra plus one class. As a corollary we extend thisfact to the cohomology of the homotopy derivations of the BV opeard in theorem 3: H ( Der ( BV , ∞ )) ∼ = grt := grt ⋊ R . Preliminaries and basic notation
General notation.
In this paper we always work over the ground field R . The degree of anelement x of a graded or differential graded (dg) vector space V will be denoted by | x | and the the r-folddesuspension by V [ r ]. For dg vector spaces we use cohomological convention, i.e. all differentials havedegree one. Furthermore, we will always use a lexicographic ordering of odd objects of graded vectorspaces, i.e. odd components of objects are ordered according to the appearance of the object in theformula from left to right. e denote the completed symmetric product space of a vector space V by S ( V ) = R ⊕ S + ( V ) = R ⊕ Y j ≥ (cid:0) V ⊗ j (cid:1) S j where the symmetric group S n acts by permutations of the factors.We will consider the tensor coalgebra S + ( V ) equipped with the deconcatenation coproduct ∆ : S + ( V ) → S + ( V ) ⊗ S + ( V ) given by(1) ∆( v · · · v n ) := n − X i =1 v · · · v i ⊗ v i +1 · · · v n . Concerning S -modules as well as operads we use the conventions from the textbook [LV12] by Lodayand Vallette. We denote the n -ary operations of an operad P by P ( n ). The operadic r -fold desuspension P{ r } is an operad with P{ r } ( n ) = P ( n ) ⊗ sgn ⊗ rn [( n − r ]where sgn n is the sign representation of the symmetric group S n .As defined in section 5.1.4 of [LV12] the composition of two S -modules M and N is given as(2) M ◦ N = M k ≥ M ( k ) ⊗ S k N ⊗ k . Furthermore, corollary 5.1.4 of [LV12] states that in arity n the composition is given by( M ◦ N )( n ) = M k ≥ M ( k ) ⊗ S k (cid:16)M Ind S n S i ×···× S i ( N ( i ) ⊗ · · · ⊗ N ( i k )) (cid:17) where the sum extends over all the nonegative k -tupels ( i , . . . , i k ) with i + · · · + i k = n . Hence, thespace ( M ◦ N )( n ) is spanned by the equivalence classes of the elements( µ ; ν , . . . ν k ; σ )where µ ∈ M ( k ) , ν ∈ N ( i ) , . . . , ν k ∈ N ( i k ) , σ ∈ Sh ( i , . . . , i k ), as described in section 5.1.7 of [LV12].Let P be a symmetric operad with composition map γ : P ◦ P → P constituted by the linear maps γ ( i , . . . , i k ) : P ( k ) ⊗ P ( i ) ⊗ · · · ⊗ P ( i k ) → P ( i + · · · + i k ) . Furthermore, let µ ∈ P ( m ) and ν ∈ P ( n ) be two operations. The partial composition ( µ, ν ) µ ◦ i ν ∈P ( m − n ) is defined by µ ◦ i ν := γ ( µ ; id, . . . , , id, ν, id, . . . , id ) . Additionally to the composition of S -modules we use ◦ also to denote the composition of maps. It willbe clear from the context, which notation is meant.2.2. BV operad and its Koszul dual operad.
In order to study the homotopy derivations of the BV n operad we will consider its cofibrant resolution, obtained by the cobar construction for the Koszuldual cooperad according to the Koszul duality theory for inhomogeneous quadratic operads introducedby Gálvez-Carrillo, Tonks and Vallette in [GCTV12]. Therefore, let us review the Kozul dual operad ofthe BV n operad, as described in the sections 13.7.4 and 7.8.7 of [LV12].Let P be an operad, we denote its Koszul dual cooperad by P ¡ and its Koszul dual operad by P ! .The B V n operad has a representation as an inhomogeneous quadratic operad with the three generators m = · ∧ · and c = [ · , · ] in arity two as well as ∆ in arity one. R m is a trivial representation of S in degree0, R c is a trivial representation of S in degree 2 n −
1, and R ∆ is a one dimensional graded vector spacein degree 2 n − S -module the B V n operad is isomorphic to B V n ∼ = e n ◦ T (∆) / (∆ ) ∼ = Com ◦ L ie {− n + 1 } ◦ T (∆) / (∆ )where ◦ refers to the composition of S -modules as defined in equation 2 and e n is the operad that governs e n -algebras.Since the BV n operad is an inhomogeneous quadratic operad its koszul dual operad is a qudraticoperad with a differential d BV BV !2 n = (cid:0) qBV !2 n , d BV (cid:1) here the quadratic operad is isomorphic as S -module to qBV !2 n ∼ = S ( u ) ◦ e n {− n } ∼ = S ( u ) ◦ C om {− n } ◦ L ie {− } with | u | = 2 n . Let u d L ∧ · · · ∧ L N denote a homogeneous element of BV !2 n with L i being a L ie word,then the BV differential reads as(3) d BV (cid:0) u d L ∧ · · · ∧ L N (cid:1) = X ≤ i The homotopy derivations of the BV n operad will be identified with a convolution dg Lie algebra. Concerning convolution dg Lie algebraswe refer to section 6.4 of [LV12].Let C be a coaugmented cooperad with C (1) one dimensional, C (0) = 0 and P an augmented operad,we denote the convolution dg Lie algebra byHom S ( C , P ) = Y N ≥ Hom S N ( C ( N ) , P ( N )) . Ω( C ) will denote the quasi free opearad obtained by the cobar construction. A homomorphism of dgoperads α : Ω( C ) → P determines a Maurer-Cartan element in the convolution dg Lie algebra, alsodenoted by α . We twist by this Maurer-Cartan element to obtain a Lie algebraDef(Ω( C ) α → P )and call it deformation complex of the map α .The convolution dg Lie algebra Hom S ( C , P ) is isomorphic to(4) Hom S ( C , P ) ∼ = C ∗ ˆ ⊗ S P := Y N ≥ C ∗ ( N ) ˆ ⊗ S N P ( N )where the completion of the tensor product is with respect to the cohomological filtration.Let f = f ⊗ f and g = g ⊗ g be two elements of the convolution dg Lie algebra in the form (4),then the pre-Lie bracket is given by { f, g } = { f ⊗ f , g ⊗ g } = X i ( − | f || g | ( f ◦ i g ) ⊗ ( f ◦ i g )and the corresponding Lie bracket reads as(5) [ f, g ] = { f, g } − ( − | f || g | { g, f } . Here, the composition ◦ i refers to the partial operadic composition of the operad C ∗ and P respectively.2.4. Graph complexes. We will consider different kinds of graph complexes. First we consider M.Kontsevich’s graph complex GC n as defined in section 3 of [Wil15]. Elements of this graded vectorspace are depicted as undirected graphs with one kind of unlabelled black vertices of valence greater orequal three and with no tadpols. Edges have an odd degree of 1 − n , i.e. one needs to choose an orderingof the edges. In order to obtain an element of GC n one has to sum over all possible ways of assigninglabels to the vertices and divide by the order of the symmetry group of the graph. The vertices of graphsin GC n have to be at least trivalent. If we also allow bivalent vertices we denote the correspondinggraph complex by GC ≥ n . Furthermore, we consider only graphs without tadpoles, loops with one vertex,as depicted in the following picture.We define the operation Γ • Γ , which means that we sum over all vertices of Γ , each time insert Γ in the chosen vertex of Γ and sum over all possible ways to reconnect the incident edges of the chosenvertex of Γ to one of the vertices of Γ .Let Γ be a graph in GC n . We introduce the following notation.(6) Γ e sum over all vertices of Γ and connect for each term the incident edge to the corresponding vertex. Ifthere are several incident edges, we sum over all possible ways to connect the incident edges to verticesof Γ. ΓThe following picture shows an example. • = 4The Lie bracket on GC n can then be described in the following way[Γ , Γ ] = Γ • Γ − ( − | Γ || Γ | Γ • Γ . Using this notation, the differential on the graph complex GC n is defined as(7) ˜ δ Γ = h , Γ i = • Γ − ( − | Γ | Γ • . Note that the differential ˜ δ does neither split nor connect two connected components, the graphcomplex can be written as completed symmetric product of connected graphs GC n = S + ( GC n,conn ) . The cohomological degree of a connected component corresponds to 2n ( Graphs n as described in definition 3.6 of [Wil15]. Generatorsof this graded vector space are depicted by graphs with two kinds of vertices, external vertices which arenumbered and depicted as empty (white) dots as well as internal vertices which are depicted as blackdots and are indistinguishable and thus unnumbered.We consider only graphs without tadpoles, a loop with one vertex, at internal and external nodes asdepicted in the following picture.Furthermore, we consider only graphs with all internal vertices at least trivalent and with no connectedcomponents consisting entirely of internal vertices. Edges have an odd degree of 1 − n , i.e. one needsto choose an ordering of the edges.An operation in arity N has N numbered external vertices. The following picture shows connectedgraphs with 3 and 2 external vertices respectively.Let Γ be a graph in Graphs n . We introduce again the following notation.(8) ΓThis time we sum over all internal and external vertices of Γ and connect for each term the incident edgeto the corresponding vertex.The partial operadic composition Γ ◦ i Γ corresponds to insert Γ at the external vertex i of Γ andsum over all possible ways to reconnect the incident edges of vertex i of Γ to one of the vertices (internalor external) of Γ . The following picture shows an example of a partial operadic composition. ◦ =Let Γ be an operation in arity N and Γ be an operation in arity N then the Lie bracket on Graphs n induced by the operadic composition is defined as[Γ , Γ ] = X i ∈ N Γ ◦ i Γ − ( − | Γ || Γ | X i ∈ N Γ ◦ i Γ . The differential on the graph complex Graphs n is defined in the following way(9) δ Γ = h , Γ i − ( − | Γ | 12 Γ • = ◦ Γ + ( − | Γ | X i ∈ int. vert. of Γ Γ ◦ i − ( − | Γ | 12 Γ • . Finally, we consider the operad BV Graphs n . It differs from the operad Graphs n by allowingtadpoles at external vertices and is isomorphic as S -module to BV Graphs n ∼ = Graphs n ◦ S (∆) / (∆ ) , where the generator ∆ in arity one has degree 1 − n and corresponds to a tadpole at an external vertex.The differential on BV Graphs n is the same as the one on Graphs n . Since it doesn’t alter tadpoles theisomorphism is one of dg S -modules(10) ( BV Graphs n , δ ) ∼ = ( Graphs n , δ ) ◦ S (∆) / (∆ ) . For further details to the considered graph complexes we refer the reader to [Wil15].Let us consider a graph Γ ∈ GC n with one external vertex attached to itΓ . The differential δ does not have any influence on the external vertex and its incident edge, i.e.(11) δ Γ = ˜ δ Γwhere ˜ δ refers to the differential on GC n as defined in equation 7.3. Homotopy derivations of the BV n operads Homotopy derivations as deformation complex. We identify the complex of homotopy deriva-tions of the BV n operad with the deformation complex(12) Der( BV n, ∞ ) := Y N ≥ Hom S N (cid:0) BV ¡2 n ( N ) , BV n, ∞ ( N ) (cid:1) [1] ∼ = Def( BV n, ∞ id → BV n, ∞ )[1]where BV n, ∞ = Ω( BV ¡2 n ) denotes the quasi-free resolution of the BV n operad obtained by the cobarconstruction of the Koszul dual cooperad BV ¡2 n as introduced by Gálvez-Carrillo, Tonks and Vallettein [GCTV12] and described in chapter 7.8.7 of [LV12]. Since the projection p : Ω( BV ¡2 n ) ։ BV n is aquasi-isomorphism, the deformation complex is quasi-isomorphic toDef( BV n, ∞ p → BV n )[1] . There is an injective map of operads i : BV n ֒ → BV Graphs n which maps the generating operations tographs in the following way · ∧ · 7→ [ · , · ] ∆ . (13)As proven by Kontsevich [Kon99] as well as Lambrechts and Volić [LV14] the restriction of this map to e n ֒ → Graphs n is a quasi-isomorphism, which is also stated in proposition 3.9 in [Wil15]. From thiswe deduce that the Cohomology of the BV Graphs n operad corresponds to the BV n operad due toequation 10 and the operadic Künneth formula H • ( BV Graphs n ) ∼ = H • ( Graphs n ) ◦ S (∆) / (∆ ) ∼ = e n ◦ S (∆) / (∆ ) ∼ = BV n . ence, the inclusion BV n ֒ → BV Graphs n is a quasi-isomorphism. Therefore, the considered deforma-tion complex is quasi-isomorphic to(14) Def( BV n, ∞ α → BV Graphs n )[1] , where α = i ◦ p : Ω( BV ¡2 n ) → BV Graphs n . Following the notation of section 2.3 this deformationcomplex can be written as Y N ≥ Hom S N (cid:0) BV ¡2 n ( N ) , BV Graphs n ( N ) (cid:1) [1] ∼ = Y N ≥ BV !2 n ( N ) ˆ ⊗ S N BV Graphs n ( N )where BV !2 n denotes the Koszul dual operad of the BV n operad as described in section 2.2. Finally, theconsidered deformation complex 12 is quasi-isomorphic to(15) Def := Y N ≥ e n {− n } ( N ) ˆ ⊗ S N BV Graphs n ( N ) [[ u ]] . Elements of this deformation complex can be visualised as graphs in the following way. We interprethomogeneous elements of the deformation complex as a direct sum of graphs with two kinds of edgesand additionally with a power series in u . External vertices are arranged on a horizontal dotted line.The BV Graphs part including internal vertices is depicted above the line using solid lines for edges.Furthermore, the map of operads e n {− n } ֒ → Graphs n which maps the generating operations asfollows · ∧ · 7→ [ · , · ] (16)allows to depict the e n part also as part of the graph using dashed edges connecting the external nodesbelow the horizontal line. Note that for example a nested double Lie [ · , [ · , · ]] bracket in the e n part willbe mapped to ⊕ . The following picture shows an example of an element of the deformation complex.(2 u + u ) ⊕ (2 u + u )We will now express the differential on the considered deformation complex 14 in the form of theconvolution dg Lie algebra 15 using the Lie bracket defined in equation 5. The differential consists ofdifferent parts. The first part of the differential originates from the Koszul dual BV !2 n of the BV n operadas described in equation 3. Let Γ denote an element of the deformation complex 14 in the form of theconvolution dg Lie algebra 15. The the differential d BV can be written as d BV Γ = u h , Γ i . In the same notation the contributions to the differential due to the twisting by the Maurer-Cartanelement α in the deformation complex 14 read as d ∧ Γ = h , Γ i d [ , ] Γ = h , Γ i d ∆ Γ = u h , Γ i . Finally, there is the contribution form the differential δ inherent to the operad BV Graphs n as definedin equation 9. Summarising, the total differential on the considered deformation complex 14 in the formof the convolution dg Lie algebra Def as defined in 15 is given by(17) d = δ + h , · i + h , · i + u h , · i + h , · i! . Hence, the homotopy derivations Der( BV n, ∞ ) of the BV n operad are quasi-isomorphic to the convolu-tion dg Lie algebra ( Def, d ) in the form of equation 15 and with differential 17. .2. A map of coalgebras. We can express the considered convolution dg Lie algebra Def as completedsymmetric product of its connected components(18) Def = S + (cid:0)(cid:0) e n {− n } ˆ ⊗ S BV Graphs n (cid:1) conn (cid:1) [[ u ]] , where connectedness refers to graphs connected via solid or dashed edges. Similarly the graph complex GC ≥ n splits into the completed symmetric product of its connected components GC ≥ n = S + (cid:16) GC ≥ n,conn (cid:17) . In the following we define a map F of the cocommutative coalgebras(19) S + (cid:16) GC ≥ n,conn ⊕ (cid:17) [[ u ]] −→ F S + (cid:0)(cid:0) e n {− n } ˆ ⊗ S BV Graphs n (cid:1) conn (cid:1) [[ u ]] . Here, the coalgebra structure corresponds to the tensor coalgebra equipped with the deconcatenationcoproduct defined in equation 1. For both coalgebras the space of cogenerators is composed of therespective connected graphs and will be denoted by(20) U := (cid:16) GC ≥ n,conn ⊕ (cid:17) V := (cid:0) e n {− n } ˆ ⊗ S BV Graphs n (cid:1) conn . The map F of cocommutative coalgebras is defined via its projections on the cogenerators F n : S n ( U ) → V. Let Γ ∈ GC ≥ n,conn , the map F : U → V, Γ ˆΓ corresponds to "adding a hair". In this context a hairconsists of one external vertex connected to the graph by a solid edge . The image of the map F equalsthe sum of graphs with one hair obtained by adding a hair to Γ in all possible ways, i.e. at every vertex.Using the notation of equation 6 we can summarise the map F as(21) Γ ˆΓ = Γ . Let Γ , Γ ∈ V be connected graphs with one and two external vertices respectively. Furthermore letΓ , Γ ∈ V be two connected graphs. We consider the compositionΓ ◦ Γ which is defined by inserting Γ into the external vertex of Γ and sum over all possible ways to connectthe incident edges of Γ to the vertices of Γ . Similarly the compositionΓ ◦ (Γ , Γ )is defined by inserting Γ into the first and Γ into the second vertex of Γ and sum over all possibleways to connect the incident edges of Γ to vertices of Γ and Γ respectively.We will use the following notation. For connected graphs Γ i ∈ U we abbreviateΓ . . . Γ n := 1 n ! X σ ∈ S n Γ σ (1) ⊗ · · · ⊗ Γ σ ( n ) ∈ S n ( U ) . Furthermore, let Γ i ∈ V for i = 0 be connected graphs with one external vertex and let Γ ∈ S n ( V ) forsome n ≥ 1, then we denote a symmetrised stack of compositions byΓ ( . . . (Γ n (Γ ))) := X σ ∈ S n Γ σ (1) ◦ (Γ σ (2) ◦ ( · · · ◦ (Γ σ ( n ) ◦ Γ ))) . Since Γ . . . Γ n − = 0vanishes due to the odd symmetry of interchanging the first two graphs we only need to consider at mostone graph . sing the introduced notation as well as the notation for connecting an edge to a graph as stated inequation 8 the projection F n is defined as F n (Γ . . . Γ n ) := u n − ˆΓ ( . . . (ˆΓ n ( ))) = u n − F n ( Γ . . . Γ n − ) := u n − ˆΓ ( . . . (ˆΓ n − ( ))) + u ˆΓ ( . . . (ˆΓ n − ( ))) ! = u n − + u n where ˆΓ denotes the image of Γ under the map F . Note that this notation includes an implicit summationover all possible ways to add a hair to a vertex of Γ as defined in equation 21 for the map F .In order to prove the following lemma we will need some properties of the proposed map F n . Firstnotice that Γ . This follows from the fact that we have to sum over alle possible ways to connect edges 1 and 2 to avertex to both the graph above, depicted as solid circle, as well as the graph below, depicted as dashedcircle. Therefore, there is an odd symmetry of interchanging the two edges 1 and 2. Hence, the graphvanishes.From this follows(22) h , ˆΓ ( . . . (ˆΓ n (Γ ))) i = n X j =1 ˆΓ ( . . . ( h , ˆΓ j i ( . . . (ˆΓ n (Γ ))))) + ˆΓ ( . . . (ˆΓ n ( h , ˆΓ i )))which is equivalent to Γ = X j Γ j + Γ . Furthermore, we have X p,q ; p + q = n X τ ∈ sh ( p,q ) ˆΓ τ (1) ( . . . (ˆΓ τ ( p ) (Γ ′ ))) ⊗ ˆΓ τ ( p +1) ( . . . (ˆΓ τ ( n ) (Γ ′′ ))) = ˆΓ ( . . . (ˆΓ n (Γ ′ ⊗ Γ ′′ ))) . rom the last two properties we deduce that(23) X p,q ; p + q = n X τ ∈ sh ( p,q ) ◦ (cid:16) ˆΓ τ (1) ( . . . (ˆΓ τ ( p ) (Γ ′ ))) , ˆΓ τ ( p +1) ( . . . (ˆΓ τ ( n ) (Γ ′′ ))) (cid:17) = ˆΓ ( . . . (ˆΓ n ( Γ ′ Γ ′′ )))which corresponds to X p,q ; p + q = n X τ ∈ sh ( p,q ) Γ ′ Γ ′′ = Γ ′ Γ ′′ . Finally we verify that(24) δ (ˆΓ ( . . . (ˆΓ n (Γ )))) = n X j =1 ˆΓ ( . . . ( d ˜ δ Γ j ( . . . (ˆΓ n (Γ ))))) + ˆΓ ( . . . (ˆΓ n ( δ Γ )))where the differential ˜ δ on the graph complex GC n is defined in equation 7 and the differential δ on thegraph complex Graphs n is described in equation 9. Indeed we have h , ˆΓ ( . . . (ˆΓ n (Γ ))) i = n X j =1 ˆΓ ( . . . ( h , ˆΓ j i ( . . . (ˆΓ n (Γ ))))) + ˆΓ ( . . . (ˆΓ n ( h , Γ i )))as well as (ˆΓ ( . . . (ˆΓ n (Γ )))) • = n X j =1 ˆΓ ( . . . (ˆΓ j • ( . . . (ˆΓ n (Γ ))))) + ˆΓ ( . . . (ˆΓ n (Γ • )))and δ ˆΓ = c ˜ δ Γ . The last identity follows from equation 11.3.3. A quasi-isomporhism. Let us consider the graph complex (cid:16) S + (cid:16) GC ≥ n,conn ⊕ (cid:17) [[ u ]] , ˜ d (cid:17) where the differential ˜ d is given by(25) ˜ d Γ := ˜ δ + u h , Γ i and ˜ δ refers to the differential on GC n as defined in equation 7. The differential ˜ d acts as a coderivationon the deconcatenation copruduct.Let U and V denote the cogenerators of the respective coalgebras as defined in equation 20. We showthe following lemma: Lemma 1. The map F of coalgebras (cid:0) S + ( U ) [[ u ]] , ˜ d (cid:1) −→ F (cid:0) S + ( V ) [[ u ]] , d (cid:1) is a map of complexes, i.e. F ◦ ˜ d = d ◦ F .Proof. Let π : S + ( V ) → V denote the projection onto the cogenerators. The projection of the differential d , defined in equation 17, splits in two parts π ◦ d = d + d with d : V → Vd : S ( V ) → V. hereby the part d is given by(26) d (Γ ⊗ Γ ) = u ( ◦ (Γ , Γ ) + ◦ (Γ , Γ )) = u Γ Γ + Γ Γ ! Projecting the relation F ◦ ˜ d = d ◦ F to the cogenerators V and restricting to S n ( U ) leads to the following two relations, which are to beshown:(27) n X j =1 F n (Γ . . . ˜ d Γ j . . . Γ n ) = d F n (Γ . . . Γ n )+ X p,q ; p + q = n ; p,q =0 X τ ∈ sh ( p,q ) d ( F p (Γ τ (1) . . . Γ τ ( p ) ) ⊗ F q (Γ τ ( p +1) . . . Γ τ ( n ) ))(28) n − X j =1 F n ( Γ . . . ˜ d Γ j . . . Γ n − ) = d F n ( Γ . . . Γ n − )+ 2 X p,q ; p + q = n − q =0 X τ ∈ sh ( p,q ) d ( F p +1 ( Γ τ (1) . . . Γ τ ( p ) ) ⊗ F q (Γ τ ( p +1) . . . Γ τ ( n − ))Let us first prove equation 27. The term d F n (Γ . . . Γ n ) = A + B + C + D + E has the following contributions A := δF n (Γ . . . Γ n ) = n X j =1 F n (Γ . . . ˜ δ Γ j . . . Γ n ) B := u n − h , ˆΓ ( . . . (ˆΓ n ( ))) i = u n − (cid:16) ◦ (ˆΓ ( . . . (ˆΓ n ( )))) − ˆΓ ( . . . (ˆΓ n ( ))) (cid:17) C := u n − h , ˆΓ ( . . . (ˆΓ n ( ))) i = u n − (cid:16) ◦ (ˆΓ ( . . . (ˆΓ n ( )))) − ˆΓ ( . . . (ˆΓ n ( ))) (cid:17) D := u n h , ˆΓ ( . . . (ˆΓ n ( ))) i = u n n X j =1 ˆΓ ( . . . ( h , ˆΓ j i ( . . . (ˆΓ n ( ))))) E = u n h , ˆΓ ( . . . (ˆΓ n ( ))) i = 0 . In the calculation for A we used equation 24, whereas for D equation 22 was applied. The terms A + D are equal to the left hand side of equation 27: n X j =1 F n (Γ . . . ˜ d Γ j . . . Γ n ) = A + D. According to equation 26 the second term on the right hand side X p,q ; p + q = n ; p,q =0 X τ ∈ sh ( p,q ) d ( F p (Γ τ (1) . . . Γ τ ( p ) ) ⊗ F q (Γ τ ( p +1) . . . Γ τ ( n ) )) = F + G has the contributions F := u n − X p,q ; p + q = n ; p,q =0 X τ ∈ sh ( p,q ) (cid:16) ˆΓ τ (1) ( . . . (ˆΓ τ ( p ) )) , ˆΓ τ ( p +1) ( . . . (ˆΓ τ ( n ) )) (cid:17) = − BG := u n − X p,q ; p + q = n ; p,q =0 X τ ∈ sh ( p,q ) (cid:16) ˆΓ τ (1) ( . . . (ˆΓ τ ( p ) )) , ˆΓ τ ( p +1) ( . . . (ˆΓ τ ( n ) )) (cid:17) = − C which cancel the terms B + C due to equation 23.Let us now verify equation 28. The term d F n ( Γ . . . Γ n − ) = AA + BB + CC + DD + EE as the following contributions AA := δF n ( Γ . . . Γ n − ) = n − X j =1 F n ( Γ . . . ˜ δ Γ j . . . Γ n − ) BB := u n − h , ˆΓ ( . . . (ˆΓ n − ( ))) + u ˆΓ ( . . . (ˆΓ n − ( ))) i = u n − (cid:16) ◦ (ˆΓ ( . . . (ˆΓ n − ( )))) − ( . . . (ˆΓ n − ( ))) (cid:17) + u n (cid:16) ◦ (ˆΓ ( . . . (ˆΓ n − ( )))) − ( . . . (ˆΓ n − ( ))) (cid:17) CC := u n − h , ˆΓ ( . . . (ˆΓ n − ( ))) + u ˆΓ ( . . . (ˆΓ n − ( ))) i = u n − (cid:16) ◦ (ˆΓ ( . . . (ˆΓ n − ( )))) − ( . . . (ˆΓ n − ( ))) (cid:17) + u n (cid:16) ◦ (ˆΓ ( . . . (ˆΓ n − ( )))) − ( . . . (ˆΓ n − ( ))) − ( . . . (ˆΓ n − ( ))) (cid:17) DD : = u n h , ˆΓ ( . . . (ˆΓ n − ( ))) + u ˆΓ ( . . . (ˆΓ n − ( ))) i = u n n − X j =1 (cid:16) ˆΓ ( . . . ( h , ˆΓ j i ( . . . (ˆΓ n − ( ))))) + ˆΓ ( . . . ( h , ˆΓ j i ( . . . (ˆΓ n − ( ))))) (cid:17) EE : = u n h , ˆΓ ( . . . (ˆΓ n − ( ))) + u ˆΓ ( . . . (ˆΓ n − ( ))) i = 2 u n ˆΓ ( . . . (ˆΓ n − ( ))) . Again in the calculation for AA we used equation 24, whereas for DD equation 22 was applied. Theterms AA + DD cancel again the left hand side of equation 28: n − X j =1 F n ( Γ . . . ˜ d Γ j . . . Γ n − ) = AA + DD. Following equation 26 the second term on the right hand side2 X p,q ; p + q = n − q =0 X τ ∈ sh ( p,q ) d ( F p +1 ( Γ τ (1) . . . Γ τ ( p ) ) ⊗ F q (Γ τ ( p +1) . . . Γ τ ( n − )) = F F + GG has the following contributions F F : = 2 u n − X p,q ; p + q = n − q =0 X τ ∈ sh ( p,q ) (cid:16) ˆΓ τ (1) ( . . . (ˆΓ τ ( p ) ( ))) + u ˆΓ τ (1) ( . . . (ˆΓ τ ( p ) ( ))) , ˆΓ τ ( p +1) ( . . . (ˆΓ τ ( n − ( ))) (cid:17) = − BBGG : = 2 u n − X p,q ; p + q = n − q =0 X τ ∈ sh ( p,q ) (cid:16) ˆΓ τ (1) ( . . . (ˆΓ τ ( p ) ( ))) + u ˆΓ τ (1) ( . . . (ˆΓ τ ( p ) ( ))) , ˆΓ τ ( p +1) ( . . . (ˆΓ τ ( n − ( ))) (cid:17) = − ( CC + EE )which cancel the terms BB + CC + EE due to equation 23. (cid:3) Using the fact that the map F is a map of complexes we can prove the main result of this paper: Theorem 2. The map (cid:16) S + (cid:16) GC ≥ n,conn ⊕ (cid:17) [[ u ]] , ˜ d (cid:17) −→ F ( Def, d ) is a quasi-isomorphism. ere, Def refers to the considered deformation complex in the form of equation 14 and 18 and d tothe total differential defined in equation 17. Finally, ˜ d is defined in equation 25. Proof. By lemma 1 F is a map of complexes. Let us consider the bounded above complete descendingfiltration with respect to the power p of u . The map F is compatible with the filtration and the restrictionto the associated graded vector spaces (cid:16) u p S + (cid:16) GC ≥ n,conn ⊕ (cid:17) , ˜ δ (cid:17) → (cid:16) gr p Def, δ + h , · i + h , · i(cid:17) is a quasi-isomorphism by theorem 1.3 in [Wil15]. Hence, the map F is a quasi-isomorphism, too. (cid:3) Cohomology of the homotopy derivations of the BV n operads. From theorem 2 we candeduce that the cohomology of the homotopy derivations of the BV n operads are given by H ( Der ( BV n, ∞ )) = H (cid:16) S + (cid:16) GC ≥ n,conn ⊕ (cid:17) [[ u ]] , ˜ d (cid:17) . Let us consider the tensor product over the ring R [[ u ]]. We will denote the completed symmetricproduct space of a vector space V with respect to the tensor product over R [[ u ]] by S + R [[ u ]] ( V ) . Note that the differential ˜ d as defined in equation 25 acts as coderivation on the coproduct of connectedgraphs. It does neither split connected graphs nor does it connect two connected components. Hence,we can apply the Künneth formula as well as the fact that taking comohology interchanges with takinginvariants with respect to the symmetric group S n . Therefore, we can pull the cohomology inside thecompleted symmetric product. Since wie consider the completed symmetric product with respect to R [[ u ]] also the decoration with power series in u can be interchanged with the product. Finally notethat the extra class is exact due to the odd symmetry of interchanging two edges but not closedunder the differential ˜ d . Summarised we can write the homotopy derivations of the BV n operads in thefollowing form(29) H ( Der ( BV n, ∞ )) = H (cid:16) S + (cid:16) GC ≥ n,conn ⊕ (cid:17) [[ u ]] , ˜ d (cid:17) = S + R [[ u ]] (cid:16)(cid:16) H (cid:16) GC ≥ n,conn , ˜ d (cid:17) ⊕ (cid:17) [[ u ]] (cid:17) . Zeroth cohomology isomorphic to grt . As a corollary of theorem 2 we can extend theorem1.2 in [Wil15] by Willwacher and deduce that the cohomology of the homotopy derivations of the BV operad is isomorphic to the Grothendieck-Teichmüller Lie algebra plus one class. Theorem 3. H ( Der ( BV , ∞ )) ∼ = grt := grt ⋊ R where R acts on grt by multiplication with the degree with respect to the grading on grt .Proof. Willwacher proved that the zeroth cohomology of the graph complex GC , considered as Liealgebra, is isomorphic to the Grothendieck-Teichmüller Lie algebra H ( GC ,conn ) ∼ = grt , [Wil15, Theorem 1.1], and deduced that H ( Der ( e , ∞ )) ∼ = grt , [Wil15, Theorem 1.2].Furthermore, Merkulov and Willwacher showed in [MW14] that H (cid:0) GC ,conn [[ u ]] , ˜ d (cid:1) ∼ = grt where the differential ˜ d is defined in equation 25.By proposition 3.4 in [Wil15] we have H (cid:16) GC ≥ ,conn (cid:17) = H ( GC ,conn ) ⊕ M j ≥ j ≡ R [2 − j ] . ere, the class R [2 n − j ] is represented by a loop with j edges. The differential ˜ d does neither split norglue different connected components and oops are exact but not closed under it. Hence we also have H (cid:16) GC ≥ ,conn , ˜ d (cid:17) = H (cid:0) GC ,conn , ˜ d (cid:1) ⊕ M j ≥ j ≡ R [2 − j ] . Note that the cohomological degree of a connected component corresponds to 2 ( u is an even variable with degree 2.The theorem follows form equation 3 if we consider the cohomology in degree 0. (cid:3) Acknowledgement The Author would like to thank Prof. T. Willwacher for proposing the problem question, giving inputsand ideas as well as for his kind advice and support. Furthermore, the author was partially supportedby the ERC grant GRAPHCPX (678156), awarded to Prof. T. Willwacher. References [GCTV12] Imma Gálvez-Carrillo, Andrew Tonks, and Bruno Vallette. “Homotopy Batalin-Vilkovisky al-gebras”. In: J. Noncommut. Geom. issn : 1661-6952. doi : . url : https://doi.org/10.4171/JNCG/99 .[Get94] E. 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