Formality of the homotopy calculus algebra of Hochschild (co)chains

F ormalit y of the homo top y calculus algebra of Ho c hsc hild (co)chains V asili y Dolgushev, Dmit ry T a mark i n, and Bo ris T sygan T o Mikhail Olshanetsky on the o c c asio n o f his 70th birthday. Abstract The Kon tsevic h-Soib elman solution of the cyclic v ersion of Deligne’s conjecture and the formalit y of th e op er ad of little d iscs on a cylinder pro vide us with a natural homo- top y calculus structure on th e pair ( C • ( A ) , C • ( A )) “Hochsc hild co c hains + Ho c hschild c hains” of an associativ e algebra A . W e sho w that for an arbitrary smo oth alge braic v ariety X ov er a ﬁeld K of c haracteristic zero the sheaf ( C • ( O X ) , C • ( O X )) of homotop y calculi is formal. Th is result w as announced in paper [29] by the second and the th ird author. Con ten ts 1 In tro duction 2 2 Preliminaries 3 2.1 (Co)op erads and (co)algebras . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Ho c hsc hild (co)c hain complexes . . . . . . . . . . . . . . . . . . . . . . . . . 9 3 The op erads Ho ( calc ) , Ho ( e 2 ) , and Ho ( Lie + δ ) 11 3.1 Description of the op erads Ho ( calc ) and Ho ( e 2 ) . . . . . . . . . . . . . . . 11 3.2 Description of the op erad Ho ( Lie + δ ) . . . . . . . . . . . . . . . . . . . . . . . 15 4 The Kon tsevic h-Soibelman op erad and the op erad of little discs on a cylin- der 19 4.1 The Ko n tsevic h-Soib elman op era d KS . . . . . . . . . . . . . . . . . . . . . 19 4.2 The o p erad of litt le discs on a cylinder . . . . . . . . . . . . . . . . . . . . . 23 4.3 Required results from [23] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 4.4 A useful prop erty of the op erad KS . . . . . . . . . . . . . . . . . . . . . . . 34 5 The homotop y calculus on the pair ( C • norm ( A ) , C norm • ( A )) . 37 6 F ormalit y theorem 46 6.1 En v eloping algebra o f a G erstenhab er a lg ebra . . . . . . . . . . . . . . . . . 46 6.2 Shea v es of Ho c hsc hild (co)c hains o n an algebraic v ariet y . . . . . . . . . . . 48 6.3 Morita equiv alence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 6.4 Pro of of Theorem 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 1 7 Applications and generalizations 62 1 In tro duction The standar d Carta n calculus on p olyv ector ﬁelds and exterior forms can b e natura lly ex- tended to the Ho c hsc hild cohomolog y H H • ( A, A ) a nd the Ho chsc hild homology H H • ( A, A ) of an arbitrary asso ciativ e a lg ebra A [11], [24]. This calculus is induced b y simple op era- tions on Ho chs c hild (co)c hains, and the iden tities of this a lgebraic structure ho ld fo r these op erations up to homotop y . The Kon tsevic h-Soib elman pro of of the cyclic v ersion of Deligne’s conj ecture [23] and the formalit y of the op erad of little discs on a cylinder 1 imply that this nice collection of the op erations on the pair ( C • norm ( A ) , C norm • ( A )) “(normalized) Ho c hsc hild co c hains + (normalized) Ho chs c hild c hains” can b e extende d to an ∞ - or homotopy calcu lus structure. This homot o p y calculus structure on the pair ( C • norm ( A ) , C norm • ( A )) is a natural general- ization of the homotop y Gerstenhab er algebra structure on the co chains C • norm ( A ) . In pap er [13] w e prov ed the formality of t his homotopy Gerstenhab er algebra on C • norm ( A ) fo r an ar- bitrary regular comm utativ e algebra A o v er a ﬁeld K of c haracteristic zero. In this pap er w e extend this result to the homot o p y calculus algebra on the pair ( C • norm ( A ) , C norm • ( A )) . As w ell as in [13] w e also consider the situation when the algebra A is replaced by the structure sheaf O X of a smo oth algebraic v ariety X o v er the ﬁeld K . More precisely , w e con- sider the homotop y calculus alg ebra on the pair ( C • norm ( O X ) , C norm • ( O X )) where C • norm ( O X ) and C norm • ( O X ) is, resp ectiv ely , the sheaf of (normalized) Ho c hsc hild co c hains and the sheaf of (normalized) Ho c hsc hild c hains of O X . In this pap er we show that the sheaf of homotop y calculi ( C • norm ( O X ) , C norm • ( O X )) is formal. If A is an asso ciativ e algebra (with unit), the pair ( C • norm ( A ) , C norm • ( A )) is also equipp ed with an algebraic structure deﬁned by a degree − 1 Lie brack et on C • norm ( A ) , a degree − 1 Lie mo dule structure on C norm • ( A ) o v er C • norm ( A ) , and Connes’ op erator on C norm • ( A ) whic h is compatible with the Lie mo dule structure. In the pap er we refer to suc h algebra structures as Λ Lie + δ -algebra. (See Deﬁnition 4.) In pap er [31] the t hir d author conjectured that if A is a regular comm utative algebra then this Λ Lie + δ -algebra structure on ( C • norm ( A ) , C norm • ( A )) is formal. This conjecture w as pro v ed in [33] (a t least in the case R ⊂ K ) b y Willw ac her who used the constructions of B. Shoikhet [25] and the ﬁrst author [12]. In general H o (Λ Lie + δ )-part of the homotop y calculus structure o n ( C • norm ( A ) , C norm • ( A )) deriv ed from [23] may not coincide with the Λ Lie + δ -algebra on the pair ( C • norm ( A ) , C norm • ( A )) . Ho w ev er, w e sho w that this homotop y calculus algebra on ( C • norm ( A ) , C norm • ( A )) is quas i- isomorphic to another homotopy calculus alg ebra on ( C • norm ( A ) , C norm • ( A )) whose H o (Λ Lie + δ )- part is the o rdinary Λ Lie + δ -algebra giv en b y the ab ov e Lie brac k et on C • norm ( A ), the Lie a l- gebra mo dule on C norm • ( A ) o v er C • norm ( A ) and Connes’ op erator on C norm • ( A ) . In this sense, the for ma lity of the homotopy calculus algebra on ( C • norm ( A ) , C norm • ( A )) is a generalization of Willw ac her’s cyclic formalit y t heorem [33]. The o rganization of the pap er is as follows. In Section 2 we ﬁx the nota tion and recall required results ab out (co)o p erads and (co)algebras. Section 3 is dev oted to ∞ - or homotop y v ersions for the algebras ov er the op erads calc , e 2 , a nd Lie + δ . In Section 4 w e recall the 1 See Pro po sition 11.3.3 on page 50 in [23]. 2 Kon tsevic h-Soib elman o p erad and the op era d Cyl of little discs on a cylinder. W e show that the homology operad H −• (Cyl , K ) of Cyl with the rev ersed grading is the op erad of calculi. Finally w e recall required r esults from [23] and prov e a useful prop erty o f the Kon tsevic h- Soib elman op erad. Section 5 is dev oted to prop erties of the homotop y calculus algebra on the pair ( C • norm ( A ) , C norm • ( A )) . In Section 6 w e formulate and prov e t he main result of this pap er. (See Theorem 5 on page 49.) In the concluding section we discussion applications and generalizations of Theorem 5. W e also discuss recen t articles related to our main result. Ac kno wledgmen t. A bigger part o f this work was done when V.D. was a Boas Assistan t Professor of Mathematics D epart ment at North w estern Univ ersit y . During these tw o y ears V.D. b eneﬁted from w orking at North w estern so muc h that he feels a s if he ﬁnished one more graduate sch o ol. V.D. cordially thanks Mathematics Departmen t at Nort hw este rn Univ ersit y for this time. The results of this work were presen ted at the conference P oisson 20 0 8 in Lausanne. W e w ould lik e to thank the participan ts of t his conference for questions and useful commen ts. V.D. w ould lik e to thank P a v el Snop ok for show ing him a v ery con v enien t dra wing program “Inkscap e”. D.T. and B.T. are supp orted b y NSF gr a n ts. The w ork of V.D. is partia lly supp orted b y the Gra n t for Supp ort of Scien tiﬁc Sc ho ols NSh-8065.200 6.2. 2 Preliminaries 2.1 (Co) op erads and (co)algebras Most of the notation and conv en tions for (co)op erads and their (co)algebras a r e b orrow ed from [13]. Dep ending on a con text our underlying symmetric monoidal category is either the cate- gory of graded vec tor spaces, or the category of c hain complexes , or the category of compactly generated top olo g ical spaces, or the category o f sets. By susp ension s V of a graded v ector space (o r a c hain complex) V w e mean ε ⊗ V , where ε is a one-dimensional v ector space placed in degree +1 . F or a v ector v ∈ V w e denote b y | v | its degree. The symmetric group of p erm utations of n letters is denoted b y S n . The underlying ﬁeld K has c haracteristic zero. F or an op erad O we denote b y Alg O the category of algebras o v er the op erad O . Dually , for a co op erad C w e denote by Coalg C the catego ry o f nilp oten t 2 coalgebras o v er the co op erad C . By c or estriction we mean t he canonical map ρ V : F C ( V ) → V (2.1) from the free coalgebra F C ( V ) to the v ector space of its cogenerators V . W e often omit the subscript in the nota tion ρ V for the corestriction. F or a p olynomial functor P w e denote b y T ( P ) (resp. T ∗ ( P )) the free op erad (resp. the free co op erad) (co)generated by P . The notation • is reserv ed for the monoidal pro duct o f the p olynomial functors. Th us, if P and Q are p olynomial functors then P • Q ( n ) = M k 1 + ··· + k m = n P ( m ) ⊗ S m ( Q ( k 1 ) ⊗ · · · ⊗ Q ( k m )) . (2.2) This formula can b e easily generalized to the colored p olynomial functors. 2 F or the deﬁnition of nilp otent coalge br a see section 2 . 4 . 1 in [19]. 3 By “ suspension” of a (co)o p erad O of graded v ector spaces (or c hain complexes) we mean the (co)op erad Λ( O ) whose m -th v ector space is Λ( O )( m ) = s 1 − m O ( m ) ⊗ sgn m , (2.3) where sgn m is the sign represen tation of the symmetric group S m . F or a comm utativ e algebra B and a B -mo dule V we denote b y S B ( V ) the symme tric algebra of V ov er B . S m B ( V ) stands for the m -th compo nen t of this algebra. If B = K then B is omitted from the no t ation. The a bbreviation “DGLA” stands for diﬀeren tial graded Lie algebra. W e denote by ∗ the p olynomial functor ∗ ( n ) = ( K , if n = 1 , 0 , otherwise . (2.4) This functor carries the unique structure of the op erad (resp. the co o p erad) suc h that ∗ is the initial (resp . the terminal) ob ject in the category of op erads (resp. co op erads) o f graded v ector spaces or c hain complexes. There is an ob vious generalization of ∗ (2.4) to the category of sets and to the category of top olo g ical spaces. Ho w ev er, w e will need ∗ only for linear (co)op erads, i.e. the (co)op erads in the category of gra ded v ector spaces or the category of c hain complexes. All the linear op erads (resp. linear co o p erads), w e consider, a r e equipped with an aug- men tation (resp. coaugmen tation). In o t her w ords, for ev ery op erad O w e will hav e a chose n morphism of op erads: τ : O → ∗ . (2.5) Dually for ev ery co op erad C w e will hav e a chosen morphism of co o p erads κ : ∗ → C . (2.6) W e are going to deal with 2-colored (co)op erads. Throughout the pap er w e lab el the tw o colors of all 2- colo red (co)o p erads b y c and a . F or example, the notation Lie + is reserv ed for the 2 -colored op erad whic h gov erns the pair s “Lie algebra V and a Lie algebra mo dule W ov er V .” V ectors of the Lie alg ebra V are colored b y c a nd v ectors of the mo dule W are colored by a . F or a linear 2- colored op erad O w e will denote by O c ( n, k ) (resp. O a ( n, k )) t he v ector space of op era t ions pro ducing a v ector with the color c (resp. a ) from n vectors with the color c and k v ectors with the color a . W e use the same not a tion for the linear 2-colored co op erads a nd fo r top ological 2-colored op erads. The p olynomial functor ∗ (2.4) has the o b vious generalization to t he category of linear 2-colored (co)op erads: ∗ c ( n, k ) = ( K , if ( n, k ) = (1 , 0) , 0 , otherwise . ∗ a ( n, k ) = ( K , if ( n, k ) = (0 , 1 ) , 0 , otherwise . (2.7) 4 F or a linear o p erad O w e denote by B ar ( O ) its bar construction. D ua lly , for a linear co op erad C w e denote by C obar ( C ) its cobar construction. W e recall t ha t, as a co op erad of graded vec tor spaces , B ar ( O ) is freely generated b y the p olynomial functor s − 1 O , where O is the k ernel of the augmen tation ( 2.5). Dually , as an op erad of graded v ector spaces, C obar ( C ) is freely generated b y the p olynomial functor s C , where C is cok ernel of the coaugmen tation (2.6). The diﬀeren tial ∂ B ar on the op erad B ar ( O ) is deﬁned using the m ultiplication of the op erad O and the diﬀeren tial ∂ C obar on the co op erad C obar ( C ) is deﬁned using the com ultiplication o f the co op erad C . See Chapter 3 in [15] or Section 2 in [17] for details. F or a quadratic o p erad O there is a natural sub-co op erad O ∨ of B ar ( O ) whic h satisﬁes the prop ert y: ∂ B ar    O ∨ = 0 . The details of the construction of O ∨ can b e found in Section 5.2 in [1 5 ]. F ollowing [18] w e call O ∨ the Koszul dual co op erad of O . F or a linear op erad O (resp. linear co op erad C ) and a v ector space V w e denote by F O ( V ) (resp. by F C ( V )) the free algebra (resp. free coalgebra) o v er the o p erad O (resp. co op erad C ). F or a linear 2-color ed (co)op erad O the functor 3 F O splits a ccording to the colors c a nd a as F O ( V , W ) = F O ( V , W ) c ⊕ F O ( V , W ) a , where F O ( V , W ) c = M n,k O c ( n, k ) ⊗ S n × S k V ⊗ n ⊗ W ⊗ k , and F O ( V , W ) a = M n,k O a ( n, k ) ⊗ S n × S k V ⊗ n ⊗ W ⊗ k . W e need to recall some facts ab out algebras o v er the op erad C obar ( C ) for a coaugmen ted co op erad C . Since C obar ( C ) is freely generated by the susp ension s C of the cok ernel C of the coaug- men tation (2.6) a C obar ( C )-algebra structure on a chain complex V is uniquely determined b y the restriction of the m ultiplication map µ : F C obar ( C ) ( V ) → V to the subspace F s C ( V ) ⊂ F C obar ( C ) ( V ) . In other w ords, a C obar ( C )-algebra structure on V is uniquely determined by a degree 1 map from F ( C )( V ) to V . It turns o ut that the maps from F ( C )( V ) to V hav e a elegan t des cription in terms of co deriv at ions of the free coalgebra F C ( V ) . T o recall this description w e in tro duce the L ie subalgebra Co der ′ ( F C ( V )) = { Q ∈ Co der( F C ( V )) | Q    V = 0 } , (2.8) 3 F O is called the Sch ur functor. 5 where V is considered a s a subspace of C (1) ⊗ V via the coaugmentation (2.6 ) . In ot her w ords, the elemen ts of Co der ′ ( F C ( V )) a re co deriv ations of the f ree coalgebra F C ( V ) which can b e factored thr o ugh the pro jection F C ( V ) → F C ( V ) . It is no t hard to see that the subspace (2.8 ) is closed under the comm utator and t he dif- feren tials coming from C and V . Th us t he graded v ector space Co der ′ ( F C ( V )) is in fact a DGLA. Let us recall from [17] the follow ing prop osition Prop osition 1 ( Prop osition 2.14 [17 ]) F or a c o augmente d c o op er ad C the c omp osition with the c or estriction (2.1) ρ V : F C ( V ) → V induc es an isom orphism of gr ade d ve ctor sp ac es C oder ′ ( F C ( V )) ∼ = Hom ( F C ( V ) , V ) , (2.9) wher e, as a b o v e , C is the c o kernel of the c o a ugm entation (2.6) of C . Due to this prop o sition a C obar ( C )-algebra structure on a chain complex V is uniquely determined by a degree 1 co deriv ation Q ∈ C oder ′ ( F C ( V )) . (2.10) According to Prop osition 2.15 from [17] the compatibilit y of the C obar ( C )-algebra struc- ture on V with the total diﬀerential on C obar ( C ) and the diﬀeren tial on V is equiv alen t t o the Maurer-Carta n equation for the corresponding deriv atio n (2 .10): [ ∂ C + ∂ V , Q ] + 1 2 [ Q, Q ] = 0 , (2.11) where ∂ C is the diﬀerential on F C ( V ) induced b y the one o n the co op erad C and ∂ V comes from that on V . In other w ords, Prop osition 2 ( Prop osition 2.15, [17 ]) Ther e is a na tur al bije ction b etwe en the Maur er- Cartan elemen ts of the DGLA C od er ′ ( F C ( V )) and an d the C obar ( C ) -algebr a structur es on V . If we ha v e a map µ : C 1 → C 2 (2.12) of coaugmente d co op erads then the corresp onding map b et w een the cobar constructions C obar ( µ ) : C obar ( C 1 ) → C obar ( C 2 ) allo ws us to pull C obar ( C 2 )-algebra structure on V to a C obar ( C 1 )-algebra o n V . W e claim that Prop osition 3 I f Q 1 is a Maur er-Cartan element of the DGLA C od er ′ ( F C 1 ( V )) c orr esp ond - ing to a C obar ( C 1 ) -algebr a s tructur e on V and Q 2 is a Maur er-Cartan element of the DGLA C oder ′ ( F C 2 ( V )) c orr esp on d ing to a C obar ( C 2 ) -algebr a structur e o n V then ρ V ◦ Q 1 = ρ V ◦ Q 2 ◦ F ( µ ) , (2.13) wher e the map F ( µ ) : F C 1 ( V ) → F C 2 ( V ) is induc e d by (2.12) . 6 Pro of. Let ν 2 : F C obar ( C 2 ) ( V ) → V b e the C o bar ( C 2 )-algebra structure on V . Then the C obar ( C 1 )-algebra structure on V ν 1 : F C obar ( C 1 ) ( V ) → V is obtained b y comp osing the map ν 2 with the map F ( C obar ( µ )) : F C obar ( C 1 ) ( V ) → F C obar ( C 2 ) ( V ) . It is no t hard to see that the restriction of ν 1 to the subspace F s C 1 ( V ) coincides with the comp osition of the ma ps F s C 1 ( V ) F ( µ ) → F s C 2 ( V ) and ν 2    F s C 2 ( V ) : F s C 2 ( V ) → V . Th us the prop osition follow s from the equation ν i    F s C i ( V ) = ρ V ◦ Q i ◦ σ , where ρ V is the corestriction (2.1) and σ is the susp ension isomorphism σ : F s C i ( V ) → F C i ( V ) , and i = 1 , 2 .  W e will freely use Prop o sitions 1, 2 and 3 for colored co o p erads. W e remark that a ll 2-colored (co)op erads, w e consider, satisfy the following prop ert y: an ar gume nt with the c olor a c an enter an op er ation at most onc e. If an ar gument with this c o l o r enters a n op er ation then the r esulting c olor is also a . Otherwise the r es ulting c olor is c . In o ther words, for every n O c ( n, k ) = O a ( n, k ) = { 0 } , ∀ k > 1 , O a ( n, 0) = { 0 } , O c ( n, 1) = { 0 } (2.14) for the (co)op erads of graded v ector spaces or chain complexes a nd O c ( n, k ) = O a ( n, k ) = ∅ , ∀ k > 1 , O a ( n, 0) = ∅ , O c ( n, 1) = ∅ (2.15) for the (co)op erads of top olo gical spaces or sets. It is not hard to see that bar and cobar constructions the (co)op erads of gr a ded ve ctor spaces or c hain complexes preserv e prop ert y (2.14). Let us recall that Deﬁnition 1 (M. Gerstenhab er, [16]) A g r ad e d ve ctor sp ac e V is a Gerstenhab e r algebr a if it is e quipp e d with a c ommutative and ass o c i a tive pr o duct ∧ of de gr e e 0 and a Lie br acket [ , ] of de gr e e − 1 . These op er ations have to b e c omp atible in the sense of the fol lo w ing L eibniz rule [ a, b ∧ c ] = [ a, b ] ∧ c + ( − 1) ( | a | +1) | b | b ∧ [ a, c ] , (2.16) wher e a, b, c ar e homo g ene ous ve ctors of V . 7 Deﬁnition 2 A pr e c alculus is a p air of a Gerstenhab er a lgebr a ( V , ∧ , [ , ]) and a gr ade d ve ctor sp a c e W to gether wi th • a mo dule structur e i • : V ⊗ W 7→ W of the gr ade d c ommutative algebr a V on W , • an a c tion l • : s − 1 V ⊗ W 7→ W of the Lie algebr a s − 1 V on W which ar e c omp atible in the sense of the fol lowing e quations i a l b − ( − 1) | a | ( | b | +1) l b i a = i [ a,b ] , (2.17) and l a ∧ b = l a i b + ( − 1) | a | i a l b . (2.18) F urthermore, Deﬁnition 3 A c alculus is a pr e c alculus ( V , W , [ , ] , ∧ , i • , l • ) with a de gr e e − 1 unary op er ation δ on W such that δ i a − ( − 1) | a | i a δ = l a , (2.19) and 4 δ 2 = 0 . (2.20) W e call l and i the Lie deriv ativ e and the contraction, resp ectiv ely . W e will use the fo llowing list of (co)op erads: • Lie (resp. coLie ) is the op erad of Lie algebras ( r esp. the coo p erad of Lie coa lgebras), • comm (resp. coc omm ) is the op erad of comm utativ e (associative) alg ebras (resp. the op erad of co comm utativ e coasso ciativ e coalgebras), • e 2 denotes the op erad o f Gerstenhab er algebras, (see D eﬁnition 1), • KS denotes the op erad of M. Kontse vic h and Y. Soib elman. This op era d 5 is describ ed in sections 11.1, 11 .2 and 1 1 .3 of [23], • Lie + (resp. coLie + ) denotes the 2-colored op erad of pairs “Lie algebra + its mo dule” (resp. the 2-colored co op erad of pairs “Lie coalgebra + its como dule”), • comm + (resp. co c omm + ) denotes the 2-colored op erad of pairs “ comm utativ e algebra + its mo dule” (resp. the 2- colored co op erad of pairs “co comm utativ e coalgebra + its como dule”), • p calc denotes the 2-colored o p erad of precalculi, (see Deﬁnition 2 ) , • calc denotes the 2- colored op era d of calculi, (see D eﬁnition 3 ) , • asso c is the no n-symmetric op erad of sets con trolling unital monoids; eac h set asso c ( n ), n ≥ 0, is a p oint. 4 Although δ 2 = 0 , the o p e r ation δ is never consider e d as a par t of the diﬀerential o n W . 5 In [23] this op er ad is denoted by P . 8 It is not hard to show that for the v ector space of the free calculus algebra generated b y the pair ( V , W ) w e ha v e F calc ( V , W ) ∼ = F comm + ( F Λ Lie + ( V , W ⊕ s − 1 W )) . (2.21) In other w ords, for the colo r comp onen ts w e ha v e the isomorphisms of graded ve ctor spaces: F calc ( V , W ) c ∼ = F comm ( F Λ Lie ( V )) , (2.22) and F calc ( V , W ) a ∼ = F comm + ( F Λ Lie ( V ) , F Λ Lie + ( V , W ⊕ s − 1 W ) a ) a . (2.23) 2.2 Ho c hsc hild (co)c hain c omp lexes F or an asso ciative algebra A C • ( A ) = Hom ( A ⊗• , A ) denotes the Ho ch sc hild co c hain complex and C • ( A ) = A ⊗ A ⊗ ( −• ) stands for the Ho chsc hild chain complex o f A with the rev ersed g rading. F or the normalized v ersions of these complexes we reserv e the notatio n: C • norm ( A ) = { P ∈ H om ( A ⊗• , A ) | P ( . . . , 1 , . . . ) = 0 } and C norm • ( A ) = A ⊗ ( A/ K 1) ⊗ ( −• ) . • The notation ∂ H och is reserv ed b oth fo r the Ho c hsc hild cob oundary op erator on C • norm ( A ) and Ho chs c hild b oundary op erator on C norm • ( A ) ( ∂ H och P )( a 0 , a 1 , . . . , a k ) = a 0 P ( a 1 , . . . , a k ) − P ( a 0 a 1 , . . . , a k )+ P ( a 0 , a 1 a 2 , a 3 , . . . , a k ) − . . . +( − 1) k P ( a 0 , . . . , a k − 2 , a k − 1 a k ) + ( − 1) k +1 P ( a 0 , . . . , a k − 2 , a k − 1 ) a k ∂ H och ( a 0 , a 1 , . . . , a m ) = ( a 0 a 1 , a 2 , . . . , a m ) − ( a 0 , a 1 a 2 , a 3 , . . . , a m ) + · · · + ( − 1) m − 1 ( a 0 , . . . , a m − 2 , a m − 1 a m ) + ( − 1) m ( a m a 0 , a 1 , a 2 , . . . , a m − 1 ) , a i ∈ A , P ∈ C k norm ( A ) . • The nota tion ∪ is reserv ed for the cup-pro duct on C • norm ( A ) P 1 ∪ P 2 ( a 1 , a 2 , . . . , a k 1 + k 2 ) = P 1 ( a 1 , . . . , a k 1 ) P 2 ( a k 1 +1 , . . . , a k 1 + k 2 ) , (2.24) P i ∈ C k i norm ( A ) . 9 • [ , ] G stands f or the Gerstenhab er brack et on C • norm ( A ) [ Q 1 , Q 2 ] G = k 1 X i =0 ( − 1) ik 2 Q 1 ( a 0 , . . . , Q 2 ( a i , . . . , a i + k 2 ) , . . . , a k 1 + k 2 ) − ( − 1) k 1 k 2 (1 ↔ 2) , (2.25) Q i ∈ C k i +1 norm ( A ) . • I P ( c ) is the con traction of a Ho chs c hild co c hain P ∈ C k norm ( A ) with a Ho c hsc hild c hain c = ( a 0 , a 1 , . . . , a m ) I P ( a 0 , a 1 , . . . , a m ) = ( ( a 0 P ( a 1 , . . . , a k ) , a k +1 , . . . , a m ) , if m ≥ k , 0 , otherwise . (2.26) • L Q ( c ) denotes the Lie deriv ativ e of a Ho c hsc hild c hain c = ( a 0 , a 1 , . . . , a m ) along a Ho c hsc hild co c hain Q ∈ C k +1 norm ( A ) L Q ( a 0 , a 1 , . . . , a m ) = m − k X i =0 ( − 1) k i ( a 0 , . . . , Q ( a i , . . . , a i + k ) , . . . , a m )+ (2.2 7) m − 1 X j = m − k ( − 1) m ( j +1) ( Q ( a j +1 , . . . , a m , a 0 , . . . , a k + j − m ) , a k + j +1 − m , . . . , a j ) . • B : C norm • ( A ) → C norm •− 1 ( A ) denotes Connes’ op erator B ( a 0 , a 1 , . . . , a m ) = m X i =0 ( − 1) mi (1 , a i , . . . , a m , a 0 , a 1 , . . . , a i − 1 ) . (2.28) The notation H H • ( A ) (resp. H H • ( A )) is used for the Ho chsc hild cohomolo gy (resp. homology groups) of A with co eﬃcien ts in A H H • ( A ) = H • ( C • norm ( A ) , ∂ H och ) , H H • ( A ) = H • ( C norm • ( A ) , ∂ H och ) . T o describ e alg ebraic structures on pairs ( C • norm ( A ) , C norm • ( A )) and ( H H • ( A ) , H H • ( A )) w e use the the language of op erads. Th us, the G erstenhab er brack et [ , ] G equips the co c hain complex C • norm ( A ) with an algebra structure ov er the o p erad Λ Lie and the Lie deriv a - tiv e (2.27) equips the pair ( C • norm ( A ) , C norm • ( A )) with the algebra structure o v er the op erad Λ Lie + . In order to add Connes’ op erator (2.28) into t his o p eradic picture w e giv e one more deﬁnition 10 Deﬁnition 4 We say that the p air of gr ade d ve ctor sp ac es ( V , W ) is an algebr a ove r the op er ad Lie + δ if V is a Lie algeb r a, W is a mo dule over V and W is e quipp e d with a de gr e e − 1 unary op er ation δ satisfying the e quations δ 2 = 0 , (2.29) and [ δ , l a ] = 0 , ∀ a ∈ V , (2.30) wher e l is the action of V on W . Adding Connes’ op erator B in to the picture w e ma y say that the pair ( C • norm ( A ) , C norm • ( A )) is a Λ Lie + δ -algebra. The op erations (2.24), (2.25), (2.26), (2.27) and (2.28) are closed with res p ect to the (co)b oundary op erator ∂ H och . According to [16] the op erations ∪ (2.24 ) and [ , ] G (2.25) induce o n H H • ( A ) the structure of a Gerstenhaber alg ebra. F urthermore, it is know n [11] that the op erations (2.24), (2.25 ), (2.26), (2.2 7) and (2.28) induce on the pair ( H H • ( A ) , H H • ( A )) the structure of the calculus algebra. 3 The op erads Ho ( calc ) , Ho ( e 2 ) , and Ho ( Lie + δ ) In this section w e describ e the homotop y v ersions for the alg ebras ov er the op erads calc , e 2 , and Lie + δ . 3.1 Descriptio n of the op erads Ho ( calc ) and Ho ( e 2 ) T o describ e the homot o p y v ersion of calc -algebras w e use the canonical coﬁbran t resolution C obar ( B ar ( calc )) . In other w ords, we set Ho ( calc ) = C obar ( B ar ( calc )) . (3.1) The co op erad B ar ( calc ) will b e used throughout the pap er. F or this reason w e reserv e a short-hand notation B = B ar ( calc ) (3.2) for this co op erad. Recall that, as a co op erad of graded v ector spaces, B = B ar ( c alc ) is freely generated b y the p olynomial functor s − 1 calc , where calc is the k ernel of the augmen tation. W e represen t elemen ts of the free coalgebra F B ( V , W ) and elemen ts of the co op erad B graphically . Th us Figures 1, 2 represen t the simples t elemen ts of F B ( V , W ) c with γ 1 and γ 2 b eing ve ctors in V . Figures 3, 4 sho w the simplest elemen ts of F B ( V , W ) a with γ ∈ V and c ∈ W . Figures 5 and 6 represen t simple elemen ts of B a (0 , 1) . The dashed line in ﬁgures 3, 4, 5, and 6 is used to lab el the argumen ts of the color a and the solid line is used to lab el the arg uments of the color c . Using this graphical notation we ma y p erform simple computations in the coalg ebra F B ( V , W ). F or example, using equation (2.19), we presen t on Figure 7 a simple computation with the bar diﬀeren tial ∂ B ar . Here γ ∈ V and c ∈ W . 11 γ 1 γ 2 ∧ [ , ] γ 1 γ 2 Figure 1: The pro duct ∧ ∈ calc c (2 , 0) is used Figure 2: The brac k et [ , ] ∈ calc c (2 , 0) is used c γ i c γ l Figure 3: The commutativ e mo dule struc- ture i ∈ calc a (1 , 1) is used Figure 4: The Lie algebra mo dule struc- ture l ∈ calc a (1 , 1) is used 12 P S f r a g r e p la c e m e n t s δ P S f r a g r e p la c e m e n t s δ δ δ Figure 5: The unary op eration δ ∈ calc a (0 , 1) is used Figure 6: The num b er o f δ ’s on the ﬁgure is m F or the op erad e 2 w e use a resolution whic h is simpler than the canonical one C obar ( B ar ( e 2 )) . More precisely , as in [13], w e set Ho ( e 2 ) = C obar ( e 2 ∨ ) . (3.3) Due to koszulit y of the op erad e 2 the inclusions ι e 2 : e 2 ∨ ֒ → B ar ( e 2 ) (3.4) and C obar ( ι e 2 ) : C obar ( e 2 ∨ ) ֒ → C obar ( B ar ( e 2 )) (3.5) are quasi-isomorphisms of co op erads and op erads, resp ectiv ely . It is the second quasi- isomorphism (3.5) whic h allows us to replace t he canonical resolution C obar ( B ar ( e 2 )) b y (3.3). T o get a more tractable description of a lgebras o v er the op erads Ho ( calc ) and Ho ( e 2 ) w e introduce the f ollo wing DGLAs Co der ′ ( F e 2 ∨ ( V )) = { Q ∈ Co der( F e 2 ∨ ( V )) | Q    V = 0 } , (3.6) Co der ′ ( F B ( V , W )) = { Q ∈ Co der( F B ( V , W )) | Q    V ⊕W = 0 } , (3.7) where Co der( F e 2 ∨ ( V )) (resp. Co der( F B ( V , W )) ) is the DG LA o f co deriv a tions of the free coalgebra F e 2 ∨ ( V ) (resp. the free coalgebra F B ( V , W )). F urthermore, V (resp. V ⊕ W ) is considered as a subspace of F e 2 ∨ ( V ) (resp. F B ( V , W )) via the corr esp o nding coaugmen tation. According to Prop osition 2 the Ho ( e 2 )-algebra structures on V are in bijection with the Maurer-Cartan elemen ts of the DGLA Co der ′ ( F e 2 ∨ ( V )) . Similarly , the Ho ( calc )-a lgebra 13 P S f r a g r e p la c e m e n t s δ δ δ i i δ ∂ B ar ∂ B ar + = = γ γ γ γ γ c c c c c − ( − 1) | γ | l i i Figure 7 : A simple computation with ∂ B ar 14 structures on the pa ir ( V , W ) are in bijection with the Maurer-Cartan elemen t s of t he DG L A Co der ′ ( F B ( V , W )) . Moreo v er, due to Prop osition 1 the Maurer-Cartan elemen t Q of the DGLA (3.6 ) (resp. the DGLA (3.7)) is uniquely determined b y its comp o sition ρ V ◦ Q (resp. ρ V , W ◦ Q ) with the corestriction ρ V : F e 2 ∨ ( V ) → V (resp. the corestriction ρ V , W : F B ( V , W ) → V ⊕ W ) . The v ector space of the free coalgebra F B ( V , W ) splits according to the t w o colors ( c , a ) as F B ( V , W ) = F B ( V , W ) c ⊕ F B ( V , W ) a , (3.8) where F B ( V , W ) c = F B ar ( e 2 ) ( V ) . (3.9) Th us fo r ev ery Ho ( calc )-alg ebra ( V , W ) the graded v ector space V is an a lgebra ov er the op erad C obar ( B ar ( e 2 )) . Using this algebra structure o v er C obar ( B ar ( e 2 )) and the em b edding (3.5) we get a Ho ( e 2 )-algebra structure on V . T o describe the relatio nship b et w een these algebras we denote b y Q V , W the Maurer- Cartan elemen t of the DGLA Co der ′ ( F B ( V , W )) corresp o nding to the Ho ( calc )- algebra structure on ( V , W ) . Next, we denote b y Q V the Maurer- Cartan elemen t of the DGLA Co der ′ ( F e 2 ∨ ( V )) corresp onding to the Ho ( e 2 )-algebra structure on V . Prop osition 3 implies that ρ V ◦ Q V = ρ V ⊕W ◦ Q c V , W ◦ F ( ι e 2 ) , (3.10) where ι e 2 is the em b edding (3.4) and Q c V , W = Q V , W    F B ( V , W ) c . Due to Prop osition 1 the co deriv ation Q V (resp. the co deriv ation Q V , W ) is uniquely determined by the comp o sition ρ V ◦ Q V (resp. ρ V , W ◦ Q V , W ) . Th us equation (3.10) indeed describes the relationship b et w een the Ho ( calc )-algebra structure on ( V , W ) and the Ho ( e 2 )- algebra structure on V . Remark. The v ector space of op erations of the co o p erad B with no a rgumen ts havin g color c is B a (0 , 1) = K [ u ] , (3.11) where u is an auxiliary v ariable of degree − 2 . The monomial u m corresp onds to the elemen t of B a (0 , 1) whic h is dra wn on Figure 6 (See pag e 13). 3.2 Descriptio n of the op erad Ho ( Lie + δ ) The canonical coﬁbran t resolution C o bar ( B ar ( Lie + δ )) can b e simpliﬁed. In this subsection w e construct a sub-co o p erad ( Lie + δ ) ∨ of B ar ( Lie + δ ) suc h that the em b edding of o p erads C obar (( Lie + δ ) ∨ ) ֒ → C obar ( B ar ( Lie + δ )) is a quasi-isomorphism. This construction go es along the lines of [15 ], [18 ]. (See also Deﬁni- tion 3.2.1 in [19].) It allows us to set Ho ( Lie + δ ) = C obar (( Lie + δ ) ∨ ) . 15 Let us ﬁrst recall that algebras o v er the op erad Lie + δ are pairs ( V , W ) where V is a L ie algebra W is a Lie algebra mo dule o v er V and W is equipp ed with degree − 1 unary op eration δ whic h satisﬁes the identities δ 2 = 0 . (3.12) and δ l a − ( − 1) | a | l a δ = 0 , (3.13) where l : V ⊗ W → W is the action of V on W . Th us the op erad Lie + δ is generated b y the elemen tary op erations [ , ] , l and δ , where [ , ] denotes the Lie brac k et. T hese op erat ions are sub ject to the homogeneous quadratic relations: the Jacobi iden tity for the Lie brac k et [ , ] , and the compatibility equation b etw een l and [ , ] l a l b − ( − 1) | a | | b | l a l b = l [ a,b ] (3.14) and, ﬁnally , equations (3.12) a nd (3.13) . T o construct the co op erad ( Lie + δ ) ∨ w e in tro duce the p olynomial functor S spanned lin- early by the elemen tary op erat ions [ , ] , l , and δ of the op erad Lie + δ . W e also in tro duce the linear span R of the homogeneous quadratic relatio ns of Lie + δ b et w een the elemen tary op erations. Next, we consider the fr ee co op erad T ∗ ( s − 1 S ) generated b y t he p olynomial functor s − 1 S . The co op erad T ∗ ( s − 1 S ) ma y b e view ed as a sub-co op erad of B ar ( Lie + δ ) if w e forget ab o ut the diﬀeren tial ∂ B ar . Let us remark that, the co op erad T ∗ ( s − 1 S ) is equipp ed with the natural gr a ding T ∗ ( s − 1 S ) = ∞ M m =0 T ∗ m ( s − 1 S ) , T ∗ 0 ( s − 1 S ) = ∗ , (3.15) where ∗ is the terminal ob ject (2.7) in the category o f 2-colo red co op erads a nd T ∗ m ( s − 1 S ) consists of the elemen ts of degree m in the elemen ta ry op erations. Thus , since the relations b et w een the elemen tary op erations are quadrat ic, s − 2 R is a subspace of T ∗ 2 ( s − 1 S ) . First, w e construct the co op erad ( Lie + δ ) ∨ as a sub-co op erad of T ∗ ( s − 1 S ) and then w e will show that ( Lie + δ ) ∨ b elongs to the k ernel o f the bar diﬀeren tial ∂ B ar . W e construct ( Lie + δ ) ∨ b y induc tion on the degree m in (3.15) . Th e base of the induction is give n b y the equations ( Lie + δ ) ∨ ∩ T ∗ 0 ( s − 1 S ) ⊕ T ∗ 1 ( s − 1 S ) = T ∗ 0 ( s − 1 S ) ⊕ T ∗ 1 ( s − 1 S ) , (3.16) ( Lie + δ ) ∨ ∩ T ∗ 2 ( s − 1 S ) = s − 2 R , (3.1 7) and the step is giv en b y the condition: a ve ctor v ∈ T ∗ m ( s − 1 S ) b elongs to ( Lie + δ ) ∨ pr ov ide d e ∆( v ) ∈ ( Lie + δ ) ∨ • ( Lie + δ ) ∨ . Here ∆ is the copro duct: ∆ : T ∗ ( s − 1 S ) → T ∗ ( s − 1 S ) • T ∗ ( s − 1 S ) , and e ∆( v ) = ∆( v ) − v ⊗ (1 ⊗ · · · ⊗ 1) − 1 ⊗ ( v ⊗ 1 ⊗ · · · ⊗ 1) − 1 ⊗ (1 ⊗ v ⊗ 1 · · · ⊗ 1) − . . . 16 − 1 ⊗ (1 ⊗ · · · ⊗ 1 ⊗ v ) . By construction ( Lie + δ ) ∨ is a sub-co o p erad of T ∗ ( s − 1 S ) . Equation (3.1 7 ) imply immediately that ∂ B ar v = 0 , ∀ v ∈ ( Lie + δ ) ∨ ∩ T ∗ 2 ( s − 1 S ) . Then the compatibilit y of ∂ B ar with the copro duct ∆: ∆ ∂ B ar = ( ∂ B ar ⊗ (1 ⊗ · · · ⊗ 1) + 1 ⊗ ( ∂ B ar ⊗ 1 ⊗ · · · ⊗ 1) + . . . ) ∆ and the inductiv e deﬁnition of ( Lie + δ ) ∨ imply t ha t ∂ B ar v = 0 , ∀ v ∈ ( Lie + δ ) ∨ . (3.18) Th us ( Lie + δ ) ∨ b elongs to the k ernel o f the bar diﬀeren tial ∂ B ar in B ar ( Lie + δ ) . The fo llowing prop osition giv es us a description o f the coa lgebras o v er the co op erad ( Lie + δ ) ∨ Prop osition 4 A p air ( V , W ) of gr ade d ve ctor sp ac es fo rms a c o al g ebr a over the c o op er ad ( Lie + δ ) ∨ if ( V , W ) is a c o algebr a over the c o op er ad Λ co comm + and W is e quipp e d with a de gr e e 2 endomorphism δ ∨ : W → W satisfying the e quation l ∨ ◦ δ ∨ = (1 ⊗ δ ∨ ) l ∨ , wher e l ∨ is the c o action o f V on W l ∨ : W → s − 1 ( V ⊗ W ) . Pro of. Let us consider the r estricted dual v ector space [ T ∗ ( s − 1 S )] ∗ = Hom restr ( T ∗ ( s − 1 S ) , K ) (3.19) of the free co op erad T ∗ ( s − 1 S ) with resp ect to the grading (3.15) . It is not hard t o see that [ T ∗ ( s − 1 S )] ∗ = T ( s S ∗ ) is the free operad T ( s S ∗ ) generated b y the susp ension s S ∗ of the linear dual S ∗ of the p olynomial functor S . F rom the construction o f ( Lie + δ ) ∨ it follows that the restricted dual [( Lie + δ ) ∨ ] ∗ of the co op erad ( Lie + δ ) ∨ is the quotien t of the free op erad T ( s S ∗ ) with resp ect to the ideal generated b y the p olynomial functor of dual relatio ns R ∗ = { r ∈ Hom ( T ∗ 2 ( s − 1 S ) , K ) , | r    R = 0 } . (3.20) Let { [ , ] ∗ , l ∗ , δ ∗ } b e the basis of S ∗ whic h is dual to the basis { [ , ] , l , δ } of S . Dualizing the Jacobi relation for [ , ] and the compatibilit y (3.1 4 ) of l with [ , ] w e see that t he op eration s [ , ] ∗ satisﬁes the axioms of an asso ciativ e comm utativ e pro duct and the op eration s l ∗ satisﬁes the a xiom of a mo dule ov er an asso ciative and comm utativ e algebra. 17 Dualizing the relation (3.1 3) w e see that s l ∗ and s δ ∗ are compatible in the sense of the follo wing relation s δ ∗ s l ∗ = s l ∗ (1 ⊗ s δ ∗ ) . (3.21) Finally the presence of the relation (3.12) implies that we sh ould not imp ose any additional condition on s δ ∗ b esides (3 .2 1) . Th us a pair ( e V , f W ) is an algebra o v er the op erad [( Lie + δ ) ∨ ] ∗ if ( e V , f W ) is a Λ − 1 comm + - algebra a nd f W is equip p ed with a degree 2 endomorphism s δ ∗ whic h is compatible with the action of e V on f W in the sense of (3.21). T aking the dual partner of an algebra ov er the op erad [( Lie + δ ) ∨ ] ∗ w e get the statemen t of the prop osition.  Prop osition 4 implies that a free coalgebra ov er the co op erad ( Lie + δ ) ∨ generated b y a pair ( V , W ) is F ( Lie + δ ) ∨ ( V , W ) = F Λ co comm + ( V , W [[ u ]]) , (3.22) where u is an auxiliary v ariable of degree − 2 . W e claim that Prop osition 5 Th e op er ad Lie + δ is Koszul. In other wor ds the emb e dding C obar (( Lie + δ ) ∨ ) → C obar ( B ar ( Lie + δ )) is a quasi-isom orphism of op er ads. Pro of. The criterion of G inzburg and Kapranov [18] (theorem 4 . 2 . 5) reduces this question to computation o f the homology of a fr ee Lie + δ -algebra. More precisely , w e need to show that for ev ery pair ( V , W ) of v ector spaces the complex F ( Lie + δ ) ∨ ◦ F Lie + δ ( V , W ) (3.23) has nontrivial cohomology only in degree 0 . Here t he diﬀerential on the complex ( 3 .23) is deﬁned along the lines of [17 ] using the t wisting co c hain of the pair ( Lie + δ , ( Lie + δ ) ∨ ) . (See Section 2.4 in [17] for more details.) If we split the complex (3.23) according to the colors c and a and use equation (3 .22) then w e get tw o complexes: F ( Lie + δ ) ∨ ◦ F Lie + δ ( V , W ) c = F Λ co comm ◦ F Lie ( V ) , (3.24) and F ( Lie + δ ) ∨ ◦ F Lie + δ ( V , W ) a = F Λ co comm + ( F Lie ( V ) , T ( V ) ⊗ ( W ⊕ δ W )[[ u ]]) a , (3.25) where T ( V ) denotes the tensor algebra of V , δ is the unary op eration of Lie + δ and u is an auxiliary v ariable of degree − 2 . The ﬁrst complex is exactly the Harrison complex of the free Lie algebra generated by V and it is kno wn that this complex has nontrivial cohomology only in degree 0 . The second complex is the tensor pro duct of the Harrison complex of the free mo dule generated by W ov er the f r ee Lie algebra F Lie ( V ) and the De Rham complex ( K [[ u ]] ⊕ δ K [[ u ]] , δ ∂ ∂ u ) 18 of the algebra K [[ u ]] . Thu s the second complex also has nontriv ial cohomo lo gy only in degree 0 .  This Prop osition implies immediately tha t the em b edding C obar (Λ( Lie + δ ) ∨ ) ֒ → C obar ( B ar (Λ Lie + δ )) is a quasi-isomorphism of op erads. Th us we ma y set Ho ( Λ Lie + δ ) = C obar (Λ ( Lie + δ ) ∨ ) . (3.26) W e w ould also lik e to remark that equation (3.22) implies that F Λ( Lie + δ ) ∨ ( V , W ) = F Λ 2 co comm + ( V , W [[ u ]]) , (3.27) where u is an auxiliary v ariable of degree − 2 . 4 The Kon tsevic h-Soib elman op erad and the op erad of lit t le discs on a cylin d er 4.1 The Kon tsevic h-Soib elman op erad KS Let us describe the auxiliary op erad H (of sets) of “natura l” 6 op erations on the pa ir ( C • ( A ) , C • ( A )) . This op erad is go ing to ha v e a coun table set o f colors Ξ = Z + ⊔ Z − , (4.1) where Z + (resp. Z − ) denotes the set of nonnegative (resp. nonpositive) in tegers. The n um b ers fr o m the set Z + lab el the degrees of the Ho chs c hild co c hains and the n um b ers from the set Z − lab el the degrees of Ho chs c hild c hains. Using H w e construct the DG op erad KS o f Ko n tsevic h and Soib elman. The latter op erad 7 is describ ed in sections 11.1, 1 1.2 and 11.3 o f [23]. F or the Ξ-colored o p erad H w e only allow the op erations in whic h a c hain may en ter as an argumen t a t most once. If a c hain en ters t hen the result of the op eration is a lso a c hain. Otherwise the result is a co c hain. W e denote the set of op erat ions pro ducing a co c hain from n co chains b y H ( n, 0) . The set of op eratio ns pro ducing a chain from n co c hains and 1 chain is denoted b y H ( n, 1) . H ( n, 0) is the set of equiv alence classes o f ro oted 8 planar trees T with mark ed vertice s. The equiv alence relation is the ﬁnest one in whic h tw o suc h tr ees are equiv alen t if one o f them can b e o btained fro m the o ther by either: • the con traction of an edge with unmarked ends or 6 W e are not sure if these o p er ations are natural in the se ns e of category theory . 7 In [23] this op er ad is denoted by P . 8 Recall that a tree ca lled r o ote d if if its ro ot v ertex has v alency 1 . 19 P S f r a g r e p la c e m e n t s Q P a 1 a 2 Figure 8 : • remo ving an unmarked vertex with only one edge orig inating from it and jo ining the t w o edges adjacent to this v ertex in to one edge. If a marke d v ertex is in ternal then it is reserv ed for a co c hain whic h en ters as an argumen t of the opera t ion. The n um ber of the incoming edges of suc h v ertex is the degree of the corresp onding cochain. If a mark ed verte x is terminal then it is reserv ed either for a co c hain of degree 0 or for an a r g umen t of the co c hain pro duced by the op eration. The unmark ed vertice s (b oth internal and terminal) are reserv ed for the op erations of the non-symm etric op erad assoc whic h con trols unital monoids. F or example, an unmark ed terminal v ertex is reserv ed for unit of A , an unmarke d v ertex o f v alency 2 is reserv ed fo r the identit y transfor ma t ion on A , and an unmarked v ertex of v alency 3 is reserv ed for the asso ciativ e pro duct on A . The ro ot v ertex is sp ecial. Since our trees are ro o ted this v ertex has alw a ys v a lency 1. It is a lw a ys mark ed and reserv ed for the outcome of the co c hain pro duced b y the op erat io n corresp onding to the tree. The tr ee on ﬁgure 8 represen ts an o p eration which pro duces the 2- co c hain: a 1 ⊗ a 2 → Q ( a 1 , a 2 , 1) P from a degree 0 co chain P and a degree 3 co c hain Q . Marke d v ertices in this ﬁgure are lab eled by small circles. The unmark ed t erminal ve rtex correspo nds to the insertion of the unit into Q ( a 1 , a 2 , 1) . The unmarke d 3- v alent v ertex giv es the pro duct o f P and Q ( a 1 , a 2 , 1) . Let us denote by H m a m r ( n, 1) the set of op erations pro ducing a c hain in C − m r ( A ) from n co c hains a nd a c hain in C − m a ( A ) . H m a m r ( n, 1) is describ ed using for ests of r o oted trees drawn on the standard cylinder Σ = S 1 × [0 , 1] (4.2) and sub ject to the follow ing conditions: 20 1. ev ery tree of the forest has its ro ot vertex on the b oundary S 1 × { 0 } , 2. all vertice s of the forest lying on the b oundary of the cylinder are mark ed: • the ve rtices lying on the b oundary S 1 × { 1 } are mark ed b y integers 0 , 1 , . . . , m a in the counterc lo c kwise or der; these v ertices are reserv ed fo r t he comp onen ts of the chain whic h en ters as an arg umen t, • the ro ots are mark ed b y in tegers 0 , 1 , . . . , m r in the same coun terclo ckw ise order; they a re reserv ed for comp o nents of the resulting c hain, 3. all ot her mark ed v ertices of the forest lie on the lateral surfa ce S 1 × (0 , 1) of the cylinder and there are exactly n suc h mark ed v ertices. On the set of t hese forests w e in tro duce the ﬁnest equiv alence relation in whic h tw o such forests are equiv alen t if one of them can b e obtained from the other by either: • isotopy , or • the con traction of an edge with unmarked ends, or • remo ving an unmarked vertex with only one edge orig inating from it and jo ining the t w o edges adjacent to this v ertex in to one edge. H m a m r ( n, 1) is the set of the corresp onding equiv alence classes. As w e see fro m the conditions, all unmark ed v ertices lie on t he lateral surface S 1 × (0 , 1) of the cylinder. As ab ov e, these v ertices are reserv ed for op erations of asso c . The mark ed v ertices lying on the lateral surface S 1 × (0 , 1 ) are reserv ed for co c hains. W e allow forests with no mark ed v ertices lying on the lateral surface S 1 × (0 , 1) . Such forests represen t op erations whic h pro duce a c hain f rom a c hain. Figure 9 giv es an example o f an o p eration o f H (2 , 1) whic h pro duces the c hain ( b 0 , b 1 , b 2 , b 3 ) = ( P a 3 , Q ( a 0 , 1 , a 1 ) , 1 , a 2 ) (4.3) from a degree 0 co chain P , a degree 3 co chain Q and a degree − 3 c hain ( a 0 , a 1 , a 2 , a 3 ) . Mark ed v ertices in this ﬁgure are lab eled b y small circles. The unmark ed 3-v alen t vertex giv es the pro duct of P and a 3 , the t w o unmark ed terminal v ertices giv e units of A and the unmarke d 2-v alen t vertex giv es the iden tity op eratio n on A . The v ertices lying o n the b oundary S 1 × { 1 } are mar ked b y the comp onen ts of the chain ( a 0 , a 1 , a 2 , a 3 ) and the ro ots are mark ed b y the comp o nents of t he ch ain (4.3). It is clear how the op erad H acts on t he pair ( C • ( A ) , C • ( A )) . F rom this action it is also clear how to comp ose the op erations. F or example, the comp osition of op erations from H ( n 1 , 1) and H ( n 2 , 1) corresp onds to putting one cylinder on the top of the other matc hing the ro ots of the ﬁrst cylinder with the v ertices lying on the upp er circle of the second cylinder, and then shrinking the resulting cylinder to the required heigh t. Recall that the op erad H is colored by degrees of the co chains and degrees of the c hains. It is not hard to see tha t H ( n, 0) is a cosimplicial set with resp ect to the degree of the res ulting co c hain and a p olysimplicial set with resp ect to the degrees of the co chains en tering as argumen ts. 21 P S f r a g r e p la c e m e n t s Q P a 0 a 1 a 2 a 3 b 0 b 1 b 2 b 3 Figure 9: An op eration pro duces which the c hain ( 4.3) 22 Similarly , H ( n, 1) is a cosimplicial set with resp ect to the degree of the c ha in en tering as an argumen t and a p olysimplicial set with resp ect to the degrees of the co chains en tering as argumen ts and the degree of the resulting c hain. These p oly-simplicial/cosimplicial structure is compatible with the comp ositions and w e get Deﬁnition 5 The DG op er ad KS is the r e alization of the op er ad H in the c ate gory of chain c o m plexes. It follows f rom the construction that KS is a 2- colored op erad whic h acts o n the pair ( C • norm ( A ) , C norm • ( A )) . It is not hard to see that the op erations ∪ (2.24), [ , ] G (2.25), I (2.26), L (2.27) and B (2.28) come from the action of the op erad KS . Remark 1. The op erad KS with its action on ( C • norm ( A ) , C norm • ( A )) w as in tro duced b y Kontsev ic h and Soib elman in 9 [23] in the case when A is an A ∞ -algebra. Here w e recall the construction of KS in the case when A is simply an asso ciative a lgebra. It is this assumption on A which allo ws us to utilize the natural cosimplicial/simplicial structure on ( C • norm ( A ) , C norm • ( A )) . Remark 2. If we restrict o urselv es t o the subspace of op erations of KS whic h do not in v olv e c hains then w e g et the minimal op erad of Kon tsevic h and Soib elman describ ed in [2 2]. 4.2 The op erad of litt le d iscs on a cylinder A “top ological partner” of K S is the op erad Cyl of discs on a cylinder [23], [29]. As w ell as the op erad of Kontse vic h and So ib elman Cyl is a 2-colored op erad satisfying the pro p ert y (2.15). The spaces Cyl c ( n, 0), n ≥ 1 are t he spaces of the little disc o p erad. T o introduce the space Cyl a ( n, 1) for n ≥ 1 w e consider cylinders S 1 × [ a, c ] for a, c ∈ R , a < c with the natura l ﬂat metric and deﬁne the top olog ical space g Cyl n . A p oin t of the space g Cyl n is a cylinder S 1 × [ a, c ] together with a conﬁguration of n ≥ 1 discs on the lateral surface S 1 × ( a, c ) and a p osition of tw o p oin ts b and t lying o n the b oundaries S 1 × a and S 1 × c , resp ectiv ely . The top olog y on the space g Cyl n is deﬁned in the obvious w a y using the ﬂat metric on the cylinder. The space g Cyl n is equipped with a free action of the gro up S 1 × R . The subgroup S 1 ⊂ S 1 × R simu ltaneously rotates all the cylinders a nd the subgroup R ⊂ S 1 × R acts by parallel shifts S 1 × [ a, c ] → S 1 × [ a + l , c + l ] , l ∈ R . The space Cyl a ( n, 1) fo r n ≥ 1 of the o p erad Cyl is the quotien t Cyl a ( n, 1) = g Cyl n /S 1 × R . (4.4) 9 See sections 11 .1, 1 1.2, and 11.3 in [23]. 23 The space Cyl a (0 , 1) is the space of conﬁguratio ns of tw o (p ossibly coinciding) p oints b and t on the circle S 1 considered mo dulo rotations. Although it is o b vious that Cyl a (0 , 1) is homeomorphic to the circle S 1 w e still deﬁne Cyl a (0 , 1) using the conﬁguration space in order to b etter visualize the op erations o f the op erad. The insertions of t he t yp e Cyl c ( n, 0) × Cyl c ( m, 0) → Cyl c ( n + m − 1 , 0) are deﬁned in the same as for the op erad o f little squares. The op erations of the t yp e Cyl c ( n, 0) × Cyl a ( m, 1) → Cyl a ( n + m − 1 , 1) are insertions o f the conﬁguration of little discs of Cyl c ( n, 0) in a little disc on the la teral surface of a cylinder. Finally the op erations of the type Cyl a ( n, 1) × Cyl a ( m, 1) → Cyl a ( n + m, 1) corresp ond to putting the ﬁrst cylinder under the second one while t he second cylinder is rotated in suc h a w a y that the p oin t b of the second cylinder coincides with the p oin t t of the ﬁrst cy linder. The composition inv olving degenerate conﬁgurations of Cyl a (0 , 1) are deﬁned in the ob vious wa y . T o describe the op erad of homology groups of Cyl w e will need some results ab out the conﬁguration spaces of distinct p oints on the punctured plane R 2 \ { 0 } . Let us denote b y Conf n ( R 2 \ { 0 } ) the conﬁgurat ion space of n distinct p oin ts on the punctured plane R 2 \ { 0 } and consider the following pro jections p k : Conf n ( R 2 \ { 0 } ) → R 2 \ { 0 } , (4.5) p k ( x 1 , . . . , x n ) = x k . Due to E. F adell and L. Neuw irth [14] w e hav e Theorem 1 (Theorem 1, [14]) F or every k = 1 , 2 , . . . , n the map p k is a lo c al l y trivial ﬁbr a tion. Using the ideas o f E. F adell a nd L. Neu wirth [14] w e sho w tha t Prop osition 6 Th e map p : Conf n ( R 2 \ { 0 } ) → Conf n − 1 ( R 2 \ { 0 } ) , p ( x 1 , x 2 , . . . , x n ) = ( x 2 , . . . , x n ) . (4.6) is a lo c al ly trivial ﬁbr ation. F urthermor e, the ﬁb er F n of p i s F n = R 2 \ { 0 , q 2 , . . . , q n } , (4.7) wher e q 2 , . . . , q n ar e n − 1 distinct p oints of the punctur e d plane R 2 \ { 0 } . 24 Pro of. F or the op en unit disc D 1 on R 2 cen tered at the orig in there exists a con tin uous map θ : D 1 × ¯ D 1 → ¯ D 1 (4.8) satisfying the follo wing prop erties: – for all x ∈ D 1 the map θ ( x , ) : ¯ D 1 → ¯ D 1 is a a homeomorphism ha ving ∂ ¯ D 1 ﬁxed. – for all x ∈ D 1 w e hav e θ ( x , x ) = 0 . F or distinct p oin ts q 2 , . . . , q n on the punctured plane R 2 \ { 0 } we c ho ose op en discs D q 2 , D q 3 , . . . , D q n , (4.9) whic h a re cen tered at q 2 , q 3 , . . . , q n , resp ectiv ely . Eac h disc ¯ D q j a v oids the orig in 0 and for eac h i 6 = j ¯ D q i ∩ ¯ D q j = ∅ . Let us denote resp ective ly b y r 2 , . . . , r n the ra dii of the discs (4.9) and let U b b e the follo wing neigh b orho o d of Conf n − 1 ( R 2 \ { 0 } ): U b = { ( x 2 , . . . , x n ) | x 2 ∈ D q 2 , x 3 ∈ D q 3 , . . . , x n ∈ D q n } . (4.10) The desired homeomorphism h from p − 1 ( U b ) → F n × U b is give n b y the formula: h ( x 1 , x 2 , . . . , x n ) =          ( x 1 , x 2 , . . . , x n ) , if x 1 / ∈ n [ j =2 D q j , ( q j + r j θ  x j − q j r j , x 1 − q j r j  , x 2 , . . . , x n ) , if x 1 ∈ ¯ D q j , (4.11) where r j is the radius of the j - th disc D q j . Since the op en subsets of the form U b (4.10) co v er Conf n − 1 ( R 2 \ { 0 } ) the map p (4.6) is indeed a lo cally trivial ﬁbration.  Fib er F n of p (4.6) is homotopy equiv alen t to the w edge sum ∨ n S 1 of n circles. Hence, the homolog y groups of F n (4.7) are H • ( F n , K ) =      K , if • = 0 , K n , if • = 1 , 0 , otherwise . (4.12) Let us sho w that Prop osition 7 Th e fundam ental g r oup π 1 (Conf n − 1 ( R 2 \ { 0 } )) a c ts trivial ly on the homolo gy gr oups H • ( F n , K ) of F n . Pro of. It is o b vious that w e only need to consider the action of π 1 (Conf n − 1 ( R 2 \ { 0 } )) on H 1 ( F n , K ) . T o get the cycles represen ting elemen ts of a basis for H 1 ( F n , K ) w e c ho ose closed discs D 0 , D q 2 , D q 3 , . . . , D q n , (4.13) whic h are cen tered at 0 , q 2 , q 3 , . . . , q n , resp ectiv ely . The discs (4.13) are c hosen in suc h a w a y that their closures ¯ D 0 , ¯ D q 2 , ¯ D q 3 , . . . , ¯ D q n 25 are pairwise disjoin t. The boundaries of the discs (4.1 3) are cycles represen ting the elemen ts of a basis for H 1 ( F n , K ) . Let us iden tify R 2 with the complex plane C and consider the following lo op in Conf n − 1 ( R 2 \ { 0 } ) f ( t ) = ( e 2 π it q 2 , e 2 π it q 3 , . . . , e 2 π it q n ) : [0 , 1] → Conf n − 1 ( R 2 \ { 0 } ) , (4.14) where ( q 2 , q 3 , . . . , q n ) is a ﬁxed collection of the distinct p oints of R 2 \ { 0 } . The lo op (4.14) lif t s to the f o llo wing lo op in Conf n ( R 2 \ { 0 } ) e f ( t ) = ( e 2 π it x 1 , e 2 π it q 2 , e 2 π it q 3 , . . . , e 2 π it q n ) : [0 , 1] → Conf n ( R 2 \ { 0 } ) . (4.15) As w e go around the lo op (4.15) the p oin t x 1 of the ﬁb er F n returns to its original p o sition. Th us the eleme nt [ f ] ∈ π 1 (Conf n − 1 ( R 2 \ { 0 } )) represen ted b y the lo op f (4.1 4) acts trivially on H • ( F n , K ) . Let now g b e an arbitrary lo op in Conf n − 1 ( R 2 \ { 0 } ) . T o ﬁnd the action of the homotopy class [ g ] o n H • ( F n , K ) w e need to lift the map γ ( y , t ) = g ( t ) : F n × [0 , 1] → Conf n − 1 ( R 2 \ { 0 } ) (4.16) to a map e γ ( y , t ) : F n × [0 , 1] → Conf n ( R 2 \ { 0 } ) (4.17) whic h ma kes the dia g ram F n × { 0 } ֒ → Conf n ( R 2 \ { 0 } ) ↓ e γ ր ↓ p F n × [0 , 1] γ → Conf n − 1 ( R 2 \ { 0 } ) (4.18) comm utativ e. T o construct the lift e γ w e divide the segmen t [0 , 1] in to small enough subsegmen ts [ t i , t i +1 ] satisfying the prop ert y g ([ t i , t i +1 ]) ⊂ V , (4.19) where V is an op en subset of Conf n − 1 ( R 2 \ { 0 } ) of the form (4.10) . Then eac h individual lift e γ    F n × [ t i ,t i +1 ] : F n × [ t i , t i +1 ] → Conf n ( R 2 \ { 0 } ) (4.20) can b e constructed using the trivialization (4 .1 1). With this construction in mind w e consider the comp ositions p k ◦ g of g with the pro- jections p k (4.5) fo r k ∈ { 2 , . . . , n } . Since the image p k ◦ g ([0 , 1]) of the segmen t [0 , 1] is a compact subset in R 2 \ { 0 } we ma y c ho ose the disc D 0 in suc h a w a y that the closure ¯ D 0 a v oids the p oints of the images p k ◦ g ([0 , 1]) for all k ∈ { 2 , . . . , n } . Therefore t he lift e γ (4.17) can b e chose n in suc h a w a y that e γ ( y , t ) = y , ∀ y ∈ ¯ D 0 ⊂ F n . Hence the action of the homotopy class [ g ] on the homology class represen ted by the b oundary of the disc D 0 is trivial. 26 Let us examine the lo op p k ◦ g in R 2 \ { 0 } more closely . Since the fundamen tal group of R 2 \ { 0 } is generated b y the homotop y class of the lo op l ( t ) = e 2 π it q k : [0 , 1] → R 2 \ { 0 } around t he origin there exists an integer N ∈ Z suc h that the lo op p k ◦ ( g ∗ f N ) is n ull- homotopic. Here f is the lo o p (4.14) in Conf n − 1 ( R 2 \ { 0 } ) . But the class of f acts trivially o n the homology of the ﬁb er F n . Therefore, without loss of generality , w e ma y assume tha t the comp o sition p k ◦ g is n ull-homoto pic. Th us, due to Theorem 1 of F adell and Neu wirth w e ma y further assume that p k ◦ g is a constan t map p k ◦ g ( t ) ≡ q k . (4.21) In other w ords, the k -t h p oin t x k do es no t mov e as w e go a long the lo op g . Since for eac h j 6 = k the ima g e p j ◦ g ([0 , 1]) of the segmen t [0 , 1 ] is a compact subset in R 2 \ { 0 } we ma y c ho ose the disc D q k in suc h a wa y that the closure ¯ D q k a v oids t he p oints of the images p j ◦ g ([0 , 1]) for all j 6 = k . Th us, using the partition of t he segmen t [0 , 1] satisfying the prop ert y (4.19) and con- structing the lift (4 .17) using the trivializations o f the form (4.1 1) w e see that the lift e γ can b e c hosen in suc h a w a y that e γ ( y , t ) = y , ∀ y ∈ ¯ D q k ⊂ F n . Therefore the action o f the homotopy class [ g ] on t he homology class represen ted b y the b oundary of the disc D q k is trivial. The pro p osition is pro v ed.  Generalizing the result of F. R. Cohen [9] w e get the follo wing Theorem 2 The hom olo gy op er a d H −• (Cyl , K ) of Cyl with the r evers e d gr ading is the op- er ad c alc of c alculi. Pro of. Since the op erad Cyl has t w o colors algebras ov er H −• (Cyl , K ) are pairs of g raded v ector spaces ( V , W ) . The comp onents Cyl c ( n, 0) form the op era d of little discs. Thus , due t o Theorem 1.2 in 10 [9] V is a Gerstenhab er algebra. The space Cyl c (0 , 0) is empt y , the space Cyl c (1 , 0) is a po in t and the space Cy l c ( n, 0) for n > 1 is homoto p y equiv alen t to the space Conf n ( R 2 ) o f conﬁgurations of n distinct p oints on R 2 . The map Cyl c ( n, 0) ∼ → Conf n ( R 2 ) (4.22) whic h establishe s the equiv a lence asso ciates with a conﬁguration of disjoin t discs the conﬁg- uration of their cen ters. The space Conf 2 ( R 2 ) is, in turn, homotop y eq uiv alen t to S 1 . Thus the generator of H 0 ( S 1 ) represen ts the comm utativ e pro duct on V and the generator o f H 1 ( S 1 ) represen ts the brac k et on V . This bra ck et has degree − 1 b ecause we use the r evers ed gra ding on the homology groups. 10 W e only need the op era tions whic h surv ive in c haracteris tic zer o. 27 The op erations without inputs of color c corresp ond to homolog y classes of the space Cyl a (0 , 1) . Since this space is ho meomorphic to the circle S 1 w e ha v e H • (Cyl a (0 , 1) , K ) = ( K , if • = 0 , 1 , 0 , otherwise . The g enerator of H 0 (Cyl a (0 , 1)) corresp onds to the identit y transformation o f W a nd the generator δ o f H 1 (Cyl a (0 , 1)) corresp onds to a unary op eration on W . The operation δ has degree − 1 b ecause we use the rev ersed grading on the homolog y groups. The identit y δ 2 = 0 follo ws immediately from the fact that H 2 (Cyl a (0 , 1) , K ) = 0 . Let us construct a homotopy equiv alence b etw een the space Cyl a ( n, 1) for n ≥ 1 and the space Conf n ( R 2 \ 0 ) × S 1 , (4.23) where Conf n ( R 2 \ 0 ) is the conﬁguration space of n distinct p oin ts on the punctured plane R 2 \ 0 . First, we kill the rotation symmetry by ﬁxing the p osition of the p oint t on the upp er b oundary S 1 × { c } of the cylinder S 1 × [ a, c ] . Second, w e kill translation symmetry b y setting c = 0 . Next we assign to eac h conﬁguration o f discs on the lateral surface S 1 × ( a, 0) the con- ﬁguration of cen ters of the discs. In this w a y w e get a homo t o p y equiv alence b etw een the space Cyl a ( n, 1) and the space Conf n ( S 1 × ( a, 0)) × S 1 , (4.24) where the p oints on the fa ctor S 1 corresp ond t o p ositions of the p oint b on S 1 × { a } . Finally , using t he map χ : S 1 × ( a, 0) → R 2 \ 0 , χ ( ϕ, y ) =  y a − y cos( ϕ ) , y a − y sin( ϕ )  w e get the desired homotop y equiv alence Cyl a ( n, 1) ≃ Conf n ( R 2 \ 0 ) × S 1 . (4.25) Let us consider the homology groups of Cyl a (1 , 1) in more details. The space Cyl a (1 , 1) is homotop y equiv alen t to R 2 \ 0 × S 1 and the latter space is, in turn, homotopy equiv alent to S 1 × S 1 . Th us Cyl a (1 , 1) ≃ S 1 × S 1 . (4.26) 28 P S f r a g r e p la c e m e n t s b t Figure 10: Ho w to get the cycle in Cyl a (1 , 1) represen ting the op eration i δ Therefore H −• (Cyl a (1 , 1) , K ) = K ⊕ s − 1 K 2 ⊕ s − 2 K . T o iden tify the cycles represen ting the homology classes w e parameterize the circle S 1 b y the angle v ariable ϕ ∈ [0 , 2 π ] and the torus S 1 × S 1 b y the pair of angle v ariables ϕ 1 , ϕ 2 ∈ [0 , 2 π ] . The zeroth homology space H 0 (Cyl a (1 , 1) , K ) is one-dimens ional and its generator cor r e- sp onds to the contraction i of elemen ts of V with elemen ts of W . The second homolo gy space H 2 (Cyl a (1 , 1) , K ) is also one-dimensional. Its generator corresp onds to the comp osition δ i δ . The cycle ϕ → (0 , ϕ ) : S 1 ֒ → S 1 × S 1 (4.27) represen ts the homolo gy class corresp onding to the composition i δ . In order to get this cycle in Cyl a (1 , 1) w e need to revolv e the p oin t b on the low er b oundary ab out t he ve rtical axis as it is shown on Figure 10 . The comp osition δ i is, in turn, represen t ed b y t he diagona l ϕ → ( ϕ, ϕ ) : S 1 → S 1 × S 1 (4.28) of the torus. T o get this cycle w e need to r evolv e sim ultaneously the disc and the p oint b ab out t he v ertical axis as it is shown on Figure 11 . The homo lo gy classes δ i a nd iδ form a basis of H 1 (Cyl a (1 , 1) , K ) . W e w ould lik e to remark that the homolog y class represen ted b y the cycle ϕ → ( ϕ, 0) : S 1 ֒ → S 1 × S 1 (4.29) equals to the combin ation δ i − i δ . 29 P S f r a g r e p la c e m e n t s b t Figure 11: Ho w to get the cycle in Cyl a (1 , 1) represen ting the op eration δ i Indeed it is easy t o see tha t t he cycles (4.27), (4.28), and (4.29) form the b oundary of the follo wing 2-simplex in S 1 × S 1 { ( ϕ 1 , ϕ 2 ) | ϕ 1 ≤ ϕ 2 } ⊂ S 1 × S 1 . Th us the cycle (4.29) represen ts the homology class corresp o nding t o the Lie deriv ativ e l . T o get this cycle in Cyl a (1 , 1) w e need to revolv e the disc ab out the v ertical a xis as it is sho wn on Figure 12 . In general, fo r n ≥ 1 t he homology of the space Cyl a ( n, 1) can b e computed with the help of the homological v ersion of Lemma 6.2 from [9] . Due to this lemma w e hav e H • (Conf n ( R 2 \ 0 ) , K ) = n O j =1 H • ( ∨ j S 1 , K ) . (4.30) Using the homotop y equiv alence ( 4 .25) and the K ¨ unneth formula, w e deduce that H • (Cyl a ( n, 1) , K ) = n O j =1 H • ( ∨ j S 1 , K ) ⊗ H • ( S 1 , K ) . (4.31) Let us sho w that the op erad H −• (Cyl , K ) is generated by op erations of H −• (Cyl c (2 , 0) , K ), H −• (Cyl a (0 , 1) , a nd H −• (Cyl a (1 , 1) , K ) . Since the case of the op erad of little discs was already considered by F. Cohen [9] we should only consider the op erations of H −• (Cyl a ( n, 1) , K ) . Due to homotop y equiv alence (4.25) the homology classes of Cyl a ( n, 1) ar e of the forms α ⊗ 1 , (4.32) 30 P S f r a g r e p la c e m e n t s b t Figure 1 2: How to get the cycle in Cyl a (1 , 1) represen ting the op eration l and α ⊗ φ , (4.33) where α ∈ H −• (Conf n ( R 2 \ 0 ) , K ) , 1 is the generator of H 0 ( S 1 , K ) and φ is t he generator of H 0 ( S 1 , K ) . It is ob vious that homology classes of the form (4 .33) are obta ined b y comp osing the homology classes of the form (4.32) with the generator δ of H 1 (Cyl a (0 , 1) , K ) . T o analyze the homology classes (4 .3 2) w e consider the Serre sp ectral sequence corre- sp onding to the ﬁbra t io n (4 .6). D ue to Prop osition 7 the E 2 term of the sequence is E 2 p,q = H p (Conf n − 1 ( R 2 \ 0 ) , K ) ⊗ H q ( F n , K ) , (4.34) where F n is the ﬁb er (4 .7) of p . Since F n is homotopy equiv alen t to the w edge ∨ n S 1 of n circles equation (4.30) implies that the v ector spaces M p E 2 p, •− p and H • (Conf n ( R 2 \ 0 ) , K ) hav e the same dimension. Th us, using the fact that sp ectral sequence cor r esp o nding to the ﬁbration (4.6) con v erges to H • (Conf n ( R 2 \ 0 ) , K ) , w e deduce that this spectral sequence degenerates at E 2 and H • (Conf n ( R 2 \ 0 ) , K ) = M p + q = • H p (Conf n − 1 ( R 2 \ 0 ) , K ) ⊗ H q ( F n , K ) (4.35) Using equation (4.12) we reduce this expression further to H • (Conf n ( R 2 \ 0 ) , K ) = 31 H • (Conf n − 1 ( R 2 \ 0 ) , K ) ⊗ H 0 ( F n , K ) ⊕ H •− 1 (Conf n − 1 ( R 2 \ 0 ) , K ) ⊗ H 1 ( F n , K ) . (4.36) Let v 1 , v 2 , . . . , v n ∈ V and w ∈ W b e the arg uments of an op eration corresp o nding to a homology class in (4.36). It is not hard to see that the generator of H 0 ( F n , K ) = K corresp onds to the con traction i with v 1 and the generators of H 1 ( F n , K ) = K n corresp ond 11 to the brac k ets [ v 1 , v j ] for j ∈ { 2 , 3 , . . . , n } and the Lie deriv ativ e l v 1 along v 1 . Th us the homo lo gy classes of H −• (Cyl a ( n, 1) , K ) are all pro duced b y the op eradic comp o- sitions of the classes in H −• (Cyl a ( n − 1 , 1) , K ) , H −• (Cyl a (1 , 1) , K ) , and H −• (Cyl c (2 , 0) , K ) . This inductiv e argumen t allo ws us to conclude that the op erad H −• (Cyl , K ) is generated b y the classes i ∈ H 0 (Cyl a (1 , 1) , K ) , l ∈ H 1 (Cyl a (1 , 1) , K ) , δ ∈ H 1 (Cyl a (0 , 1) , K ) , ∧ ∈ H 0 (Cyl c (2 , 0) , K ) , [ , ] ∈ H 1 (Cyl c (2 , 0) , K ) . (4.37) Using equation (2.23) and the fact [21] that dim Lie ( n ) = ( n − 1)! it is not hard to sho w that calc a ( n, 1) ∼ = calc a ( n − 1 , 1) ⊗ ( K ⊕ s − 1 K n ) as graded v ector spaces. On the other hand, equation (4.31) giv es us the same isomorphism H −• (Cyl a ( n, 1) , K ) ∼ = H −• (Cyl a ( n − 1 , 1 ) , K ) ⊗ ( K ⊕ s − 1 K n ) of graded v ector spaces for H −• (Cyl , K ) . Therefore, the dimensions of graded comp onents of H −• (Cyl a ( n, 1) , K ) and calc a ( n, 1) are equal. Th us, in order the complete t he pro of o f the theorem, w e need to sho w the op erat ions (4.37) satisfy the iden tities of calculus algebra (see D eﬁnition 3). The iden tit ies of the Gerstenhab er algebra w ere already c hec ked in [9]. The iden tit y (2.19) was c hec k ed ab ov e. On Figure 13 we sho w ho w to c hec k the identit y [ i, l ] = i [ , ] . (4.38) The remaining iden tities can b e c hec k ed in the similar wa y . Theorem 2 is prov ed.  4.3 Re quired results from [23] W e will need Theorem 3 (M. Kontsevic h, Y. Soib elman, [23]) The op er ad KS is quasi - i s o morphic to the op er ad of singular ch ains of the top o lo g ic a l op er ad Cyl . The homol o gy op er ad H −• ( KS , K ) is gener ate d by the classes of the op er ations ∪ (2.24), [ , ] G (2.25), I (2.26), L (2.27) a n d B (2.28) . F urthermore, (See Prop osition 11 .3 .3 on pa ge 50 in [23]) 11 This picture is very reminiscent of the consider a tion of Ho chschild-Serre sp ectra l sequence in the pro o f of Prop osition 4.1 in [28]. 32 P S f r a g r e p la c e m e n t s b b b b t t t t + ∼ ∼ Figure 1 3: How to c hec k iden tity (4.38) 33 Prop osition 8 Th e op er ad of si n gular chains of the top olo gic al op er ad Cyl is formal. Com bining these t w o statemen ts with Theorem 2 we get the follow ing corollary . Corollary 1 The p air ( C • norm ( A ) , C norm • ( A )) is a homotopy c alculus algebr a. The op er ations of this algebr a structur e ar e exp r ess e d in terms of op er ations of H . The induc e d c alculus structur e on the p air ( H H • ( A ) , H H • ( A )) c oincides with the one in [11] . 4.4 A useful pr op ert y of the op erad KS F or the 2-colored op erad K S of c hain complexes w e ha v e the following Prop osition 9 Th e elements of K S c ( k , 0 ) have de gr e es deg ≥ 1 − k (4.39) and the elemen ts of KS a ( k , 1 ) have de gr e es deg ≥ − 1 − k . (4.40) Pro of. Let us start with KS c ( k , 0 ) fo r k = 1 . Op erations o f KS c (1 , 0) pro duce a Ho chs c hild co c hain from a Ho chsc hild co c hain. In order to prov e that these op erations ha v e no nnega t ive degrees w e need to show that the op eratio ns from H (1 , 0) which low er the n um b er the n um b er of arguments of the co c hain do not con tribute to the realization of H . Let T be a tree represen ting an op eration from H (1 , 0) and let v a b e the v ertex adjacen t to the ro ot. If v a is marke d then it is mark ed b y the only co c hain P whic h en ters as an argumen t. All other mark ed vertice s are necessarily terminal and they are reserv ed for the argumen ts of the co c hain pr o duced by the op eration. In order to low er the n um b er of argumen ts, we need to insert the unit into the co chain P . The insertion of the unit is a degeneracy of the simplicial structure o n H (1 , 0) . Hence all op erations f r om H (1 , 0) whic h low er the degree of the co c hain do not contribute to the realization. If v a is unmarke d then, starting with the mark ed v ertex v P reserv ed for the co c hain P , w e can form the prop er maximal subtree with v P b eing the v ertex adjacen t t o the ro ot. In order to con tribute to the realization the op eration corresp onding to this subtree has to ha v e a nonnegative degree. Hence, so do es the op eration corresp o nding to the whole tree. W e pro v ed (4.39) fo r k = 1 . Let us tak e it as a base of the induction and a ssume that (4.39) is pro v ed for all m < k . W e consider a tree T whic h represen ts an op eration from H ( k , 0) and denote the v ertex adjacen t to the roo t of T by v a . Let us consider the case whe n the v ertex v a is mark ed. Sa y , v a is reserv ed for a co c hain P 1 of degree q 1 . Then the tree T has exactly q 1 maximal prop er subtrees whose ro ot v ertex is v a . W e denote these subtrees by T 1 , T 2 , . . . , T q 1 . The n um b er q 1 splits into the sum q 1 = p n + p y , (4.41) where p n is the n um b er of the subtrees with no v ertices reserv ed for co c hains a nd p y is the n um b er of the subtrees in whic h at least one v ertex is reserv ed for a co ch ain. 34 Let P denote the co chain pro duced by the o p eration in qu estion and let r by t he n um b er of a rgumen ts of this co c hain. W e will ﬁnd an estimate f o r r using the inductiv e assumption. Ev ery subtree with no v ertices reserv ed for co c hains has t o giv e at least one ar gumen t for the co c hain P . Otherwise , w e ha v e to insert the unit as an argumen t of the co c hain P 1 . In this case t he op eration in question is a obtained from anot her op eratio n by degeneracy . Therefore this op eration w ould not con tribute to the realization o f H . If T j is a subtree with exactly k j v ertices reserv ed for co c hains then, ob viously , k j < k . Hence, applying the assumption of the induction, w e get that the num ber of argumen ts of P coming from T j is greater or equal (1 − k j ) + q j , where q j is the t otal degree of all co c hains en tering a s argumen ts of the op eratio n corre- sp onding to the subtree T j . Th us the n um b er r of the arguments of the co chain P pro duced b y the operat io n in question can b e estimated b y r ≥ p n + p y X j =1 (1 − k j + q j ) . (4.42) This inequalit y can b e rewritten as r − ( p n + p y + p y X j =1 q j ) ≥ − p y X j =1 k j . Due to equation (4.4 1) the sum p n + p y + p y X j =1 q j is the total degree of all co c hains en tering as argumen ts of the op eratio n. F urthermore, since the ve rtex v a is reserv ed for o ne of the co chains p y X j =1 k j = k − 1 and the desired inequality (4.39) is prov ed in this case. Let us no w consider the case when the v ertex v a is unmark ed. The v alency of t he v ertex v a has to b e at least 2 . Otherwise t he op eration will ha v e the empt y set of argumen ts. If the v alency of this v ertex is 2 then w e remo v e it using the equiv alence transformation. Th us, without loss of generalit y , we ma y assume that the v ertex v a has at least 2 incoming edges. Let us denote by s the num ber of t hese incoming edges and let T 1 , . . . , T s b e the maximal prop er subtrees of T whose ro ot v ertex is v a . If the v ertices reserv ed for co c hains b elong to only one of these subtrees T i then excising the subtrees T j for j 6 = i w e get ano ther op eration whose degree is less or equal the degree of the original op eratio n. Since in the new tree the unmarked v ertex v a has the v alency 2 w e may remo v e this ve rtex by the corresp onding equiv alence transformation. 35 If in this mo diﬁed tree the v ertex adjacent t o the ro ot is mark ed then w e deduce the desired inequality to the case considered ab o v e. Otherwise, w e should only consider the case whe n the v ertex v a is unmark ed, its v alency is at least 3 and eac h maximal prop er subtree T j of T with ro ot v ertex v a has at least one v ertex reserv ed for a co chain. Let s b e the n um ber of the maximal prop er subtrees of T whose ro ot v ertex is v a . Since the n umber of these subtrees is greater or eq ual than 2 therefore ev ery subtree T j represen ts an op erat io n with the num b er of argumen ts k j < k . Let deg( T j ) b e the degree of the op erat ion corresp onding to the j -th subtree T j . Applying the assumption of the induction we get the inequalit y deg( T j ) ≥ 1 − k j . (4.43) It is clear that the degree deg( T ) of the op eration correspo nding to the tree T is the sum of degrees of the op erations corresp onding to the subtrees T 1 , . . . , T s . Therefore deg( T ) ≥ s X i =1 (1 − k i ) . On the other hand s X i =1 k i = k . There fore, deg( T ) ≥ s − k and inequalit y (4.39) follo ws fr om the fact that s ≥ 2 . T o prov e the second inequalit y (4.40) w e denote b y H m a m r ( k , 1 ) the set of op erations pro- ducing a c hain ( b 0 , b 1 , . . . , b m r ) (4.44) in C − m r ( A ) fr om k co c hains and a c hain ( c 0 , c 1 , . . . , c m a ) (4.45) in C − m a ( A ) . Let F b e a forest on the cylinder (4.2) represen ting an op eration from H m a m r ( k , 1 ) whic h con tribute to the realization of H ( k , 1) . Our purp ose is to pro v e the inequalit y − m r ≥ − m a + q − 1 − k , (4.46) where q is the total degree of a ll k co c hains o f the op eration. This inequality is equiv alen t to m a ≥ m r + q − 1 − k . (4.47) By construction the forest F has exactly m r trees. Let us denote these trees b y S 1 , . . . , S m n r , T 1 , . . . , T m y r where the trees S 1 , . . . , S m n r ha v e no vertic es reserv ed for co chains and eac h tree T i has at least one v ertex reserv ed for a co c hain. Ob viously , m r = m n r + m y r . (4.48) 36 The ro ots of the trees S 1 , . . . , S m n r , T 1 , . . . , T m y r are mark ed b y compo nents of the c hain (4.44). If t he ro ot of the tree S i is mark ed by the comp onen t b j of the for j 6 = 0 then S i has to hav e at least one terminal v ertex marked b y a comp onen t of the c hain (4.4 5). Otherwise w e ha v e to inse rt the unit as the j -th component of (4.44) for j 6 = 0 . In this case the op eration in question is a comp osition of the another op eration and a degeneracy . Hence, this op eration w ould not contribute to the realization of H ( k , 1) . Let us denote by m i the n um b er of the terminal vertice s of the tree T i mark ed b y com- p onen ts of the c hain (4.45). If the tree T i has exactly k i v ertices reserv ed for the co c hains then T i represen ts an op eration from H ( k i , 0) . F urthermore, if the o p eration corresp onding to the forest F contributes to t he realization then so do es the op eration corresp o nding t o the tree T i . Henc e, the num ber m i can b e estimated using the inequalit y (4 .39) m i ≥ q i + 1 − k i , where q i is the total degree of all co chains of the op eration correspo nding to the tree T i . Th us we get the follow ing inequalit y for m a m a ≥ ( m n r − 1) + m y r X i =1 ( q i + 1 − k i ) , (4.49) where the ﬁrst term ( m n r − 1) in the right hand side comes f r om estimate of the n um b er of the marke d terminal v ertices of the tr ees S 1 , . . . , S m n r . Inequalit y (4.49) can b e rewritten as m a ≥ m n r + m y r + q − 1 − k , (4.50) where q is the total degree of a ll k co c hains o f the op eration in question. Due to equation (4.48) inequalit y (4.50) coincides exactly with the desired inequalit y (4.47). The pro p osition is pro v ed.  5 The homot op y calculus on the pair ( C • norm ( A ) , C norm • ( A )) . Since the op erad Λ Lie + δ is a subop erad of calc , Corollary 1 implies that the pair ( C • norm ( A ) , C norm • ( A )) carries a natural H o ( Λ Lie + δ )-algebra structure. In this section w e show that the homotop y calculus on ( C • norm ( A ) , C norm • ( A )) can be mo diﬁed in su c h a w a y that its H o (Λ Lie + δ )-algebra part b ecomes the Λ Lie + δ -algebra structure giv en by the op erations [ , ] G (2.25), L (2 .2 7), and B (2.28). Due to Prop osition 2 a homotopy calculus structure on the pa ir ( C • norm ( A ) , C norm • ( A )) is a Maurer-Cartan elemen t Q ∈ Co der ′ ( F B (C • norm (A) , C norm • (A))) . (5.1) In other w ords, Q is a degree 1 co deriv ation of the coalgebra F B ( C • norm ( A ) , C norm • ( A )) satisfying the condition Q    C • norm ( A ) ⊕ C norm • ( A ) = 0 , 37 and the equation [ ∂ B ar + ∂ H och , Q ] + Q ◦ Q = 0 , (5.2) where C • norm ( A ) ⊕ C norm • ( A ) is considered as subspac e of F B ( C • norm ( A ) , C norm • ( A )) via coaug- men tation and ∂ H och is the diﬀeren tial coming from the Ho chs c hild (co)b oundary o p erator on ( C • norm ( A ) , C norm • ( A )) . It is con v enien t to reserv e a notatio n for the Lie algebra Co der ′ ( F B (C • norm (A) , C norm • (A))) L = Co der ′ ( F B (C • norm (A) , C norm • (A))) . (5.3) Prop osition 1 implies that the co deriv at io n Q is uniquely determined b y its comp osition with the corestriction ρ : F B ( C • norm ( A ) , C norm • ( A )) → C • norm ( A ) ⊕ C norm • ( A ) q = ρ ◦ Q : F B ( C • norm ( A ) , C norm • ( A )) → C • norm ( A ) ⊕ C norm • ( A ) , (5.4) while equation (5.2) is equiv alen t to [ ∂ H och , q ] + q ◦ ∂ B ar + q ◦ Q = 0 . (5.5) The coalg ebra F B ( C • norm ( A ) , C norm • ( A )) is equipp ed with a nat ur a l increasing ﬁltration 12 C • norm ( A ) ⊕ C norm • ( A )[[ u ]] = F 1 F B ( C • norm ( A ) , C norm • ( A )) ⊂ F 2 F B ( C • norm ( A ) , C norm • ( A )) ⊂ . . . F m F B ( C • norm ( A ) , C norm • ( A )) c = M k ≤ m B c ( k , 0 ) ⊗ S k ( C • norm ( A )) ⊗ k , F m F B ( C • norm ( A ) , C norm • ( A )) a = M k +1 ≤ m B a ( k , 1 ) ⊗ S k ( C • norm ( A )) ⊗ k ⊗ C • ( A ) , (5.6) where u is an auxiliary v ariable of degree − 2 . Using (5.6) w e endo w the Lie alg ebra (5.3) with the follo wing decreasing ﬁltration L = F 0 L ⊃ F 1 L ⊃ F 2 L ⊃ . . . F m L = { Y ∈ L | Y    F m F B ( C • norm ( A ) ,C norm • ( A )) = 0 } . (5.7) Since L = lim m L / F m L the Lie subalgebra F 1 L 0 is pronilp oten t. Therefore F 1 L 0 in tegrates to a pro unip oten t group G = exp( F 1 L 0 ) (5.8) whic h a cts on the Maurer-Cartan elemen ts of L . This action is deﬁned b y the form ula: exp( Y ) Q = exp([ , Y ]) Q + f ([ , Y ]) [ ∂ B ar + ∂ H och , Y ] , (5.9) where f is the p o w er series of the function f ( x ) = e x − 1 x 12 See Equatio n (3.11). 38 at the p oin t x = 0 . Let Q b e a Maurer-Cartan elemen t of the DG LA L whic h corresp onds to the homotopy calculus structure on the pair ( C • norm ( A ) , C norm • ( A )) which comes from the action of the op erad KS . F or eve ry Y ∈ F 1 L the homotop y calculus on the pair ( C • norm ( A ) , C norm • ( A )) correspond- ing to the Maurer-Cartan elemen t exp( Y ) Q is quasi-isomorphic to the homotopy calculus corresp onding the original Maurer-Cartan elemen t Q . Indeed, the desired Ho ( calc )-quasi- isomorphism is expressed in terms of Y as exp([ , Y ]) : ( F B ( C • norm ( A ) , C norm • ( A )) , Q ) → ( F B ( C • norm ( A ) , C norm • ( A )) , exp( Y ) Q ) . (5.10) Th us w e get a family of m utually quasi-isomorphic homotop y calculus structures on ( C • norm ( A ) , C norm • ( A )) . Let us denote this family b y S calc . W e claim that Theorem 4 The family S calc c o n tains a homotopy c alculus structur e whose H o (Λ Lie + δ ) - algebr a p art is the Λ Lie + δ -algebr a structur e given by the op er a tions [ , ] G (2.25), L (2 . 27), and B (2.28). Pro of. According to Prop osition 2 and equation (3.26) the Ho (Λ Lie + δ ) is giv en b y a Maurer- Cartan elemen t M o f the DG LA Co der ′ ( F Λ( Lie + δ ) ∨ (C • norm (A) , C norm • (A)) ) . (5.11) Due t o Prop o sition 1 this Maurer- Cartan elemen t is, in turn, uniquely determined b y the comp osition with the corestriction ρ on to C • norm ( A ) ⊕ C norm • ( A ) m = ρ ◦ M : F Λ( Lie + δ ) ∨ ( C • norm ( A ) , C norm • ( A )) → C • norm ( A ) ⊕ C norm • ( A ) . (5.12) Finally , Proposition 3 shows that the map m is r elat ed to the map q (5 .4) by the equation m = q    F Λ( Lie + δ ) ∨ ( C • norm ( A ) ,C norm • ( A )) , (5.13) where Λ( Lie + δ ) ∨ is considered as a sub-co op erad of B = B ar ( calc ) via the c hain of em b ed- dings: Λ( Lie + δ ) ∨ ֒ → B ar (Λ Lie + δ ) ֒ → B ar ( calc ) . The Ho (Λ Lie )-algebra structure on C • norm ( A ) is induced by the Ho ( e 2 )-algebra structure whic h, in turn, comes fro m the action of the minimal o p erad of Kontse vic h and Soib elman [22] on C • norm ( A ) . It w as pro v ed in [1 3] (see Theorem 2) that this Ho ( Lie )-algebra is in f act a gen uine Lie a lgebra giv en b y [ , ] G . Th us it remains to take care ab o ut the op erations inv olving a c hain. Due to (3.2 7) all the op erations of the Ho (Λ Lie + δ )-algebra in v olving a c hain are com bined in to a single degree 1 map m a = m    F Λ 2 co comm + ( C • norm ( A ) ,C norm • ( A )[[ u ]]) a : F Λ 2 co comm + ( C • norm ( A ) , C norm • ( A )[[ u ]]) (5.14) → C norm • ( A ) . 39 suc h t ha t m a    C norm • ( A ) = 0 . (5 .15) The latter equation fo llo ws from the fa ct that the coderiv ation M b elongs to the DGLA (5.11). In other w ords, we hav e the inﬁnite collection of op erations: m a k ,n : S k ( C • norm ( A )) ⊗ C norm • ( A ) → C norm • ( A ) (5.16) of degrees 1 − 2 k − 2 n , where S k ( C • norm ( A )) is t he k -t h comp onen t of the sy mmetric algebra S ( C • norm ( A )) o f C • norm ( A ) . Due to Prop osition 9 the op eration m a k ,n v anish if k + 2 n > 2 . F urthermore, due to equation (5.15) w e ha v e m a 0 , 0 = 0 . Thus w e need to analyze only there op erations: one unary op eration m a 0 , 1 : C norm • ( A ) → C norm • ( A ) (5.17) of degree − 1 , one binary op eration m a 1 , 0 : C • norm ( A ) ⊗ C norm • ( A ) → C norm • ( A ) (5 .18) of degree − 1 and one ternary op eration m a 2 , 0 : S 2 ( C • norm ( A )) ⊗ C norm • ( A ) → C norm • ( A ) (5.19) of degree − 3 . Theorems 2 and 3 imply that the op eratio n (5.1 7) diﬀers from Connes’ op erator B b y a n exact op era t io n. Namely , m a 0 , 1 ( c ) = B ( c ) + ∂ H och β ( c ) − β ( ∂ H och c ) , where β is an op eration in KS β : C norm • ( A ) → C norm • ( A ) of degree − 2 . Using Prop osition 9 w e deduce that β is zero. Hence m a 0 , 1 = B . Due to Prop osition 3 the op eration (5.1 8) is express ed in terms o f q (5.4) as m a 1 , 0 ( P , c ) = q ( b 1 ) , (5.20) where P ∈ C • norm ( A ) , c ∈ C norm • ( A ) , and t he elemen t b 1 ∈ F B ( C • norm ( A ) , C norm • ( A )) is de- picted on Figure 14. Theorems 2 and 3 imply that m a 1 , 0 diﬀers from the actio n L of co c hains on chains b y an exact op era t io n. In other words, m a 1 , 0 ( P , c ) = − ( − 1) | P | L P c + ∂ H och ψ ( P , c ) − ψ ( ∂ H och P , c ) − ( − 1) | P | ψ ( P , ∂ H och c ) , (5.21 ) where | P | is the degree of P and ψ : C • norm ( A ) ⊗ C norm • ( A ) → C norm • ( A ) 40 P S f r a g r e p la c e m e n t s b 1 b 2 P P c c = = l i Figure 1 4: Elemen ts b 1 and b 2 P S f r a g r e p la c e m e n t s b 3 b 4 P P c c = = l δ i δ Figure 1 5: Elemen ts b 3 and b 4 is an op eration in KS a (1 , 1) of degree − 2 . W e remark that ψ may b e considered as a map ψ : Λ( Lie + ) ∨ (1 , 1) ⊗ C • norm ( A ) ⊗ C norm • ( A ) → C norm • ( A ) (5.22 ) of degree 0 . Our purpo se is to extend ψ “b y zero” to the whole ve ctor space of the coalgebra F B ( C • norm ( A ) , C norm • ( A )) . This extension depends on the c hoice of basis in calc a (1 , 1) . W e c ho ose the basis { l , i, i δ, l δ } (5.23) and extend ψ to B a (1 , 1) ⊗ C • norm ( A ) ⊗ C norm • ( A ) ψ : B a (1 , 1) ⊗ C • norm ( A ) ⊗ C norm • ( A ) → C norm • ( A ) , (5.24) as ψ ( b 1 ) = ψ ( P , c ) , ψ ( b 2 ) = ψ ( b 3 ) = ψ ( b 4 ) = 0 , ψ ( b λ ) = 0 , where the elemen ts b 1 , b 2 , b 3 , b 4 , b λ ∈ B a (1 , 1) ⊗ C • norm ( A ) ⊗ C norm • ( A ) are depicted on Figures 14, 15, 16 , λ is an arbit r a ry elemen t of the basis ( 5.23), a nd P ∈ 41 P S f r a g r e p la c e m e n t s b λ δ δ δ δ δ P c = λ Figure 1 6: The n um b er of δ ’s is ≥ 1 42 C • norm ( A ) , c ∈ C norm • ( A ) . Next w e extend ψ by zero to the whole v ector space of the coalgebra F B ( C • norm ( A ) , C norm • ( A )) = M n B c ( n, 0) ⊗ S n ( C • norm ( A )) ⊗ n ⊕ M n B a ( n, 1) ⊗ S n ( C • norm ( A )) ⊗ n ⊗ C norm • ( A ) . (5.25) Then according to Prop osition 1 the equation ρ ◦ Ψ = ψ deﬁnes a deriv ation Ψ o f t he coalgebra (5.25). The deriv atio n Ψ has degree 0 since so do es the map ψ . F urthermore, it is ob vious that Ψ ∈ F 1 L . Applying the elemen t exp( − Ψ) of the g roup G (5.8) to the Maurer-Cartan elemen t Q (5.1) w e adjust the comp onen t m a 1 , 0 b y killing this additional exact term ∂ H och ψ ( P , c ) − ψ ( ∂ H och P , c ) − ( − 1) | P | ψ ( P , ∂ H och c ) in (5.21) . In doing this we do not change the unary op erations b ecause Ψ ∈ F 1 L . Th us we are left with o nly one non- v anishing op eration (5.19) . The Maurer-Carta n equation (5.5) implies that m a 2 , 0 should b e closed with resp ect to the diﬀeren tial ∂ H och . Since the degree of m a 2 , 0 is − 3 , using Theorems 2 a nd 3 , w e deduce that up to ∂ H och -exact terms the op eration m a 2 , 0 is made of the follo wing “ building blo c ks”: L [ P 1 ,P 2 ] G B c , L P 1 L P 2 B c , L P 2 L P 1 B c , where P 1 , P 2 ∈ C • norm ( A ) a nd c ∈ C norm • ( A ) . Using the symmetry in the argumen ts P 1 , P 2 and the compatibility with ∂ H och it is not hard to sho w that (up to ∂ H och -exact terms) the most general express ion for m a 2 , 0 is m a 2 , 0 ( P 1 , P 2 , c ) = ( − 1) | P 1 | µL [ P 1 ,P 2 ] G B c , (5.26) where µ ∈ K , P 1 , P 2 ∈ C • norm ( A ) a nd c ∈ C norm • ( A ) . If necessary , w e apply the ab o v e tric k with the action (5.9) of the group (5 .8) to mo dify Q (5.1) so that equation (5.2 6) indeed ho lds. T o kill m a 2 , 0 w e will in tro duce the map y : B a (1 , 1) ⊗ C • norm ( A ) ⊗ C norm • ( A ) → C norm • ( A ) . (5.27) This map is deﬁned b y t he equations y ( b 1 ) = − µL P B c , y ( b 2 ) = y ( b 3 ) = y ( b 4 ) = 0 , y ( b λ ) = 0 , where the elemen ts b 1 , b 2 , b 3 , b 4 , b λ ∈ B a (1 , 1) ⊗ C • norm ( A ) ⊗ C norm • ( A ) 43 are depicted on Figures 14, 15, 16, λ is an arbitrary elemen t of the basis (5.23) in calc a (1 , 1) and P ∈ C • norm ( A ) , c ∈ C norm • ( A ) . Then w e extend y b y zero to the whole v ector space of t he coalgebra (5.25). It is not hard to see that y is of degree 0 . According to Prop osition 1 the equation ρ ◦ Y = y deﬁnes a degree 0 co deriv ation Y of the coalgebra (5.25) . F urthermore, [ ∂ H och , Y ] = 0 , (5.28) and Y ∈ F 1 L . (5.29) Applying the elemen t exp( Y ) of the group G (5.8) to the Maurer-Cartan elemen t Q (5.1) we get another homotop y calculus structure on ( C • norm ( A ) , C norm • ( A )) . This homotop y calculus structure is determined b y the Maurer-Carta n elemen t exp( Y ) Q . Let us denote b y ˜ m a 0 , 1 , ˜ m a 1 , 0 , and ˜ m a 2 , 0 the o p erations (5.17), (5.18), (5.19) of the H o (Λ Lie + δ )-algebra corresponding to the new homotop y calculus exp( Y ) Q . Since Y ∈ F 1 L the unary op eration cannot c hange. Th us ˜ m a 0 , 1 = m a 0 , 1 = B . F or the binary op eration w e hav e ˜ m a 1 , 0 ( P , c ) = ρ ◦ exp( Y ) Q ( b 1 ) , (5.30) where ρ is the pro jection fro m F B ( C • norm ( A ) , C norm • ( A )) on to C • norm ( A ) ⊕ C norm • ( A ) and the elemen t b 1 ∈ F B ( C • norm ( A ) , C norm • ( A )) a is depicted on F ig ure 14. Using (5.9) , (5.28), and (5.29) w e simplify equation (5.30) as fo llows ˜ m a 1 , 0 ( P , c ) = m a 1 , 0 ( P , c ) + [( q ◦ Y − y ◦ Q ) + y ◦ ∂ B ar ] b 1 . It is ob vious that ∂ B ar b 1 = 0 . F urthermore, it is not hard t o see that q ◦ Y ( b 1 ) = 0 and y ◦ Q ( b 1 ) = 0 . Th us the binary op eration (5.18) is also unc hanged. F or the ternary op eration ˜ m a 2 , 0 w e ha v e ˜ m a 2 , 0 ( P 1 , P 2 , c ) = ρ ◦ exp( Y ) Q ( b ) = = m a 2 , 0 ( P 1 , P 2 , c ) + ( q ◦ Y − y ◦ Q )( b ) + y ◦ ∂ B ar ( b ) − y ◦ ∂ B ar ◦ Y ( b ) + 1 2 y ◦ Y ◦ ∂ B ar ( b ) , where the elemen t b ∈ F B ( C • norm ( A ) , C norm • ( A )) a is depicted on Figure 17 and, as a b ov e, w e used (5.28) and (5.2 9). It is obv ious that ∂ B ar ( b ) = 0 and it is not hard t o c hec k tha t ∂ B ar ◦ Y ( b ) = 0 . F urther- more a direct computation sho ws tha t ( q ◦ Y − y ◦ Q )( b ) = − ( − 1) | P 1 | µL [ P 1 ,P 2 ] G B c . Th us ˜ m a 2 , 0 = 0 (5.31) and Theorem 4 is pro v ed.  44 P S f r a g r e p la c e m e n t s [ , ] − P 1 P 1 P 1 P 2 P 2 P 2 c c c l l l l l ( − 1) | P 1 | − ( − 1) | P 1 || P 2 | Figure 1 7: Here P 1 , P 2 ∈ C • norm ( A ) a nd c ∈ C norm • ( A ) 45 6 F ormalit y the orem 6.1 En v eloping algebra of a Gerstenhab er algebra Let ( V , W ) b e a pair of g r aded ve ctor spaces. F or o ur purp o ses we will need to consider calc -algebras on ( V , W ) with a ﬁxed Gerstenhaber algebra structure ( V , ∧ , [ , ]) on V . F or suc h c alc -algebras w e call W a c alc -m o dule over ( V , ∧ , [ , ]). The category of calc -mo dules ov er V is equ iv alen t to a category of o rdinary mo dules o v er the en veloping a lg ebra [30] of the Gerstenhaber algebra V . In this section we recall the construction of this env eloping alg ebra and describe its prop erties. Let us start with the fo llo wing deﬁnition: Deﬁnition 6 F or a Gerstenhab er algebr a ( V , ∧ , [ , ]) we deﬁ n e a n asso ciative algebr a Y 0 ( V ) which is gener ate d by two sets of symb ols l v , i v v ∈ V (6.1) of de gr e es | i v | = | v | , | l v | = | v | − 1 . (6.2) These symb ols ar e K -line ar in v and they ar e subje ct to the fol lowin g r ela tion s i v 1 · v 2 = i v 1 i v 2 , [ i v 1 , l v 2 ] = i [ v 1 ,v 2 ] , l v 1 · v 2 = l v 1 i v 2 + ( − 1) | v 1 | i v 1 l v 2 , [ l v 1 , l v 2 ] = l [ v 1 ,v 2 ] . (6.3) F urthermore, Deﬁnition 7 ([30]) If ( V , · , [ , ]) is a Gerstenhab er a l g e br a then the ass o ciative alge b r a Y ( V ) is gen er a te d by symb ols (6 .1) a n d an element δ of de gr e e − 1 . The symb ols (6.1) ar e K -line ar in v a n d they ar e subje c t to the fol lowin g r elations δ 2 = 0 , [ δ , i v ] = l v , i v 1 · v 2 = i v 1 i v 2 , [ i v 1 , l v 2 ] = i [ v 1 ,v 2 ] . (6.4) It is obvious that the category of calc -mo dules ov er V is equiv alent to the category of ordinary mo dules ov er the asso ciativ e algebra Y ( V ) . Let us giv e the follo wing natural deﬁnition Deﬁnition 8 A DG c om mutative algebr a V is c al le d r e gular if the m o dule Ω 1 ( V ) of its K¨ ahler diﬀer e n tials is c o ﬁbr ant. Remark. Notice t ha t if V is a comm utativ e algebra concen trated in degree 0 then the ab ov e condition of regularit y means that the V - mo dule Ω 1 ( V ) is pro jective . W e claim that Prop osition 10 L et V b e a D G Gerstenhab er algebr a and R → V b e its c oﬁbr a n t r esolution. If the c orr esp onding DG c ommutative algebr a V is r e gular then the induc e d map Y ( R ) → Y ( V ) (6.5) is a quasi-isom orphism of DG asso ciative algebr as. 46 Pro of. Due to the ob vious equalit y of c hain complexes Y ( V ) = Y 0 ( V ) ⊕ Y 0 ( V ) δ (6.6) it suﬃces to sho w that the map Y 0 ( R ) → Y 0 ( V ) (6.7) is a quasi-isomorphism. F or this purp ose we in tro duce the fo llo wing Lie-Rinehart algebra structure [24] on the pair ( V , Ω 1 ( V )) , where Ω 1 ( V ) is the mo dule of K¨ ahler diﬀerentials of the D G comm utativ e algebra V . The Lie brack et { , } on Ω 1 ( V ) and the a ction l of Ω 1 ( V ) on V are deﬁned in terms of the Lie brac k et o n V as follows { a 1 da 2 , b 1 db 2 } = ( − 1) | a 2 | +1 a 1 [ a 2 , b 1 ] db 2 + ( − 1) ( | a 2 | +1) | b 1 | a 1 b 1 d ([ a 2 , b 2 ])+ (6.8) ( − 1) ( | a 1 | + | a 2 | +1)( | b 1 | + | b 2 | +1)+ | b 2 | b 1 [ b 2 , a 1 ] da 2 , l a 1 da 2 ( b ) = ( − 1) | a 2 | +1 a 1 [ a 2 , b ] . (6.9) The identities of a Gerstenhab er algebra imply that equations (6.8) and (6.9) indeed deﬁne a Lie-Rinehart algebra on the pair ( V , Ω 1 ( V )) . Next w e remark that for ev ery (DG) Gerstenhab er algebra V the asso ciativ e alg ebra Y 0 ( V ) is nothing but the env eloping algebra of the Lie-R inehart alg ebra ( V , Ω 1 ( V )) . Indeed, the required isomorphism is deﬁned on generators as a 7→ i a , db 7→ l b , a, b ∈ V . Then the PBW-ﬁltration on Y 0 ( V ) is V ∼ = F 0 Y 0 ( V ) ⊂ F 1 Y 0 ( V ) ⊂ F 2 Y 0 ( V ) ⊂ . . . (6.10) where F k Y 0 ( V ) is spanned b y monomials in which the n um b er of sym b ols l v , v ∈ V is less or equal to k . Since the DG comm utativ e a lg ebra V is regular w e can apply the PBW-theorem [24] t o the Lie-Rinehart algebra ( V , Ω 1 ( V )) . Usin g this theorem (see Theorem 3.1 in [24]) w e conclude that the asso ciated graded algebra is isomorphic to the symmetric algebra S V (Ω 1 ( V )) M k F k Y 0 ( V ) / F k − 1 Y 0 ( V ) ∼ = S V (Ω 1 ( V )) . (6.11 ) Since R is a coﬁbrant resolution of V t he same arg umen t with PBW theorem from [24] w orks for Y 0 ( R ) . The map (6.7) is ob viously compatible with the ﬁltrations (6.10) on Y 0 ( R ) and Y 0 ( V ) . F urthermore, these ﬁltra t ions are co complete Y 0 ( V ) = colim k F k Y 0 ( V ) , Y 0 ( R ) = colim k F k Y 0 ( R ) . Hence, in order to prov e that the map (6.7) is a quasi-isomorphism, w e need to show that so is the map S R (Ω 1 ( R )) → S V (Ω 1 ( V )) . (6.12) This statemen t follows from the regularit y of V . Th us the prop osition is pro v ed.  47 6.2 Shea v es of Ho c hsc hild (co)c hains on an algebraic v ariet y Let X b e a smo oth algebraic v ariet y o v er K with the structure sheaf O X . W e denote b y V • X the sheaf of p olyv ector ﬁelds and b y Ω −• X b e the sheaf of exterior fo rms with reve rsed grading. D X denotes the sheaf of diﬀeren tial op erato rs on X and D Ω −• X denotes the sheaf of diﬀeren tial o p erators o n the sheaf of (graded) comm utativ e alg ebras Ω −• X . In the aﬃne case X = Sp ec(A) w e will use the short-hand notation for the corresp onding mo dules of global sections V • ( A ) = Γ( X , V • X ) , Ω −• ( A ) = Γ( X , Ω −• X ) , D ( A ) = Γ( X , D X ) , and ﬁnally D (Ω −• ( A )) = Γ( X , D Ω −• X ) . The pair ( V • X , Ω −• X ) is a calculus algebra with respect t o the op erat ions: the exterior pro duct ∧ on V • X , the Sc houten-Nijenh uis brac k et [ , ] S N on V • X , the contraction I of a form with a p olyv ector, the Lie deriv ative L of a form a long a p olyv ector, and ﬁnally the de Rham diﬀeren tial d on the exterior forms. Using the con t raction I w e deﬁne the natural O X -linear pa ir ing h , i b etw een the sheav es V • X and Ω −• X h , i : V • X ⊗ O X Ω −• X → O X h γ , η i = ( I γ η , if | η | = −| γ | , 0 , otherwise . (6.13) Here γ and η are lo cal sections of V • X and Ω −• X , resp ectiv ely . An appropria t e v ersion of Ho c hsc hild co c hain complex for the structure sheaf O X is the sheaf of p olydiﬀeren tial op erato rs [27], [34]. W e will denote this sheaf b y C • ( O X ) . F or example C 0 ( O X ) is the structure sheaf O X and C 1 ( O X ) is the sheaf D X of diﬀeren tial op erators o n X . Let us a lso denote b y C • norm ( O X ) the sheaf of normalized p olydiﬀeren tial op erators. These are the p olydiﬀeren tial op erators satisfying the prop ert y P ( . . . , 1 , . . . ) = 0 . Similarly an appropriate v ersion of Hochs c hild c hain complex for the structure sheaf O X is the sheaf of p olyjets [31]: C • ( O X ) = H om O X ( C −• ( O X ) , O X ) , (6.14) where H om denotes the she af- Ho m and C • ( O X ) is conside red with its natural left O X - mo dule structure. F or example C 0 ( O X ) is the structure sheaf O X and C − 1 ( O X ) is the sheaf of ∞ - j ets. There are natural a nalogs of the degenerate Ho c hsc hild c hains c = ( c 0 , . . . , 1 , . . . ) and these degenerate c hains form a subsheaf C degen • ( O X ) of C • ( O X ) . F urthermore the sub- sheaf C degen • ( O X ) is closed with respect to the Ho c hsc hild b oundary op erator ∂ H och . W e deﬁne the sheaf C norm • ( O X ) of normalized Ho c hsc hild c hains as the quotien t sheaf C • ( O X ) /C degen • ( O X ) . It is not hard to show that C norm • ( O X ) = H om O X ( C −• nor m ( O X ) , O X ) . (6.15) 48 As w ell as f or Ho c hsc hild complexes of an asso ciativ e algebra the inclusion C • norm ( O X ) ֒ → C • ( O X ) and the pro jection C • ( O X ) → C norm • ( O X ) are quasi-isomorphism s of complexes of shea v es. F urthermore, the action of the Ko n tsevic h- Soib elman op erad KS on the pair of shea v es ( C • norm ( O X ) , C norm • ( O X )) is w ell-deﬁned 13 . Th us the pair ( C • norm ( O X ) , C norm • ( O X )) is a sheaf of homotopy calculi. Let us recall that the em b edding  c : V • X ֒ → C • norm ( O X ) (6.16) is called the Ho c hsc hild-Kostan t-Rosen b erg map. It is kno wn [20] that  V is a quasi- isomorphism of complexes of shea v es wh ere the sheaf V • X is considered with the zero dif- feren tial. The corresp o nding quasi-isomorphism fo r Ho c hsc hild c hains  a : C norm • ( O X ) → Ω −• X (6.17) is called the Connes-Ho chs c hild-Kostant-Rosenberg map. This map is deﬁned b y the equa- tion h γ ,  a ( c ) i = c (  c ( γ )) , (6.18) where c is a lo cal section of C norm − m ( O X ) , γ is a lo cal section of V m X and t he pairing h , i is deﬁned in (6.13). It is kno wn [10] that the maps  c and  a are compatible with the op erations of the Cartan calculus on the pair ( V • X , Ω −• X ) and the op eratio ns ∪ (2.24), [ , ] G (2.25), I (2.2 6), L (2.27) and B (2 .2 8) on the pair ( C • norm ( O X ) , C norm • ( O X )) up to homo topy . W e upgrade this observ ation to the follo wing theorem. Theorem 5 If X is a smo oth algebr aic variety over a ﬁeld K of char acteristic zer o then the she af ( C • norm ( O X ) , C norm • ( O X )) of hom otopy c alculi i s quasi-isomorp h ic to the she a f ( V • X , Ω −• X ) of c a lculi. The pro o f of this theorem is give n in Subsection 6.4. 6.3 Morita equiv alence In this subsection w e will sho w that the sheaf of a lg ebras Y ( V • X ) is Morita equiv alen t t o the sheaf D X [ d ] / ( d 2 ) where d is an auxiliary v ariable of degree − 1 comm uting with all the diﬀeren tial o p erators. First we remark that Y 0 ( V • X )-mo dule structure on the sheaf Ω −• X giv es us a natural map Y 0 ( V • X ) → D Ω −• X (6.19) b et w een the shea v es of asso ciativ e algebras. W e claim that 13 Obvious ex tensions of the opera tio ns on Ho chsc hild chains to the op erations on p olyjets is discussed in details in [2]. 49 Prop osition 11 Th e map (6.1 9) is an isomorphis m of she aves of asso ciative algebr as. Pro of. W e need to show that ( 6 .19) gives us an isomorphism on stalks Y 0 ( V • X ) x → ( D Ω −• X ) x for ev ery p oint x ∈ X . Th us, since X is smo oth, it suﬃces to show that the map Y 0 ( V • ( A )) → D Ω −• ( A ) (6.20) is an isomorphism for ev ery lo cal regular (comm utativ e) algebra A o v er K . It is easy to see that the asso ciative algebra Y 0 ( V • ( A )) is generated by sym b ols: i a , i v , l a , l v , (6.21) where a ∈ A and v ∈ V 1 ( A ) . Under the map (6.2 0) the sym b ols go to i a → I a , i v → I v , l a → L a , l v → L v . Since the ima g es of the sy m b o ls (6.21) satisfy the same relations t herefore the map (6.20) is injectiv e. T o sho w that (6.2 0) is surjectiv e w e remark that the algebra D Ω −• ( A ) is generated b y diﬀeren tial o p erators o f the form a · , d b · , a, b ∈ A and deriv a t io ns Der K (Ω −• ( A )) of Ω −• ( A ) . Since a · = I a and d b · = L b it remains to show that ev ery deriv ation W ∈ Der K (Ω −• ( A )) b elongs to the imag e of the map (6 .20). The r egula r ity of A implies that the A -mo dules Ω 1 ( A ) a nd Ω −• ( A ) a re free. More pre- cisely , if x 1 , . . . , x n is a regular system of par ameters in A then the A -module Ω 1 ( A ) is freely generated by the 1 -forms d x i , i = 1 , 2 , . . . , n , (6.22) and the A -mo dule Ω −• ( A ) is freely generated b y the forms dx i 1 dx i 2 . . . dx i k , 1 ≤ i 1 < i 2 < · · · < i k ≤ n . (6.23) Dually t he A -mo dule V 1 ( A ) = Der K ( A ) is freely generated b y e 1 , e 2 , . . . , e n , (6.24) where e i is a deriv atio n of A deﬁned by the equation I e i ( dx j ) = δ j i . Since the A -mo dule Ω −• ( A ) is freely generated by the fo r ms (6.23) therefore eve ry deriv a- tion W ∈ Der K (Ω −• ( A )) is uniquely determined b y its v alues on the elemen ts o f A and the 1-forms (6.2 2) . In general w e hav e W ( a ) = X 1 ≤ i 1 0 there exists the collection of maps Υ q 1 (6.75) for q < m satisfying equation (6.77) for q < m − 1 . Then, due to equation (6.77) for q = m − 2 the map ˇ ∂ Υ m − 1 1 is closed with resp ect to the diﬀeren tial ∂ G : ∂ G ( ˇ ∂ Υ m − 1 1 ) = 0 . But ˇ ∂ Υ m − 1 1 ∈ Hom  D X [ d ] / ( d 2 ) ⊗ O X , ˇ C m ( G − m )  . Th us, using the fact that the cohomology of the stalk G x is concen trated in degree 0 we conclude that there exists the next map Υ m 1 in (6.75) satisfying equation (6.77) fo r q = m − 1 . W e pro v ed the existence of the map Υ 1 in (6.7 1) satisfying equation (6.72) for k = 0 . No w we pro ceed b y induction on k in (6.71) and (6.72). Let us assume that Υ k (6.71) are constructed for k < m and equation (6.72) holds for k < m − 1 . Then equation ( 6.72) for k = m − 2 implies that t he elemen t D H och Υ m − 1 ∈ Hom  ( D X [ d ] / ( d 2 )) ⊗ ( m − 1) ⊗ O X , ˇ C • ( G )  (6.78) is closed with resp ect to the diﬀeren tial ∂ G + ˇ ∂ . Since the sheaf ˇ C • ( G ) is acyclic with resp ect to the functor of global sections the map e ν (6.63) induces the quasi-isomorphism b et w een the c hain complex Hom  ( D X [ d ] / ( d 2 )) ⊗ ( m − 1) ⊗ O X , ˇ C • ( G )  (6.79) and the c hain complex Hom  ( D X [ d ] / ( d 2 )) ⊗ ( m − 1) ⊗ O X , ˇ C • ( O X )  . (6.80) 61 It is ob vious that the cohomology of the latter complex is concen trated only in non-negativ e degrees. On the other hand the co cycle (6.78) has the negative degree − m + 1 . Hence there exists the next map Υ m satisfying equation (6.72) for k = m − 1 . Prop osition 16 is pro v ed.  Th us the sheaf (6 .64) of D X [ d ] / ( d 2 )-mo dules is quasi-isomorphic to t he sheaf G = D X ⊗ O X V • X [ d ] / ( d 2 ) ⊗ Y ( F e 2 (Ξ X )) B R a . Com bining this observ a tion with Prop osition 14 we see t ha t the shea v es Ω −• X and Y ( V • X ) ⊗ Y ( F e 2 (Ξ X )) B R a are quasi-isomorphic as shea v es of Y ( V • X )-mo dules. Theorem 5 is prov ed.  7 Applicatio n s and generalizati o ns Let, as a b ov e, X b e a smo oth algebraic v ariety ov er a ﬁeld K of c haracteristic zero. The homotop y calculus algebra on the pair ( C • norm ( X ) , C norm • ( X )) giv es us a comm + -mo dule structure o n the pair ( V • X , Ω −• X ) . Theorem 3 implies that this comm + -mo dule structure on ( V • X , Ω −• X ) is giv en b y the ∧ -pro duct of polyve ctor ﬁelds and con traction of p o lyve ctors with forms. According to [27] and [34] the Ho ch sc hild cohomolog y H H • ( X ) o f the v ariet y X is the h yp ercohomology of the sheaf C • norm ( O X ): H H • ( X ) = H • ( C • norm ( O X )) . F urthermore, according to [6], the Ho c hsc hild ho mology H H • ( X ) of the v ariety X is the h yp ercohomology of the sheaf C norm • ( O X ) H H • ( X ) = H • ( C norm • ( O X )) . Th us, using Theorem 5 w e get the follo wing generalization of Corollary 2 from [1 3] Corollary 2 F or every sm o oth algebr aic variety X over a ﬁ e ld K of cha r acteristic zer o the comm + -algebr as ( H • ( X , V • X ) , H • ( X , Ω −• X )) and ( H H • ( X ) , H H • ( X ) ) ar e is o morphic.  62 This statemen t is the existence part of Caldara ru’s conjecture [7] on the Ho chs c hild structure of a smo o t h algebraic v ariet y . The cohomological part of this conjecture w as pro v ed in [4]. As far as w e kno w, D. Calaque, C. Rossi, and M. V an den Bergh are curren tly writing a n article [3] with a pro of of homological part of Caldararu’s conjecture. Com bining Theorem 4 with Theorem 5 we deduce the statemen t of cyclic formality conjecture ( see Conjecture 3.3.2 in [31]) from [31 ] fo r a n arbitr ary smo oth algebraic v ariety o v er a ﬁeld K of characteristic zero: Corollary 3 (T. Willwac her, [33]) If X is a smo oth algebr a i c variety a ﬁ e l d K of char- acteristic zer o then the she af of Λ Lie + δ -algebr as ( C • ( O X ) , C • ( O X )) is formal.  Remark. Strictly sp eaking the metho ds used b y T. Willw ach er in [33] require an additional assumption R ⊂ K . Theorems 4 and 5 a llo w us to remov e the assumption R ⊂ K from the statemen t of Corollary 3. The pro of of Theorem 5 can b e easily mo diﬁed for the follo wing t w o cases: • X is complex manifold with O X b eing the sheaf of holomorphic functions, • X is a real manifold with O X b eing the sheaf of C ∞ functions. Th us we get the follow ing ob vious mo diﬁcation of Theorem 5 Theorem 6 If X is a c o m plex manifold (r esp. r e al mani f o ld) with O X b ei ng the she af of holomorphic functions (r esp. the she af of C ∞ r e al functions) then the she af ( C • norm ( O X ) , C norm • ( O X )) of homotopy c alculi is quasi-isom orphic to the she af ( V • X , Ω −• X ) of c alculi. F or C ∞ real case w e a lso get the following statemen t Corollary 4 If X is a r e al m anifold with O X b ei ng the she af of C ∞ functions then the homotopy c alculus algebr a  Γ( X , C • norm ( O X )) , Γ( X , C norm • ( O X ))  is quasi-isomorphic to the c alculus algebr a  Γ( X , V • X ) , Γ( X , Ω −• X )  . Pro of. In the C ∞ real case the c hain of quasi-isomorphisms connecting the shea v es ( C • norm ( O X ) , C norm • ( O X )) and ( V • X , Ω −• X ) consists of soft shea v es. Hence, a pplying the f unctor Γ( X , ) of global sections w e get t he desired result.  W e w ould lik e to men tion recen t pap ers [5] and [8]. In pap er [8] A. Cattaneo and G. F elder consider the DG Lie algebra mo dule C C − −• ( O X ) of negativ e cyclic chains o v er the 63 DGLA C • ( O X ) of Ho c hsc hild co c hains on a C ∞ real manifold equipp ed with a v olume form. Using an in teresting mo diﬁcation of the P oisson sigma mo del A. Cattaneo and G . F elder construct a curious L ∞ morphism (not a quasi-isomorphism!) from this DG Lie alg ebra mo dule to a DG Lie algebra mo dule mo deled on p olyv ector ﬁelds using the v olume fo rm. A. Cattaneo and G. F elder also apply this result to a construction of a speciﬁc trace on the deformation quan tization algebra of a unimo dular P oisson manif o ld. Although this trace can b e constructed using the f ormalit y quasi-isomorphism for Ho c hsc hild c hains [25], [33] the relation of the L ∞ morphism of A. Cattaneo and G . F elder to the formality quasi- isomorphism is a my stery . P ap er [5] is dev oted t o t he pro of of Kontse vic h’s cyclic formality conjecture for co chains form ulated in pap er [26]. W e susp ect that the statemen t of this conjecture ma y b e related to Theorem 5 and Corollary 3 via t he V an den Bergh dualit y [32] b et w een Ho c hsc hild coho- mology and Ho c hsc hild homology . References [1] J.M. Boardman a nd R.M. V ogt, Homotop y in v ariant algebraic structures on top olog ical spaces, L ecture Notes in Mathematics, 347 , Springer - V erlag, Berlin - New Y ork, 1973. [2] D. Calaque, V. D olgushev, and G. Halb out, F o rmalit y theorems for Ho chsc hild c hains in the Lie algebroid setting, J. Reine Angew. Math. 612 (2 007) 8 1–127; arXiv:math/0504372 [3] D. Calaque, C. Rossi, and M. V an den Bergh, in preparation. [4] D. Calaque and M. V an den Bergh, Ho c hsc hild cohomology and Atiy ah classes, [5] D. Calaque and T. 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Formality of the homotopy calculus algebra of Hochschild (co)chains

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