Proof of Swiss Cheese Version of Delignes Conjecture

PR OOF OF SWISS CHEESE VERSIO N OF DELIGNE’S C ONJECTURE V.A. DOLGUSHEV, D.E. T AMARKIN, AND B.L. TSYGAN T o the b e autiful c ountry of Confo e der atio Helvetic a. Abstra ct. F or an associative algebra A we consider the pair “the Ho c hschild co chain complex C • ( A, A ) and the algebra A ”. There is a natural 2-colored op erad whic h acts on t his p air. W e sho w that this operad is qu asi-isomorphic to th e singular chain op erad of V orono v’s Swiss Cheese operad. This statement is th e Swiss Cheese versi on of th e Deligne conjecture form ulated by M. Kontsevic h in [22]. 2000 MSC: 18D50, 16E40. Contents 1. In tro d uction 2 1.1. Remarks on higher dimensional v ersions 3 1.2. Organization and la y out of the p ap er 4 2. The op erad Ø of n atural op erations on the ob jects C n ( A, A ) , n = 0 , 1 , 2 , . . . ; A ) 5 3. Tw o com b in atorial descriptions of the op erad Ø 7 3.1. Finite ordinals instead of natural n umbers 7 3.2. Planar trees 7 3.3. Replacing trees with sequences 10 3.4. The whole op erad Ø in terms of sequences 13 3.5. Op eradic structure on s Ø 13 4. Review of 2-op erads 15 4.1. Colored 2-operads 16 4.2. Unital colored 2-operads 17 4.3. Symmetrization 18 4.4. Batanin’s theorem 19 5. Swiss Cheese (SC) Op erads 19 5.1. Symmetric Swiss Cheese t yp e op erads 20 5.2. SC 2-operads 21 5.3. Desymmetrization 23 5.4. Symmetrization 23 6. Linking the op erad s Ø with 2 op erads: a 2-op erad seq 24 6.1. 2-op erad seq (cochain part) 24 6.2. SC v ersion of seq 25 6.3. T otaliz ation: A d g 2-o p erad | seq | and a dg op er ad | s Ø | 25 6.4. Extension to the SC-setting: a dg SC 2-operad SC seq and an SC op e rad | s Ø | 26 7. The SC 2-o p erad br and the SC op erad braces 27 7.1. An increasing ﬁltration on the colo r ed 2-oper ad seq 27 7.2. Extension of the ﬁltration on to SC seq 27 1 2 V.A. DOLGUSHEV, D.E. T AMARKIN, AND B.L. TSYGAN 7.3. Compatibilit y of the ﬁltration with the op eradic structure 28 7.4. Deﬁnition of the (SC) 2-operad br 31 7.5. An increasing ﬁltration on s Ø 32 7.6. Deﬁnition of the (SC) op erad braces 34 8. Pro of of Theorem 7.9 35 8.1. Pro of of Prop osition 8.1 37 9. Pro of of Theorem 2.1 46 App end ix 49 References 51 1. Introduction The in terest to v arious ve r s ions [6], [8], [18], [20 ], [23], [24], [25], [27], [31 ], [34], [36] of the Deligne conjecture on Ho chsc hild complex is motiv ate d by generalizations [11], [12], [30], [33], [35] of the famous K on tsevic h’s form alit y th eorem [21]. Thus, in recent prepr in t [24 ] M. Kon tsevic h and Y. Soib elman prop ose d a pro of of the c hain v ersion of Deligne’s conjecture f or Hochsc hild complexes of an A ∞ -algebra. T his is an imp ortan t step in pro ving the form alit y f or the homotop y calculus algebra of Hochsc hild (co)c hains [12]. Let A b e an asso ciativ e alg ebr a and C • ( A, A ) b e the Hoc hschild co c hain complex of A . The original ve rs ion o f D eligne’s co n j ecture says that the op er ad of nat u r al op erations on C • ( A, A ) is qu asi-isomorphic to the sin gu lar c hain op erad of the op erad E 2 of little discs [10], [26]. This statemen t is not very precise b ecause there are diﬀerent choi ces of what one ma y call “the op erad of natural op erations on C • ( A, A ) .” One ma y u s e the so-called min im al op erad of M. K ontsevic h and Y. Soib elman [23] or the op erad of br aces [16], [19] as in [27] an d [37] or the “big op erad” of M. Batanin and M. Markl [5]. Due to works of v arious p eople [5], [8], [23], [27], [32], and [37] it is no w kno wn that all these op erads are quasi-isomorphic to the singular c hain op erad of the op er ad E 2 . The top ologica l op erad E 2 of little discs admits a n atural extension to a 2-co lored to p ological op erad whic h is calle d the S wiss Cheese oper ad SC 2 . This op erad was p r op osed by A. V orono v in [38]. In [38] A. V orono v also describ ed the h omology op erad H −• (SC 2 ) . More precisely , he sho wed that an alge br a o v er the op erad H −• (SC 2 ) is a pair of graded v ector spaces ( V 1 , V 2 ), where V 1 is a Gerstenhab er algebra 1 , and V 2 is an asso ciativ e alge b ra equipp ed with a mo d ule stru cture ov er the comm utativ e algebra V 1 (1.1) V 1 ⊗ V 2 → V 2 , satisfying the follo wing condition (1.2) ( u 1 · v 1 ) . . . ( u n · v n ) = ( u 1 . . . u n ) · ( v 1 . . . v n ) , where u i ∈ V 1 , v i ∈ V 2 , and for the m ultiplication of the corresp ondin g elements we u se either the asso ciativ e algebra structure in V 2 or the comm utativ e algebra stru cture in V 1 . It is not hard to pr o v e the follo win g prop osition: Prop osition 1.1. If A is an asso ciative algebr a and H H • ( A, A ) is its Ho chschild c ohomolo gy then the p air ( H H • ( A, A ) , A ) forms an algebr a over the op er ad H −• ( SC 2 ) . 1 In particular, it means that V 1 is a commutativ e algebra. PROOF OF SWISS CHEESE VERSION OF DELIGNE’S CONJECTURE 3 Pr o of. Ind eed the a sso ciativ e algebra stru cture on A is already giv en. H H • ( A, A ) is a Gerstenhab er algebra due to [14 ]. Finally , to deﬁne th e mo du le str ucture on A ov er th e comm utativ e algebra H H • ( A, A ) we use the fact that the zeroth Ho c hschild cohomology H H 0 ( A, A ) is the cen ter Z ( A ) of A . Namely , we declare z · a = ( z a , if z ∈ H H 0 ( A, A ) = Z ( A ) , 0 , otherwise . Equation (1.2) is nontrivial only when u i ∈ H H 0 ( A, A ). In this case the required condition is automatica lly satisﬁed since u i ’s are elemen ts of the cen ter Z ( A ) of A .  In this pap er we pro ve the Swiss C heese version of Deligne’s conjecture wh ic h extend s Prop osition 1.1 to th e level of cochai n s. T o form ulate this v ersion of Deligne’s conjecture we , ﬁrst, construct a 2-colored DG op erad Λ of natural op erations on the pair ( C • ( A, A ); A ) . Roughly sp eaking, this op er ad is generated b y the insertions of a co chain into a co chain, the cup-pro duct of co chains an d th e in s ertions of elemen ts of the alge b r a A in to a co c hain. The precise description of Λ is give n in Section 2. The main result of th is pap er is th e follo wing theorem Theorem 1.2. The 2-c olor e d DG op er ad Λ of natur al op er ations on the p air ( C • ( A, A ) , A ) is quasi- isomorph ic to the singular chain op er ad of V or onov’s Swiss Che ese op er ad SC 2 . The induc e d action of the homol o gy op er ad H −• ( SC 2 ) on the p air ( H H • ( A ) , A ) r e c overs the one fr om Pr op osition 1.1 . W e prov e this theorem using ideas from [32] and Batanin’s theorem [2] whic h iden tiﬁes the homotop y t yp e of V orono v’s Swiss Cheese op erad with that of the symmetrization of a contrac tible coﬁbran t S wiss Cheese t yp e 2-op erad . T h e requ ir ed facts ab out 2-op erads are reviewe d in Sections 4,5 1.1. Remarks on higher dimensional v ersions. V oronov’s Swiss Ch eese op erad admits the ob vious higher dimensional analogue SC d ( d ≥ 2) . This op erad extends the op erad of d -c u b es in the same w a y as the op erad SC 2 extends the op er ad of little disk s . F rom this p oint of view, Theorem 1.2 is a 2-dimensional case of the follo wing conjecture formulated b y M. Kont sevic h in [22]: the DG op er ad of natur al op er ations on the p air “ a d -algebr a 2 and its Ho chschild c omplex” is quasi-isomorphic to the singular chain op er ad of SC d +1 . In [22, Section 2.5] M. Kon tsevic h also conjectures that th e Ho c hsc h ild co c hain complex of a d -algebra is a ﬁnal ob ject in an approp r iate catego ry of “Swiss Cheese algebras”. In our p ap er, this question ab out universalit y is n ot addressed. In [13] J .N.K. F r an cis show ed that an appropriate d eformation complex for a d -algebra A is an extension of its Ho chsc hild complex by A . In the s p irit of this resu lt the ab o ve version of Deligne’s conjecture can b e reformulat ed as follo ws: the DG op er ad of natur al op er ations on the deformation c omp lex of a d -algebr a is quasi-isomorphic to the singular chain op er ad of SC d +1 . Notation a nd con ven tions. W e denote b y k the ground ﬁ eld and b y “(co)c hain complexes” w e mean (co)c hain complexes of v ector spaces o ver k . A is a unital associativ e algebra o v er k and C • ( A, A ) is th e n ormalized Hochsc hild co c hain complex of A w ith co eﬃcien ts in A (1.3) C • ( A, A ) = hom(( A/ k ) ⊗ • , A ) . The abbreviation SMC stands for “symmetric monoidal categ ory” and the notation 1 is reserv ed for the u nit of a symmetric monoidal cate gory . W e also u se the abbr eviation SC for “Swiss Cheese 2 Recall from [17 ] that a d -algebra is an algebra ov er the homology op erad H −• ( E d ) of t he op erad of little d -cub es E d . 4 V.A. DOLGUSHEV, D.E. T AMARKIN, AND B.L. TSYGAN t yp e” when w e discuss the Swiss Cheese t yp e s y m metric op erads, 2-op erads, sets, ordinals, and 2-trees. 1.2. Organization and la yo ut of the pap er. All argument s of the p ap er can b e restricted onto the setting w hen w e only care ab out op erations on C • ( A, A ) (and n ot on the p air ( C • ( A, A ) , A ). Throughout th e pap er we use terms lik e ’non SC part’ or ’co c hain p art’ to in dicate that w e r estrict to C • ( A, A ) only . The exp osition is organized so that most of the constructions are ﬁrst introduced in the non SC s etting and then extended to the whole SC picture. A s a rule, this SC extension is rather straightfo r w ard. In our exp osition we tried to isolate th e sp ots dealing with the SC setting; w e hop e that the reader in terested in proving Deligne’s conjecture only will b e able to easily recognize these sp ots and d rop th em without an y harm to understanding. Let us no w go o ver the con ten t of the pap er. W e start (Sec 2) w ith deﬁnin g an op erad Ø of natural op erations on the inﬁnite coll ection of ob jects (1.4) C n ( A, A ) , n = 0 , 1 , 2 , . . . ; A. Next, we explain h ow, using the functors of p olysimplicial/cosimplicial totaliz ation (whic h are called c ondensation in [4]), we can conv ert the op erad Ø in to a dg op erad | Ø | whic h acts on the pair of complexes: C • ( A, A ) and A . The op erad | Ø | is th e same as the op erad Λ in Theorem 1.2. In Sec 3 we giv e a com binatorial description of Ø in terms of trees and then reformulate it in terms of sequences. The latt er descirption is used in the rest of the p ap er. Next, we inv ok e Batanin’s 2-op erad th eory: in Sec 4 we review the basic notions of the theory and in S ec 5 w e discuss an SC version of these notions (also due to Batanin). Th is section is not needed for the co c hain (Deilgne’s) part of the SC conjecture. In Section 6 w e deﬁne a 2-op eradic v ersion seq of the op erad Ø. In Section 6.3 we apply the totaliza tion (=condensation) pro cedur e to the op erads seq and s Ø. As a result we get an op erad | s Ø | acting on the complex C • ( A, A ) as w ell as its 2-op eradic v ersion | seq | . A t this moment the adv an tage of the 2-op eradic approac h can b e seen: the 2-op erad | seq | turns out to b e cont r actible, con trary to | s Ø | . W e conclude the s ection with extending the ab o v e menti oned constructions to the SC setting. W e obtain a con tractible SC 2-op erad | SC seq | whic h acts on the pair ( C • ( A, A ) , A ) . If this op erad satisﬁed a tec hn ical condition of b eing r e duc e d , Batanin’s theory wo uld imply an action of V oronov’s SC op erad on ( C • ( A, A ) , A ). But | SC seq | happ ens to b e n on-reduced whic h causes us to ﬁnd a reduced cont r actible s ub-op erad br of | SC seq | , see Sec 7. Using a similar appr oac h w e also construct a sub op erad braces of | s Ø | . The action of the op erad | braces | on C • ( A, A ) seems to b e equiv alen t to th e celebrated br ac e structur e on C • ( A, A ) ([15], [16], [19]). Batanin’s theory can n o w b e app lied to | br | ; w e get an action on ( C • ( A, A ) , A ) of a certain op erad E which is homotop y equiv alent to V orono v’s SC op erad (the op er ad E is the symmetrization of a coﬁbran t resolution of br , i.e. E := sym R br , see (7.25 )). It also foll ows that this ac tion passes through the action of braces that is we ha ve a map of op erads E → braces . W e pr o v e that this map is a w eak equiv alence, see Th eorem 7.9; th e p ro of of this theorem o ccupies the whole Sec 8. W e are no w ready for pro ving th e SC conjecture (Sec. 9). There is an App endix whic h con tains a certain con tractibilit y s tatement needed for proving Lemma 7.2. Ac kno wledgment. W e would lik e to thank M. Batanin and J. Bergner for useful discussions. W e also thank anon ymous referees for carefully reading the pap er and many u s eful remarks and suggestions. A b ig part of this wo r k w as done when V.D. w as a Boas Assistan t Professor of Mathematics Department at North we stern Univ ersity . During these t wo yea r s V.D. b eneﬁted f rom PROOF OF SWISS CHEESE VERSION OF DELIGNE’S CONJECTURE 5 w orking at Northw estern so muc h th at he feels as if he ﬁ nished one more graduate school. V.D. cordially than k s Mathematics Departmen t at North w estern Univ ersit y for th is time. Th e results of this w ork w ere pr esented at the famous Su lliv an’s Einstein Chair S eminar. W e w ould lik e to thank th e p articipan ts of this semin ar for questions and useful comments. W e esp ecially thank D. Su lliv an for his r emarks which m otiv ated us to rewrite the form ulation of our main result in this p ap er. D.T. and B.T. are supp orted by NSF gran ts. V.D. is supp orted by th e NSF gran t DMS 0856196, Regen t’s F acult y F ello ws hip, and the Gran t for Supp ort of Scienti ﬁ c Schools NSh- 8065.2 006.2. A p art of this work wa s done when V.D. liv ed in Ir vine and participated in th e v anp o ol program to comm ute to the UCR campus. V.D. w ould lik e to thank T ran s p ortation and P arking Services of the UC R iverside for their work. 2. The ope rad Ø of n a tural opera tions on the obje cts C n ( A, A ) , n = 0 , 1 , 2 , . . . ; A ) Let A b e a unital monoid in s ome tensor (not necessarily symm etric) category (for example, in the cate gory of complexes o v er a ﬁeld). Consider the full nonsym metric end omorphism op erad of A C A ( n ) := hom( A ⊗ n ; A ) , n ≥ 0 . It is clear that A is n aturally a C A -algebra. The asso ciativ e unital s tr ucture on A giv es rise to a map of nonsymmetric op erads asso c → C A , where asso c is the nons ymmetric op erad of sets con trolling un ital monoids; eac h space assoc ( n ), n ≥ 0, is a p oin t. W e ﬁx a set of colors X ρ := N ⊔ { a } and deﬁn e a X ρ -colored symmetric op erad Ø in the category of sets as an op erad whose alg ebr a structure on an X ρ -family of ob j ects ( C ( n ) , n ∈ N ; A ) is: — a nonsymmetric op erad structure on the collection of ob jects C ( n ); — a map of n onsymmetric op erads asso c → C ; — a C -algebra stru cture on A . The op erad Ø has the follo w ing sets of op erations: — Ø( k ) n n 1 ,n 2 ,...,n k := Ø (( n 1 , n 2 , . . . , n k ) 7→ n )) where all the entries are in N ; — Ø( k, N ) n 1 ,n 2 ,...,n k := Ø(( n 1 , n 2 , . . . , n k , a , a , . . . a | {z } N ) 7→ a ) , N ≥ 0 . The op eradic sets for other colorings are empt y . The sets Ø( k ) n n 1 ,n 2 ,...,n k form a N -colored op erad in the ob vious wa y . Call this op er ad the c o chain p art of Ø . An algebra o v er this op erad is a non-symmetric operad C equipp ed with a map (o f non-symmetric op erads) asso c → C . Later on (see 3.2) an exp licit com binatorial description of the op erad Ø w ill b e giv en. 2.0.1. The unary op erations in the colo red op erad Ø endo w the set of co lors with the follo wing catego ry structure: — hom( n, a ) = ∅ for all n > 0, hom(0 , a ) is a one-point s et; — hom( a , n ) = ∅ for all n ∈ N ; — hom( n, m ) = hom ∆ ([ n ] , [ m ]) for all n, m ∈ N ; — hom( a , a ) = { Id } . This im p lies that the op eradic sets of our colored op erad Ø ha ve a n atural p olysimplicial/cosimplicial structure, namely: 6 V.A. DOLGUSHEV, D.E. T AMARKIN, AND B.L. TSYGAN the collecti on of sets Ø( k ) n n 1 ,n 2 ,...,n k , as n i , n ru n th r ough N , is a fu nctor Ø( k ) : (∆ op ) k × ∆ → Sets , (the functor is simplicial in eac h of the argumen ts n 1 , n 2 , . . . , n k and cosimplicial in n ); lik ewise, for eac h N , the collecti on of sets Ø( k, N ) n 1 ,n 2 ,...,n k forms a f unctor Ø( k , N ) : (∆ op ) k → Set s . 2.0.2. Let S b e a cosimplicial complex giv en by S ([ n ]) • := C −• (∆ n , k ) , where the complex on the righ t h and side is the normalized chain complex of the simplex ∆ n put in the n on-p ositiv e d egrees. Using this complex, w e can con v ert p olysimplicial/cosimplicial sets into complexes. Namely , let F : (∆ op ) k → Set s b e a fun ctor. Set | F | := k [ F ] ⊗ (∆ op ) k S ⊠ k , where S ⊠ k : ∆ k → complexes : S ⊠ k ([ n 1 ] , [ n 2 ] , . . . , [ n k ]) := k O i =1 S ([ n i ]) . Giv en a functor G : (∆ op ) k × ∆ → Sets , denote b y G n the ev aluation at [ n ] ∈ ∆ so that G n : (∆ op ) k → Sets and n 7→ G n is a functor from ∆ to the category of k -simplicial sets. Set | G | := h om ∆ ( S • , | G • | ) . 2.0.3. Set | Ø | ( k ) := | Ø( k ) | ; | Ø | ( k , N ) := | Ø( k , N ) | . W e see that these spaces form a 2-colo red DG op erad. Denote this t wo-c olored op erad by | Ø | . No w let A b e a unital associativ e algebra ov er the ﬁeld k . It is easy to see th at the n ormalized Ho c hsc h ild cochain complex C • ( A, A ) (1.3) can b e wr itten as C • ( A, A ) := hom ∆ ( S ∗ , C A ( ∗ )) . Therefore the DG op erad | Ø | acts on the pair ( C • ( A, A ) , A ) . This t wo -colored DG op erad | Ø | is the desired op erad Λ of natur al op erations on the pair ( C • ( A, A ) , A ) and our Theorem 1.2 can b e reform ulated as Theorem 2.1. The op er ad | Ø | is we akly e quivalent to the singular c hain op er ad of V or onov’s Swiss Che ese op er ad SC 2 . The induc e d action of the homolo gy op er ad H −• ( SC 2 ) on the p air ( H H • ( A ) , A ) r e c overs the one fr om P r op osition 1.1 . PROOF OF SWISS CHEESE VERSION OF DELIGNE’S CONJECTURE 7 W e p ro v e this theorem in Section 9. Remark. Our m etho d also wo rks in the top ologica l s etting: one can apply the top ologica l real- ization f unctors to the p olysimplicial/cosimplicia l sets f rom 2.0.1 so as to get a top ologica l colored op erad | Ø | top . This op erad can b e pr o v en to b e weakly equiv alen t to V oronov’s Swiss Cheese op erad. 3. Two combina t orial descriptions of t he oper a d Ø Our ﬁrst description w ill b e in terms of planar trees. Next, w e will explain a transition from th e tree description to an other one, in terms of sequen ces. Eac h construction will b e ﬁr st introdu ced for the cochain part of Ø and then extended to the whole op erad. These extensions for b oth constructions are rather s tr aigh tforw ard. W e w ill start with ﬁxin g a more con venien t language. 3.1. Finite ordinals instead of natural num b ers. Recall th at our set of colors is N ⊔ { a } , and that an Ø-algebra structure on the collection of spaces ( C ( n ) , n ∈ N ; A ) is the same as a nonsymmetric op erad stru cture on the collectio n of spaces C ( n ) , n ∈ N , a map of op erads asso c → C , and a C -algebra structure on A . The deﬁnition of a nonsymmetric op erad implies that we ha ve a total order on the set of arguments so that it is b etter to r eplace the natur al num b ers with isomorphism classes of ﬁnite ord inals: the num b er n gets replaced w ith the ordinal < n > = { 1 < 2 < · · · < n } . Giv en ﬁnite sets S , S c , an S -family { I s } s ∈ S and an S c -family { I s } s ∈ S c of ﬁnite (p ossibly emp t y) ordinals, an ord inal J , and a set S a , w e then ha ve th e f ollo win g op eradic sets: (3.1) Ø( S ) J { I s } s ∈ S ; (3.2) Ø( S c , S a ) { I s } s ∈ S c , where in (3.1) the set of arguments is S and the coloring of s ∈ S is I s , the result has the color J . In (3.2), the set of argumen ts is S c ⊔ S a the argument s ∈ S c has color I s and all argument s from S a ha ve color a . Th e r esult also has colo r a . 3.2. Planar trees. 3.2.1. The c o chain p art of Ø via planar tr e es. F or a ﬁnite set S and ord inals I s , s ∈ S ; J , we describ e Ø( S ) J { I s } s ∈ S as the set of equiv alence classes of planar trees T w ith the follo wing structure: — a subs et of th e set of v ertices of a tree T is identiﬁed with S ⊔ J in s uc h a wa y that with elemen ts of J we may only id en tify the terminal v ertices of T . W e call the v ertices identiﬁed with elemen ts of S ⊔ J mark ed. — the ordered set of edges originating at the ve rtex marked b y s ∈ S is iden tiﬁed with I s . Notice that, the subset of vertice s iden tiﬁed with J acquires from J a natural lin ear order. W e require th at this linear ord er coincides with the order whic h is obtained b y going around the tree in the clo ckwise direction starting from the root vertex. The equiv alence relation is the ﬁnest one in whic h tw o suc h trees are equiv alen t if one of them can b e obtained from th e other by either: the con traction of an edge with unmarke d ends 8 V.A. DOLGUSHEV, D.E. T AMARKIN, AND B.L. TSYGAN or: remo ving an unmark ed vertex with only one edge originating from it and joining the t w o edges adjacen t to this v ertex into on e edge. Example. The p lanar tree T in ﬁgure 1 represents an elemen t in Ø ( S ) J { I s } s ∈ S with S = { s 1 , s 2 } , J = { j 1 , j 2 , j 3 } , I s 1 = ∅ , and I s 2 = < 3 > . In all the ﬁgures we use circles to denote the ve r tices s 1 s 2 j 1 j 2 j 3 a 1 a 2 a 3 a 4 Figure 1. T ree T mark ed b y elemen ts of S and arro ws to denote v ertices mark ed b y elemen ts of J . Thus, in ﬁgur e 1 the vertice s a 1 , a 2 , a 3 , and a 4 are unm ark ed. The vertice s a 1 and a 2 corresp ond to the p ro duct in asso c (2), a 3 corresp onds to the id en tit y op eration in asso c (1), and a 4 corresp onds to the unit in asso c (0) . In ﬁgures 2 and 3 we dep ict the trees T 1 and T 2 whic h are equiv alent to the original tree T . s 1 s 2 j 1 j 2 j 3 a 1 a 2 a 4 Figure 2. T ree T 1 The tree T 1 is obtained f rom T by r emoving the u n mark ed ve rtex a 3 and joining th e t w o edges adjacen t to this ve rtex int o one edge. The tree T 2 is obtained fr om T by con tracting th e edge w ith the un mark ed ends a 1 and a 2 . The u nmark ed vertex a of the tree T 2 (ﬁgure 3) corresp onds to the unique elemen t of asso c (3) . Applying b oth of the equiv alence op erations to the tree T in ﬁgu r e 1 w e obtain the tree T 3 depicted in ﬁgure 4. Although the tree T 3 has unmarke d v ertices a and a 4 , it is no longer p ossible to apply any equiv alence operation to T 3 . W e call suc h trees minimal . It is ob vious that every equiv alence class of Ø( S ) J { I s } s ∈ S con tains at least one minimal tree. The equiv alence class con taining all these planar trees T , T 1 , T 2 , and T 3 corresp onds to the op eration whic h sends a Ho chsc hild co c hain P 1 ∈ C A (0) and a Ho c h sc hild co c hain P 2 ∈ C A (3) to PROOF OF SWISS CHEESE VERSION OF DELIGNE’S CONJECTURE 9 s 2 j 2 j 3 a 4 a s 1 j 1 a 3 Figure 3. T ree T 2 s 2 j 2 j 3 a 4 a s 1 j 1 Figure 4. T ree T 3 the Hochsc hild co c hain Q ∈ C A (3) deﬁned b y the form ula Q ( b 1 , b 2 , b 3 ) = P 2 ( b 1 , 1 , b 2 ) b 3 P 1 , b 1 , b 2 , b 3 ∈ A . 3.2.2. The whole op er ad Ø in terms of planar tr e es. Let us no w describ e the set Ø( S c , S a ) { I s } s ∈ S c , where w e use the same notation as ab o v e. Eac h elemen t of this set can b e r epresen ted b y a p lanar tree T with the f ollo win g additional structure: — a subset of the s et of vertice s of T is iden tiﬁed with S c ⊔ S a in su c h a w ay that with elemen ts of S a w e may only identify the terminal v ertices of T . W e call the vertice s identiﬁed with elemen ts of S c ⊔ S a mark ed; — the ordered set of edges originating at the ve rtex marked b y s ∈ S c is iden tiﬁed with I s . The equiv alence relation on the set of isomorphism classes of such trees is deﬁned in the same w a y as in the previous section. This description implies the follo wing ident iﬁcation: Ø( S c , S a ) { I s } s ∈ S c = G > ∈ ord ( S a ) Ø( S c ) S a ,> { I s } s ∈ S c , where ord ( S a ) is the set of all total orders on S a . 10 V.A. DOLGUSHEV, D.E. T AMARKIN, AND B.L. TSYGAN Let us also describ e the degenerate cases. In the case S is the empty set ∅ w e hav e Ø( ∅ ) J = asso c ( J ) . If S a = ∅ then Ø( S c , ∅ ) { I s } s ∈ S c = Ø( S c ) ∅ { I s } s ∈ S c . Finally , if S c is empt y then Ø( ∅ , S a ) = G > ∈ ord ( S a ) asso c ( S a , > ) . 3.3. Replacing trees with sequences. W e will put in to a corresp ondence to any planar tree from the previous subs ection a certain sequence whic h will lead to another description of Ø. W e start with the co chain part of Ø. 3.3.1. Co chain p art of Ø in terms of se quenc es, I. W e need the follo wing notation. Giv en a vertex v of a planar tree marke d by an element s ∈ S , let u s d ra w a little circle cen tered at this v ertex. This circle gets split in to sectors, the set of these sectors is totally ordered in the clo c kwise order. Denote this ordered set b y I ′ s . Th e set of edges orig inating at v is naturally identiﬁed with − → I ′ s , where − → I ′ s is the set of pairs − − → i 1 i 2 , wh ere i 2 is an im m ediate successor of i 1 and i 1 , i 2 ∈ I ′ s . W e see that I ′ s is the next ord inal after I s . Belo w, give n an ordinal K , we denote by K ′ its next ordinal. Giv en a planar tree T whic h d eﬁnes an element T ∈ Ø( S ) J { I s } s ∈ S , let us consider its small tubular neigh b orh o o d an d let u s w alk along its b oundary starting fr om th e ro ot v ertex of our tree in th e clockwise direction. On our wa y , we will meet the v ertices mark ed by elemen ts of S and v ertices mark ed by elemen ts of J . (The latter ones are termin al according to our requiremen t.) Ev ery time w e appr oac h a v ertex v marked b y s ∈ S , we are at a certain sector from I ′ s . Thus, give n a planar tree T representi n g an element T ∈ Ø( S ) J { I s } s ∈ S , w e obtain a total order > T on the set G s ∈ S I ′ s ⊔ J . Example. Let us sh o w ho w we obtain the order for the tree T 3 giv en in ﬁgur e 4. This tree represent s an elemen t in Ø( { s 1 , s 2 } ) { j 1 ,j 2 ,j 3 } I s 1 ,I s 2 where I s 1 is empt y and I s 2 = < 3 > . Th is means that the v ertex lab eled by s 1 (see ﬁgure 5) is surroun ded by a sin gle sector s 1 1 , wh ile the vertex lab eled by s 2 is surrou n ded by f our s ectors s 1 2 , s 2 2 , s 3 2 , s 4 2 whic h we num b er in the clo c kwise direction. W alking s 1 1 s 1 2 s 2 2 s 3 2 s 4 2 j 1 j 2 j 3 Figure 5. T ree T 3 PROOF OF SWISS CHEESE VERSION OF DELIGNE’S CONJECTURE 11 along the b oun dary of a small tubular neigh b orh o o d of T 3 , as it is s h o wn on ﬁgur e 5, w e get the follo wing order on the set { s 1 1 , s 1 2 , s 2 2 , s 3 2 , s 4 2 , j 1 , j 2 , j 3 } : s 1 2 < j 1 < s 2 2 < s 3 2 < j 2 < s 4 2 < j 3 < s 1 1 . F or eve r y p lanar tree T represen ting an element T ∈ Ø( S ) J { I s } s ∈ S the corresp onding total order > T satisﬁes: 1) let T 1 , T 2 b e planar trees r ep resen ting the same elemen t T 1 = T 2 ∈ Ø( S ) J { I s } s ∈ S Then > T 1 = > T 2 . Hence, for eac h T ∈ Ø( S ) J { I s } s ∈ S w e hav e a w ell deﬁned ord er, to b e denoted b y > T ; 2) The tota l order > T = > T agrees with the existing ord ers on I ′ s , J ; 3) giv en distinct s 1 , s 2 ∈ S it is imp ossible to ﬁnd i 1 , j 1 ∈ I ′ s 1 ; i 2 , j 2 ∈ I ′ s 2 suc h that i 1 < T i 2 < T j 1 < T j 2 . Let us denote the s et of all total orders satisfying conditions 2) and 3) b y Ord( S ) J { I s } s ∈ S . Prop osition 3.1. The set Ord( S ) J { I s } s ∈ S is in 1-to-1 c orr esp ondenc e with Ø ( S ) J { I s } s ∈ S . Pr o of. Let us describ e an inductiv e construction which assigns to eac h total ord er π on (3.3) G s ∈ S I ′ s ⊔ J satisfying conditions 2) and 3) a minimal tree T which reco v ers the order π b y walking along a small tubular neigh b orho o d of T . The induction goes by the ord er | S | of the set S . F or S = ∅ the s et Ord( S ) J { I s } s ∈ S consists of a single elemen t. That is the giv en order on J . In this case it is v ery easy to ﬁ n d a minimal tree whic h reco v ers this order. It is also easy to see that suc h a tree is unique. Let us su pp ose that w e can constr u ct a desir ed minim al tree for all element s of Ord( S 0 ) J { I s } s ∈ S 0 if | S 0 | < | S | . W e need to present a construction for ev ery π ∈ Ord( S ) J { I s } s ∈ S . Condition 3) implies that for an arbitrary pair s , ˜ s ∈ S exactly one of the follo win g options realizes: (1) all elemen ts of I ′ ˜ s are smaller than elemen ts of I ′ s , (2) all elemen ts of I ′ ˜ s are greater than elemen ts of I ′ s , (3) I ′ ˜ s splits into t w o non-empty subsets such that all elemen ts of the ﬁ rst subs et are smaller than all elements of I ′ s while all the elements of the second subset are greater than elemen ts of I ′ s (4) same as (3) with s and ˜ s in terc han ged. If the th ird (r esp . four th) option realize s w e sa y that s < ˜ s (resp. ˜ s < s ). Thus w e get a partial order on the set S . Since S is ﬁnite, it has at least one minimal elemen t. L et us d enote this elemen t b y s min and in tro du ce the interv al e I s min of the ordin al (3.3) b et wee n the minimal elemen t of I ′ s min and the maximal element of I ′ s min . It is ob vious that e I s min consists of elemen ts of I ′ s min and s ome elemen ts of J . Let us consider the set (3.4) G s ∈ S (1) I ′ s ⊔ J (1) , 12 V.A. DOLGUSHEV, D.E. T AMARKIN, AND B.L. TSYGAN where S (1) = S \ { s min } and J (1) is obtained fr om J by attac hing the elemen t s min and remo ving those elemen ts of J whic h b elong to the in terv al e I s min . In other w ords, (3.5) J (1) = J ⊔ { s min } \ ( J ∩ e I s min ) . Notice that, the set (3.4) is obtained fr om (3.3) by replacing the interv al e I s min b y a single element s min . Hence, (3.4) acquires a natural tot al order. Let us denote this order by π (1) . It is not hard to see that π (1) satisﬁes conditions 2) and 3) and hence is an elemen t of the set Ord( S (1) ) J (1) { I s } s ∈ S (1) . Since | S (1) | < | S | we can assign to π (1) a minimal tree T (1) whic h reco v ers the order π (1) on (3.4) . T o construct th e desired tree T we observ e that the elemen t s min is identiﬁed with an external v ertex v of T (1) . S o, w e dra w from this ve rtex v edges lab eled b y elemen ts − − → i 1 i 2 of − → I ′ s min . R ecall that − → I ′ s min consists of pairs − − → i 1 i 2 , where i 2 is an immediate successor of i 1 and i 1 , i 2 ∈ I ′ s min . Let us denote b y t i 1 i 2 the terminal v ertex of the edge corresp onding to − − → i 1 i 2 . If there are no elemen ts of J b et ween i 1 and i 2 then w e lea v e t i 1 i 2 as an unmark ed terminal v ertex of the tree T . If th er e is only one element j of J b et wee n i 1 and i 2 w e lea v e t i 1 i 2 as a terminal ve r tex of T and mark it b y j . Finally , if we ha v e elements j 1 , . . . , j m ∈ J ( m > 1) b et ween i 1 and i 2 , then we dra w from the v ertex t i 1 i 2 exactly m terminal edge s. W e lea ve t i 1 i 2 unmarked and mark the corresp onding terminal v ertices by j 1 , . . . , j m in the clockwise direction. Let us denote the r esu lting tree by T . It is not hard to see that, since T (1) reco v ers the order π (1) on (3.4) the tree T reco v ers the ord er π on (3.3) . I t is also obvious that, sin ce the tree T (1) is minimal, so is T . W e al ready ha v e a map from the set Ø( S ) J { I s } s ∈ S to the set Ord( S ) J { I s } s ∈ S whic h is deﬁn ed by assigning the tota l order to a tr ee. Let us denote this m ap b y ν ord ν ord : Ø( S ) J { I s } s ∈ S → O rd( S ) J { I s } s ∈ S . The ab o v e constru ction pro vides us with the map in the opp osite direction: ν tree : Ord( S ) J { I s } s ∈ S → Ø( S ) J { I s } s ∈ S . It is clear from the construction that th e comp osition ν ord ◦ ν tree is the iden tit y on Ord( S ) J { I s } s ∈ S . It is n ot hard to v erify th at if w e start with a minimal tree T r epresen ting an element T ∈ Ø( S ) J { I s } s ∈ S , and assign to T the total ord er π from Ord( S ) J { I s } s ∈ S , then the ab ov e constr u ction giv es us bac k exactly th e same minimal tree T . T his implies that the comp osition ν tree ◦ ν ord is the iden tit y on the set Ø( S ) J { I s } s ∈ S and the prop osition follo ws 3 .  Remark. The constru ction p resen ted in the p ro of is reminiscent of Kontsevic h -Soib elman p airs of complemen tary orders [23]. 3 In particular, it implies th at in eac h equiv alence class of trees th ere is exactly one minimal tree. PROOF OF SWISS CHEESE VERSION OF DELIGNE’S CONJECTURE 13 3.3.2. Co chain p art of Ø via se quenc e s, II: mo diﬁc ation. Giv en a total order as ab o ve , we can construct a map Q : G s ∈ S I ′ s → J ′ as follo ws. W e iden tify J ′ = J ⊔ { M } , where M > J . Set Q ( x ) = j if j is the minimal element from J s u c h that j > x ; if there is no su c h j , set Q ( x ) = M . Th u s, giv en a total order as in the previous sub s ection, we obtain the follo wing data: — a tota l order on the set I := G s ∈ S I ′ s ; a non-decreasing map I → J ′ . These data should satisfy: i) the order on I agrees with those on eac h I ′ s ; ii) same as condition 3) f rom Sec 3.3. Denote the set of suc h ob jects b y s Ø( S ) J ′ { I ′ s } s ∈ S . This set is in 1-to-1 corresp ondence with the set of total ord ers from the previous sub s ection, hence w e ha ve a bijectio n with the set Ø( S ) J { I s } s ∈ S : (3.6) s Ø( S ) J ′ { I ′ s } s ∈ S → Ø ( S ) J { I s } s ∈ S . 3.4. The whole op erad Ø in t erms of sequences. Lik ewise, one ident iﬁes the set Ø( S c , S a ) { I s } s ∈ S c with the s et of total orders on G s ∈ S c I ′ s ⊔ S a satisfying: — the total order agree s with those on eac h I ′ s ; — same as condition 3) fr om Sec. 3.3. Denote the set of suc h total orders b y s Ø( S c , S a ) { I s } s ∈ S c . The construction of the 1-to -1 corresp ondence (3.7) s Ø( S c , S a ) { I ′ s } s ∈ S c → Ø ( S c , S a ) { I s } s ∈ S c is the same as in the pr evious sub section. 3.5. Op e ra dic structure on sØ. Let N ′ b e the set of isomorphism classes of n on-empt y ﬁ n ite ordinals. The identiﬁcatio ns (3.6 ), (3.7) imp ly that the colored op erad structure on Ø indu ces a colored op erad structure on the collecti on of spaces s Ø. It turns out that this op eradic structure can b e naturally form ulated in terms of s Ø. W arning. W e will not u se the sy mb ol ′ an ymore when talking ab out ordinals from N ′ . The reason is that in the s equ el, instead of the op erad Ø, the isomorphic op erad s Ø will b e used. 14 V.A. DOLGUSHEV, D.E. T AMARKIN, AND B.L. TSYGAN 3.5.1. Op er adic structur e on the c o c hain p art of s Ø. L et T b e a ﬁnite set and let S t b e a T -family of ﬁnite sets. Let S := ⊔ t S t and p : S → T b e th e m ap whic h sends S t to t . Supp ose we are give n ordinals I s , s ∈ S ; J t , t ∈ T , and J . Describ e the op eradic comp osition s Ø( T ) J { J t } t ∈ T × Y t ∈ T s Ø( S t ) J t { I s } s ∈ S t → s Ø( S ) J { I s } s ∈ S . Let u ∈ s Ø( T ) J { J t } t ∈ T and u t ∈ s Ø( S t ) J t { I s } s ∈ S t . Let us describ e the comp osition v of these elemen ts. 1) the total order > v is deﬁned as a u nique one — whic h agrees with the orders > u t on ⊔ s ∈ S t I s ⊂ ⊔ s ∈ S I s for eac h t ∈ T ; — for whic h the map (3.8) ⊔ t ∈ T F u t : ⊔ s ∈ S I s → ( ⊔ t ∈ T J t , > u ) is non-decreasing. 2) the map F v is just the comp osition of (3.8) with the map F u . Remark. The n on-SC part of the op er ad s Ø as w ell as its totaliz ation was considered in earlier pap ers on Deligne’s conjecture and its v ariations. Th us, the n on-SC part of s Ø is isomorphic to the second ﬁltration stage of the lattice path op erad introdu ced by M. Batanin and C. Berger in [4]. The non-SC part of the totalization of s Ø w as considered in pap ers [28] and [29] b y J. E. McClure and J. H. Smith. 3.5.2. Op er adic structur e on the whole op er ad s Ø. T o describ e the remaining comp osition maps we consider sets S c , S a , T c , T a and let P : S c ⊔ S a → T c ⊔ T a b e a map suc h th at P − 1 T c ⊂ S c . F or t ∈ T a w e set ( P − 1 t ) a := P − 1 t ∩ S a ; ( P − 1 t ) c := P − 1 t ∩ S c . Let { I s } s ∈ S c ; { J t } t ∈ T c ; b e n on-empt y ordinals. W e n eed to deﬁ ne the follo wing comp osition map: s Ø( T c , T a ) { J t } t ∈ T c × Y t ∈ T c s Ø( P − 1 t ) J t { I s } s ∈ P − 1 t × Y t ∈ T a s Ø(( P − 1 t ) c , ( P − 1 t ) a ) { I s } s ∈ ( P − 1 t ) c → s Ø( S c , S a ) { I s } s ∈ S c . Cho ose elemen ts v ∈ s Ø( T c , T a ) { J t } t ∈ T c ; u t ∈ s Ø( P − 1 t ) J t { I s } s ∈ P − 1 t ; t ∈ T c , u t ∈ s Ø(( P − 1 t ) c , ( P − 1 t ) a ) { I s } s ∈ ( P − 1 t ) c ; t ∈ T a and denote th eir comp osition b y w . Let us set I w := G s ∈ S c I s ⊔ S a . and deﬁne a m ap F : I w → I v , where I v = G t ∈ T c J t ⊔ T a , PROOF OF SWISS CHEESE VERSION OF DELIGNE’S CONJECTURE 15 as follo ws: — If t ∈ T c then the restriction of F to the subset I u t := G s ∈ P − 1 t I s , should coincide with the m ap F u t : I u t → J t ; — if t ∈ T a then the r estriction of F to the subset I u t := G s ∈ ( P − 1 t ) c I s ⊔ ( P − 1 t ) a should send ev ery elemen t to t . W e deﬁne the order > w as the unique one for whic h the map F is non-decreasing and whic h agrees with the orders > u t on I u t , t ∈ T c ⊔ T a . 4. Review of 2-operads W e are going to remin d th e basic deﬁn itions from Batanin’s theory of 2-oper ad s whic h w ill b e used b elo w. Next, we r eview Batanin’s deﬁnition of S C 2-operad. An or dinal is a ﬁn ite totally ordered set. Another name for ordinals is a 1-tree. A 2-tr e e t is a p air of ordinals S , T along with an order preserving map t : S → T . A 2-tree is calle d prune d if the map t is sur jectiv e. A map of 2-tr e es P : ( t : S → T ) → ( t 1 : S 1 → T 1 ) is a p air of maps P S : S → S 1 ; P T : T → T 1 suc h that t 1 P S = P T t ; P T is order p reserving; P S preserve s the order on eac h set t − 1 t , t ∈ T . This w a y , 2-trees f orm a category 2-tree s . Giv en s 1 ∈ S 1 , w e deﬁne a 2-tree P − 1 s 1 as follo ws: t    ( P S ) − 1 s 1 : ( P S ) − 1 s 1 → ( P T ) − 1 t 1 ( s 1 ) . A 2-op er ad in a symmetric monoidal categ ory (SMC) C is deﬁned as: — a functor O : 2-trees × → C , where 2-t rees × is the group oid of isomorphisms of 2-trees (note that ev ery ob ject in this group oid has the trivial automorphism group); — for ev ery map of 2-trees P : t → t 1 , w here t : S → T ; t 1 : S 1 → T 1 , th ere should b e giv en a map O ( t 1 ) ⊗ O s 1 ∈ S 1 O ( P − 1 s 1 ) → O ( t ) called the op er adic c omp osition map . These map s should satisfy a certain asso ciativit y p rop erty . In order to formulate it let us deﬁne the ob jects O ( P ), where P : t → t 1 is a map of 2-trees as follo ws: O ( P ) := O s 1 ∈ S 1 O ( P − 1 s 1 ) . The op eradic insertion maps can b e r ewr itten as O ( t 1 ) ⊗ O ( P ) → O ( t ) . Giv en a c hain of m ap s of 2-trees t P → t 1 Q → t 2 , the op eradic insertion maps naturally giv e rise to a map (4.1) O ( Q ) ⊗ O ( P ) → O ( QP ) . 16 V.A. DOLGUSHEV, D.E. T AMARKIN, AND B.L. TSYGAN Indeed, for ev ery s 2 ∈ S 2 , where t 2 : S 2 → T 2 , the map P naturally restricts to a map of 2-trees P s 2 : ( QP ) − 1 s 2 → Q − 1 s 2 and w e ha v e O ( P ) ∼ = O s 2 ∈ S 2 O ( P s 2 ) . W e then deﬁ ne the map (4.1) as follo ws: O ( Q ) ⊗ O ( P ) ∼ = O s 2 ∈ S 2 O ( Q − 1 s 2 ) ⊗ O ( P s 2 ) → O s 2 ∈ S 2 O (( QP ) − 1 s 2 ) = O ( QP ) . The asso ciativit y axiom r equires that the m ap (4.1) b e asso ciativ e: the follo wing maps should coincide: O ( R ) ⊗ O ( Q ) ⊗ O ( P ) → O ( RQ ) ⊗ O ( P ) → O ( RQP ) and O ( R ) ⊗ O ( Q ) ⊗ O ( P ) → O ( R ) ⊗ O ( QP ) → O ( RQ P ) . Remark. 1-trees are simply ordinals and the deﬁn ition of a 1-operad based on 1-trees coincides with the d eﬁnition of a nonsymmetric op erad. 4.1. Colored 2-op erads. 4.1.1. Color e d 2-tr e es. Fix a set of colors X ρ . Deﬁne a colored 2-tree τ as: — a 2-tree t τ : S τ → T τ ; — a map χ τ : S τ → X ρ ; — an elemen t c τ ∈ X ρ . 4.1.2. Giv en colo r ed 2-trees τ 1 , τ 2 w e deﬁn e their map P : τ 1 → τ 2 as follo ws: — if c τ 1 = c τ 2 , then it is j ust a map P : t τ 1 → t τ 2 of the underlyin g 2-trees; — if c τ 1 6 = c τ 2 , then w e declare that there are no maps τ 1 → τ 2 . This w a y colored 2-trees f orm a cate gory . Giv en su c h a map and s 2 ∈ S τ 2 the 2-tree P − 1 s 2 naturally receiv es a coloring as follo ws. Recall that the 2-tree P − 1 s 2 is deﬁned as t τ 1    ( P S ) − 1 s 2 : ( P S ) − 1 s 2 → ( P T ) − 1 t τ 2 s 2 . W e then deﬁ ne χ P − 1 s 2 : P − 1 S s 2 → X ρ as the restriction of χ τ 1 and set c P − 1 s 2 := χ τ 2 ( s 2 ) . 4.1.3. W e th en deﬁne a colored 2-op erad in a SMC C as: — a functor O from the isomorphism group oid of the category of colored 2-trees to the category C ; — for every map P : τ 1 → τ 2 of colored 2-trees there s hould b e giv en the op eradic comp osition map O ( τ 2 ) ⊗ O s 2 ∈ S τ 2 O ( P − 1 s 2 ) → O ( τ 1 ) . Next, giv en a map P : τ 1 → τ 2 w e deﬁn e O ( P ) := O s 2 ∈ S τ 2 O ( P − 1 s 2 ) PROOF OF SWISS CHEESE VERSION OF DELIGNE’S CONJECTURE 17 and observ e that the op eradic comp osition maps naturally pro duce maps O ( Q ) ⊗ O ( P ) → O ( QP ) , where P : τ 1 → τ 2 , Q : τ 2 → τ 3 . Lastly w e require the asso ciativit y of this map in the same wa y as for the non-colored 2-op erads. 4.2. Unital colored 2-op erads. 4.2.1. Giv en a color c ∈ X ρ , consider a sp ecial 2-tree u c : pt → pt such that χ u c sends pt to c and c u c := c . F or ev ery isomorphism P : τ 1 → τ 2 of colored 2-trees eve ry pr e-image P − 1 s 2 , s 2 ∈ S τ 2 , is isomorphic (canonically) to u c , where c = χ τ 2 ( s 2 ). F ur thermore, for every co lored 2-tree τ the pre-image Q − 1 pt of th e p oin t for a unique map Q : τ → u c τ is equal to τ . Let 1 b e the unit of the underlying sym metric monoidal category . Deﬁne a unital 2-op er ad as a colored 2-operad O along with maps (4.2) 1 → O ( u c ) for eac h c ∈ X ρ satisfying: — for ev ery isomorphism P : τ 1 → τ 2 , the map O ( τ 2 ) ∼ = O ( τ 2 ) ⊗ 1 ⊗ S τ 2 → O ( τ 2 ) ⊗ O s 2 ∈ S τ 2 O ( P − 1 s 2 ) → O ( τ 1 ) coincides with the map O ( τ 2 ) → O ( τ 1 ) induced b y P − 1 from the deﬁnition of O as a functor f rom the isomorphism group oid of the categ ory of colored 2-trees. — for ev ery colored 2-tree τ the comp osition O ( τ ) ∼ = 1 ⊗ O ( τ ) → O ( u c τ ) ⊗ O ( τ ) → O ( τ ) is the iden tit y on O ( τ ) . 4.2.2. Prune d c olor e d 2-op er ads. Let P : τ 1 → τ 2 b e a m ap of colored 2-trees. Acco rd ing to M. Batanin [2] P is called a ful l inje ction if P S : S τ 1 → S τ 2 is a color-preservin g isomorphism and P T : T τ 1 → T τ 2 is an injection. Next, let τ b e a colored 2-tree with its underlying 2-tree t : S → T . W e sa y th at τ is prune d if the map t : S → T is surj ectiv e. Let O b e a unital colored 2-o p erad. Consid er the comp osition map associated with P : O ( τ 2 ) ⊗ O s 2 ∈ S 2 O ( P − 1 s 2 ) → O ( τ 1 ) . It is clear that eac h P − 1 s 2 is a 2-tree of the form u c , c ∈ X ρ . Hence w e h a v e unital maps 1 → O ( P − 1 s 2 ). Pre-comp osition with these maps giv es rise to a map (4.3) O ( τ 2 ) → O ( τ 1 ) . Deﬁnition 4.1. We c al l O a prune d 2-op er ad if f or every ful l inje ction P the map (4.3) is an isomorph ism. F or every colored 2-tree τ there exists a unique (up-to an isomorph ism ) prun ed colored 2-tree τ ′ together with a full in jection τ ′ → τ . Th us, a pr uned 2-op erad is completel y d etermin ed b y prescribing its spaces for eac h pru ned 2-tree. 18 V.A. DOLGUSHEV, D.E. T AMARKIN, AND B.L. TSYGAN 4.2.3. R e duc e d 2-op er ads. In this section all 2-op erads are non-colored. Let us ﬁrst deﬁ ne the trivial 2-op erad triv by setting triv ( τ ) = 1 for all 2-trees τ . Here 1 is the unit of the S MC and the op eradic comp osition maps are the canonical maps sending tens or pro ducts of 1 to 1 . W e say th at a pruned 2-op erad O is r e duc e d if — all the u nit maps 1 → O ( u ), 1 → O ( u ) are isomorphism s; — O ( τ ) = 1 when ever | S τ | ≤ 1 so that w e ha ve an identiﬁcati on O ( τ ) = t riv ( τ ) f or all suc h τ ; — for every map P : τ 1 → τ 2 where | S τ 1 | , | S τ 2 | ≤ 1 the corresp onding op eradic comp osition la w coincides with that of triv . Note that, equiv alentl y , one can only require that the conditions are th e case for pruned 2-tree s τ . 4.2.4. Desymmetrization. Giv en a colored symmetric op erad O , one can deﬁne a colored 2-op erad des O by setting des O ( τ ) = O ( S τ ) , where the coloring on the right hand side is determined by th at of τ , and the op eradic comp osition maps are in herited from those of O . 4.3. Symmetrization. If the S MC C has small colimits then the functor des has a left adjoin t sym . F or m an y categories of higher op erads the fu nctor sym can b e elegan tly expressed using colimits [2]. Here w e recall from [2] a description of the functor sym for the category of redu ced 2-op erads. F or ev ery set S we deﬁne a category J ( S ) . The ob jects of J ( S ) are prun ed 2-trees of the form t : S → T . Morphisms are the maps b et we en 2-trees w hic h induce the ident ity map on S . Notice that, although elements of a set S are not ordered, c h o osing an ob ject of the catego ry J ( S ) w e equip S with a total order. Remark. It is not hard to show that for ev ery set S the categ ory J ( S ) is a p oset whose opp osite is calle d the Milgram p oset [1]. Let O b e a reduced 2-operad. F or every set S the 2-op erad O giv es us an obvio u s (contra v arian t) functor from the category J ( S ) to the underlyin g SMC C . W e denote this functor by O S . According to Theorem 4.3 from [2 ] w e ha ve (4.4) sym O ( S ) = colim J ( S ) O S . The op eradic multiplicatio ns of sym O can b e easily obtained from th ose of O u sing the p rop erties of coli mits. 4.3.1. Mo del structur e . Let u s consider the category of reduced 2-op erads in the category of com- plexes o v er the groun d ﬁ eld k (i.e. a dg 2-op erad). Acco rd ing to T heorem 5.3 fr om [2] th is catego ry h as a closed mo del structur e un iquely determined by the conditions that the class of ﬁ- brations (resp . weak equiv alences) should consist of all maps f : O 1 → O 2 satisfying: giv en an y 2-tree t , the indu ced map of complexes f : O 1 ( t ) → O 2 ( t ) is comp onent- wise surjectiv e (resp. is quasi-isomorphism). Same Theorem 5.3 from [2] implies a mo del structure in the categ ory of top ologica l redu ced 2-op erads. PROOF OF SWISS CHEESE VERSION OF DELIGNE’S CONJECTURE 19 Let no w J ( S ) ˜ b e the category of con tra v arian t functors from J ( S ) to the category of complexes o v er k . One has a mo d el stru cture on J ( S ) ˜ w hic h is d eﬁned in a similar wa y: Let F 1 , F 2 ∈ J ( S ) ˜ . The class of ﬁbrations (resp. w eak equiv alences) by deﬁn ition consists of all maps f : F 1 → F 2 satisfying: giv en any 2-tree t ∈ J ( S ), the indu ced map of complexes f : F 1 ( t ) → F 2 ( t ) is comp onen t-wise sur jectiv e (resp. is quasi-isomorphism). A functor F ∈ J ( S ) ˜ is called c oﬁbr ant if the natur al map from the initial ob ject 0 → F is coﬁbran t. Lemma 4.2. F or every dg r e duc e d 2-op er ad O ther e exists a c oﬁbr ant op er ad RO and a we ak e quivalenc e f : RO → O such that for every ﬁnite set, the functor RO S ∈ J ( S ) ˜ is c oﬁbr ant. Pr o of. See the pro of of Theorem 7.1 in [3].  4.3.2. Algebr as over c olor e d 2-op er ads. Giv en an X ρ -colored 2-op erad O and an X ρ -family of ob- jects X c ∈ C , c ∈ X ρ , w e deﬁ n e an O -algebra str u cture on { X c } c ∈ X ρ as a m ap f : O → des full ( { X c } c ∈ X ρ ) , where full ( X ) is the fu ll colored symmetric endomorphism op erad of X . If O is unital, then w e additionally require that f matc hes the units. 4.4. Batanin’s theorem. W e observe that the op erad triv is red u ced and set R triv → triv to b e its coﬁbran t resolution in the cate gory of reduced 2-op erads. Theorem 4.3 (Theorem 7.2, 7.3, [2]) . The symmetric op er ad sym R triv is we akly e quivalent to the op er ad of little discs if C is the c ate gory of top olo gic al sp ac es, and to the singular chain op er ad of little discs if C is the c ate gory of chain c omplexes of k -ve ctor sp ac es. Let us sk etc h its p ro of for C b eing the category of top ological sp aces. First, we obs er ve, that the 2-op erad R triv can b e replaced with an y weakl y equiv alen t one. Batanin uses the Getzle r-Jon es 2-op erad GJ . This 2-operad is constructed in [2] as a sub 2-op erad of the desymmetrization de s ( FM ) of the F ulton-MacPherson v ersion FM of little d iscs op erad Then, since the desymmetrization functor des admits the left adjoin t sym , the inclusion GJ ֒ → des ( FM ) pro du ces the follo wing map sym ( GJ ) → FM whic h can b e shown to b e an isomorph ism, h ence a we ak equiv alence. T h is completes th e pro of. The case when C is th e catego ry of chain complexes is treated by applying th e singular c hain functor. 5. Swiss Che ese (SC) Op erads In this section we discu ss SC-mo diﬁcations of the n otions of colored op erad and colored 2-op erad. W e conclude with form u lating th e S C version of Batanin’s theorem on the symmetrization of the trivial 2-operad. 20 V.A. DOLGUSHEV, D.E. T AMARKIN, AND B.L. TSYGAN 5.1. Symmetric Swiss Cheese t yp e operads. In this sub section w e recall from [2] the notion of the symmetric Swiss C heese t yp e op erads. Here, we call them symmetric SC op erads for short. Let X ρ := { a , c } b e the set of colors. An SC-set is a collection of the follo win g data: —a ﬁnite set S ; —a map χ S : S → X ρ ; —an elemen t c S ∈ X ρ . These data should satisfy: — if a ∈ χ ( S ), then c S = a . A map of SC-sets is a usu al map P : S 1 → S 2 satisfying: if s 1 ∈ S 1 is su c h that χ S 1 ( s 1 ) = a , then χ S 2 ( P ( s 1 )) = a . Giv en suc h P and s 2 ∈ S 2 , P − 1 s 2 is natur ally an SC-set: χ P − 1 s 2 is the restriction of χ S 1 ; c P − 1 s 2 = χ S 2 ( s 2 ). An SC-op erad in a sy m metric m on oidal category C is a f u nctor O from the group oid of SC-sets and their colo r preserving bijections to C . F or ev ery map P : S 1 → S 2 of SC-sets, there sh ould b e given a comp osition map O ( S 2 ) ⊗ O s 2 ∈ S 2 O ( P − 1 s 2 ) → O ( S 1 ) . These comp ositions sh ould satisfy the asso ciativit y la w whic h is similar to that for usual op erads. It is clear h o w to deﬁn e u nital sym m etric SC op erads. In this pap er all our symmetric SC op erads are unital. 5.1.1. R e duc e d symm etric SC op er ads. W e sa y that a un ital symmetric SC op erad O is r e duc e d if for ev ery SC set S with at most one elemen t O ( S ) ∼ = 1 . F or th e one elemen t SC sets these isomorp h isms should coincide with the unit maps. F urther- more, the op eradic comp ositions of zero-ary and unary op er ations send p ro ducts of 1 to 1 via the corresp onding isomorphism of the symmetric m on oidal category . 5.1.2. Color e d symmetric SC-op er ads. Fix tw o sets of colors X ρ c and X ρ a . A c olor e d SC-set S is a map χ S : S → X ρ c ⊔ X ρ a and an elemen t c S ∈ X ρ c ⊔ X ρ a satisfying: if χ − 1 S X ρ a is non-empt y , then c S ∈ X ρ a . W e declare th at there are n o maps b et w een colored S C-sets S 1 and S 2 if c S 1 6 = c S 2 . On the other hand if c S 1 = c S 2 then a map from S 1 to S 2 is a m ap of sets P : S 1 → S 2 satisfying th e prop ert y: for any s 2 ∈ S 2 , the set P − 1 s 2 along with the map χ S 1 | P − 1 s 2 : P − 1 s 2 → X ρ c ⊔ X ρ a and the element c P − 1 s 2 := χ S 2 ( s 2 ) is a c olo r e d SC-se t. Th us, giv en a map of colored SC-sets P : S 1 → S 2 and s 2 ∈ S 2 , w e hav e a colored SC -set P − 1 s 2 . A c olor e d SC-op er ad O in a SM C C is a functor O fr om the isomorphism group oid of colored SC-sets to C along with the comp osition maps: giv en a map P : S 1 → S 2 of colored sets, one sh ould ha ve a map O ( S 2 ) ⊗ O s 2 ∈ S 2 O ( P − 1 s 2 ) → O ( S 1 ) satisfying the associativit y pr op ert y as ab o v e. The op erad s Ø is an example of colored SC op erad. In deed, let X ρ c := N and X ρ a := { a } and let S b e a colored SC -set. Let S c := χ − 1 X ρ c and S a := χ − 1 X ρ a . In the case c S ∈ X ρ c , set s Ø( S ) := s Ø( S ) c S { χ ( s ) } s ∈ S ; if c S = a , we set s Ø( S ) := s Ø( S c , S a ) { χ ( s ) } s ∈ S c . PROOF OF SWISS CHEESE VERSION OF DELIGNE’S CONJECTURE 21 5.2. SC 2-op erads. Let us review Batanin’s deﬁnition of a Swiss C heese t yp e (or simply S C) 2-op erad from [2]. This n otion is obtained via mo difying the deﬁn ition of a usu al 2-operad as follo ws: 1) An SC-or dinal is any n on-empt y ordinal; its minimum is considered to b e mark ed. 2) A map of SC-or dinals is a m onotonous map preserving the min ima. 3) An SC 2-tr e e t is a monotonous map t : S → T where S is a usual ordinal and T is an SC-ordinal. A map of SC 2-trees ( t : S → T ) → ( t 1 : S 1 → T 1 ) is a map of sets P S : S → S 1 as w ell as a map of SC-ordinals P T : T → T 1 suc h th at t 1 P S = P T t 2 ; the map P S m ust preserve the order on eac h set t − 1 t , t ∈ T . Give n s 1 ∈ S 1 suc h that t 1 ( s 1 ) is not the minimum of T 1 , we deﬁne a usu al 2-tree P − 1 s 1 in the same wa y as for the usu al 2-trees (see the b eginning of Sec. 4); in the case t 1 ( s 1 ) is the minim u m of T 1 , we naturally get an SC 2-tree P − 1 S s 1 . 4) W e deﬁn e an SC 2-op er ad in a symmetric monoidal c ate gory C as: — a functor O : 2-trees × ⊔ SC 2-trees × → C ; — for ev ery map of 2-trees or SC 2-t r ees P : t → t 1 , there should b e giv en a map O ( t 1 ) ⊗ O s 1 ∈ S 1 O ( P − 1 s 1 ) → O ( t ) . These maps should s atisfy the asso ciativit y pr op ert y which is similar to that for usual 2-op erads. 5.2.1. Unital SC 2-op er ads. In this pap er all SC 2-op erads are assumed to b e unital. T o introdu ce th e n otion of u nital SC 2-op erads we d eﬁne u c to b e the ord inary 2-tree pt → pt . W e also deﬁne u a to b e an S C 2-tree in which a 1-elemen t ordinal is mapp ed in to a one-elemen t SC ordinal. F or ev ery isomorphism P : t 1 → t 2 of 2-trees or S C 2-trees eve r y p r e-image P − 1 s 2 , s 2 ∈ S t 2 , is either u c or u a . F or ev ery 2-tree t the pre-image Q − 1 c pt of the p oin t for a uniqu e m ap Q c : t → u c is equal to t . F urthermore, for ev ery SC 2-tree t the p re-image Q − 1 a pt of the p oint for a un ique map Q a : t → u a is also equal to t . Deﬁne a unital SC 2-op er ad as an SC 2-op erad O along with maps 1 → O ( u c ) and 1 → O ( u a ) . satisfying: — for ev ery isomorphism P : t 1 → t 2 , of 2-tree s or SC 2-trees th e map O ( t 2 ) ∼ = O ( t 2 ) ⊗ 1 ⊗ S t 2 → O ( t 2 ) ⊗ O s 2 ∈ S t 2 O ( P − 1 s 2 ) → O ( t 1 ) coincides with the map O ( t 2 ) → O ( t 1 ) induced by P − 1 from the deﬁnition of O as a functor from the corresp onding group oid. — for ev ery 2-tree t the comp osition O ( t ) ∼ = 1 ⊗ O ( t ) → O ( u c ) ⊗ O ( t ) → O ( t ) is the iden tit y on O ( t ) . — for ev ery SC 2-tree t the comp osition O ( t ) ∼ = 1 ⊗ O ( t ) → O ( u a ) ⊗ O ( t ) → O ( t ) is the iden tit y on O ( t ) . 22 V.A. DOLGUSHEV, D.E. T AMARKIN, AND B.L. TSYGAN 5.2.2. W e d eﬁne the tr ivial SC 2-operad t riv b y setting triv ( t ) = 1 , for all 2-trees and S C 2-trees t . Here 1 is th e unit of the SMC and the op eradic multiplica tions are the canonica l maps sending tensor pro ducts of 1 to 1 . 5.2.3. Color e d SC 2-op e r ads. W e deﬁn e a colored SC 2-op erad as follo ws. Fix 2 sets of colors: X ρ c and X ρ a . Deﬁne a coloring of an SC 2-tree t τ : S τ → T τ as follo w s. First decomp ose S τ = S τ ,a ⊔ S τ ,c , where S τ ,a is the t τ -preimage of the minim um of T τ , and S τ ,c is the complemen t. A ( X ρ c , X ρ a ) -c olor ing of τ is a prescription of maps χ τ ,c : S τ ,c → X ρ c ; χ τ ,a : S τ ,a → X ρ a and an elemen t c τ ∈ X ρ a . As w ell as for ordinary colo red 2-tree s w e declare that th ere are no map s b etw een colored SC 2-trees τ and τ 1 if c τ 6 = c τ 1 . On the other h and, if c τ = c τ 1 then a map P : τ → τ 1 is just the map of the und erlying SC 2-trees. Then it is clear that for ev ery s 1 ∈ S τ 1 suc h that t τ 1 s 1 is the minim u m , the SC 2-tree P − 1 s 1 is natur ally ( X ρ c , X ρ a )-colored. F urther m ore, for ev ery s 1 ∈ S τ 1 suc h that t τ 1 s 1 is not the min im um, the 2-tree P − 1 s 1 is naturally X ρ c -colored. W e d eﬁne a ( X ρ c , X ρ a )-colored SC 2-op erad as a fu nctor O from th e disjoint un ion of the group oid of X ρ c -colored 2-trees and the group oid of ( X ρ c , X ρ a )-colored S C 2-trees to C . Giv en a map P : τ → τ 1 of X ρ c -colored 2-trees or ( X ρ c , X ρ a )-colored SC 2-trees there should b e giv en a map O ( τ 1 ) ⊗ O s 1 ∈ S 1 O ( P − 1 s 1 ) → O ( τ ) . The asso ciativit y axiom s hould b e satisﬁed. 5.2.4. Unital c olor e d SC 2-op er ads. As well as SC 2-op erads all colored SC 2-operad s are assumed to b e unital. T o introd uce the notion of un ital colored SC 2-op erads we deﬁne u c , c ∈ X ρ c , b e the colored 2-tree pt → pt for which the p oint pt has the color c and c u c = c . S imilarly , we d eﬁne u a , a ∈ X ρ a to b e the colored SC 2-tree in wh ic h the one-elemen t ord in al is mapp ed into the one-elemen t SC ordinal and all colorings are a . F or ev ery isomorphism P : τ 1 → τ 2 of X ρ c -colored 2-tree s or SC 2-trees ev ery pre-image P − 1 s 2 , s 2 ∈ S τ 2 , is either u c or u a . F or every colored 2-tree or colored SC 2-tree τ the pre-image Q − 1 τ pt of the p oin t for a unique map Q τ : τ → u c τ is equal to τ . Deﬁne a unital c olor e d SC 2-op er ad as a colored S C 2-op erad O along with m aps 1 → O ( u c ) and 1 → O ( u a ) for all c ∈ X ρ c and a ∈ X ρ a satisfying: — for ev ery isomorphism P : τ 1 → τ 2 , of colo red 2-trees or colored SC 2-trees the m ap O ( τ 2 ) ∼ = O ( τ 2 ) ⊗ 1 ⊗ S τ 2 → O ( τ 2 ) ⊗ O s 2 ∈ S τ 2 O ( P − 1 s 2 ) → O ( τ 1 ) coincides with the map O ( τ 2 ) → O ( τ 1 ) induced b y P − 1 from the deﬁnition of O as a functor f rom the corresp onding group oid. — for ev ery colored 2-tree or colo r ed SC 2-tree τ the comp osition O ( τ ) ∼ = 1 ⊗ O ( τ ) → O ( u c τ ) ⊗ O ( τ ) → O ( τ ) is the iden tit y on O ( τ ) . PROOF OF SWISS CHEESE VERSION OF DELIGNE’S CONJECTURE 23 5.2.5. Prune d SC 2-op er ads. A (colored) SC 2-tree τ is called pr uned if Im( t τ ) ⊃ T τ \ m T τ , where m T τ is the mark ed minimum of T τ . F or ev ery colored SC 2-tree τ there exists a unique u p to isomorp hism pru ned colored SC 2-tree τ ′ and a map P : τ ′ → τ su c h that P S : S τ ′ → S τ is a b ijection; P T is injectiv e, and P ind u ces an isomorphism of colorings. F or ev ery s uc h P the pr e-images P − 1 s are of th e form u c or u a , therefore, giv en a unital op erad O , w e hav e a m ap (5.1) O ( τ ′ ) → O ( τ ) . By analogy with ordinary 2-op erads (see Su bsection 4.2.2) O is called prune d if all s u c h maps (5.1) are isomorphisms. 5.2.6. R e duc e d SC 2-op er ads. W e sa y that a pru ned (non-colored) SC 2-op erad O is r e duc e d if — all the u nit maps 1 → O ( u c ), 1 → O ( u a ) are isomorphisms; — O ( τ ) = 1 when ever | S τ | ≤ 1 so that w e ha ve an identiﬁcati on O ( τ ) = t riv ( τ ) f or all suc h τ ; — for every map P : τ 1 → τ 2 where | S τ 1 | , | S τ 2 | ≤ 1 the corresp onding op eradic comp osition la w coincides with that of triv . Note that, equiv alentl y , one can only require that the conditions are th e case for pruned 2-tree s τ . 5.3. Desymmetrization. Given a sym m etric S C-op erad Q , Batanin deﬁn es its desymmetrization des Q by setting des Q ( t ) := Q ( S t ) for all 2-trees and SC 2-trees t . Here S t is treated as an SC-set as follo w s: — if t is a us u al 2-tree th en w e deﬁne all the colorings to b e c ; — if t is an SC 2-tree, we set c S t := a and w e giv e the p reimage of mark ed elemen t of T t the color a , th e remaining elemen ts of S t receiv e color c . Giv en a reduced symmetric SC-op erad Q , its desymmetrization des Q is a reduced SC 2-op erad so that des is a functor f rom the category of r educed symmetric SC op erads to that of r educed SC 2-op erads. 5.3.1. In the same spirit, one d eﬁnes the desymmetrization of a colored symmetric SC-op erad O . Let τ b e a colored SC 2-tree ; w e then see that S τ is a colo red SC-set in the natural w a y: the map χ S τ    S τ ,c := χ τ ,c and χ S τ    S τ ,a := χ τ ,a . Finally , c S τ := c τ . W e then set des ( O )( τ ) := O ( S τ ) with the comp osition la w d etermined b y that in O . 5.4. Symmetrization. The con tent of this section is a straightfo rward SC generalization of Sec 4.3. Under an assu mption that S MC C has small colimits, the f unctor des has a left adjoint sym . W e ha ve a d escrip tion of the functor sym for the category of reduced SC 2-op erads wh ic h is similar to that for r educed 2-op er ad s (see. Sec 4.3) . F or ev ery SC set S we deﬁ ne a cat egory J ( S ) . If c S = c , then the category J ( S ) is the same as in Sec 4.3: th e ob jects of J ( S ) are pruned 2-trees of the form t : S → T . Morphisms are the maps b et we en 2-trees w hic h induce the ident ity map on S If c S = a then ob jects of J ( S ) are prun ed SC 2-trees t : S → T suc h that the preimage of the minimal element of T coincides with S a = χ − 1 ( a ) . Morphisms are the m aps b etw een SC 2-trees whic h indu ce the identi ty map on S . 24 V.A. DOLGUSHEV, D.E. T AMARKIN, AND B.L. TSYGAN As in the non S C case, all categ ories J ( S ) are in fact p osets. Let O be a redu ced SC 2-op erad. F or ev ery SC set S the op erad O giv es u s an obvious (con- tra v arian t) functor from the category J ( S ) to the un derlying SMC C . W e denote this functor b y O S . According to Theorem 9.1 from [2 ] w e ha ve (5.2) sym O ( S ) = colim J ( S ) O S . The op eradic multiplicatio ns of sym O can b e easily obtained from th ose of O u sing the p rop erties of coli mits. 5.4.1. Mo del structur e. T he categ ory of dg prun ed S C 2-op erads has a mod el stru cture whic h is deﬁned in the same w a y as in Sec 4.3.1 Same is true for the mo d el structure on the categ ory J ( S ) ˜ of con tra v arian t fu nctors from J ( S ) to the category of comp exes o v er k . Lemma 4.2 holds true in the S C con text. Lemma 5.1. F or e very dg r e duc e d SC 2-op er ad O ther e e xi sts a c oﬁbr ant dg r e duc e d SC 2-op er ad RO and a quasi-isomorphism f : RO → O such that the functors RO S ∈ J ( S ) ˜ a r e c oﬁbr ant for any SC set S . Pr o of. Similar to the pro of of Lemma 4.2.  5.4.2. Batanin ’s the or em. Theorem 5.2 (Theorem 9.2, 9.4, [2]) . The symmetric SC op er ad sym R triv is we akly e quivalent to V or onov’s Swiss Che e se op er ad if C is the c ate gory of top olo gic al sp ac es, and to the singular chain op er ad of V or onov’s Swiss Che e se op er ad if C is the c ate gory of chain c ompl exes of k -ve c tor sp ac es. Batanin’s pro of goes along the same lines as his pro of of Theorem 4.3. 6. Linking the operad s Ø with 2 oper a ds: a 2-ope rad seq In this section we will deﬁn e a 2-sub-op erad seq ⊂ des s Ø. Next, w e deﬁne the S C-v ersion of seq . 6.1. 2-op erad seq ( cochain part) . Let us ﬁr st r ecall the deﬁnition of the co chai n part (i.e. ’non-SC part’) of the N -colored op erad s Ø (Sec 3.3.2 ) with the notation s lightly c hanged. Let S b e a ﬁ nite set and J ; I s , s ∈ S b e n on-empt y ﬁ nite ordinals. Eac h elemen t u of the op eradic space s Ø( S ) J { I s } s ∈ S is deﬁned b y means of the follo wing data: — a tota l order > u on the set I := G s ∈ S I s ; a non-decreasing map Q u : I → J. These data should satisfy: i) the order > u on I agrees with those on eac h I s ; ii) same as condition 3) f rom Sec 3.3. The comp osition la w for the op erad s Ø w as deﬁned in Sec 3.5. Let u s no w deﬁne a colored sub -2-op erad seq of des s Ø. Let τ b e a N -colored 2-tree, whic h is deﬁned by means of a 2-tree t : S → T and its N -coloring such that an s ∈ S has color I s , wh er e I s is a non-empt y ﬁnite ordinal, and the color of th e r esult is J . More formally , χ τ ( s ) = I s ; c τ = J . Let us deﬁne subsets seq ( τ ) := seq ( t ) J { I s } ,s ∈ S ⊂ s Ø( S ) J { I s } ,s ∈ S PROOF OF SWISS CHEESE VERSION OF DELIGNE’S CONJECTURE 25 whic h consists of all elemen ts u ∈ s Ø( S ) J { I s } ,s ∈ S satisfying: — if i, k ∈ I s 1 , j ∈ I s 2 , s 1 6 = s 2 and i < u j < u k , then t ( s 2 ) < t ( s 1 ); — if s 1 , s 2 ∈ S , s 1 < s 2 , and t ( s 1 ) = t ( s 2 ), then I s 1 < u I s 2 . One can c hec k that thus deﬁn ed su bspaces are closed un der all 2-op eradic comp osition maps so that seq ⊂ des s Ø is a colored sub-2-op erad. Th us deﬁned colored s u b-2-op erad coincides with th e colored 2-op erad seq as deﬁned in Sec 6.1 of [32]. The c hec k that seq is closed und er th e 2-op eradic comp ositions follo ws fr om the observ ation that the 2-op eradic comp osition maps inherited from s Ø are the same as in lo c. cit. 6.2. SC vers ion of seq. Let us deﬁne a colored SC 2-op erad SC seq by mo difyin g the d eﬁnition of seq as follo w s. First of all w e ﬁx the sets of colors: — the set X ρ c is the same as the set of colors of seq , i.e. N ; — the set X ρ a is the one element set { a } ; we ident ify a u nique elemen t of X ρ a with the ordinal consisting of 1 element. — Giv en a usu al colored 2-tree τ we set SC seq ( τ ) := seq ( τ ); — give n a colored S C 2-tree τ , let us construct a usual colored 2-tree τ ′ with the underlying 2-tree t ′ = t : S → T . Deﬁn e a map χ τ ′ : S → X ρ c = N b y setting a) if s ∈ S and t ( s ) is the minimum of T , then w e set χ τ ′ ( s ) to b e the one-elemen t ordinal; b) if s ∈ S and t ( s ) is not th e minimum of T , then we set χ τ ′ ( s ) = χ τ ,c ( s ) , where χ τ ,c is a deﬁning map of the coloring for τ (see Sec. 5.2.3) . Lastly , we set c τ ′ to b e the one-elemen t ordinal. W e then deﬁ ne SC seq ( τ ) := seq ( τ ′ ). Note that w e ha ve natural inclusions SC seq ( τ ) ⊂ s Ø( S τ ) = ( des s Ø)( τ ) , where S τ is the colored SC-set corresp ondin g to the colored 2-tree or the colored SC 2-tree τ as deﬁned in Sec 5.3. Th us SC seq is a colo red SC 2-sub op erad of des s Ø. 6.3. T otalization: A dg 2-op erad | seq | and a dg op erad | sØ | . Using the fun ctor of (co)- simplicial totaliza tion w e w ill con ve rt a colored op erad s Ø and a colored 2-op erad seq into diﬀer- en tial graded op erads. Th e S C versions will b e co vered in the next section 6.4. The spaces of unary op erations in seq and s Ø giv e a category structur e on N hom( I 1 , I 2 ) = seq ( t 0 ) I 2 I 1 = s Ø I 2 I 1 , where t 0 : pt → pt . This category is isomorphic to the simplicial category ∆. The action of these un ary op erations d eﬁnes a p olysimplicial/cosimplicial str ucture on the col- lection of op er ad ic sets. Giv en a 2-tree t : S → T , the collect ion of sets seq ( t ) J { I s } s ∈ S , where I s , J are non -emp t y ﬁnal ordinals, forms a functor seq ( t ) : ∆ × (∆ op ) S → Set s , where J ∈ ∆ and I s ∈ ∆ op . 26 V.A. DOLGUSHEV, D.E. T AMARKIN, AND B.L. TSYGAN Using the functor S : ∆ → complexes we can tak e the total complexes of these p olysimp licial (cosimplicial) sets, in the same wa y as in Subsection 2.0.2 . W e set | seq | ( t ) := | seq ( t ) | ; The complexes | seq | ( t ) au tomatically form a d g-op erad. Similarly , the s ets s Ø( S ) J { I s } s ∈ S form a fu nctor s Ø( S ) : ∆ × (∆ op ) S → Sets so that we can deﬁne | s Ø | ( S ) := | s Ø( S ) | . W e h a v e a dg-op erad structure on | s Ø | . Th e embedd ing seq ⊂ des s Ø induces a map | seq | → des | s Ø | . 6.4. Extension to the SC-setting: a dg SC 2-op era d SC seq and a n SC op erad | sØ | . Let u s no w extend the construction of | seq | and | s Ø | to the S C case. Giv en an SC 2-tree t : S → S 1 , let us decomp ose S = S c ⊔ S a , where S a is the pr e-image of the minim u m of S 1 . Su pp ose we are gi ven ordinals I s , s ∈ S c . Using these data, w e naturally get a colored SC 2-tree τ := τ ( t , { I s } s ∈ S c ), where the coloring sets are X ρ c = N and X ρ a = { a } . Eac h elemen t of s ∈ S c receiv es color I s ; eac h elemen t of S a gets colored in a ; w e set c τ = a . W e set SC seq ( t ) { I s } s ∈ S c := SC seq ( τ ) . Th u s, giv en an SC 2-tree t : S → S 1 , w e get a p olysimplicial set SC seq ( t ) : (∆ op ) S c → Set s . Set | SC seq | ( t ) := | SC seq ( t ) | . F or t b eing a 2-tree w e set | SC seq | ( t ) := | seq | ( t ). This wa y the dg 2-op erad | seq | extends to an S C 2-op erad | SC seq | . Giv en an SC set S = S c ⊔ S a and ordinals I s , s ∈ S c w e get a N -colored SC-set which determines the op eradic set s Ø( S c , S a ) { I s } s ∈ S c . These sets form a fu n ctor s Ø( S ) : (∆ op ) S c → Set s . and we can set | s Ø | ( S ) := | s Ø( S ) | thereb y getting an SC symmetric op erad | s Ø | whic h is an SC exstension of the symmetric op erad | s Ø | from the p revious section 6.3. The map SC seq → des s Ø of SC 2-op erads indu ces a map (6.1) | SC seq | → des | s Ø | of dg SC 2-op erads. Since the operad s Ø is isomorphic to Ø th e DG op erad | s Ø | is isomorphic to the DG op erad Λ = | Ø | of natural op erations on the pair ( C • ( A, A ); A ) . Th us w e h a v e a map from the S C 2-op erad | SC seq | to des Λ . PROOF OF SWISS CHEESE VERSION OF DELIGNE’S CONJECTURE 27 7. The SC 2-operad br a nd the SC op erad braces It is not hard to see that the SC 2-operad | SC seq | is pru ned. How ev er, neither | SC seq | nor | s Ø | is reduced. In this section w e constru ct a reduced SC 2-op erad br wh ic h is quasi-isomorph ic to the S C 2-op erad | SC seq | . Similarly , w e construct a redu ced SC op erad braces wh ich is qu asi-isomorphic to the S C op erad | s Ø | . Both br and braces are obtained as sub op erads of | SC seq | and | s Ø | , resp ectiv ely . As usual w e will ﬁ rst mak e all deﬁnition for the non-SC part and then extend th em to the SC-setting 7.1. An increasing ﬁltration on the colored 2-op erad seq. Let t b e a 2-tree t : S → T and v ∈ seq ( t ) J { I s } s ∈ S . Consider the order > v on (7.1) I := I S := G s ∈ S I s . Call t w o elemen ts i 1 , i 2 ∈ I S elementary e quivalent if i 1 , i 2 ∈ I s for s ome s ∈ S and for every i ∈ I S b et wee n i 1 and i 2 with resp ect to the order < v the elemen t i b elongs to I s . In this w a y we get an equiv alence relation on I S . Denote by | v | the n umb er of equiv alence classes with resp ect to this relation. Let F N seq ( t ) J { I s } s ∈ S b e the subset consisting of all elemen ts v with | v | ≤ N + | S | . Roughly sp eaking, the diﬀerence | v | − | S | coun ts ho w man y times the order < v cuts the ordinals I s , s ∈ S in to sub ordinals. 7.2. Extension of the ﬁltration on to SC seq. Let t : S → T b e an SC 2-tree. As ab o v e, we set S a to b e the p re-image of th e minim um of T and S c := S \ S a . Recall that an elemen t v ∈ SC seq ( t ) { I s } s ∈ S c is nothing else but a total order > v on (7.2) G s ∈ S c I s ⊔ S a sub ject to certain conditions. In order to deﬁne the elemen tary equiv alence r elation on (7.2) we r eplace (7.2) by the isomorph ic set (7.3) I S c ⊔ S a = G s ∈ S I s , where I s is the one element ordinal for ev ery s ∈ S a . Using the total ord er > v on I S c ⊔ S a and the constru ction fr om the p revious subsection we get the elemen tary equiv alence relation on the set I S c ⊔ S a and h ence on (7.2 ). On the set (7.2) the elementa ry equiv alence relation can b e describ ed as follo ws. Th e restriction of this r elatio n onto S a coincides with the iden tit y relation, there is n o elemen t of S a whic h is equiv alen t to an elemen t i ∈ G s ∈ S c I s . Finally w e call t wo element s i 1 , i 2 ∈ G s ∈ S c I s elemen tary equiv alen t iﬀ 28 V.A. DOLGUSHEV, D.E. T AMARKIN, AND B.L. TSYGAN — i 1 , i 2 ∈ I s for some s ∈ S c , — for eve ry elemen t i of the set (7.2) b et wee n i 1 and i 2 with resp ect to the order < v w e hav e i ∈ I s . W e d enote the num b er of equiv alence classes in (7.3) I S c ⊔ S a b y | v | and deﬁne the su bset F N SC seq ( t ) { I s } s ∈ S c ⊂ SC seq ( t ) { I s } s ∈ S c to consist of all elements v with | v | ≤ N + | S | . 7.3. Compatibility of the ﬁltration with the op eradic structure. Lemma 7.1. The ﬁltr ation F is c omp atible with op er adic c omp ositions on SC seq . Pr o of. Let us ﬁrst prov e Lemma for the ﬁltration on 2-op erad seq (i.e. the ’non-SC’-part). Consider op eradic comp ositions of the f ollo wing t yp e: Let P : t 1 → t 2 b e a map of 2-trees, where t 1 : S 1 → T 1 and t 2 : S 2 → T 2 . Let P S : S 1 → S 2 b e the induced map. F or ev ery s 2 ∈ S 2 , w e ha ve a pre-image P − 1 S ( s 2 ) ⊂ S 1 . Let I s 1 , s 1 ∈ S 1 ; J s 2 , s 2 ∈ S 2 ; J b e non-empty ordinals. Let w ∈ seq ( t 2 ) J { J s 2 } s 2 ∈ S 2 ; v s 2 ∈ seq ( P − 1 s 2 ) J s 2 { I s 1 } s 1 ∈ P − 1 S ( s 2 ) . Let us denote b y z the comp osition of these elemen ts and estimate | z | . Su p p ose that J σ ⊂ ( G s 2 ∈ S 2 J s 2 , > w ) for σ ∈ S 2 is split in to | σ | equiv alence classes. Consider the map I σ := G s 1 ∈ P − 1 σ I s 1 → J σ . It is clear that the num b er of equiv alence classes of I σ ⊂ ( G s 1 ∈ S 1 I s 1 , > z ) do es not exceed | v σ | + | σ | − 1. Therefore | z | ≤ X σ ∈ S 2 ( | v σ | + | σ | − 1) = | w | − | S 2 | + X σ | v σ | . Hence, | z | − | S 1 | ≤ | w | − | S 2 | + X σ ( | v σ | − | P − 1 σ | ) whic h means that this comp osition is compatible with the ﬁltration F . This concludes the p ro of for seq . The extension to SC seq is straigh tforw ard.  This Lemma, in particular implies that the p olysimplicial/co simp licial str u cture on SC seq is com- patible with the ﬁltration F . Therefore, the ﬁ ltration F descends on to the lev el of total complexes so that we ha v e an increasing ﬁltration on eac h op eradic complex | SC seq | ( t ): F N | SC seq | ( t ) ⊂ | SC seq | ( t ). Lemma 7.2. The ﬁltr ation F on | SC seq | satisﬁes the fol lowing pr op erties: (1) The op er adic c omp ositions i n | SC seq | ar e c omp atible with the ﬁltr ation. (2) The c omp lex F N | SC seq | ( t ) is c onc entr ate d in the de gr e es ≥ − N . PROOF OF SWISS CHEESE VERSION OF DELIGNE’S CONJECTURE 29 (3) The quotient F N | SC seq | ( t ) . F N − 1 | SC seq | ( t ) only has c ohomo lo gy c onc entr ate d in de gr e e − N . Pr o of. (1) F ollo ws from the p revious lemma. (2) Let us ﬁrst consider the non-SC p art of the statemen t (that is, we will p ro v e the statemen t for seq ). Let t : S → T b e a 2-tree. Let us consid er the simplicial realization with resp ect to the lo wer indices for (7.4) seq ( t ) J { I s } s ∈ S . Let v ∈ seq ( t ) J { I s } s ∈ S . According to Subs ection 7.1 the order > v on (7.5) I S := G s ∈ S I s deﬁnes on I S an equiv alence relation. If (7.6) | v | + | J | − 1 < X s ∈ S | I s | . then there exist t w o diﬀerent bu t equiv alen t elements of I S whic h go to the same elemen t in J . In this case th e elemen t v is obtained from another elemen t by applying a degeneracy . Th u s if in equ alit y (7.6) holds for v then v do es not con tribute to the realizatio n of (7.4). Therefore if v con tributes to the realization then | J | − 1 − X s ∈ S ( | I s | − 1) ≥ −| v | + | S | and hence the complex F N | seq | ( t ) is concen trated in degrees ≥ − N . This ﬁnishes the pro of for | seq | . The general SC-case is similar. Let t : S → T b e an SC 2-tree. Let S a b e the pre-image of the minimal elemen t of T and S c = S \ S a . An elemen t v of (7.7) SC seq ( t ) { I s } s ∈ S c is a tota l order > v on (7.8) G s ∈ S c I s ⊔ S a sub ject to certain conditions. According to S ubsection 7.2 the order > v giv es us the elemen tary equiv alence relation on the set (7.8). If at least one equiv alence class in (7.8) conta ins more than 1 elemen t then th e corresp onding elemen t v in (7.7) is obtained from another element b y applying a degeneracy . Ind eed, only the equiv alence classes in G s ∈ S c I s 30 V.A. DOLGUSHEV, D.E. T AMARKIN, AND B.L. TSYGAN ma y con tain more than one element. And if at least one class con tains more than 1 element then there are d istinct elemen ts i 1 , i 2 ∈ I s for some s ∈ S c suc h that one of them go es r ight after another in the ord inal (7.8). Therefore, if v con tributes to the realization of (7.7) then X s ∈ S c | I s | + | S a | = | v | . Hence X s ∈ S c ( | I s | − 1) = | v | − | S a | − | S c | or equiv alent ly − X s ∈ S c ( | I s | − 1) = | S | − | v | . If v ∈ F N SC seq ( t ) { I s } s ∈ S c then the right hand side of the latter equation is ≥ − N . Thus th e complex F N | SC seq | ( t ) is concen trated in degrees ≥ − N . (3) Let us ﬁrst consider th e co c hain complex (7.9) F N | SC seq | ( t ) . F N − 1 | SC seq | ( t ) in the case when t : S → T is a usu al 2-tree. If an elemen t v in (7.4) rep r esen ts a non-zero vec tor in (7.9) then the set I S (7.5) has exactly N + | S | equiv alence classes. The total ord er on I S giv es a total ord er on the set of th ese equiv alence classes. Hence the set of equiv alence classes in I S can b e identiﬁed with the ordin al { 1 , 2 , . . . , N + | S |} . F ur thermore eac h equiv alence class is a sub set of I s for some s ∈ S . Th u s to every suc h elemen t v in (7.4) we assign a surjection (7.10) σ : { 1 , 2 , . . . , N + | S |} → S from the ordinal { 1 , 2 , . . . , N + | S |} to the set 4 S . Not all su c h surjections can b e gotten fr om the elemen ts of (7.4) repr esenting non -zero v ectors in (7.9). The 2-tree t : S → T , the deﬁnition of SC seq , and the deﬁnition of the elemen tary equiv alence relation imp ose the follo wing conditions on the p ossible surjections (7.10) : A σ ( i ) 6 = σ ( i + 1) ∀ i = 1 , 2 , . . . , N + | S | − 1 , B if s 6 = ˜ s and j 1 < i < j 2 for i ∈ σ − 1 ( s ) and j 1 , j 2 ∈ σ − 1 ( ˜ s ) then t ( s ) < t ( ˜ s ) in T , C if t ( s ) = t ( ˜ s ) and s < ˜ s then all elemen ts of σ − 1 ( s ) are sm aller than all elemen ts of σ − 1 ( ˜ s ) . Let us denote by D ( t , N ) the set of all surjections (7.10) satisfying ab o v e conditions A , B , and C . It is n ot h ard to see that the elemen ts of (7.4) representing non-zero v ectors in (7.9) an d cor- resp ondin g to the same su rjection (7.10) span a sub complex of (7.9). F urther m ore for eve ry map (7.10) this sub complex is isomorphic to the co chain complex | Ξ N + | S | | • + N , where | Ξ k | • are the complexes describ ed in the App endix. Th u s (7.9 ) is isomorphic to the dir ect sum of identica l co chain complexes (7.11) F N | SC seq | ( t ) . F N − 1 | SC seq | ( t ) ∼ = M σ ∈ D ( t ,N ) | Ξ N + | S | | • + N . 4 Recall that S is also equipp ed with a total order but in general σ is not a map of ordinals. PROOF OF SWISS CHEESE VERSION OF DELIGNE’S CONJECTURE 31 Therefore, due to Prop osition 9.2 f rom the App endix w e ha v e, (7.12) H •  F N | SC seq | ( t ) . F N − 1 | SC seq | ( t )  =      M σ ∈ D ( t ,N ) k , if • = − N , 0 , otherwise . Let us no w consider the co chain complex (7.13) F N | SC seq | ( t ) . F N − 1 | SC seq | ( t ) in the case when t : S → T is an S C 2-tree. As ab o ve S a is the pre-image of th e m in imal elemen t of T and S c = S \ S a . If an elemen t v of (7.7) represents a non-zero v ector in (7.13) then the set (7.14) I S c ⊔ S a = G s ∈ S c I s ⊔ S a has exactly N + | S | equiv alence classes. The total order on I S c ⊔ S a giv es us a total order on the set of its equiv alence classes. Hence the set of the equiv alence classes can b e id en tiﬁed with the standard ord in al { 1 , 2 , . . . , N + | S |} . F urthermore, eac h equiv alence class is either a subset of I s for some s ∈ S c or a one elemen t subset of S a . Th us we get a surj ection (7.15) σ : { 1 , 2 , . . . , N + | S |} → S from the ordinal { 1 , 2 , . . . , N + | S |} to the set S . As w ell as in th e case of the usual 2-tree this surjection satisﬁes ab o ve conditions A , B , and C . Let us remark that, since S a is the pre-image of the m in imal elemen t of T , conditions A , B , and C imp osed on the surjection (7.15) imply that f or ev ery s ∈ S a the pre-image σ − 1 ( s ) is a one elemen t set. As ab o v e w e denote by D ( t , N ) the set of all su rjections (7.15) satisfying ab ov e conditions A , B , and C . Similarly to the case of a u sual 2-tree th e set of elements of (7.7) representing non-zero vec tors in (7.13) s plits into the d isj oin t un ion of su bsets, corresp onding surjections σ ∈ D ( t , N ) . And similarly the elemen ts of (7.7) repr esen ting n on-zero v ectors in (7.13) and corresp onding to the same map (7.15) sp an a sub complex of (7.13). These sub complexes are all isomorph ic to the co c hain complex | Ξ N + | S c | | • + N , 0 , where the bicomplexes | Ξ k | • , • are d escrib ed in the App endix. It is not hard to see that the complex | Ξ N + | S c | | • , 0 consists of the ﬁeld k placed in degree 0 . Th u s for an SC 2-tree t w e hav e (7.16)  F N | SC seq | ( t ) . F N − 1 | SC seq | ( t )  • ∼ =      M σ ∈ D ( t ,N ) k , if • = − N , 0 , otherwise and statemen t (3) holds in this case too.  7.4. Deﬁnition of t he (SC) 2-op erad br. Usin g th is ﬁltration w e giv e the follo wing deﬁn ition. Deﬁnition 7.3. We deﬁne the dg (SC) 2-op er ad br as a sub op er ad of | SC seq | with (7.17) br ( t ) = M N ≥ 0 G N | SC seq | ( t ) , 32 V.A. DOLGUSHEV, D.E. T AMARKIN, AND B.L. TSYGAN wher e G N | SC seq | ( t ) =  v ∈ F N | SC seq | ( t ) − N    dv ∈ F N − 1 | SC seq | ( t )  , and t is either a 2-tr e e or an SC 2-tr e e. Lemma 7.2 implies that the inclus ion br ֒ → | SC seq | is a quasi-isomorphism. F urther m ore, Prop osition 7.4. The SC 2-op er ad br is r e duc e d. Pr o of. Let t : S → T b e a 2-tree or an SC 2-tree with | S | ≤ 1 . T he condition | S | ≤ 1 implies that the ﬁltration F on | SC seq | ( t ) is trivial: F − 1 | SC seq | ( t ) = 0 and F N | SC seq | ( t ) = | SC seq | ( t ) for all N ≥ 0. Therefore, br ( t ) is s imply the v ector space of degree 0 cocycles in | SC seq | ( t ) (7.18) br ( t ) = | SC seq | ( t ) 0 ∩ ker d . Due to Lemma 7.2 th e complex | SC seq | ( t ) is concen trated in nonnegativ e d egrees. Hence H 0  | SC seq | ( t )  = | SC seq | ( t ) 0 ∩ ker d . On the other hand, equations (7.12 ) and (7.16 ) imply that H 0  | SC seq | ( t )  = k [ D ( t , 0) ] and it is easy to see that if | S | ≤ 1 then D ( t , 0) is a one elemen t set. Th u s br ( t ) is ind eed isomorphic to k . It is not hard to c hec k that the isomorphisms k ∼ = br ( u c ) and k ∼ = br ( u a ) are give n by the unit maps.  7.5. An increasing ﬁltration on sØ. W e will now deﬁne an an alogue of the ﬁ ltration F from the previous subsection for the (SC ) op erad s Ø . Let us ﬁrst consider the n on-SC case. Let S b e a ﬁn ite set and J, I s , s ∈ S b e non-empt y ﬁn ite ordin als. Ev ery elemen t u ∈ s Ø J { I s } s ∈ S giv es us a total order > u on the set I S := G s ∈ S I s . F ollo wing Su b section 7.1 this order give s us the elemen tary equ iv alence relation on I S . W e denote the n umb er of equiv alence classes in I S b y | u | and deﬁne F N s Ø J { I s } s ∈ S as th e su bset consisting of all elemen ts u ∈ s Ø J { I s } s ∈ S with | u | ≤ N + | S | . Let us no w extend this deﬁnition for the SC-case. Let S b e an SC with c S = a (the case c S = c corresp onds to the non-S C part and has jus t b een considered). W e sp lit S as S = S c ⊔ S a where S c = χ − 1 ( c ) and S a = χ − 1 ( a ) . By deﬁnition an elemen t u ∈ s Ø( S c , S a ) { I s } s ∈ S c is a tota l order on th e set I S c ⊔ S a = G s ∈ S c I s ⊔ S a sub ject to certain conditions. F ollo wing Subs ection 7.2 this order giv es us the elemen tary equiv alence relation on I S c ⊔ S a . PROOF OF SWISS CHEESE VERSION OF DELIGNE’S CONJECTURE 33 Let us denote the num b er of the equiv alence classes in I S c ⊔ S a b y | u | and deﬁne F N s Ø( S ) { I s } s ∈ S c as the set of all element s u ∈ s Ø( S ) { I s } s ∈ S c with | u | ≤ N + | S | . W e claim that Lemma 7.5. The ﬁltr ation F is c omp atible with op er adic c omp ositions on s Ø. Pr o of. Similar to pro of of Lemma 7.1.  This lemma imp lies that the ﬁ ltration F on s Ø is compatible w ith th e p olysimpicial/co simp licial structure. Therefore, the form u la F N | s Ø | ( S ) = | F N ( s Ø)( S ) | deﬁnes an increasing ﬁltration on th e d g SC oper ad | s Ø | . Lemma 7.6. The ﬁltr ation F on | s Ø | satisﬁes the fol lowing pr op erties: (1) The op er adic c omp ositions i n | s Ø | ar e c omp atible with the ﬁltr ation F . (2) The c omp lexes F N | s Ø | ( S ) ar e c onc e ntr ate d in the de gr e es ≥ − N . (3) The c ohomolo gy of the quotient F N | s Ø | ( S ) /F N − 1 | s Ø | ( S ) is c onc entr ate d in the de gr e e − N . Pr o of. Since the p ro of is v ery similar to th at of L emm a 7.2 w e will only brieﬂy outline the p ro of of (3). Let us ﬁr st treat the non-SC part. Let S b e a ﬁnite set. As well as for the 2-op erad | seq | the co c hain complex F N | s Ø | ( S ) /F N − 1 | s Ø | ( S ) is isomorphic to a direct sum of id en tical complexes (7.19) F N | s Ø | ( S ) . F N − 1 | s Ø | ( S ) ∼ = M σ ∈ D ( S,N ) | Ξ N + | S | | • + N , where the complexes | Ξ k | are describ ed in the App endix and D ( S, N ) is the set of surjections (7.20) σ : { 1 , 2 , . . . , N + | S |} → S satisfying the follo wing conditions I σ ( i ) 6 = σ ( i + 1) ∀ i = 1 , 2 , . . . , N + | S | − 1 , I I if s 6 = ˜ s ∈ S then it is imp ossible to hav e i 1 , i 2 ∈ σ − 1 ( s ) , and j 1 , j 2 ∈ σ − 1 ( ˜ s ) suc h that i 1 < j 1 < i 2 < j 2 . Th u s Prop osition 9.2 implies statemen t (3) in the case c S = c . Let us n o w consider the SC-case. I f S is an SC set with c S = a , S c = χ − 1 ( c ) and S a = χ − 1 ( a ) then the complex F N | s Ø | ( S ) /F N − 1 | s Ø | ( S ) is isomorphic to a direct sum of id en tical complexes (7.21) F N | s Ø | ( S ) . F N − 1 | s Ø | ( S ) ∼ = M σ ∈ D ( S,N ) | Ξ N + | S c | | • + N , 0 , where the bicomplexes | Ξ k | • , • are describ ed in the App endix and D ( S, N ) is the set of surjections (7.20) satisfying ab o v e cond itions I , I I and the additional condition: I I I if s ∈ S a then σ − 1 ( s ) consists of exact ly one elemen t. Since the complex | Ξ N + | S c | | • , 0 consists of the ﬁeld k placed in degree 0 , statemen t (3) follo ws in this case to o.  34 V.A. DOLGUSHEV, D.E. T AMARKIN, AND B.L. TSYGAN W e w ould like to remark that condition B in the pr o of of Lemma 7.2 implies condition I I in the pro of of Lemma 7.6. Therefore, for ev ery 2-tree t : S → T we ha ve the inclusion (7.22) D ( t , N ) ⊂ D ( S, N ) . Similarly , if t is an SC 2- tree th en conditions A , B , and C imply conditions I , II , and I I I . Therefore, w e ha ve the in clusion (7.2 2) for SC 2-trees t as well. W e will use this inclusion later. 7.6. Deﬁnition of the ( SC ) op erad braces. W e no w deﬁne a useful sub op erad of | s Ø | Deﬁnition 7.7. We deﬁne the dg SC op er ad braces as a sub op e r ad of | s Ø | with (7.23) braces ( S ) = M N ≥ 0 G N | s Ø | ( S ) , wher e G N | s Ø | ( S ) =  v ∈ F N | s Ø | ( S ) − N    dv ∈ F N − 1 | s Ø | ( S )  , and S is an SC set. Lemma 7.6 implies that the inclus ion braces ֒ → | s Ø | is a quasi-isomorphism. Prop osition 7.8. The dg SC op er ad braces is r e duc e d. Pr o of. Let S b e an SC set with | S | ≤ 1 . It is not hard to construct a pru n ed 2-tree or a pr u ned SC 2-tree t : S → T with S b eing the source ordinal. It is easy to see that if | S | < 1 then br ( t ) = braces ( S ) as coc h ain complexes. Th u s the d esir ed statemen t follo ws immediately fr om Prop osition 7.4 .  Let us no w consider a coﬁbrant r esolution R br → br of br in the closed mo del categ ory of reduced dg (SC) 2-operads. It is clear fr om the deﬁ n itions of br and braces that w e h a v e the embedd ing of dg (SC) 2-op erads br ֒ → de s bra ces . Since sym is the left adjoint f unctor for des this emb edding pro duces the m ap (7.24) sym br → braces . Comp osing (7.24) with the map sym R br → sym br w e get the map (7.25) sym R br → braces . W e claim that Theorem 7.9. The map (7.25) is a quasi-isomorphism of dg (SC) 2-op er ads. This theorem pla ys a crucial role in pro ving our main result (Th eorem 2.1). W e dev ote the next section to th e pro of of this theorem. PROOF OF SWISS CHEESE VERSION OF DELIGNE’S CONJECTURE 35 8. Proof of Theorem 7.9 W e n eed to sh o w that for ev ery (SC) set S the map (8.1) ( sym R br )( S ) → braces ( S ) is a quasi-isomorphism of co chai n complexes. Due to the symmetrization form ula (see equation (5.2)) sym R br ( S ) = colim J ( S ) R br S . As R br is a coﬁbr an t resolution of br , Lemma 5.1 implies that the functor R br S is coﬁbran t. Hence, the natural m ap ho colim J ( S ) R br S → colim J ( S ) R br S is a w eak equiv alence. Hence we ha ve a zig-zag weak equiv alence sym R br ( S ) ∼ → ho colim J ( S ) br S . Th u s w e need to sho w that the map ho colim J ( S ) br S → braces ( S ) is a quasi-isomorphism of co chai n complexes. F or this, it su ﬃces to sh o w that so is th e map (8.2) ho colim J ( S ) ( F N br /F N − 1 br ) S → F N braces ( S ) /F N − 1 braces ( S ) for ev ery N . Equations (7.12), (7.16) and statement (2) of Lemma 7.2 imp ly that for ev ery 2-tree or SC 2-tree t , (8.3) F N br ( t )  F N − 1 br ( t ) = k [ D ( t , N )][ N ] , where k [ D ( t , N )][ N ] is considered as a co chain complex w ith the zero d iﬀeren tial. Similarly , equations (7.19), (7.21) and statemen t (2) of Lemma 7.6 imp ly that for ev ery SC set S (8.4) F N braces ( S )  F N − 1 braces ( S ) = k [ D ( S, N )][ N ] , where k [ D ( S, N )][ N ] is considered as a co chain complex w ith the zero diﬀerentia l. Let us recall that for ev ery (SC) set S and for ev ery t ∈ J ( S ) we ha ve the inclusion D ( t , N ) ⊂ D ( S, N ) . Let S b e a ﬁn ite (SC) set. Then for σ ∈ D ( S, N ) we set J ( σ ) ⊂ J ( S ) to b e the full sub category of all 2-trees t su ch that σ ∈ D ( t , N ) . Recall that for ev ery (SC) set S the category J ( S ) is a p oset. It is not hard to see that f or eve r y morphism P : t → e t in the category J ( S ) w e hav e the inclusion (8.5) D ( e t , N ) ⊂ D ( t , N ) . F urth ermore, th e morphism F N br ( e t )  F N − 1 br ( e t ) → F N br ( t )  F N − 1 br ( t ) corresp onding to P : t → e t is giv en by this in clusion. 36 V.A. DOLGUSHEV, D.E. T AMARKIN, AND B.L. TSYGAN Com binin g this observ ation with equations (8.3) and (8.4) w e conclude that (8.6) ho colim J ( S ) ( F N br /F N − 1 br ) S = M σ ∈ D ( S,N ) ho colim J ( σ ) k , ( F N /F N − 1 ) braces ( S )( N ) = M σ ∈ D ( S,N ) k , and (8.2) is in duced b y the natural maps (8.7) ho colim J ( σ ) k → k , where, b y abuse of notation, k denotes b oth the und erlying ﬁeld and th e fun ctor which assigns k to ev ery ob ject of J ( σ ) . Th u s it suﬃces to sh ow that the map (8.7) is a quasi-isomorphism for ev ery σ ∈ D ( S, N ) . The ob vious top ological counte rp art of this statemen t can b e formula ted as Prop osition 8.1. F or every (SC) set S and every element σ ∈ D ( S, N ) the natur al map (8.8) ho colim J ( σ ) pt → pt is a we ak e quivalenc e. In what follo ws, by abuse of notation, w e denote a constan t functor from J ( σ ) to another catego ry by the und erlying ob ject. F or example, in (8.8) pt denotes b oth the one-p oint space and the functor fr om J ( σ ) to th e catego ry of top ological spaces whic h assigns pt to ev ery ob ject of J ( σ ) . Let us p ostp one the p r o of of Prop osition 8.1 to the end of the section and sho w that this prop osition indeed implies that (8.7 ) is a quasi-isomorphism. W e, ﬁ r st, use the adj unction (8.9) | | top : sSets ← → T op : C sing ∗ b et wee n the category T op of top ological s paces and the category sSets of sim p licial sets. Here | | top denotes the realiza tion functor and C sing ∗ is the singular c hain functor. Using the fact that the adjunction (8.9) giv es a Q uillen equiv alence b etw een T op and sSets it is not hard to deduce fr om Prop osition 8.1 its coun terpart for simplicial sets. Namely , Pr op osition 8.1 implies that for every σ ∈ D ( S, N ) the natural map (8.10) ho colim J ( σ ) △ 0 → △ 0 is a w eak equiv alence of s im p licial s ets, where △ 0 = hom ∆ ( , [0 ]) is the terminal ob j ect of the category sSets . Therefore, for the simplicial Ab elian group Z △ 0 , the natural map (8.11) ho colim J ( σ ) Z △ 0 → Z △ 0 is a w eak equiv alence. Notice that, via the Dold-Kan corresp ondence, (8.11) can b e view ed as a m ap of co c hain 5 com- plexes of Ab elian groups. F urtherm ore, to say that (8.11) is a weak equiv alence of simplicial Ab elian groups is to s ay that (8.11) is a quasi-isomorphism of the corresp ond ing co c hain complexes. Recall that the forgetful functor Ψ : k − V ect → Ab 5 Here we revers e the standard grading of the Dold-Kan corresp ondence. PROOF OF SWISS CHEESE VERSION OF DELIGNE’S CONJECTURE 37 from the category k − V ect of k -v ector spaces to the category Ab of Ab elian groups admits the left adjoin t fun ctor k ⊗ Z : Ab → k − V ect . Using this adjunction and the qu asi-isomorphism (8.11 ) we deduce that the natural map (8.12) ho colim J ( σ ) k △ 0 → k △ 0 is a quasi-isomorphism of co chai n complexes of k -v ector spaces. Here k △ 0 is the cochain complex · · · id → k 0 → k id → k 0 → k with the righ t most term placed in degree 0 . Th is complex is ob viously quasi-isomorphic to k placed in d egree 0 . And hence the map (8.7) is indeed a quasi-isomorphism of cochain complexes. In order to complete the p ro of of Theorem 7.9 it remains to prov e Prop osition 8.1. 8.1. Pro of of Prop osition 8.1. W e need a coﬁb r an t resolutio n of the trivial fun ctor from th e p oset J ( σ ) to the category of top ological spaces. The closed mo del structur e on the catego ry of functors from J ( σ ) to T op is obtained from that on top ological sp aces u sing the transfer principle 6 of C. Berger and I. Moerdijk [9]. In other words, ﬁbrations (resp. wea k equiv alences) b et wee n functors from J ( σ ) are ob ject-wise ﬁbr ations (resp. ob ject-wise w eak equiv alences). In order to construct the resolution, giv en a ﬁnite set S , w e co n s ider the conﬁguration space Conf ( S ) of distinct p oin ts on R 2 lab eled b y elemen ts of S . It is known that the space Conf ( S ) admits a cellular sub d ivision into the F o x-Neu wir th cell s [7], [17], [37]. Eac h F o x-Neu wir th cell FN t corresp onds to a pru ned 2-tree t : S → T and it can b e deﬁned as the s p ace of all in jectiv e maps from the 2-tree t to the generalized 2-tree: ( x, y ) → x : R 2 → R , where on R 2 w e use the lexicographic order. In other words, a conﬁ guration { ( x s , y s ) } s ∈ S b elongs to FN t iﬀ the follo wing conditions are satisﬁed: — if t ( s ) = t ( ˜ s ) and s < ˜ s th en x s = x ˜ s and y s < y ˜ s , — if t ( s ) < t ( ˜ s ) then x s < x ˜ s . An example of a conﬁguration from FN t 1 for the 2-tree t 1 : { 1 , 2 , 3 , 4 , 5 } → { 1 , 2 , 3 } t 1 (1) = t 1 (2) = 1 , t 1 (3) = t 1 (4) = 2 , t 1 (5) = 3 is depicted in ﬁgure 6 This construction can b e easily generaliz ed to pruned SC 2-trees. Namely , if t : S → T is a prun ed SC 2-tree with S a b eing the preimage of the minimal elemen t of T and S c = S \ S a then FN t consists of conﬁgurations { ( x s , y s ) } s ∈ S satisfying the follo wing conditions: — if s ∈ S a then x s = 0 ; if s ∈ S c then x s > 0 , — if t ( s ) = t ( ˜ s ) and s < ˜ s th en x s = x ˜ s and y s < y ˜ s , — if t ( s ) < t ( ˜ s ) then x s < x ˜ s . Recall that for pru n ed SC 2-trees the range t ( S ) do es not in general include the minimal element . In other w ords, the subset S a ma y b e empty . In this case w e still require that x s > 0 f or s ∈ S c . 6 The transfer p rinciple can be applied in this case b ecause J ( σ ) is a ﬁnite p oset and the Quillen’s path-ob ject argument obviousl y wo rks for top ological spaces. 38 V.A. DOLGUSHEV, D.E. T AMARKIN, AND B.L. TSYGAN P S f r a g r e p l a c e m e n t s 1 2 3 4 5 Figure 6. A t yp ical p oin t of FN t 1 If S is an (SC) set then for ev ery map P : t → e t of prun ed (SC) 2-trees in the category J ( S ) w e hav e the ob vious inclusion (8.13) FN e t ֒ → ∂ FN t , where ∂ FN t denotes the b oundary of th e F o x-Neu wir th cell F N t . F or example, w e ma y consider the 2-tree t 2 : { 1 , 2 , 3 , 4 , 5 } → { 1 , 2 } t 2 (1) = t 2 (2) = 1 , t 2 (3) = t 2 (4) = t 2 (5) = 2 with a (unique) map in J ( { 1 , 2 , 3 , 4 , 5 } ) P : t 1 → t 2 , P S = id , P T (1) = 1 , P T (2) = P T (3) = 2 . A conﬁ gurations f rom F N t 2 consists of a pair of distinct v ertical lines; the left line carries p oin ts 1 and 2 such that 1 is b elo w 2; the righ t line carries p oin ts 3, 4, 5 whic h are put in th e order fr om the b ottom to th e top. (See ﬁgure 7.) It is clear that FN t 2 b elongs to the b oundary of FN t 1 . P S f r a g r e p l a c e m e n t s 1 2 3 4 5 Figure 7. A t yp ical p oin t of FN t 2 Let σ ∈ D ( S, N ) and J ( σ ) b e the sub-p oset of J ( S ) deﬁned ab o ve . Usin g the inclusion (8.13) w e up grad e the corresp ondence (8.14) t → Φ σ ( t ) = [ e t ∈J ( σ ); t → e t FN e t PROOF OF SWISS CHEESE VERSION OF DELIGNE’S CONJECTURE 39 to the f u nctor Φ σ : J ( σ ) → T op . The u nion in (8.14) is taken ov er all the prun ed (SC) 2-trees e t ∈ J ( σ ) f or w hic h w e ha ve a map from t to e t . Example 8.2. W e consider S = { α, β , γ , δ } with c S = a , χ ( α ) = χ ( γ ) = χ ( δ ) = c , χ ( β ) = a , and σ b eing the follo wing map σ : { 1 , 2 , 3 , 4 , 5 , 6 } → S σ (1) = α , σ (2) = δ , σ (3) = γ , σ (4) = β , σ (5) = γ , σ (6) = δ . The map σ is an elemen t of D ( S, 2) and the SC 2-tree t : { β < γ < α < δ } → { 1 , 2 , 3 , 4 } t ( β ) = 1 , t ( γ ) = 2 , t ( α ) = 3 , t ( δ ) = 4 is an ob ject of J ( σ ) . There are exactly thr ee pruned SC 2-trees e t ∈ J ( σ ) for wh ich there is a m ap t → e t . The ﬁrst one is e t 1 = t and the second one is e t 2 : { β < γ < α < δ } → { 1 , 2 , 3 } e t 2 ( β ) = 1 , e t 2 ( γ ) = 2 , e t 2 ( α ) = e t 2 ( δ ) = 3 . The third SC 2-tree e t 3 : { β < α < γ < δ } → { 1 , 2 , 3 } e t 3 ( β ) = 1 , e t 3 ( α ) = e t 3 ( γ ) = 2 , e t 3 ( δ ) = 3 . So the space Φ σ ( t ) consists of conﬁgur ations { ( x s , y s ) } s ∈{ α,β ,γ ,δ } satisfying the follo wing condi- tions: — x β = 0 < x γ ≤ x α ≤ x δ , and x γ < x δ , — if x α = x γ then y α < y γ , — if x α = x δ then y α < y δ . Prop osition 8.3. L et S b e an (SC) set and σ ∈ D ( S, N ) . Then the functor Φ σ (8.14) is a c oﬁbr ant r esolution of the trivial functor fr om J ( σ ) to the c ate gory of top olo gic al sp ac es. Pr o of. Let S b e an (SC) set and σ ∈ D ( S, N ) . Let us sho w that Φ σ ( t ) is contract ible for ev ery prun ed (SC) 2-tree t : S → T for which σ ∈ D ( t , N ) . W e giv e a detailed pro of of con tractibilit y of Φ σ ( t ) in the case w hen c S = c (i.e. S is a us u al, non-SC, set) and h ence t is a pr uned (non-S C) 2-tree. The SC case c S = a is v ery similar. The 2- tree t : S → T giv es u s a total order on th e set S . So we iden tify S with the ordinal { 1 , 2 , 3 , . . . , | S |} and d en ote b y ( x i , y i ) the coordin ates of the p oin t lab eled by i ∈ { 1 , 2 , 3 , . . . , | S |} . Next, w e consider the follo wing sequence of su bspaces Φ σ ( t ) = F 0 ⊃ F 1 ⊃ F 2 ⊃ · · · ⊃ F | S | where F k consists of conﬁgurations { ( x i , y i ) } ∈ Φ σ ( t ) with y i = i , ∀ i ≤ k . Let us s h o w that F k +1 is a deformation retract of F k for all k = 0 , 1 , 2 , . . . , | S | − 1 . A deformation retraction f : F k × [0 , 1] → F k of F k on to F k +1 is giv en by th e f ormula: (8.15) f ( { ( x i , y i ) } , t ) = { ( x i , y i ( t )) } , where y i ( t ) = ( i , if i ≤ k , (1 − t ) y i + t ( k + 1 + y i − y k +1 ) , if i > k . 40 V.A. DOLGUSHEV, D.E. T AMARKIN, AND B.L. TSYGAN W e need to sh o w that for all t ∈ [0 , 1] and for all conﬁgurations { ( x i , y i ) } ∈ F k the p oint f ( { ( x i , y i ) } , t ) b elongs to Φ σ ( t ) . More precisely , w e need to c h ec k that if x i = x j and i < j then y i ( t ) < y j ( t ) for all t ∈ [0 , 1] . First, it is ob vious that if i < j ≤ k the y i ( t ) < y j ( t ) r egardless of whether x i equals x j or not. Second, it is n ot hard to see that if k < i < j and y i < y j then y i ( t ) < y j ( t ) for all t ∈ [0 , 1] . Finally , if i ≤ k < j and x i = x j then the conﬁguration { ( x i , y j ) } b elongs to the F o x-Neuwirth cell FN e t corresp onding to a pruned 2-tree e t for whic h e t ( i ) = e t ( j ) . The latter implies that e t ( i ) = e t ( i + 1) = · · · = e t ( j − 1) = e t ( j ) and h ence x i = x i +1 = · · · = x j − 1 = x j . Therefore y i < y i +1 < · · · < y j − 1 < y j and, in particular 7 , y j ≥ y k +1 > y k = k . Using these inequalities w e conclude that for all t ∈ [0 , 1] y j ( t ) = (1 − t ) y j + t ( k + 1) + t ( y j − y k +1 ) > k + t ( y j − y k +1 ) ≥ k . On the other h an d y i ( t ) ≤ k . Thus, if x i = x j and i < j then y j ( t ) > y i ( t ) for all t ∈ [0 , 1] . F urth ermore, if y i = i for all i ≤ k + 1 then y i ( t ) ≡ i for all i ≤ k + 1 . Thus f is indeed a deformation retraction of F k on to F k +1 . Let us n o w iden tify T with the stand ard ordin al { 1 , 2 , 3 , . . . , | T |} . Next w e n ote that if e t ∈ J ( σ ) admits a m ap t → e t then equ alit y t ( i ) = t ( j ) imp lies the equalit y e t ( i ) = e t ( j ) . Hence, if t ( i ) = t ( j ) then x i = x j for ev ery conﬁguration { ( x i , y i ) } ∈ Φ σ ( t ) . Therefore the function i → x i factors through t : { 1 , 2 , 3 , . . . , | S |} → { 1 , 2 , 3 , . . . , | T |} and hence, w e ma y describ e conﬁgurations from Φ σ ( t ) using the colle ctions of co ordin ates { z l , y i } , z l , y i ∈ R where l ∈ { 1 , 2 , 3 , . . . , | T |} and i ∈ { 1 , 2 , 3 , . . . , | S |} . F or ev ery conﬁguration { z l , y i } from Φ σ ( t ) w e ha ve (8.16) z 1 ≤ z 2 ≤ z 3 ≤ · · · ≤ z | T | and if z l = z m for l 6 = m th en the corresp onding conﬁguration b elongs to the F o x-Neu wir th cell FN e t of a 2-tree e t 6 = t . T o sh ow the contrac tibilit y of F | S | w e consider the follo wing sequence of su bspaces: F | S | = G 0 ⊃ G 1 ⊃ G 2 ⊃ · · · ⊃ G | T | ∼ = pt . where G k consists of conﬁgurations { z l , y i } ∈ F | S | satisfying the follo wing condition z l = l , ∀ l ≤ k . In terms of the original co ordinates ( x i , y i ) the latt er condition reads x i = t ( i ) , if t ( i ) ≤ k . W e sh o w that for all k ≤ | T | − 1 the space G k +1 is a deformation retract of G k . 7 In this case y j = y k +1 only if j = k + 1 PROOF OF SWISS CHEESE VERSION OF DELIGNE’S CONJECTURE 41 The desired deformation retraction is deﬁned b y the form ula (8.17) g ( { z l , y i } , t ) = { z l ( t ) , y i } , where z l ( t ) = ( l , if l ≤ k , (1 − t ) z l + t ( k + 1 + z l − z k +1 ) , if l > k . T o pr o v e that the conﬁguration { z l ( t ) , y i } b elongs to F | S | for all t ∈ [0 , 1] w e need to c hec k that inequalities (8.18) z 1 ( t ) ≤ z 2 ( t ) ≤ z 3 ( t ) ≤ · · · ≤ z | T | ( t ) hold f or all t ∈ [0 , 1] . F urtherm ore we need to c hec k that if z l < z m then z l ( t ) < z m ( t ) for all t ∈ [0 , 1] . In the case l < m ≤ k we simply hav e the in equalit y z l ( t ) < z m ( t ) . Also it is not hard to see that in the case k < l < m the inequ alit y z l ( t ) ≤ z m ( t ) (resp. z l ( t ) < z m ( t )) follo ws from z l ≤ z m (resp. z l < z m ) . Th u s it remains to consider the case l = k and m = k + 1 . In this case w e ha ve z k ( t ) ≡ z k = k . F urther m ore, due to (8.16) we h a v e z k +1 ≥ k and h ence z k +1 ( t ) = (1 − t ) z k +1 + t ( k + 1) ≥ (1 − t ) k + tk = k . It is also obvious that if z k +1 > k then z k +1 ( t ) = (1 − t ) z k +1 + t ( k + 1) > (1 − t ) k + tk = k = z k ( t ) for all t ∈ [0 , 1] . Finally , it is clear that if z k +1 = k + 1 then z k +1 ( t ) ≡ k + 1 . Th u s g (8.17) is indeed a deformation retraction of G k on to G k +1 . Since G | T | is a one-p oin t space w e conclude that F | S | , and hence, the space Φ σ ( t ) is contract ible. The pro of of the f act that Φ σ is a coﬁbrant ob ject in the category of f u nctors f rom J ( σ ) to T op is v ery similar to the pro of of Theorem 7.2 fr om [2 ]. F ollo wing the argument s of [2] w e deﬁ n e the follo wing sequence of functors Φ m σ , m ∈ Z , m ≥ 0 . On the lev el of ob j ects the functor Φ m σ op erates as (8.19) Φ m σ ( t ) =        Φ σ ( t ) , if | S | + | T | < m , and t is a 2 − tree , Φ σ ( t ) , if | S | + | T | − 1 < m , and t is an S C 2 − tr ee , ∅ , otherwise . W e wo u ld lik e to remark that the num b er | S | + | T | (resp. | S | + | T | − 1) for a 2-tree t : S → T (resp. for an SC 2-tree t : S → T ) is the dimen sion of the F o x-Neuwirth cell FN t . Thus the collection Φ m σ ma y b e considered as a ﬁltration of Φ σ b y dimension. W e h a v e th e obvious sequence of n atural transformations Φ 0 σ → Φ 1 σ → Φ 2 σ → . . . and the functor Φ σ is the sequen tial colimit Φ σ = colim m Φ m σ . 42 V.A. DOLGUSHEV, D.E. T AMARKIN, AND B.L. TSYGAN Similarly to the pro of of Theorem 7.2 from [2] we sho w th at for ev ery m the natural transfor- mation Φ m σ → Φ m +1 σ is a cell u lar extension generated b y a coﬁbration. Th u s Φ σ is ind eed a coﬁbrant ob ject in the category of fu nctors from J ( σ ) to the category of top ologica l spaces. This completes the pro of of Pr op osition 8.3.  No w that we ha v e a coﬁbr an t r esolution Φ σ of the trivial functor fr om J ( σ ) to T op we pro ve Prop osition 8.1 b y sho win g th at the sp ace (8.20) X σ = colim J ( σ ) Φ σ is con tractible for ev ery su r jection σ ∈ D ( S, N ) , wh ere S is an (SC) set. It is easy to see that (8.21) X σ = [ t ∈J ( σ ) FN t . T o get a more explicit description of the sp ace X σ (8.21) we r ecall that th e set D ( S, N ) consists of surjections σ : { 1 , 2 , 3 , . . . | S | + N } → S from th e s tand ard ord inal { 1 , 2 , 3 , . . . | S | + N } to the set S ; the surjections σ should satisfy t wo conditions I and I I from th e pro of of Lemma 7.6; if, in addition c S = a , th en we should also imp ose on σ condition I I I from the pro of of the same lemma. Let us also recall that a 2-tree (an SC 2-tree) t : S → T b elongs to J ( σ ) iﬀ the follo wing conditions are met: — if for s 6 = ˜ s there exist i 1 , i 2 ∈ σ − 1 ( ˜ s ) and i ∈ σ − 1 ( s ) suc h that i 1 < i < i 2 then t ( s ) < t ( ˜ s ) in the (SC) ordinal T , — if t ( s ) = t ( ˜ s ) and s < t ˜ s then all elemen ts of σ − 1 ( s ) are smaller than all elemen ts of σ − 1 ( ˜ s ) . Here < t is the tota l order on S coming fr om th e s tr ucture of the (S C ) 2-tree t . Th u s the sp ace X σ (8.21) consists of the conﬁgurations { ( x s , y s ) } from Conf ( S ) satisfying the follo wing conditions: C1 if ∃ i 1 , i 2 ∈ σ − 1 ( ˜ s ) and i ∈ σ − 1 ( s ) suc h that i 1 < i < i 2 then x s < x ˜ s C2 if x s = x ˜ s and all elemen ts of σ − 1 ( s ) are smaller than all elemen ts of σ − 1 ( ˜ s ) then y s < y ˜ s . If c S = a th en w e ha ve to imp ose on th e conﬁguration { ( x s , y s ) } the additional condition C3 if χ ( s ) = a then x s = 0 and if χ ( s ) = c then x s > 0 . Remark. Let S b e a us u al (non-SC) set. It can b e shown that ev ery su rjection σ ∈ D ( S, N ) give s us a pair of complemen tary orders on the set S in th e sense of M. Kon tsevic h and Y. Soib elman [23]. (See also Section 2 in [2] ab out complementary orders and higher trees.) T o a pair of complement ary orders > 0 and > 1 M. Kont sevic h and Y. Soib elman assign a sub space X > 0 ,> 1 [23] of the compactiﬁed conﬁguration space of p oin ts on R 2 lab eled b y elements of S . Our space X σ is an uncompactiﬁed v ersion of the subsp ace considered by M. Kontse vich and Y. S oib elman in [23]. PROOF OF SWISS CHEESE VERSION OF DELIGNE’S CONJECTURE 43 8.1.1. Contr actibility of X σ . W e giv e a detailed pro of of the con tractibilit y of X σ (8.21) in th e S C case when the color c S of the SC set S is a . The n on SC case ( c S = c ) is very similar. Although eve ry SC 2-tree t ∈ J ( σ ) giv es us a total order > t on S , we equip the set S with y et another tota l order whic h we denote by < σ . Namely , w e set s < σ ˜ s iﬀ — either ∃ i 1 , i 2 ∈ σ − 1 ( ˜ s ) and i ∈ σ − 1 ( s ) suc h that i 1 < i < i 2 or — all elemen ts of σ − 1 ( s ) are smaller than all elemen ts of σ − 1 ( ˜ s ) . W arning. In general, the order total > t on S coming from th e s tructure of an SC 2-tree t ∈ J ( σ ) do es not coincide with the order > σ . Thus, in Example 8.2, the map σ induces on the SC set S the order α < β < γ < δ . On the other hand w e hav e a prun ed SC 2-t r ee t : { β < γ < α < δ } → { 1 , 2 , 3 , 4 } whic h b elongs to J ( σ ) . A similar example can b e found for an SC set S with c S = c . Using the total order > σ w e identify S with the standard ordinal { 1 < 2 < 3 < · · · < | S |} . Next w e deﬁne the follo wing functions on C onf ( S ) (8.22) µ k ( { ( x s , y s ) } ) = min( y k , y k +1 , . . . , y | S | ) whic h are ob viously contin uous. Then w e introdu ce the sequence of sub spaces X σ = Y 0 ⊃ Y 1 ⊃ · · · ⊃ Y | S | , where Y k consists of conﬁgurations { ( x s , y s ) } ∈ X σ satisfying the prop erties (8.23) y s = y 1 + s − 1 , ∀ s ≤ k , (8.24) µ k +1 ( { ( x s , y s ) } ) = y k + 1 . Let us sho w that Y k +1 is homotop y equiv alen t to Y k for all k < | S | . F or this purp ose we introd uce an in termediate subs pace Z k Y k ⊃ Z k ⊃ Y k +1 . This subspace consists of conﬁgurations { ( x s , y s ) } ∈ Y k satisfying the prop erty (8.25) y k +1 = µ k +1 ( { ( x s , y s ) } ) . Let us consider the map h : Y k × [0 , 1] → Y k (8.26) h ( { ( x s , y s ) } , t ) = { ( x s , y s ( t )) } , where y s ( t ) = ( y s , if s 6 = k + 1 , (1 − t ) y k +1 + tµ k +1 ( { ( x s , y s ) } ) , if s = k + 1 . In order to sh o w that h ( { ( x s , y s ) } , t ) ∈ Y k w e only need to c hec k condition C 2 for all t ∈ [0 , 1] . It is clear that (8.27) y k +1 ≥ y k +1 ( t ) ≥ µ k +1 ( { ( x s , y s ) } ) , ∀ t ∈ [0 , 1] . Since { ( x s , y s ) } ∈ Y k w e hav e µ k +1 ( { ( x s , y s ) } ) > y s , ∀ s ≤ k and hence y k +1 ( t ) > y s , ∀ s ≤ k , t ∈ [0 , 1] . 44 V.A. DOLGUSHEV, D.E. T AMARKIN, AND B.L. TSYGAN F urth ermore, since cond ition C2 is satisﬁed for { ( x s , y s ) } we conclude th at all p oin ts ( x s , y s ) with s > k + 1 and x s = x k +1 lie ab o v e the p oin t ( x k +1 , y k +1 ) . Combining this observ ation with inequalit y (8.27) we conclude that if s > k + 1 and x s = x k +1 then y k +1 ( t ) < y s for all t ∈ [0 , 1] . It is clear that h ( { ( x s , y s ) } , 1) ∈ Z k and for all { ( x s , y s ) } ∈ Z k w e hav e h ( { ( x s , y s ) } , t ) = { ( x s , y s ) } , ∀ t ∈ [0 , 1] . Th u s h is a deformation retraction of Y k on to Z k . It is clear that the subspace Y k +1 consists of conﬁgur ations { ( x s , y s ) } ∈ Z k satisfying th e add i- tional prop ert y µ k +2 ( { ( x s , y s ) } ) = y k +1 + 1 . So w e consider the map h Z : Z k × [0 , 1] → Z k (8.28) h Z ( { ( x s , y s ) } , t ) = { ( x s , y s ( t )) } , where y s ( t ) = ( y s , if s ≤ k + 1 , y s + t  y k +1 + 1 − µ k +2 ( { ( x s , y s ) } )  , if s > k + 1 . In order to sh o w that h Z lands in Z k w e need to c hec k condition C2 and condition (8.2 5). Since min( y k +2 ( t ) , y k +3 ( t ) , . . . , y | S | ( t )) = min( y k +2 , y k +3 , . . . , y | S | ) + t  y k +1 + 1 − µ k +2 ( { ( x s , y s ) } )  = (1 − t ) µ k +2 ( { ( x s , y s ) } ) + t ( y k +1 + 1) ≥ µ k +1 ( { ( x s , y s ) } ) w e conclude that µ k +1 ( { ( x s , y s ( t )) } ) do es not d ep end on t . Thus condition (8.25) is satisﬁed. Next, if s ≥ k + 2 then y s ( t ) ≥ µ k +2 ( { ( x s , y s ) } ) + t  y k +1 + 1 − µ k +2 ( { ( x s , y s ) } )  = (1 − t ) µ k +2 ( { ( x s , y s ) } ) + t ( y k +1 + 1) > y k +1 for all t ∈ (0 , 1] b ecause µ k +2 ( { ( x s , y s ) } ) ≥ y k +1 and y k +1 + 1 > y k +1 . Hence y s ( t ) > y ˜ s for all s ≥ k + 2, ˜ s ≤ k + 1 and t ∈ (0 , 1] . F urth ermore, if for s, ˜ s ≥ k + 2 we ha v e y s > y ˜ s then obviously y s ( t ) > y ˜ s ( t ) for all t ∈ [0 , 1] . Th u s we conclud e th at condition C2 is satisﬁed for ev ery conﬁ guration h Z ( { ( x s , y s ) } , t ) . It is not hard to see that for all { ( x s , y s ) } ∈ Z k h Z ( { ( x s , y s ) } , 1) ∈ Y k +1 and for all t ∈ [0 , 1] and { ( x s , y s ) } ∈ Y k +1 h Z ( { ( x s , y s ) } , t ) = { ( x s , y s ) } . Th u s h Z is a deformation retractio n of Z k on to Y k +1 . W e pro ve d that X σ is homotopy equ iv alen t to th e subspace Y | S | whic h consists of conﬁgur ations { ( x s , y s ) } ∈ X σ satisfying the prop erty (8.29) y s = y 1 + s − 1 , ∀ s ∈ S . T o sh ow that Y | S | is con tractible we set, as ab o ve , S a = χ − 1 ( a ) and S c = χ − 1 ( c ) . Due to Condition C3 x s = 0 for all s ∈ S a and x s > 0 f or all s ∈ S c . PROOF OF SWISS CHEESE VERSION OF DELIGNE’S CONJECTURE 45 Restricting the total order > σ from S to S c w e get an isomorphism β : S c → { 1 < 2 < 3 < · · · < | S c |} from S c to the stand ard ordinal { 1 < 2 < 3 < · · · < | S c |} . Using this isomorphism w e deﬁn e the follo wing map H : Y | S | × [0 , 1] → Y | S | (8.30) H ( { ( x s , y s ) } , t ) = { ( x s ( t ) , y s ) } , where x s ( t ) = ( 0 , if s ∈ S a , (1 − t ) x s + tβ ( s ) , if s ∈ S c . Let us sho w that H ind eed lands in X σ . Since x s ( t ) > 0 for all s ∈ S c and t ∈ [0 , 1] w e need to c hec k Condition C1 only for s, ˜ s ∈ S c . If s, ˜ s ∈ S c , s 6 = ˜ s and th ere exists i 1 , i 2 ∈ σ − 1 ( ˜ s ) and i ∈ σ − 1 ( s ) suc h that i 1 < i < i 2 then x s < x ˜ s and β ( s ) < β ( ˜ s ) according to the deﬁnition of the tota l order < σ on S . Hence (1 − t ) x s + tβ ( s ) < (1 − t ) x ˜ s + tβ ( ˜ s ) , ∀ t ∈ [0 , 1] . Condition C2 is satisﬁed automatica lly b ecause for ev ery conﬁgur ation in Y | S | w e hav e (8.29) . Condition C3 is also obviously satisﬁed. It also follo ws from the construction that H ( { ( x s , y s ) } , t ) ∈ Y | S | for all { ( x s , y s ) } ∈ Y | S | and t ∈ [0 , 1] . F urth ermore, it is cleat that H is a deformation retraction of Y | S | on to the s u bspace L of conﬁg- urations { ( x s , y s ) } ∈ X σ with y s = y 1 + s − 1 , ∀ s ∈ S , x s = 0 , ∀ s ∈ S a , and x s = β ( s ) , ∀ s ∈ S c . The subspace L is obviously homeomorphic to the r eal line R . Th u s w e conclude that Y | S | and hence X σ is con tractible. This completes the pro of of Pr op osition 8.1 and hence the p ro of of Th eorem 7.9. Example 8.4. Let us illustrate the pro of of contrac tibilit y for X σ with the m ap σ : { 1 , 2 , 3 , 4 , 5 , 6 } → { α, β , γ , δ } from Example 8.2. Recall that c S = a χ ( α ) = χ ( γ ) = χ ( δ ) = c , and χ ( β ) = a . The space X σ consists of conﬁgur ations from Conf ( { α, β , γ , δ } ) s atisfying the follo wing cond i- tions: i ) x β = 0 < x γ < x δ , ii ) x α > 0 , iii ) if x α = x γ then y α < y γ , iv ) if x α = x δ then y α < y δ . In the ﬁrst step of the ab ov e p ro of we retract X σ on to the subs p ace Z 0 of conﬁ gu r ations satisfying the prop ert y y α = min( y α , y β , y γ , y δ ) . Second, w e retract Z 0 to the subspace Y 1 of conﬁgurations satisfying in addition the prop erty min( y β , y γ , y δ ) = y α + 1 . 46 V.A. DOLGUSHEV, D.E. T AMARKIN, AND B.L. TSYGAN Next, w e retract Y 1 to the subspace Z 1 whic h consists of conﬁgurations { ( x s , y s ) } ∈ Y 1 with y β = min( y β , y γ , y δ ) . W e keep doing so until w e get the subspace Y 4 of conﬁgurations { ( x s , y s ) } ∈ X σ with (8.31) y δ = y γ + 1 = y β + 2 = y α + 3 . Then we retract the resulting sp ace Y 4 to the subsp ace L of conﬁgurations { ( x s , y s ) } ∈ X σ satisfying (8.31) and x α = 1 , x β = 0 , x γ = 2 , x δ = 3 . P erforming the latter retraction w e may need to mov e horizon tally the p oint lab eled b y α through the vertica l lines con taining the p oint s lab eled by γ and δ . In d oing so w e will not violate conditions iii ) and iv ) b ecause the inequalities y α < y γ and y α < y δ are already ac hiev ed at the pr evious steps. The subspace L is ob viously homeomorphic to the real line. Thus con tractibilit y of X σ follo ws. 9. Proof of Theorem 2.1 Let us return to th e dg (SC) 2-op erad br introd uced in Deﬁnition 7.3 and sh ow that Prop osition 9.1. F or every prune d 2-tr e e (prune d SC 2-tr e e) t 1) the c o chain c omplex br ( t ) is c ontr actible; 2) ther e exist natur al identiﬁc ations H 0 ( br ( t )) = k under which al l op er adic c omp osition maps of the op er ad H • ( br ) evaluate d on 1 ∈ k pr o duc e 1 ∈ k . Pr o of. Due to Lemma 7.2 th e inclusion br ֒ → | SC seq | is a quasi-isomorphism of dg SC 2-op erads. W e start w ith the non-SC case. W e ha v e to sho w that for ev ery prun ed 2-tree t : S → T , the co c hain complex | seq | ( t ) is con tractible. This was pro ved in Prop osition 6.4 in [32]. F or th e con venience of the reader we brieﬂy recall the argumen t. By deﬁnition, | seq | ( t ) is the realizat ion of the cosimplicial/polysimplicial set (see Section 6) (9.1) {{ I s } s ∈ S ; J } → seq ( t ) J { I s } s ∈ S in the category of co chain complexes. Th u s w e need to sh o w that realizing (9.1 ) in the categ ory of top ological s p aces w e get a con- tractible space. F or this purp ose we ﬁ x the ordinal J and consider the corresp ondin g p olysimplicial set (9.2) {{ I s } s ∈ S } → seq ( t ) J { I s } s ∈ S . It is sho wn in [32] that for every (n on-empt y) ord inal J (9.3) | seq ( t ) J • ,..., • | top ∼ = | seq ( t ) [0] • ,..., • | top × ∆ J and moreo v er the collection of homeomorphisms (9.3) giv es an isomorph ism of the corresp onding cosimplicial top ologica l spaces. Here [0] is the one elemen t ordinal. PROOF OF SWISS CHEESE VERSION OF DELIGNE’S CONJECTURE 47 Th u s, in order to p ro v e con tractibilit y of the realization of (9.1) we need to p ro ve cont ractibilit y of the topological sp ace (9.4) | seq ( t ) [0] • ,..., • | top . This sp ace admits the follo wing exp licit description. A p oin t of | seq ( t ) [0] • ,..., • | top is giv en by an equiv alence class of d ecomp ositions of the segmen t [0 , | S | ] in to a num b er of sub s egmen ts lab eled by elemen ts of S . T he lab eling should satisfy the follo wing conditions: ℵ 1) if s 1 , s 2 ∈ S and a segmen t lab eled b y s 2 lies b et wee n segments lab eled by s 1 then t ( s 1 ) > t ( s 2 ) in T , ℵ 2) if for s 1 , s 2 ∈ S we ha ve t ( s 1 ) = t ( s 2 ) and s 1 < s 2 then all segmen ts lab eled b y s 1 are on the left-hand side of all segments lab eled b y s 2 , ℵ 3) for ev ery s ∈ S the total length of all segments lab eled b y s is 1 . Tw o such decomp ositions are equiv alent if one is obtained from the other by a num b er of op er- ations of the f ollo wing t wo t yp es: a) adding in to or deleting from our decomp osition a n umber of lab eled segmen ts of length 0 , b) joining t w o neigh b oring segmen ts of our decomp osition lab eled by an elemen t s ∈ S into one segmen t lab eled by s , or the in ve rs e op eration. In [32] it was prov ed, b y induction on | T | , that the space (9.4) is a pro duct of simplices and hence (9.4 ) is con tractible. Th u s we d educe that so is the co c h ain complex | seq | ( t ) . W e no w pass to the SC-case. Let t : S → T b e a p runed SC 2-tree with S = S a ⊔ S c , where S a is the preimage of the minimal elemen t of T and S c = S \ S a . The su bset S a ma y , in prin ciple, b e empt y . Recall that | SC seq | ( t ) is the realizatio n of the p olysimplicial set (9.5) {{ I s } s ∈ S c } → SC seq ( t ) { I s } s ∈ S c in the category of co chain complexes. Eac h elemen t u of SC seq ( t ) { I s } s ∈ S c is a tota l order > u on I = G s ∈ S c I s ⊔ S a satisfying the follo wing conditions: — it agrees with the tota l order on eac h I s and w ith the total order on S a , — if i, k ∈ I s 1 , j ∈ I s 2 , s 1 6 = s 2 and i < u j < u k , then t ( s 2 ) < t ( s 1 ) , — if s 1 , s 2 ∈ S c , s 1 < s 2 , and t ( s 1 ) = t ( s 2 ), then all elements of I s 1 are strictly smaller than all elemen ts of I s 2 . As well as the s p ace (9.4) the realization | SC seq ( t ) | top of (9.5) h as the follo wing explicit description. A p oin t of | SC seq ( t ) | top is give n b y an equ iv alence class of decomp ositions of th e segmen t [0 , | S | ] into a num b er of sub segmen ts lab eled by elemen ts of S . The lab eling s hould satisfy the f ollo win g conditions: ℵ 0 ′ ) for eac h s ∈ S a there is exactly one segment lab eled by s and its length is 1; if for s 1 , s 2 ∈ S a w e hav e s 1 < s 2 then the segment lab eled b y s 1 is on the left-hand side of the segmen t lab eled by s 2 . ℵ 1 ′ ) if s 1 , s 2 ∈ S c and a segment lab eled b y s 2 lies b et wee n segmen ts lab eled by s 1 then t ( s 1 ) > t ( s 2 ) , ℵ 2 ′ ) if for s 1 , s 2 ∈ S c w e ha ve t ( s 1 ) = t ( s 2 ) and s 1 < s 2 then all segment s lab eled by s 1 are on the left-hand side of all segments lab eled b y s 2 , ℵ 3 ′ ) for ev ery s ∈ S c the total length of all segmen ts lab eled b y s is 1 . 48 V.A. DOLGUSHEV, D.E. T AMARKIN, AND B.L. TSYGAN Tw o such decomp ositions are equiv alent if one is obtained from the other by a num b er of op er- ations of the f ollo wing t wo t yp es: a) adding in to or deleting from our decomp osition a n umber of lab eled segmen ts of length 0 , b) joinin g t wo neigh b oring segmen ts of our decomp osition lab eled b y an elemen t s ∈ S c in to one segmen t lab eled by s , or the in ve rs e op eration. If w e remo ve all elements of S a from S and the minimal elemen t t min from T then w e get a usual prun ed (non-SC) 2-tree (9.6) e t = t    S c : S c → T \ { t min } . T o this 2-tree we assign the follo wing p olysimplicial set (9.7) {{ I s } s ∈ S c } → seq ( e t ) [0] { I s } s ∈ S c and the corresp onding top ologica l sp ace (9.8) | seq ( e t ) [0] • ,..., • | top whic h w as explicitly describ ed ab o v e. (The sp ace (9.8) is ob tained fr om the space (9.4) via rep lacing t b y e t .) W e h a v e th e obvious pro jection P : | SC seq ( t ) | top → | seq ( e t ) [0] • ,..., • | top whic h sends a p oin t of | SC seq ( t ) | top to a p oin t of | seq ( e t ) [0] • ,..., • | top b y collapsing eac h segmen t lab eled b y an elemen t of S a to a p oin t. Con versely , giv en: i ) a p oin t x ∈ | seq ( e t ) [0] • ,..., • | top , and ii ) a monotonous map U : S a → [0 , | S c | ] one can reconstru ct a p oin t in | seq ( t ) | top b y in serting unit segmen ts lab eled by s ∈ S a in th e place of the p oin t U ( s ) . Th u s w e conclude that | SC seq ( t ) | top ∼ = | seq ( e t ) [0] • ,..., • | top × ∆ | S a | . Due to Prop osition 6.4 from [32] the ﬁrst comp onen t | seq ( e t ) [0] • ,..., • | top is con tractible. Hence so is | SC seq ( t ) | top . Th u s w e pro ved that | SC seq | ( t ) is con tractible for ev ery p runed SC 2-tree t . The iden tiﬁcations from Part 2) of this p r op osition come fr om th e fact that the top ological s p aces | SC seq ( t ) • • ,..., • | top for pruned 2-tree s t and | SC seq ( t ) • ,..., • | top for pruned SC 2-trees t are con tractible. These top ologica l realizations inherit the op eradic com- p ositions, whence P art 2) of this prop osition.  Prop osition 9.1 im p lies that the coﬁb ran t resolution R br of br is also a coﬁbrant resolution of the trivial (SC) 2-operad triv in the categ ory of reduced (SC) 2-op er ad s ov er co c hain complexes. Therefore, due to Batanin’s theorem (Th eorem 5.2) the symmetrization sym R br of R br is quasi-isomorphic to the singular chain op erad of V oronov’s Sw iss Cheese op erad SC 2 (in particular, the n on-SC part of sym R br is quasi-isomorphic to the singular chain op erad of the little d isc op erad (Theorem 4.3)) . PROOF OF SWISS CHEESE VERSION OF DELIGNE’S CONJECTURE 49 Due to T heorem 7.9 th e SC op erad sym R br is q u asi-isomorphic to bra ces whic h is, in turn, quasi-isomorphic to the SC op erad | s Ø | b y Lemma 7.6. Finally , by construction the S C op erad | s Ø | is isomorphic to th e op erad | Ø | . Th u s we conclude that the t wo -colored op erad | Ø | is q u asi-isomorphic to the sin gular c hain op erad of V orono v’s Swiss Cheese op erad SC 2 (and the non-SC part of | Ø | is qu asi-isomoprh ic to the singular c hain op erad of the little disc op erad). It remains to show that th e indu ced actio n of H −• (SC 2 ) on the pair ( H H • ( A, A ) , A ) coincides with the on e given in Prop osition 1.1. F or this pur p ose we presen t op erations on the pair (9.9) ( C • ( A, A ) , A ) whic h come from the action of | Ø | and whic h indu ce on ( H H • ( A, A ) , A ) the H −• (SC 2 )-algebra structure from Prop osition 1.1 . These op erations are the cup-pro d uct and the Gerstenhab er b rac k et [14] on C • ( A, A ), the asso- ciativ e pro d uct on A , and the follo wing cont r action of a co c hain P with elements of th e algebra A : (9.10) i ( P , a ) = a P (1 , 1 , . . . , 1) : C • ( A, A ) ⊗ A → A . W e w ould like to remark that since C • ( A, A ) is the normalized Ho c hsc h ild complex only degree zero cochains con tribute to the con traction. These op erations induce th e desired H −• (SC 2 )-algebra structure on ( H H • ( A, A ) , A ) and they ob viously come from the action of the S C op erad | Ø | on the pair (9.9). Since the cohomology op erad H • ( | Ø | ) of | Ø | is isomorph ic to H −• (SC 2 ) w e conclude that the action of | Ø | on (9.9) ind uce the desired H −• (SC 2 )-algebra structure on ( H H • ( A, A ) , A ) . Theorem 2.1 is pro v ed.  Appendix Let [ n ] b e the standard ordinal { 0 , 1 , 2 , . . . , n } . Giv en a collectio n of k ordinals [ n 1 ] , [ n 2 ] , . . . , [ n k ] w e consider the follo wing ordinal (9.11) I n 1 ,...,n k = [ n 1 ] ⊔ [ n 2 ] ⊔ · · · ⊔ [ n k ] , where the order is deﬁned b y the follo w ing rule: for i 1 ∈ [ n l 1 ] and i 2 ∈ [ n l 2 ] i 1 < i 2 if • l 1 < l 2 or • l 1 = l 2 and i 1 < i 2 in [ n l 1 ] . Giv en ord inals J , [ n 1 ] , [ n 2 ] , . . . , [ n k ] the collecti on (9.12) (Ξ k ) J n 1 ,...,n k = hom ∆ ( I n 1 ,...,n k , J ) form a p olysimplicial/cosimplicia l set. Indeed (Ξ k ) J n 1 ,...,n k is simplicial in [ n 1 ] , [ n 2 ] , . . . , [ n k ] and cosimplicial in J . In this app endix w e sho w that Prop osition 9.2. The c o chain c omplex | Ξ k | is c onc entr ate d in nonne gative de gr e es. F urthermor e, (9.13) H • ( | Ξ k | ) = ( k , if • = 0 , 0 , otherwise . Pro of. Th e ﬁrst statemen t is v ery easy . In deed, an elemen t v ∈ hom ∆ ( I n 1 ,...,n k , J ) will n ot con tribute to the realization if it is degenerate. It is clear th at if | J | < k X i =1 ( n i + 1) − k + 1 50 V.A. DOLGUSHEV, D.E. T AMARKIN, AND B.L. TSYGAN then v is degenerate. Therefore, elemen ts v ∈ hom ∆ ( I n 1 ,...,n k , J ) with | J | − 1 − k X i =1 n i < 0 will not con tribute to the realization. Hence the co c hain complex | Ξ k | is in deed concen trated in nonnegativ e degrees. The cochain complex | Ξ k | can b e considered as b icomplex (9.14) | Ξ k | = | Ξ k | • , • . The ﬁrst d egree is the total degree of the simplicial in dices. According to our con v en tions this degree is nonp ositiv e. Th e second degree is the degree in the cosimplicial index and this d egree is nonnegativ e. Let u s denote by ∂ s the p art of the diﬀerential in | Ξ k | which comes fr om the simplicial indices and b y ∂ c the part of th e d iﬀeren tial in | Ξ k | coming from the cosimplicial stru cture. Fixing the second degree w e get the co c hain complex (9.15) | Ξ k | • ,m whic h is the realizatio n of the p olysimp licial set (9.16) ([ n 1 ] , [ n 2 ] , . . . , [ n k ]) → hom ∆ ( I n 1 ,n 2 ,...,n k , [ m ]) . It is n ot hard to see that the realization of (9.16) in the category of topological spaces is the follo wing stretc hed m -simplex: { ( x 0 , x 1 , . . . , x m ) | x i ≥ 0 , x 0 + x 1 + x 2 + · · · + x m = k } . Therefore for eac h m the complex | Ξ k | • ,m has non-trivial cohomolo gy only in degree 0 and (9.17) H 0 ( | Ξ k | • ,m ) = k . The class whic h generates H 0 ( | Ξ k | • ,m ) is represen ted b y the map (9.18) c ∈ hom ∆ ( I 0 ,..., 0 , [ m ]) , whic h sends all elemen ts of I 0 ,..., 0 to the same elemen t 0 ∈ [ m ] . All other maps in hom ∆ ( I 0 ,..., 0 , [ m ]) are cohomolog ous to the co cycle (9.18) . It is not hard to see that (9.19) Θ = M q < 0 | Ξ k | q , • ⊕ ∂ s ( | Ξ k | − 1 , • ) is a sub complex of th e b icomplex | Ξ k | . Equation (9.17) implies that eac h term of the quotien t complex | Ξ k | / Θ is k . Using the explicit co cycle (9.18) it is not h ard to see that the quotien t complex | Ξ k | / Θ is k 0 → k id → k 0 → k id → k 0 → . . . and hence (9.20) H • ( | Ξ k | / Θ) = ( k , if • = 0 , 0 , otherwise . W e see from the construction that the bicomplex Θ (9.19) is acyclic in the ﬁ rst degree. Therefore Θ is acyclic as the total complex. Th u s H • ( | Ξ k | ) = H • ( | Ξ k | / Θ) and the prop osition follo w s.  PROOF OF SWISS CHEESE VERSION OF DELIGNE’S CONJECTURE 51 Referen ces [1] C. Baltean u, Z. Fiedoro wicz, R. Sch w¨ anzl, and R. V ogt, Iterated monoidal categories, Adv. Math. 176 (2003) 277–349 . [2] M.A. Batanin, Sy mmetrisation of n -op erads and compactiﬁcation of real conﬁguration spaces, Adv. Math. 211 , 2 (2007) 684–725; m ath .CT/06 06067. [3] M.A. Batanin, Lo cally constant n -op erads as h igher braided operads, J. N on commut. Geom. 4 , 2 (2010) 237– 263; math.A T/0804.4165. 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Proof of Swiss Cheese Version of Delignes Conjecture

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