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NON-FORMALITY OF PLANAR CONFIGURATION SPACES IN CHARACTERISTIC TWO 7 1 PAOLOSALVATORE 0 2 b e F Abstract. LetFk(R2)betheorderedconfigurationspaceofkdistinctpoints in the plane. We prove that its singular cochain algebra with coefficients in 3 Z2 isnotformalasadifferentialgradedalgebrafork≥4. ] T 1. Introduction A The notion of formality of a topological space X is usually introduced in the . h senseofrationalhomotopytheoryviasomecommutativedifferentialalgebramodel t in characteristic 0 of X, like the Sullivan-deRham algebra A (X), or the alge- a PL m bra of differential forms if X is a manifold. However one can define formality over any coefficient ring R in the non-commutative context, by requiring the al- [ gebra of R-valued cochains C∗(X,R) to be quasi-isomorphic to the cohomology 2 H∗(X,R). This means that C∗(X,R) is connectedto its cohomologyH∗(X,R) by v a zig-zag of homomorphisms of differential graded associative R-algebras inducing 6 isomorphisms in cohomology. If R is a field then this property depends only on 1 8 the characteristic of R. If R is a field of characteristic 0 then the formality in the 6 associativesenseisequivalenttotheusualcommutativeformalitybyarecentresult 0 of Saleh [15]. . 1 Let us consider the euclidean configuration spaces 0 F (Rn)={(x ,...,x )∈(Rn)k|x 6=x for i6=j}. 7 k 1 k i i 1 KontsevichandLambrechts-Volichaveprovedtheirformalityincharacteristiczero. : v Theorem 1.1. [10, 11] The configuration space F (Rn) is formal over R for any i k X k,n. r a The case of the configurationspaces in the plane Fk(R2) had been provedmuch earlier by Arnold [1]. Whatcanbesaidaboutformalityovertheintegers,orinpositivecharacteristic? It is not difficult to prove the following result. Theorem 1.2. The configuration space F (Rn) is intrinsically formal over Z if k n≥k. This means that any space with the same cohomology ring as the configuration space is formal over Z. The case n < k is open in general. A special case is F (R2)≃S1×(S1∨S1) that is formal over Z. 3 Geoffroy Horel has announced a proof of Z -formality of F (Rn) for any n > 2 p k even, using étale cohomology. One might think that formality over Z holds for all configuration spaces as in the rational case. 1 2 PAOLOSALVATORE However we found the following surprising result: Theorem1.3. TheconfigurationspaceF (R2)isnotformaloverZ foranyk ≥4. k 2 This immediately implies non-formality over the integers. Corollary 1.4. F (R2) is not formal over Z for any k ≥4. k WeapproachthisquestionbyobstructiontheoryfollowingtheworkbyHalperin- Stasheff [9] for cdga (commutative differential graded algebras) in characteristic 0 andits extensionto the non-commutativecaseby ElHaouari[7]. We implemented the computation of the non-trivial obstruction in MATLAB. The relevant library is attached. In a follow up paper we will deal with configuration spaces in R3. The paper is organized as follows: in section 2 we recall the presentation of the cohomology rings of the configuration spaces and of their Koszul dual, the Yang Baxter algebras. In section 3 we define formality of an algebra and recall the obstructiontheory to it by Halperin-Stasheff via bigradedmodels. In section 4 we describe the Barratt-Eccles-Smith simplicial model for euclidean configuration spaces. In section 5 we apply the obstruction theory and prove that F (R2) is k not formal over Z for k ≥ 4. In section 6 we interpret the obstruction class in 2 Hochschildcohomology. Finally in the appendix we describe the attachedsoftware in MATLAB. 2. Cohomology of configuration spaces and Yang-Baxter algebras We recall the presentation of the cohomology ring of the configuration spaces F (Rn). Then we describe its Koszul dual, the Yang-Baxter algebra, and its geo- k metric interpretation. Consider the direction map π : F (Rn) → Sn−1 from the i-th to the j-th i,j k particle given by π (x ,...,x )=(x −x )/|x −x |. i,j 1 k j i j i Let ι∈Hn−1(Sn−1) be the standard fundamental class. We write A =π∗ (ι)∈Hn−1(F (Rn)). ij i,j k The following computation of the cohomology ring is due to Arnold in the case n=2 and Fred Cohen for n>2. Theorem 2.1. (Arnold, Cohen) [1,5] The cohomology ring H∗(F (Rn)) for n≥2 k has a presentation with (n−1)-dimensional generators A for 1 ≤ i 6= j ≤ n and ij relations (1) A =(−1)nA ij ji (2) A2 =0 ij (3) (Arnold) A A +A A +A A =0 for i6=j 6=k 6=i ij jk jk ki ki ij Thus up to the grading this ring depends only on the parity of n. Corollary 2.2. The cohomology groups H∗(F (Rn)) are torsion free, and have a k graded basis provided by the set {A ···A |j <···<j , i <j ∀t} i1j1 iljl 1 l t t NON-FORMALITY OF PLANAR CONFIGURATION SPACES IN CHARACTERISTIC TWO 3 Corollary 2.3. The Poincare series of the configuration space is k−1 P(F (Rn))= (1+mxn−1) k m=1 Y Notice that F (Rn) is the total space of a tower of fibrations with fibers homo- k topicto∨ Sn−1 form=1,...,k−1. Thusthe additivestructuredoesnotseethe m twisting of the fibrations, but the cohomology ring does. AremarkablefactisthatthecohomologyoftheconfigurationspacesisaKoszul algebra ( see example 32 in [3]) . The following holds with coefficients in Z. Proposition 2.4. The Koszul dual algebra of H∗(F (Rn)) for n≥2 is the Yang- k Baxter algebra YB(n) generated by classes B of degree n−2, for 1 ≤ i 6= j ≤ k k ij under the relations (1) B =(−1)nB ij ji (2) (Yang-Baxter) [B ,B ]=[B ,B ]=[B ,B ] for i6=j 6=k 6=i ij jk jk ki ki ij (3) [B ,B ]=0 when {i,j}∩{r,t}=∅ ij rt Clearly this algebra up to the grading depends only on the parity of n. Proposition 2.5. [6] The Yang-Baxter algebra is torsion free and has a graded basis {B ···B |j ≤···≤j , i <j ∀t} i1j1 iljl 1 l t t The Yang-Baxter algebra has the following geometric meaning for n > 2: it is the Pontrjagin ring of the loop space of the configuration space F (Rn). Notice k that under this hypothesis F (Rn) is (n−2)-connected. k Let A∗ ∈ H (F (Rn)) be the homology basis dual to A . By the Hurewicz ij n−1 k ij theorem and adjointness we have isomorphisms Hn−2(ΩFk(Rn))∼=πn−2(ΩFk(Rn))∼=πn−1(Fk(Rn))∼=Hn−1(Fk(Rn)) Let B′ ∈H (ΩF (Rn)) ij n−2 k be the class corresponding to A∗ under the composite isomorphism. ij Theorem 2.6. (Cohen-Gitler, Fadell-Husseini) [6, 8] There is an isomorphism of algebras H∗(ΩFk(Rn))∼=YBk(n) for n>2 sending B′ to B . ij ij Corollary 2.7. The Poincare series of ΩF (Rn) for n>2 is k k−1 P(ΩF (Rn))= (1−mxn−1)−1 k m=1 Y This follows also from the homotopy equivalence k−1 ΩF (Rn)≃ Ω(∨mSn−1) k 1 m=1 Y that is not multiplicative in general. 4 PAOLOSALVATORE For n = 2 the (ungraded) algebra YB = YB(2) has also a similar geometric k k meaning that we explain. Let us start from the following well known result. Proposition 2.8. The configuration space F (R2) is a classfying space of the pure k braid group on k strands Pβ . k Now consider the descending central series of Pβ defined by k G =Pβ and G =[G ,G ]. 1 k i 1 i−1 The direct sum of the subquotients forms a Lie algebra L =⊕ (G /G ) k i i i+1 under the bracket induced by taking commutators. Let U denote the universal enveloping algebra functor. Theorem 2.9. [6] There is an isomorphism YBk ∼=ULk sending Bij to the Artin generator A of the pure braid group. ij 3. Formality and bigraded models We present the definition of formality in rational homotopy theory and in the non-commutativesense. Wethenproceedtodescribethenon-commutativeversions of the bigraded models by Halperin and Stasheff, due to El Haouari. Definition 3.1. [9] A topological space X is rationally formal if there is a zig- zag of quasi-isomorphisms of commutative differential graded algebras connecting theSullivan-deRhamalgebraAPL(X)toitscohomologyH∗(APL(X))∼=H∗(X,Q) equipped with the trivial differential. If X is a manifold this is equivalent to the existence of a zig-zag of quasi- isomorphismsbetweenthealgebraofde-RhamformsΩ∗(X)anditsrealcohomology H∗(X,R). IfX isacomplexmanifoldwemightusetheringofcomplexdifferential formsandcohomologywithcomplexcoefficients. Namelythenotionofformalityin characteristic zero does not depend on the field for connected spaces of finite type (Theorem6.8in [9]). The same is true in positive characteristic,by a similar proof following the obstruction theory in [7]. Kontsevich [10] and Lambrechts-Volic [11] have proved that the configuration space F (Rn) is formal over R for any k and n (Theorem 1.1 ). k This approach uses the commutative algebra of real PA forms and proves also the formality of the little discs operads. Arnold had easily proved the formality of F (C) [1]. His quasi-isomorphism k embeds H∗(F (C),C) into the algebra of complex differential forms Ω∗(F (C)) k C k sending A to d(z −z )/(z −z ). ij i j i j We turn now to the non-commutative case. Definition 3.2. LetR be acommutativering. AtopologicalspaceX is R-formal, or formal over R, if the algebra of singular cochains C∗(X,R) is connected to its cohomology H∗(X,R) by a zig-zag of quasi-isomorphisms of differential graded R-algebras. For spaces of finite type with torsion free homology Z-formality is universal as it implies R-formality for any ring R. NON-FORMALITY OF PLANAR CONFIGURATION SPACES IN CHARACTERISTIC TWO 5 Here is an application of formality to computations: let ΩX and LX be respec- tivelythe basedandthe unbasedloopspaceofX. LetHH denote the Hochschild ∗ homology functor from differential graded R-algebras to graded R-modules. Proposition3.3. Let X bea simply connectedR-formal spaceof finitetype. Then there are isomorphisms of graded R-modules TorH∗(X,R)(R,R)∼=H∗(ΩX,R) HH∗(H∗(X,R))∼=H∗(LX,R) ThisfollowsfromthecollapseoftherelevantEilenberg-Moorespectralsequences. ThetheoryofR-formalityisstudiedbyElHaouari[7],whoextendstheobstruc- tion theory by Halperin-Stasheff to the non-commutative case. The key point is the existence of a bigraded model that is then deformed to an actual model. A bigraded module V = ⊕Vn has an upper dimensional grading, the degree, and a k lower grading, the resolution level. The tensor algebra TV on a bigraded module inherits a bigrading by additivity. We work in the category of cochain differential graded algebras over a commutative ring R with upper grading, such that the dif- ferentialishomogeneousofdegree1. AbigradedR-algebrahasadifferentialthatis alsohomogeneouswithrespecttothelowergrading,andlowersitby1. Thetensor algebraon a bigradedmodule inherits a bigradingas well, and its cohomologytoo. We saythataR-DGAAisconnectedifH0(A)=R,generatedbythe unitofA. Proposition 3.4. (2.1.1.of [7]) Let R be a field. Given a connected graded R- algebra H, there exists a bigraded R-module V and a differential d on the tensor algebra T(V) together with a quasi-isomorphism ρ:(T(V),d)→(H,0) such that the homology in positive resolution level vanishes: H (T(V),d)=0. + Moreover V = H and ρ| is the identity. The algebra (T(V),d) is unique up to 0 V0 isomorphism and is called the bigraded model of H. Remark 3.5. Proposition 3.4 works also over a commutative ring R when Hi and Tori (R,R) are free R-modules for each i. In particular it holds for R = Z and H H =H∗(F (Rn)). k Example3.6. InthecaseoftheconfigurationspacecohomologyH =H∗(F (Rn)), k the lower grading of the Yang-Baxter algebra YB(n) = H! is the length of words k in the standard generators. The suspended filtered dual V = s(YB(n))∗ generates k the bigradedmodel (T(V),d)=B(YB(n))∗ thatis the filtered dualofthe barcon- k struction. Inotherwordsthedifferentialisthederivationinducedbythecoproduct d : V −µ→∗ V ⊗V ⊂ T(V) dual to the multiplication µ of YB(n). The resolution k map Ψ:(T(V),d)→H is then chosen by Ψ(B∗)=A , ij ij Ψ((B ...B )∗)=0 ∀l >1 i1j1 iljl We do not need to consider filtered duals for n > 2 since in that case YB(n) is k of finite type, i.e. finite dimensional in each degree. 6 PAOLOSALVATORE A bigraded module A has an induced filtration level by F (A)=⊕ A∗ . We k i≤k i say that the differential lowers filtration level by h if d(F (A))⊂F (A). k k−h The key idea of Halperin-Stasheff is to deform the bigraded model of the cohomology H(A) of a DGA A in order to get a model for the algebra itself. Theorem 3.7. (2.2.2 of [7]) Let A be a connected DGA over a field R . Let (T(V),d) be the bigraded model of H∗(A). Then there is a differential D on T(V) and a quasi-isomorphism π :(T(V),D)−→A such that D −d lowers filtration level by 2. If π′ : (T(V),D′) → A is another solution to this problem, then there exists an isomorphism φ:(T(V),D)−→(T(V),D′) such that φ−id lowers filtration level by 1, and π′◦φ ≃π. Any solution is called a model of A. The following result completes the theory, compare Theorem 5.3 in [9]. Proposition 3.8. Let A,B be connected DGA’s over a field R with an isomorphism φ¯ : H(A) ∼= H(B). Let (T(V),DA),(T(V),DB) be the respective models. Then A and B are quasi-isomorphic if and only if there exists an isomorphism φ:(T(V),D )→(T(V),D ) such that φ−id lowers filtration. A B We give a first immediate application. Definition3.9. AspaceX isintrinsicallyR-formalifitisR-formal,andanyother space with the same cohomology R-algebra is also R-formal. Theorem 3.10. The configuration space F (Rn) is intrinsically Z-formal if n≥k. k Proof. We can assume n >2 since intrinsic formality of S1 ≃F (R2) is easy. The 2 generators of the bigraded model (T(V),d) for H =H∗(F (Rn),R) are in degree k n−1,2n−3,...,1+i(n−2),... since V ∼=s(YBn)∗, and respective level 0,1,2,3,...,i−1,... Considerthe model (T(V),D) for the singular cochainalgebraC∗(Y,R) of a space Y withH∗(Y,R)∼=H. WehavethatD−dlowersthefiltrationby2. Allgenerators are in degree 1 mod n−2, andso their differentials are in degree 2 mod n−2. For a generator v of level i−1, and thus degree 1+i(n−2), a homogeneous non-zero monomial can be a term of (D −d)(v) only if it has degree 2 mod n−2. But quadratic monomials v v , with v ∈ V and b ∈ V , have to be ruled out a b a a−1 b−1 because they would have degree 2+i(n−2)=[1+a(n−2)]+[1+b(n−2)], thus filtration level (a−1)+(b−1)= i−2, contradicting that D−d lowers filtration by two. The monomial is then the product of at least n generators, that is in degree at least n+n(n −2) = 2+ (n+1)(n −2), so i > n. Then d = D on V ≤ n. The top cohomology of the configuration space (and Y) is in dimension (k−1)(n−1) < n(n−1), so there is no obstruction to extending the restriction (T(V ),D)→H inductively to a quasi-isomorphism (T(V),D)→H. ≤n (cid:3) NON-FORMALITY OF PLANAR CONFIGURATION SPACES IN CHARACTERISTIC TWO 7 A special case not covered by the theorem is that of F (R2) ≃ S1×(S1∨S1), 3 which is Z-formal , since wedges and products preserve formality [9]. In order to prove the non-formality of F (R2) (Theorem 1.3) over Z for k > 3 k 2 we introduce simplicial versions of these configuration spaces in the next section. 4. The Barratt-Eccles complex We recall the Smith simplicial model for planar configuration spaces, obtained by filtering the Barratt-Eccles complex [2]. For simplicity we write F := F (R2), k k and we adopt Z -coefficients throughout although we do not state it explicitly. 2 Definition 4.1. The full Barratt-Eccles complex on k elements is the geometric bar construction of the symmetric group on k letters WΣ . k This is a simplicial set with (WΣ ) = (Σ )l+1, so any element is a string of k l k permutations. Thefaceoperatorsdeleteasinglepermutationinthestring,andthe degeneracies double a permutation in the string. The symmetric group Σ acts diagonally levelwise. k In particular WΣ has 2 non-degenerate simplices in each degree that are 2 (12|21|12|..), (21|12|21|..). Consider the sub-simplicial set F (WΣ )⊂ WΣ spanned by non-degenerate sim- t 2 2 plices of dimension at most t−1. Thegeometricrealization|F (WΣ )|hasthehomotopytypeofthe(t−1)-sphere. t 2 We can extend the filtration to any k >2 as follows. There are simplicial versions of the projections seen in section 2, that we still denote π :WΣ →WΣ , for 1≤i6=j ≤k. ij k 2 Levelwise these maps are powers of the function π :Σ →Σ defined by ij k 2 (12) if σ−1(i)<σ−1(j) π (σ)= ij ((21) if σ−1(i)>σ−1(j). Definition 4.2. The t-th stage F (WΣ ) of the filtration of the Barratt-Eccles t k complex is defined by F (WΣ ) = π−1(F (WΣ ) ) t k l ij t 2 l i,j \ Proposition 4.3. [2] The geometric realization of the simplicial set F (WΣ ) has t k the homotopy type of the configuration space F (Rt) , and the realization of π can k ij be identified up to homotopy to the projection π : F (Rt) → St−1 considered in ij k section 2. LetN∗ denotethenormalizedcochainfunctorfromsimplicialsetstodifferential graded algebras (over Z in our case). 2 Definition 4.4. The Barratt-Eccles DG-algebra with complexity t and arity k is E∗(k):=N∗(F (WΣ )). t t k 8 PAOLOSALVATORE Here k is the arity of the Barratt-Eccles operad, that is equal to the number of pointsintheconfiguration. Theupperindexindicatesthedegree. Weareinterested mainly in t=2. Observe that each Ei(k) is a free module over Z [Σ ], so that k! divides its di- t 2 k mensionoverZ (thisholdsforanycoefficientring). Forfixedtthetopdimensional 2 non-trivial vector space is in degree (t−1) k . 2 To get a flavour we present some cases of the generating polynomial (cid:0) (cid:1) Ptk(x)= dimZ2(Eti(k))xi i X P2(x)=2(1+x) 2 P3(x)=6(1+5x+6x2+2x3) 2 P4(x)=24(1+23x+104x2+196x3+184x4+86x5+16x6) 2 P2(x)=2(1+x+x2) 3 P3(x)=6(1+5x+25x2+60x3+70x4+38x5+8x6) 3 P4(x)=24(1+23x+529x2+5550x3+30214x4+97048x5+...) 3 Let us denote the elements of the symmetric group Σ by the following letters: 3 A=(123),B =(132),C =(213),D=(231),E =(312),F =(321). Then E2(3) is the free Z [Σ ]-module on the 6 simplices 2 2 3 (A|D|F)∗, (A|E|F)∗, (A|C|F)∗, (A|C|D)∗, (A|B|F)∗, (A|B|E)∗, hence it is 36-dimensional over Z . 2 By Proposition4.3 we have quasi-isomorphic DGA’s E∗(k)≃C∗(F (R2)). 2 k Proposition 4.5. Non-formality of E∗(k) for k≥4 is equivalent to Theorem 1.3. 2 Consider the 1-cochain ω =π∗((12|21)∗)∈E1(k). ij ij 2 For example for k = 3 ω is the sum of 9 generators, and for k = 4 of 144 ij generators. Consider the 1-cochain in E1(3) 2 Ar :=(F|E)∗+(E|F)∗+(E|D)∗+(E|C)∗+(E|A)∗+(E|B)∗+(D|E)∗+(C|E)∗+(A|E)∗. Lemma 4.6. A resolution of the Arnold relation for 3 points on the cochain level is given by d(Ar)=ω ω +ω ω +ω ω 13 12 23 12 23 13 Proof. Let us compute the pairing < d(Ar),X >=< Ar,∂(X) > for basis generators X ∈ E2(3)∗. The cochain Ar is the sum of all generators of the form (E|x)∗ 2 with E 6= x ∈ Σ , and of the form (x|E)∗ with x 6= B and x 6= E. Thus d(Ar) 3 vanishes on all generators not containing E, and also on chains (E|x|y), since in complexity 2 there are no chains of the form (E|x|E), and <Ar,∂(E|x|y)>=<Ar,(E|x)+(E|y)>=1+1=0 mod 2 If E is in the middle position, d(Ar) pairs non-trivially with chains of the form (B|E|x). Since 1and3swapgoingfromB to E,only 2canmovewithrespectto 1 and 3, comparing E =(312) and x, hence the possibilities are x = D = (231) and NON-FORMALITY OF PLANAR CONFIGURATION SPACES IN CHARACTERISTIC TWO 9 x = F = (321). Finally d(Ar) pairs non-trivially on chains of the form (x|B|E), and again only 2 can move from x to B so that x=A or x=C. This proves that (4.1) d(Ar)=(B|E|D)∗+(B|E|F)∗+(A|B|E)∗+(C|B|E)∗. The cochain ω ω pairs non trivially with chains (x|y|z) with (13) and (12) sub- 13 12 strings of x, (31) and (12) of y, and (31) and (21) of z. This forces x = (123) or x=(132). In the first case y =(312) and z =(321). In the second y =(312), and either z =(321) or z =(231), hence (4.2) ω ω =(A|E|F)∗+(B|E|F)∗+(B|E|D)∗. 13 12 By similar considerations we obtain (4.3) ω ω =(A|B|F)∗+(A|E|F)∗, 23 12 (4.4) ω ω =(A|B|E)∗+(A|B|F)∗+(C|B|E)∗. 23 13 Comparing the sum of equations 4.2, 4.3, 4.4 with equation 4.1 we conclude. (cid:3) 5. The bigraded model of the Barratt Eccles complex InthissectionwestartbuildingthebigradedmodeloftheBarratt-Ecclesalgebra E∗(4) and verify its non-formality. Let (TW,D) be the model of E∗(4) of Theorem 2 2 3.7 , obtained by deforming the bigraded model (TW,d) ≃ H∗(F ), and equipped 4 with a quasi-isomorphismφ:(TW,D)→E∗(4). 2 By Theorem 3.7 d=D on W and W . We can characterize it explicitly. 0 1 Lemma 5.1. On W , D =d=0. On W , D =d is determined by 0 1 (B∗)2 if i=k andj =l ij B∗B∗ +B∗B∗ =[B∗,B∗] for j <l d((B B )∗)= ij kl kl ij ij kl ij kl Bi∗jBk∗l+Bi∗lBk∗i+Bk∗lBk∗i forj =landk <i B∗B∗ +B∗B∗ +B∗ B∗ forj =landk >i ij kl il ik kl ik Proof. TheproductoftwogeneratorsBijBuv oftheYang-Baxteralgebraisastan- dard basis generator if and only if j ≤ v. Instead, for j > v, the Yang Baxter relations yield B B if {i,j}∩{u,v}=∅ uv ij B B = ij uv (BuvBij +BujBvj +BvjBuj otherwise By dualizing the formula on basis generators we obtain the result. (cid:3) We shall show that d and D do not agree on W , and there is no “gauge equiv- 2 alence” transformation fixing the problem. Following the construction ofthe bigradedmodel in [7, 9] we canassume that φ sends each generator in W ∼=H1(F ) to a cocyle representative. 0 4 Definition 5.2. Let φ:W →E∗(4) be the linear map given by 0 2 φ(B∗)=ω . ij ij We move to the next set of generators in filtration 1 10 PAOLOSALVATORE Definition 5.3. Let φ:W →E1(4) be the linear map given by 1 2 0 if (i,j)=(k,l) ω ∪ ω if j <l φ((B B )∗)= ij 1 kl ij kl πk∗il(Ar) if i>k and j =l π∗ (Ar)+ω ∪ ω if i<k and j =l. ikl il 1 kl We remark that the ∪ product of 1-cochains over Z has a simple form. Each 1 2 k-cochaincoverZ isdeterminedbyitssupportSupp(c),thesetofnon-degenerate 2 k-chains sent to 1 by c (rather than 0). Now for given 1-cochains c,c′ the relation Supp(c∪ c′)=Supp(c)∩Supp(c′)completelydeterminesc∪ c′. Theconstruction 1 1 of ∪ goes back to Steenrod, and is presented by McClure-Smith in [12]. 1 The homomorphism π : WΣ → WΣ is induced by the simplicial version of kil 4 3 the forgetful projection from 4 to 3 configuration points that we describe below (the projection to 2 points has been described in section 4). We need a function π :Σ →Σ telling how a permutation twists the indices kil 4 3 {k,i,l}. Consider the three symbols j = k,j = i,j = l. A permutation σ ∈ Σ 1 2 3 4 gives a sequence (σ(1),σ(2),σ(3),σ(4)) containing a unique ordered subsequence (j ,j ,j ), where τ ∈ Σ . We set π (σ) = τ. The construction induces τ(1) τ(2) τ(3) 3 kil a map on the W-construction still denoted π : WΣ → WΣ , and then on kil 4 3 normalized cochains π∗ =N∗(π ):E∗(3)→E∗(4). kil kil t t Lemma5.4. Thehomomorphism ΦofDefinition5.2and5.3iscompatiblewiththe differentials, and thus defines a homomorphism of DGA φ:(T(W ),d)→E∗(4). ≤1 2 Proof. We have that d((B2)∗)=(B∗)2 by lemma 5.1 ij ij In the first case of 5.3 φ(d((B2)∗))=φ((B∗)2)=ω2 =0=d(φ((B2)∗)). ij ij ij ij Namely the top-dimensional 1-cocyle (12|21)∗ ∈ E1(2) squares to zero by dimen- 2 sionalreasons,andthenso does its pullback ω ∈E1(3). In the secondcaseof5.3 ij 2 by Lemma 5.1 d((B B )∗)=[B∗,B∗] if j <l, and ij kl ij kl φ(d((B B )∗))=φ([B∗,B∗])=ω ∪ω +ω ∪ω =d(ω ∪ ω ) ij kl ij kl ij kl kl ij ij 1 kl by Steenrod’s formula [12]. By definition d(ω ∪ ω )=d(φ((B B )∗)). ij 1 kl ij kl In the third case of 5.3 j =l and i>k we have from Lemma 5.1 that d((B B )∗)=B∗B∗ +B∗B∗ +B∗ B∗ ij kl ij kl ij ki kj ki and φ maps the right hand side to ω ω +ω ω +ω ω =π∗ (d(Ar)) =d(π∗ (Ar))=dφ((B B )∗). il kl il ki kl ki kil kil ij kl In the last case of 5.3 for i<k we have the same formula d((B B )∗)=B∗B∗ +B∗B∗ +B∗ B∗ ij kl ij kl ij ik kj ik (we only swap i and k to have the term B in standard form), so ik φ(d((B B )∗))=ω ω +ω ω +ω ω =π∗ (d(Ar))+[ω ,ω ]= ij kl il kl il ik kl ik ikl il kl