The Milnor-Moore theorem for L ∞ algebras in rational homotopy theory
aa r X i v : . [ m a t h . A T ] A p r The Milnor-Moore theorem for L ∞ algebras in rationalhomotopy theory José Manuel Moreno-Fernández
Abstract
We give a construction of the universal enveloping A ∞ algebra of a given L ∞ algebra, alter-native to the already existing versions. As applications, we derive a higher homotopy algebrasversion of the classical Milnor-Moore theorem, proposing a new A ∞ model for simply connectedrational homotopy types, and uncovering a relationship between the higher order rational White-head products in homotopy groups and the Pontryagin-Massey products in the rational loopspace homology algebra. The main goal of this paper is to construct a universal enveloping A ∞ algebra for a given L ∞ algebra, alternative to the already existing versions [15, 3], and to study the consequences of suchan structure in rational homotopy theory.Let L be an L ∞ algebra. In Def. 2.4, we introduce the universal enveloping A ∞ algebra U t ( L ).It is isomorphic to the free symmetric algebra Λ L on L as a graded vector space, and arises froma transfer process. For dg Lie algebras, U t ( L ) coincides with the classical dg associative envelope U L . To motivate the definition of U t , we first prove the following result (Thm. 2.1( i )). Theorem A.
Let L and U L be a dg Lie algebra and its classical universal enveloping dg associativealgebra, respectively. Fix a contraction from L onto H = H ∗ ( L ) , and denote by { ℓ n } the induced L ∞ structure on H. Then, there is an explicit contraction from U L onto Λ H, so that denoting by { m n } the induced A ∞ algebra structure on Λ H, the antisymmetrization © m L n ª of { m n } fits into a strictL ∞ embedding ı : ( H ,{ ℓ n }) , → ³ Λ H ,{ m L n } ´ . That is, for every x i ∈ H , ı ℓ n ( x ,..., x n ) = X σ ∈ S n χ ( σ ) m n ¡ x σ (1) ⊗ ··· ⊗ x σ ( n ) ¢ = m L n ( x ,..., x n ).The result above covers the case in which L is minimal, since any such can be obtained as acontraction of the dg Lie algebra L C ( L ). In general, U t ( L ) is defined as Λ L together with an A ∞ structure inherited from a contraction from Ω C ( L ) onto Λ L . Here, C are the Quillen chains, Ω the cobar construction, and L Quillen’s Lie functor. See Section 2 for details.The original motivation for introducing the envelope we present was for extending the classicalMilnor-Moore theorem ([24]) to L ∞ algebras in the rational setting. This is Thm. 4.1. * The author has been partially supported by the MINECO grant MTM2016-78647-P and by a Postdoctoral Fellowship ofthe Max Planck Society.2010 Mathematics subject classification: 55P62, 16S30, 17B55, 18G55, 16E45, 55S30, 55Q15.Key words and phrases: Universal enveloping algebra. Rational homotopy theory. A ∞ -algebra. L ∞ -algebra. Loop spacehomology. Higher Whitehead products. Massey-Pontryagin products. heorem B. Let X be a simply connected CW-complex. Endow π ∗ ( Ω X ) ⊗ Q with an L ∞ structure { ℓ n } representing the rational homotopy type of X for which ℓ = and ℓ = [ − , − ] is the Samel-son bracket. Then, there exists an A ∞ algebra structure { m n } on the loop space homology algebraH ∗ ( Ω X ; Q ) for which m = m is the Pontryagin product, and such that the rational Hurewiczmorphism h : π ∗ ( Ω X ) ⊗ Q , → H ∗ ( Ω X ; Q ) = U t ( π ∗ ( Ω X ) ⊗ Q ) is a strict L ∞ embedding. Therefore, the L ∞ structure on the rational homotopy Lie algebra is theantisymmetrized of the A ∞ structure on H ∗ ( Ω X ; Q ) : ℓ n ( x ,..., x n ) = X σ ∈ S n χ ( σ ) m n ¡ x σ (1) ,..., x σ ( n ) ¢ .Thm. B produces a new A ∞ model for simply connected rational homotopy types, with under-lying Hopf algebra H ∗ ( Ω X ; Q ). For finite type rational spaces, this enveloping A ∞ algebra modelcan be understood as an Eckmann-Hilton or Koszul dual to Kadeishvili’s C ∞ algebra model [14],the latter starting from cohomology instead of homotopy. We explain in Section 4.2 how to explic-itly extract the Quillen and Sullivan models from such an enveloping A ∞ model. We also uncoveran interesting relationship between the higher order rational Whitehead products on π ∗ ( Ω X ) ⊗ Q and the higher order Pontryagin-Massey products of H ∗ ( Ω X ; Q ) of simply connected spaces: theformer are antisymmetrizations of the latter, whenever these are defined. This is Thm. 4.5. In it, h is the rational Hurewicz morphism. Theorem C.
Let x ,..., x n ∈ π ∗ ( Ω X ) ⊗ Q , and denote by y k = h ( x k ) ∈ H ∗ ( Ω X ; Q ) the correspond-ing spherical classes. Assume that the higher Whitehead product set [ x ,..., x n ] W and the higherMassey-Pontryagin products sets y σ (1) ,..., y σ ( n ) ® for every permutation σ ∈ S n are defined. If theA ∞ algebra structure { m i } on H ∗ ( Ω X ; Q ) provided by Theorem B has vanishing m k for k ≤ n − ,then x = εℓ n ( x ,..., x n ) ∈ [ x ,..., x n ] W , and satisfies:h ( x ) ∈ X σ ∈ S n χ ( σ ) y σ (1) ,..., y σ ( n ) ® . Here, ε is the parity of P n − j = | x j | ( k − j ) . If moreover the higher products are all uniquely defined,then the above containment is an equality of elements. The Massey-Pontryagin products should not be confused with the classical Massey products,see Section 4.3 for details. We study the homotopical properties of the envelope U t , and we com-pare it to other alternatives in the literature in Section 3. These alternative constructions havebeen developed by Lada and Markl [15] and by Baranovsky [3]. See Prop. 3.1 for a recollection ofthe statements. In particular, the classical identity U H = HU , asserting that taking homology anduniversal enveloping algebra commute, holds only up to homotopy for any sort of enveloping A ∞ algebra, and U t is quasi-isomorphic to Baranovsky’s construction. Acknowledgements:
The author is very grateful to Martin Markl, Aniceto Murillo, Peter Teichnerand Felix Wierstra for useful feedback on this project, and also to the Max Planck Institute forMathematics in Bonn for its hospitality and financial support.
In this paper, graded objects are always taken over Z , with homological grading (differentials lowerthe degree by 1). The degree of an element x is denoted by | x | , and all algebraic structures are con-sidered over a characteristic zero field.An A ∞ algebra is a graded vector space A = { A n } n ∈ Z together with linear maps m k : A ⊗ k → A ofdegree k −
2, for k ≥
1, satisfying the
Stasheff identities for every i ≥ i X k = i − k X n = ( − k + n + kn m i − k + (id ⊗ n ⊗ m k ⊗ id ⊗ i − k − n ) = differential graded algebra (DGA), is an A ∞ algebra for which m k = k ≥
3. An A ∞ algebra is minimal if m =
0. An A ∞ morphism f : A → B is a family of linear maps f k : A ⊗ k → B of degree k − i ≥ X i = r + s + ts ≥ r , t ≥ ( − r + st f r + + t ¡ id ⊗ r ⊗ m s ⊗ id ⊗ t ¢ = X ≤ r ≤ ii = i +···+ i r ( − s m r ¡ f i ⊗ ··· ⊗ f i r ¢ being s = P r − ℓ = ℓ ( i r − ℓ − f is an A ∞ quasi-isomorphism if f : ( A , m ) → ( A ′ , m ′ ) is aquasi-isomorphism of complexes. The bar construction B A of an A ∞ algebra A is the differentialgraded coalgebra (DGC, henceforth) B A = ( T ( s A ), δ ),where T ( s A ) is the tensor coalgebra on the suspension s A of A (i.e., ( s A ) p = A p − ), and δ = P k ≥ δ k is the codifferential such that δ k [ sx | ··· | sx p ] = p − k + X i = ε i [ sx | ··· | sx i | sm k + ( x i + ,..., x i + k + ) | ··· | sx p ],where ε i is the parity of 1 + P ij = | sx j | + P k + l = ( k + − j ) | sx i + l | . The bar construction turns A ∞ morphisms A → C into DGC morphisms B A → BC , and preserves quasi-isomorphisms ([16]).The cobar construction Ω C of a coaugmented DGC C is the augmented DGA Ω C = ³ T ³ s − C ´ , d ´ ,where T ³ s − C ´ is the tensor algebra on the desuspension s − C of the cokernel C = coKer ( K → C )of the coaugmentation K → C (i.e., ( s − C ) p = C p + ), and d = d + d is the differential determinedby d ¡ s − x ¢ = − s − δ x , d ¡ s − x ¢ = X i ( − | x i | s − x i ⊗ s − y i ,where δ is the codifferential of C and P i x i ⊗ y i = ∆ ( x ) − (1 ⊗ x + x ⊗
1) is the reduced comultipli-cation of x . The cobar construction extends to A ∞ coalgebras, but we are not in the need of sucha generality in this paper.An L ∞ algebra is a graded vector space L = { L n } n ∈ Z together with skew-symmetric linear maps ℓ k : L ⊗ k → L of degree k −
2, for k ≥
1, satisfying the generalized Jacobi identities for every n ≥ X i + j = n + X σ ∈ S ( i , n − i ) ε ( σ )sgn( σ )( − i ( j − ℓ j ¡ ℓ i ¡ x σ (1) ,..., x σ ( i ) ¢ , x σ ( i + ,..., x σ ( n ) ¢ = S ( i , n − i ) are the ( i , n − i ) shuffles, given by those permutations σ of n elements such that σ (1) < ··· < σ ( i ) and σ ( i + < ··· < σ ( n );and ε ( σ ),sgn( σ ) stand for the Koszul sign and the signature associated to σ , respectively. A differ-ential graded Lie algebra (DGL) is an L ∞ algebra L for which ℓ k = k ≥ L ∞ algebra is minimal if ℓ =
0. An L ∞ morphism f : L → L ′ is a family of skew-symmetriclinear maps © f n : L ⊗ n → L ′ ª of degree n − n ≥ X i + j = n + X σ ∈ S ( i , n − i ) ε ( σ )sgn( σ )( − i ( j − f j ¡ ℓ i ¡ x σ (1) ,..., x σ ( i ) ¢ , x σ ( i + ,..., x σ ( n ) ¢ = X k ≥ i +···+ i k = n τ ∈ S ( i ,..., i k ) ε ( σ )sgn( σ ) ε k ℓ ′ k ¡ f i ⊗ ··· ⊗ f i k ¢ ¡ x τ (1) ⊗ ··· ⊗ x τ ( n ) ¢ , ith ε k being the parity of P k − l = ( k − l )( i l − f is an L ∞ quasi-isomorphism if f : ( L , ℓ ) → ( L ′ , ℓ ′ ) is a quasi-isomorphism of complexes. The Quillen chains C ( L ) of an L ∞ algebra is theequivalent cocommutative DGC (CDGC, henceforth) C ( L ) = ( Λ sL , δ ) ,where Λ sL is the cofree conilpotent cocommutative graded coalgebra on the suspension sL of L ,and δ = P k ≥ δ k is the codifferential whose correstrictions are determined by the L ∞ structuremaps, i.e., δ k ¡ sx ∧ ... ∧ sx p ¢ = X i <···< i k ε s ℓ k ¡ x i ,..., x i k ¢ ∧ sx ∧ ... c sx i ... c sx i k ... ∧ sx p . (1)The sign ε is determined by the Koszul sign rule.A morphism f = { f k } of A ∞ or L ∞ algebras is strict if f k = k ≥ Quillen functor L ( C ) on a coaugmented CDGC C is the DGL L ( C ) = ³ L ³ s − C ´ , ∂ ´ ,where L ³ s − C ´ is the free graded Lie algebra on the desuspension s − C of the cokernel of thecoaugmentation, C = coKer ( K → C ), and ∂ = ∂ + ∂ is the differential determined by ∂ ¡ s − x ¢ = − s − δ ( x ), ∂ ¡ s − x ¢ = X i ( − | x i | [ s − x i , s − y i ], (2)where δ is the codifferential of C and P i x i ⊗ y i is again the reduced comultiplication of x .There is an antisymmetrization functor ( − ) L from the category of A ∞ algebras to that of L ∞ algebras which preserves quasi-isomorphisms ([15]). For a given A ∞ algebra ( A ,{ m n }), its anti-symmetrization A L has the same underlying graded vector space and higher brackets ℓ n givenby ℓ n ( x ,..., x n ) = X σ ∈ S n χ ( σ ) m n ¡ x σ (1) ⊗ ··· ⊗ x σ ( n ) ¢ .Here, S n is the symmetric group on n letters, and we shorten the notation by χ ( σ ) = ε ( σ )sgn( σ )for σ ∈ S n . We will usually denote the higher brackets ℓ n of A L by m L n .A contraction of M onto N is a diagram of the form M N , K qi where M and N are chain complexes and q and i are chain maps such that qi = id N and i q ≃ i d M via a chain homotopy K satisfying K = K i = qK =
0. We denote it by ( M , N , i , q , K ), or simply by( i , q , K ).Following [18, Def. 2.3], a morphism of contractions f : ( M , N , i , q , K ) → ( A , B , j , p , G ) is a chainmap f : M → A such that f K = G f . Denote by b f : N → B the chain map b f = p f i . Using that i q ≃ id M , it follows that in presence of a morphism of contractions f : M → A , the interior squaresin the following diagram commute: A BM N . G pjK qf i b f hat is, p f = b f q and f i = j b f .We will be concerned with the following particular instance of the homotopy transfer theorem (see [13, 23, 18, 12, 16, 5]). Theorem 1.1.
Let ( M , N , i , q , K ) be a contraction.1. If M = ¡ A , © µ n ª¢ is an A ∞ algebra, then there exists an A ∞ algebra structure { m n } on N ,unique up to isomorphism, and A ∞ algebra quasi-isomorphismsQ : ( A ,{ µ n }) ( N ,{ m n }) : Isuch that I = i , Q = q and Q I = id N .2. If M = ( L ,{ ϑ n }) is an L ∞ algebra, then there exists an L ∞ algebra structure { ℓ n } on N , uniqueup to isomorphism, and L ∞ algebra quasi-isomorphismsQ : ( L ,{ ϑ n }) ( N ,{ ℓ n }) : Isuch that I = i , Q = q and Q I = id N . The maps involved in the higher structure of Theorem 1.1 can be described in several ways. Forthe purposes of this paper, we will describe the maps using recursive algebraic formulas. We willconsistently use the following convention for the rest of the paper: contractions of an L ∞ algebrawill be denoted by ( i , q , K ), whereas contractions of an A ∞ algebra will be denoted by ( j , p , G ). Thecapital letters I , Q or J , P will stand for the corresponding induced infinity quasi-isomorphisms.The higher multiplications { m n } on N and the terms { J n } of the A ∞ quasi-isomorphism J arerecursively given as follows. Formally, set G λ = − j , and define λ n : H ⊗ n → A for n ≥ λ n ( x ,..., x n ) = n X k = m k à X i +···+ i k = n ( − α ( i ,..., i k ) G λ i ⊗ ··· ⊗ G λ i k ! ( x ⊗ ··· ⊗ x n ).Here, α ( i ,..., i k ) = P j < k i j ( i k − m n = p ◦ λ n and J n = G ◦ λ n for all n ≥ ℓ n } and the Taylor series { I n } of the L ∞ quasi-isomorphism I arerecursively given as follows. Formally, set K θ = − i , and define θ n : H ⊗ n → L for n ≥ θ n ( x ,..., x n ) = n X k = X i +···+ i k = ni ≤···≤ i k X e S ( i ,..., i k ) ( − ε σ ℓ k ¡ I i ¡ x σ (1) ,..., x σ ( i ) ¢ ,..., I i k ¡ x σ ( i k − + ,..., x σ ( n ) ¢¢ .In the equation above, e S ( i ,..., i k ) are the ( i ,..., i k ) -shuffle permutations of the symmetric group S n , whose elements are those σ ∈ S n such that σ (1) =
1, and σ (1) < ··· < σ ( i ), σ ( i + < ··· < σ ( i ), ..., σ ( i k − + < ··· < σ ( n ).The sign ε σ is determined by the Koszul convention. Then, ℓ n = q ◦ θ n and I n = K ◦ θ n for all n ≥ A ∞ algebra as a transfer We produce the universal enveloping A ∞ algebra of a given L ∞ algebra via a transfer process. Todo so, we start by showing (Thm. 2.1) that the classical adjoint pair U : DGL ⇆ DG A : ( − ) L commutes with the transfer of higher structure. See [9, Chap. 21] for a careful exposition of theadjoint pair above. After the proof of Thm. 2.1, we explain how to produce such a universal enve-lope, which turns out to coincide with Baranovsky’s construction [3] up to homotopy. heorem 2.1. Let L and U L be a DGL and its classical universal enveloping DGA, respectively. Fixa contraction from L onto H = H ∗ ( L ) , and denote by { ℓ n } the induced L ∞ structure on H. Then,there is an explicit contraction from U L onto Λ H, so that denoting by { m n } the induced A ∞ algebrastructure on Λ H:(i) The antisymmetrization © m L n ª of { m n } fits into a strict L ∞ embeddingı : ( H ,{ ℓ n }) , → ³ Λ H ,{ m L n } ´ , that is, for every x i ∈ H , ℓ n ( x ,..., x n ) = X σ ∈ S n χ ( σ ) m n ¡ x σ (1) ⊗ ··· ⊗ x σ ( n ) ¢ = m L n ( x ,..., x n ). (ii) The A ∞ algebra ¡ Λ H ,{ m L n } ¢ is isomorphic to Baranovsky’s enveloping construction on ( H ,{ ℓ n }) . The map ı : H , → Λ H above is an L ∞ version of a PBW map L , → U L . The proof of Thm. 2.1relies in the following lemma, which is elementary but interesting in itself. It will be relevant forthe enveloping A ∞ algebra as a transferred structure (Def. 2.4). Lemma 2.2.
Let ( A ,{ µ n }) and ( L ,{ ϑ n }) be an A ∞ and an L ∞ algebra, and assume that there arecontractions of A and of L onto complexes ( M A , d ) and ( M L , ∂ ) , respectively:A M A L M L . G pj K qi
If there is a morphism of contractions f : L → A which is a strict L ∞ morphism for the antisym-metrization of the A ∞ algebra structure { µ n } , then the recursive formulas { θ n } for transferring theL ∞ structure on M L map to the antisymmetrization of those { λ n } for transferring the A ∞ structureon M A . More precisely, for any n ≥ and given x ,..., x n ∈ M L , f θ n ( x ,..., x n ) = X σ ∈ S n χ ( σ ) λ n ¡ b f ( x σ (1) ),..., b f ( x σ ( n ) ) ¢ . (3) Therefore, the higher brackets are the antisymmetrization of the higher multiplications: b f ℓ n ( x ,..., x n ) = X σ ∈ S n χ ( σ ) m n ¡ b f ( x σ (1) ),..., b f ( x σ ( n ) ) ¢ , (4) the terms of the induced L ∞ quasi-isomorphisms I : M L → L are the antisymmetrization of theterms of the A ∞ quasi-isomorphism J : M A → A:f I n ( x ,..., x n ) = X σ ∈ S n χ ( σ ) J n ¡ b f ( x σ (1) ),..., b f ( x σ ( n ) ) ¢ , (5) and b f : M L → M A is a strict L ∞ morphism for the antisymmetrization of { m n } . Remark 2.3.
The analog of Lemma 2.2 for a morphism of contractions g : A → L which is a strict L ∞ morphism for the antisymmetrization of the A ∞ algebra structure on A also holds. Proof of Lemma 2.2:
For clarity of exposition, we prove the case in which A = ( A , d ) is a DGA and M A = ( H A ,0) is its homology endowed with the trivial differential; and similarly L = ( L , ∂ ) is aDGL and M L = ( HL ,0). The general case follows exactly the same proof, but with more involvedformulas that do not give any more insight. The multiplication map of A will be denoted by m .We prove equation (3) by induction on n , and deduce at each inductive step the correspondingequation for (4) and for (5).Let n =
2. Use, in the order given, the definition of θ , that f is a Lie map for the bracketsinvolved, that f i = j b f , and recognize the recursive formula for λ : f θ ( x , x ) = f [ i ( x ), i ( x )] = £ f i ( x ), f i ( x ) ¤ = £ j b f ( x ), j b f ( x ) ¤ = m ¡ j b f ( x ) ⊗ j b f ( x ) − ( − | x || x | j b f ( x ) ⊗ j b f ( x ) ¢ = ¡ m ◦ j ⊗ j ¢ ¡ b f ( x ) ⊗ b f ( x ) − ( − | x || x | b f ( x ) ⊗ b f ( x ) ¢ = λ ¡ b f ( x ) ⊗ b f ( x ) − ( − | x || x | b f ( x ) ⊗ b f ( x ) ¢ . quation (3) is therefore proven. Using that f is a morphism of contractions, and the proof of thecase n = b f ℓ ( x , x ) = b f q θ ( x , x ) = p f θ ( x , x ) = p λ ¡ b f ( x ) ⊗ b f ( x ) − ( − | x || x | b f ( x ) ⊗ b f ( x ) ¢ = m ¡ b f ( x ) ⊗ b f ( x ) − ( − | x || x | b f ( x ) ⊗ b f ( x ) ¢ ; f I ( x , x ) = f k θ ( x , x ) = G f θ ( x , x ) = G λ ¡ b f ( x ) ⊗ b f ( x ) − ( − | x || x | b f ( x ) ⊗ b f ( x ) ¢ = J ¡ b f ( x ) ⊗ b f ( x ) − ( − | x || x | b f ( x ) ⊗ b f ( x ) ¢ .Assume next that for every p ≤ n −
1, equation (3) holds. Then, (4) and (5) also hold for p ≤ n − n =
2. Let us prove thatequation (3) holds for p = n , and then also equations (4) and (5) for p = n are straightforwardconsequence of f being a morphism of contractions and the just proven case n of equation 3. Tolighten notation, we write χ ( σ ) : = ε ( σ )sgn( σ ) for any given permutation σ .Use, in the order given: the definition of θ n , that f is a Lie map for the brackets involved, theidentity f i = j b f and the induction hypothesis, and rearrange the permutations accordingly, toend up with the recursive formula of λ n evaluated at the desired elements: f θ n ( x ,..., x n ) = n − X s = X σ ∈ S ( s , n − s ) ε ( σ ) f £ I s ¡ x σ (1) ,..., x σ ( s ) ¢ , I n − s ¡ x σ ( s + ,..., x σ ( n ) ¢¤ = n − X s = X σ ∈ S ( s , n − s ) ε ( σ ) £ f I s ¡ x σ (1) ,..., x σ ( s ) ¢ , f I n − s ¡ x σ ( s + ,..., x σ ( n ) ¢¤ = n − X s = X σ ∈ S ( s , n − s ) ε ( σ ) " J s à X τ ∈ S s χ ( τ ) b f ( x τσ (1) ) ⊗ ··· ⊗ b f ( x τσ ( s ) ) ! , J n − s à X ρ ∈ S n − s χ ( ρ ) b f ( x ρσ ( s + ) ⊗ ··· ⊗ b f ( x ρσ ( n ) ) ! = n − X s = X σ ∈ S ( s , n − s ) X τ ∈ S s ρ ∈ S n − s ε ( σ ) χ ( τ ) χ ( ρ ) £ J s ¡ b f ( x τσ (1) ),..., b f ( x τσ ( s ) ) ¢ , J n − s ¡ b f ( x ρσ ( s + ),..., b f ( x ρσ ( n ) ) ¢¤ = n − X s = X σ ∈ S n ( − s + χ ( σ ) £ J s ¡ b f ( x σ (1) ),..., b f ( x σ ( s ) ) ¢ , J n − s ¡ b f ( x σ ( s + ),..., b f ( x σ ( n ) ) ¢¤ = m µ n − X s = X σ ∈ S n ( − s + χ ( σ ) ³ J s ¡ b f ( x σ (1) ),..., b f ( x σ ( s ) ) ¢ ⊗ J n − s ¡ b f ( x σ ( s + ),..., b f ( x σ ( n ) ) ¢ − ( − α J n − s ¡ b f ( x σ ( s + ),..., b f ( x σ ( n ) ) ¢ ⊗ J s ¡ b f ( x σ (1) ),..., b f ( x σ ( s ) ) ¢¶ = λ n à X σ ∈ S n χ ( σ ) b f ( x σ (1) ) ⊗ ··· ⊗ b f ( x σ ( n ) ) ! . ä Proof of Theorem 2.1:
To prove ( i ), we show that fixed a contraction of L onto HL , one can choosea contraction of U L onto its homology
HU L ∼= U HL ∼= Λ H so that the PBW map L , → U L is amorphism of contractions, and then apply Lemma 2.2. Let ( i , q , K ) be a contraction of L onto H = HL , and write L = B ⊕ ∂ B ⊕ C for the graded vector space decomposition equivalent to it. By he PBW theorem ([9, Thm. 21.1]) and some basic facts of differential graded algebra, there aregraded vector space isomorphisms U L ∼= Λ L ∼= Λ ( B ⊕ ∂ B ⊕ C ) ∼= Λ ( B ⊕ ∂ B ) ⊗ Λ C ∼= Λ ( B ⊕ ∂ B ) ⊗ U H .Since Λ ( B ⊕ ∂ B ) is acyclic, the injection j : ( U H ,0) , → ( U L , d ) into a quasi-isomorphism, j : ( U H ,0) ( Λ ( B ⊕ ∂ B ) ⊗ U H , d ) ( U L , d ) . ≃ ∼= Decompose
U L ∼= Λ ( B ⊕ ∂ B ) ⊗ U H , let p : U L → U H ∼= ⊗ U H be the projection onto
U H , and let G be the inverse of d : Λ B ∼= −→ Λ ∂ B extended to all of U L as zero in the subspace Λ B ⊗ ⊗ U H ⊆ U L .Then, ( j , p , G ) is a contraction of U L onto
U H which is a morphism of retracts for the inclusion L = B ⊕ ∂ B ⊕ C , → U L = Λ ( B ⊕ ∂ B ⊕ C ).To prove ( i i ), denote by { µ n } the A ∞ algebra structure on U H induced by Baranovsky’s con-struction, and by { m n } the induced by the contraction ( j , p , G ). Since ( L , ∂ ) is a DGL, Baranovsky’sconstruction coincides with the classical universal enveloping DGA ([3, Thm. 3]). The L ∞ quasi-isomorphism Q : ( L , ∂ ) ≃ −→ ( H ,{ ℓ n }) provided by the contraction ( i , q , K ) transforms (by [3, Thm. 3])into an A ∞ algebra quasi-isomorphism U ( Q ) : ( U L , d ) ≃ −→ ¡ U H , © µ n ª¢ . There is another A ∞ algebraquasi-isomorphism P : ( U L , d ) ≃ −→ ( U H ,{ m n }) induced by the contraction ( j , p , G ). Hence, there isa zig-zag of A ∞ quasi-isomorphisms( U H ,{ m n }) ( U L , d ) ¡ U H , © µ n ª¢ ≃ ≃ Since { m n } and { µ n } are minimal, the two A ∞ algebra structures are A ∞ -isomorphic. ä The results above motivate Def. 2.4 for the universal enveloping A ∞ algebra on an L ∞ algebra.Recall that any L ∞ algebra L is L ∞ quasi-isomorphic to the DGL L C ( L ) ([16]), and that every L ∞ algebra has a minimal model ([20, Thm. 7.9]). Here, L : CDGC ⇆ DGL : C are the adjoint functorsintroduced by Quillen ([26]), with no bounding assumptions on the underlying complexes ([11]). Definition 2.4.
Let L be an L ∞ algebra. Its universal enveloping A ∞ algebra is U t ( L ) : = ( Λ L ,{ m n }),where { m n } is any A ∞ algebra structure arising by exhibiting Λ L as a contraction of Ω C ( L ). Inparticular, if L is minimal, then the A ∞ structure on Λ L is the one given in Theorem 2.1.The definition given is basically equivalent to Baranovsky’s. The difference is that we explicitlyuse Thm. 2.1 for constructing it, hence avoiding the use of Baranovsky’s chain homotopy K [3,Thm. 1], and with explicit, more transparent formulas whenever L is minimal. A different way ofreading Def. 2.4 is as follows. For an arbitrary L ∞ algebra L , the A ∞ structure { m n } on Λ L arisesby forming the diagram: Ω C ( L ) Λ L L C ( L ) L From this point of view, we start with a contraction from
L C ( L ) onto L producing the L ∞ struc-ture of L , and then the proof of Theorem 2.1 goes through: the classical PBW map L C ( L ) , → U ( L C ( L )) = Ω C ( L )is made a morphism of contractions, where we contract Ω C ( L ) onto its homology H ∗ ( Ω C ( L )),which is isomorphic as a graded vector space to Λ L (this isomorphism follows, for example, from[3, Thm. 1]). Given f : L → L an L ∞ morphism, and once chosen contractions Ω C ( L i ) Λ L i = U t ( L i ), i = p i j i here is a uniquely defined A ∞ morphism U t ( f ) = p ◦ Ω C ( f ) ◦ j : U t ( L ) → U t ( L ),enjoying properties similar to Baranovsky’s definition on morphisms (see [3, Thm. 3]). We collect the main properties regarding the homotopy type of the several universal envelopingconstructions in Proposition 3.1.Let L be an L ∞ algebra. Denote by U B ( L ) and U t ( L ) the construction of Baranovsky and thegiven in Def. 2.4, respectively. We will consider a third universal A ∞ envelope U d ( L ), see discus-sion after Conjecture 3.3. At this point, it suffices to know that U d ( L ) is isomorphic to Λ L as agraded vector space, and carries an A ∞ structure for which there is a DGC quasi-isomorphism C ( L ) ≃ −→ BU d ( L ).The universal envelopes U B , U t and U d are homotopy equivalent (Prop. 3.1 ( i )). Quillen’s founda-tion of rational homotopy theory, as well as other deep results (see for example [1, 10, 17]), heavilyrely on the now classical fact that homology commutes with the classical universal envelopingalgebra functor over characteristic zero fields, U H = HU . (6)See [26, Appendix B]. The identity (6) holds only up to homotopy for the universal enveloping con-structions U B , U t , U d and U (Prop. 3.1 ( i i i )), where U is Lada and Markl’s universal enveloping([15]). Another classical result of Quillen ([26], see also [25]) asserts that for a given DGL L withuniversal enveloping DGA U L , there is a natural DGC quasi-isomorphism C ( L ) ≃ −→ BU L . (7)For L ∞ algebras, although C ( L ), BU t ( L ) and BU B ( L ) are DGC’s, there is usually no direct DGCquasi-isomorphism as in (7). However, these DGC’s are always weakly equivalent, which is the liftof the quasi-isomorphism (7) when dealing with infinity structures (Prop. 3.1 ( i i )). Proposition 3.1.
Let L be an L ∞ algebra. Then,(i) There are A ∞ quasi-isomorphismsU t ( L ) ≃ U B ( L ) ≃ U d ( L ). The three constructions are then the same up to homotopy, and U t ( L ) ∼= U B ( L ) are isomorphicif L is minimal.(ii) There is an A ∞ coalgebra quasi-isomorphism C ( L ) ≃ −→ BU L , where U is any of the envelopes U t , U B or U d , which is not generally a DGC map for U B or U t .(iii) Assume that H ∗ ( L ) carries an L ∞ structure induced by a contraction from L onto it. Then,there are A ∞ quasi-isomorphisms U ( H ∗ ( L )) ≃ H ∗ ( U L ), where U is any of the envelopes U t , U B , U d or U . roof. ( i ) Theorem 2.1( i i ) asserts that U B ( L ) ≃ U t ( L ). It suffices to show that U B ( L ) ≃ U d ( L ). In-deed, the construction of U d is based on the existence of a differential d on T ¡ s Λ + L ¢ = BU d ( L ) sothat there is a DGC quasi-isomorphism C ( L ) ≃ −→ BU d ( L ). By [3, Thm 4 (ii)], there is a DGA quasi-isomorphism Ω C ( L ) → Ω BU B ( L ). Since the bar construction preserves quasi-isomorphisms, andgiven that the unit of the bar-cobar adjunction is a quasi-isomorphism for conilpotent coalgebras,there is the following zig-zag of DGC quasi-isomorphisms, from which the result follows: BU d ( L ) C ( L ) B Ω C ( L ) B Ω BU B ( L ) BU B ( L ) (8)( i i ) Follows from the zig-zag just above.( i i i ) By item ( i ), it suffices to prove it for U = U B and for U = U . Let f : L → HL be an L ∞ quasi-isomorphism. Since U B preserves quasi-isomorphisms, U B ( f ) : U B ( L ) → U B ( HL ) is an A ∞ quasi-isomorphism. Thm. 1.1 provides an A ∞ algebra structure on H ( U B ( L )), as well as an A ∞ quasi-isomorphism I : H ( U B ( L )) → U B ( L ). Thus, the following composition is an A ∞ quasi-isomorphism: H ( U B ( L )) I −→ U B ( L ) U B ( f ) −−−−→ U B ( HL ).Let us prove it for U . Fix a contraction L H , K qi (9)endow H with an L ∞ structure via Thm. 1.1, and denote by { m n } the A ∞ structure on U L . Markl’sPBW-infinity theorem [19, Thm. 4.7] gives an isomorphism of A ∞ algebras S ∗ ( L ) ∼= −→ G ∗ ( L ) .Here, G ∗ ( L ) is the associated graded A ∞ algebra for the ascending filtration of U L given by F = Q , F = Q ⊕ L , and for p ≥ F p L = Span Q n m n ( x ,.,,,. x n ) | n ≥ x j ∈ F p j L , p + ··· + p n ≤ p o ,and S ∗ ( L ) = F ( L , ℓ )/ J is the quotient of the free A ∞ algebra on the chain complex ( L , ℓ ) by the ideal generated by im-posing the vanishing on L of the antisymmetrization of the A ∞ structure © µ n ª of F ( L , ℓ ) for n ≥ µ L n ( x ,..., x n ) = n ≥ x i ∈ L .Basically, S ∗ is the "free A ∞ algebra symmetrized on L " (not to be confused with a C ∞ algebra,whose structural maps vanish on the image of the shuffle products). Denote by P the dg operadwhose free algebras are given by S ∗ (an explicit description in terms of planar trees is given in [19,Prop. 4.6]). Summarizing, for any L ∞ algebra L , there is an isomorphism of A ∞ algebras U L ∼= S ∗ ( L ),where S ∗ ( L ) = P ( L ) is the free P -algebra for a certain dg operad P . Thus, after a possible changeof homotopy in the contraction from L onto H , Berglund’s generalization of the tensor trick toalgebras over operads ([5, Thm. 1.2]) applies to the contraction (9). That is, there is a contraction U L ∼= S ∗ ( L ) S ∗ ( HL ) ∼= U HL . S ∗ ( K ) S ∗ ( q ) S ∗ ( i ) To finish, choose any A ∞ quasi-isomorphism U L ≃ H ∗ ( U L ), for instance by using Thm. 1.1.Then, there are A ∞ quasi-isomorphisms U H ∗ ( L ) ≃ −→ U L ≃ −→ H ∗ ( U L ). emark 3.2. One could try to adapt Quillen’s proof for DGL’s in [26, App. B] of the identity HU = U H for U . Several subtleties arise this way, and in fact, one cannot improve Prop. 3.1 (iii) .Indeed, any "natural" map U ( HL ) → H U L passes through a previous choice of infinity struc-tures, thus one cannot expect an isomorphism. It gets even worst than that: no choice will everbe an isomorphism, except for the trivial case, given that by definition U HL carries a non-trivialdifferential, whereas H ∗ ( U L ) does not.For P a dg operad, recall that a P -algebra is formal if there exists a zig-zag of P -algebra quasi-isomorphisms connecting it to its homology ([16]). In presence of a contraction, Lemma 2.2 givesa straightforward proof of the fact that L is formal as a DGL if, and only if, U L is formal as a DGA.This result ([27]), however, has been superseded by [8, Thm. B].We conclude this section with a conjecture.
Conjecture 3.3.
Let L be an L ∞ algebra. Lada and Markl’s universal enveloping A ∞ algebra U L issuch that there is a natural DGC quasi-isomorphism C ( L ) ≃ −→ B U L .If Conjecture 3.3 is true, then the universal enveloping U d studied in this section enjoys thehomotopical properties of U . This justifies the study of U d . To finish the homotopical study of U , it suffices to prove the weaker version of Conjecture 3.3 relaxing the DGC quasi-isomorphismto a zig-zag of quasi-isomorphisms. The algebraic formalism of Section 2 has interesting applications to rational homotopy theory.The monograph [9] is an excellent resource on rational homotopy theory. In this section, all L ∞ algebras are concentrated in non-negative degrees. Let X be a simply connected complex. The classical Milnor-Moore theorem ([24]) asserts thatthe rational homotopy Lie algebra L X = π ∗ ( Ω X ) ⊗ Q embeds as the primitive elements of therational loop space Hopf algebra H ∗ ( Ω X ; Q ). Furthermore, the latter Hopf algebra is preciselythe universal enveloping algebra of L X , and the inclusion is given by the rationalization of theHurewicz morphism, h : π ∗ ( Ω X ) ⊗ Q , → H ∗ ( Ω X ; Q ) = U ( π ∗ ( Ω X ) ⊗ Q ). (10)If only the rational homotopy Lie algebra π ∗ ( Ω X ) ⊗ Q is taken into account, then non-equivalentrational spaces may share this invariant. For instance, the rationalization of C P and of K ( Z ,2) × K ( Z ,5) are not equivalent, yet both have abelian two dimensional isomorphic rational homotopyLie algebras. However, endowing an L ∞ structure to π ∗ ( Ω X ) ⊗ Q determines a unique rationalhomotopy type, even if we include the class of nilpotent finite type complexes. In this latter case,we need to restrict to finite type pronilpotent L ∞ algebras. The rational homotopy type encodedby such an L ∞ algebra L is determined by the DGL L C ( L ) in case L = L ≥ , and by the Sullivanalgebra C ∗ ( L ) in case L = L ≥ is finite type pronilpotent. Here, C ∗ = ∨ ◦ C is the linear dual ∨ ofthe Quillen chains C . See [6, Thm. 2.3] for details. By a beautiful result of Majewski, whenever X is simply connected of finite type, these two models are homotopy equivalent ([17]).Denote U = U t . The next result lifts the morphism (10) to the context of infinity algebras. Theorem 4.1.
Let X be a simply connected complex. Endow π ∗ ( Ω X ) ⊗ Q with an L ∞ structure { ℓ n } representing the rational homotopy type of X for which ℓ = and ℓ = [ − , − ] is the Samel-son bracket. Then, there is an A ∞ algebra structure { m n } on the loop space homology algebraH ∗ ( Ω X ; Q ) for which m = m is the Pontryagin product, and such that the rational Hurewiczmorphism h : π ∗ ( Ω X ) ⊗ Q , → H ∗ ( Ω X ; Q ) = U ( π ∗ ( Ω X ) ⊗ Q ) s a strict L ∞ embedding. Therefore, the L ∞ structure on the rational homotopy Lie algebra is theantisymmetrized of the A ∞ structure on H ∗ ( Ω X ; Q ) : ℓ n ( x ,..., x n ) = X σ ∈ S n χ ( σ ) m n ¡ x σ (1) ,..., x σ ( n ) ¢ . Proof.
Assume that the rational homotopy Lie algebra π ∗ ( Ω X ) ⊗ Q carries a minimal L ∞ structure{ ℓ n } governing the rational homotopy type of X for which ℓ is the Samelson bracket. For instance,from a CW-decomposition ∗ = X (1) ⊆ X (2) ⊆ ··· ⊆ [ n X ( n ) = X ,build the Quillen minimal model L = ( L ( V ), ∂ ) of X , satisfying H ∗ ( L ) ∼= π ∗ ( Ω X ) ⊗ Q as graded Lie algebras. The choice of a contraction from L onto π ∗ ( Ω X ) ⊗ Q gives an L ∞ structureas in the statement. The rational Hurewicz homomorphism of equation (10) is, after the choice ofan ordered basis of L , the PBW map from L into U L . Therefore, h can be chosen to be h = b ı = pıi in the following diagram, which is under the hypotheses of Theorem 2.2: T V = U ( L ( V )) H ∗ ( Ω X ; Q ) L ( V ) π ∗ ( Ω X ) ⊗ Q G pjK qı i h
An application of Theorem 2.2 finishes the proof.
Remark 4.2.
Let U t ( L ) = ( Λ L ,{ m n }) be the universal enveloping A ∞ algebra of ( L ,{ ℓ n }). For each n , the composition L ⊗ n ( Λ L ) ⊗ n Λ L i n m L n has its image in L ⊆ Λ L . Let π : Λ L → L be the projection. The primitives of Λ L for the standard co-product are precisely P ∗ ( Λ L ) = L . Thus, the original L ∞ structure can be recovered by performingtwo natural operations to U t ( L ): antisymmetrizaton and restriction to primitives.( Λ L ,{ m n }) ³ P ∗ ( Λ L ), π ◦ m L n ◦ i n ´ = ( L ,{ ℓ n }).Detecting when a given cocommutative Hopf algebra is the universal envelope of its primitivesis a difficult problem. This has been studied, among others, by Anick, Cartier, Halperin, Kostant,Milnor and Moore. See for example [10]. The classical name of this sort of result is the Cartier-Milnor-Moore theorem . Does a similar statement hold in the infinity setting?
Conjecture 4.3.
Let A be an A ∞ algebra over a characteristic zero field such that there is a co-commutative, conilpotent coproduct ∆ on A which is a strict A ∞ morphism A → A ⊗ . Then, theprimitives for the coproduct L = Ker( ∆ ) = P ∗ ( A ) form an L ∞ algebra, and the inclusion L , → A extends to an isomorphism of A ∞ algebras U L ∼= −→ A which respects the Hopf structure.In the conjecture above, we expect U to be Lada and Markl’s envelope, and maybe the diagonal ∆ needs to come from a "Hopf algebra up to homotopy", so that the isomorphism might be notonly of A ∞ algebras, but of homotopy Hopf algebras. If X is a simply connected complex, and H ∗ ( Ω X ; Q ) carries a universal enveloping A ∞ structure, then H ∗ ( Ω X ; Q ) is a rational model for X .Indeed, by Remark 4.2, P ∗ ( H ∗ ( Ω X ; Q )) = π ∗ ( Ω X ) ⊗ Q is a fully-fledged L ∞ algebra capturing the rational homotopy type of X . .2 Examples. Recovering the Sullivan and Quillen models We explicitly record several examples of universal enveloping A ∞ algebras of the sort U t ( π ∗ ( Ω X ) ⊗ Q ,{ ℓ n }) = ( H ∗ ( Ω X ; Q ),{ m n }).1. The simply connected sphere S n .• For odd n , it is Λ x with | x | = n −
1, with trivial differential and trivial higher multiplica-tions of all orders.• For even n , it is Λ ( x , y ) with | x | = n − | y | = n −
2, with a unique non-trivial multipli-cation map given by m ( x , x ) = y .2. A finite product of simply-connected Eilenberg-Mac Lane spaces Q ki = K ( Q , n i ).It is given by ( Λ x ,..., x k ), where each | x i | = n i − The complex projective spaces C P k , for k ≥ . It is given by ¡ Λ x , y ¢ , with | x | = | y | = k and its only non-trivial higher multiplication is m k + ( x ,..., x ) = k + y .Indeed, an L ∞ model L = π ∗ ( Ω C P k ) ⊗ Q of C P k has a basis { x , y } with | x | = | y | = k with asingle non-vanishing higher bracket, given by ℓ k + ( x ,..., x ) = k + y . The result then follows,since the sign χ ( σ ) in the sum below is always positive:1( k + y = ℓ k + ( x ,..., x ) = X σ ∈ S k + χ ( σ ) m k + ( x ,..., x ) = ( k + m k + ( x ,..., x ).4. Coformal spaces .The universal enveloping A ∞ algebra model of any coformal space can be chosen to be theclassical universal enveloping algebra of it. Indeed, if X is coformal, then L = π ∗ ( Ω X ) ⊗ Q together with ℓ given by the Samelson product is an L ∞ model of X . Since L is a DGLwith trivial differential, the universal enveloping A ∞ algebra of it coincides with the classicalenvelope, having the latter trivial differential as well. This includes examples 1 and 2.Let U t ( L ) = ( Λ L ,{ m n }) be universal enveloping A ∞ model of a simply connected complex X .Let L = P ∗ ( H ∗ ( Ω X ; Q )) be the primitives for the natural diagonal (Rmk. 4.2). Then, one recovers:• Provided X is of finite type, a (not necessarily minimal) Sullivan model ( Λ V , d ) of X by set-ting V = ( sL ) ∨ and d = P n ≥ d n determined by the pairing 〈 d n ( v ), sx ∧ ... ∧ sx n 〉 = ε X σ ∈ S n χ ( σ ) v ; sm n ¡ x σ (1) ,..., x σ ( n ) ¢® , (11)where ε is the parity of P n − j = ( n − j ) | x j | .• A (not necessarily minimal) Quillen model by setting( L ( U ), ∂ ) = ¡ L ¡ s − Λ + sL ¢ , ∂ + ∂ ¢ = L C ³ P ∗ ( H ∗ ( Ω X ; Q )),{ m L n } ´ .The quadratic part ∂ of the differential is the standard induced by the reduced coproductof C ( L ) (see formula (2)), and ∂ is explicitly given on generators by ∂ ¡ s − ¡ sx ∧ ... ∧ sx p ¢¢ = p X k = X i ≤···≤ i k X σ ∈ S k ε σ ( i ,..., i k ) s − ³ sm k ¡ x i σ (1) ,..., x i σ ( k ) ¢ ∧ sx ... d sx i ... d sx i k ... ∧ sx i p ´ .The sign ε σ ( i ,..., i k ) = − ε ( − n i ik χ ( σ ) is given by the Koszul sign rule and the involved maps. .3 Higher Whitehead products and Pontryagin-Massey products Several authors have related the (ordinary, as well as higher) Whitehead products [ − , − ] on π ∗ ( X )with the Pontryagin product ∗ on H ∗ ( Ω X ; R ). For instance, the main result in [28] states that thetwo-fold Whitehead product of x ∈ π n + and y ∈ π m + is an antisymmetrized Pontryagin product: h [ x , y ] = ( − n ¡ h ( x ) ∗ h ( y ) − ( − nm h ( y ) ∗ h ( x ) ¢ .Here, h : π ∗ ( X ) ∼= −→ π ∗− ( Ω X ) → H ∗− ( Ω X ; Z ) is the Hurewicz morphism precomposed with an iso-morphism. In [2, Thm 3.3], it is shown that under some hypothesis, certain higher order White-head product sets [ x ,..., x k ] W ⊆ π ∗ ( X ) are non-empty, and contain an element which is a sort ofgeneralized k -fold Pontryagin product.In the rational case, Thm. 4.1 is the most general form of these sort of statements. Assumingthe existence of non-trivial higher products in a sense to be explained, one can go a step furtherand extract an interesting relationship. For space considerations, and since this section is aboutan application of the main results of this work, we omit a (necessarily lengthy) explanation of thenecessary background. Instead, we refer the reader to [29] for background on the (rational) higherorder Whitehead products, and to [4] for an account of their relationship with L ∞ structures. Westart with the following observation. Proposition 4.4.
Let X be a simply connected complex. The A ∞ algebra structures on H ∗ ( Ω X ; Q ) arising from exhibiting H ∗ ( Ω X ; Q ) as a contraction of the chains DGA C ∗ ( Ω X ; Q ) and by takinguniversal enveloping A ∞ algebra of an L ∞ model on π ∗ ( Ω X ) ⊗ Q are A ∞ quasi-isomorphic.Proof. Let L = ( π ∗ ( Ω X ) ⊗ Q ,{ ℓ n }) be the L ∞ model of X , and assume without loss of generalitythat L arises as a contraction of the Quillen model ( L ( U ), ∂ ) of X . Denote by { m n } the A ∞ structureon H ∗ ( Ω X ; Q ) arising from Thm. 2.1. There is a square U ( L ( V )) H ∗ ( Ω X ; Q ) L ( V ) π ∗ ( Ω X ) ⊗ Q ≃≃ whose horizontal top and bottom arrows are A ∞ and L ∞ quasi-isomorphisms, respectively. Sincethere is a DGL quasi-isomorphism L ( U ) ≃ −→ λ ( X ) onto the Quillen construction λ ( X ) ([26]), andthe classical enveloping functor U preserves quasi-isomorphisms ([9, Thm. 21.7]), there is a DGAquasi-isomorphism U L ( U ) ≃ −→ U λ ( X ). Since U λ ( X ) is weakly equivalent to C ∗ ( Ω X ; Q ) as a DGA,there is an A ∞ quasi-isomorphism U λ ( X ) ≃ −→ ( H ∗ ( Ω X ; Q ),{ m ′ n }) for { m ′ n } induced by exhibiting H ∗ ( Ω X ; Q ) as a contraction of C ∗ ( Ω X ; Q ).The Massey products of a space X are certain higher order operations on the cohomology al-gebra H ∗ ( X ; R ). These arise from relations between the cup product and the differential in thesingular cochains C ∗ ( X ; R ), see [21, 22]. The Massey products and the A ∞ structures on H ∗ ( X ; R )are tightly related, see [7] for details. Both, the Massey products and A ∞ structure, exist in thehomology H of any DGA A - one needs not consider these operations only when A is the singularcochain algebra of a space. So, given that H ∗ ( Ω X ; R ) is the homology of the DGA C ∗ ( Ω X ; R ) for thePontryagin product, it makes sense to consider the algebraic Massey products on H ∗ ( Ω X ; R ). Wecall these higher products on H ∗ ( Ω X ; R ) arising from relations between the Pontryagin productand the differential of the DGA C ∗ ( Ω X ; R ) the higher Massey-Pontryagin products of X . This way,we avoid the confusion with the classical Massey products of X . Again for space considerations,we refer the reader to the works mentioned in this paragraph for the necessary background onMassey products and A ∞ structures.Denote by h : π ∗ ( Ω X ) ⊗ Q → H ∗ ( Ω X ; Q ) the rational Hurewicz morphism. heorem 4.5. Let x ,..., x n ∈ π ∗ ( Ω X ) ⊗ Q , and denote by y k = h ( x k ) ∈ H ∗ ( Ω X ; Q ) the correspond-ing spherical classes. Assume that the higher Whitehead product set [ x ,..., x n ] W and the higherMassey-Pontryagin products sets y σ (1) ,..., y σ ( n ) ® for every σ ∈ S n are defined. If the A ∞ algebrastructure { m k } on H ∗ ( Ω X ; Q ) provided by Thm. 4.1 has vanishing m k for k ≤ n − , then x = εℓ n ( x ,..., x n ) ∈ [ x ,..., x n ] W , and satisfies:h ( x ) ∈ X σ ∈ S n χ ( σ ) y σ (1) ,..., y σ ( n ) ® . Here, ε is the parity of P n − j = | x j | ( k − j ) . If moreover the involved higher products are all uniquelydefined, then the above containment is an equality of elements. Since the particular case n = m = Corollary 4.6.
Let x , x , x ∈ π ∗ ( Ω X ) ⊗ Q , and denote by y k = h ( x k ) ∈ H ∗ ( Ω X ; Q ) the correspond-ing spherical classes. Assume that the triple Whitehead product [ x , x , x ] W and the triple Masseyproducts y σ (1) , y σ (2) , y σ (3) ® , σ ∈ S , are defined. Then x = εℓ ( x , x , x ) ∈ [ x , x , x ] W , and satis-fies: h ( x ) ∈ X σ ∈ S χ ( σ ) y σ (1) , y σ (2) , y σ (3) ® . If moreover the triple products are all uniquely defined, then the above containment is an equalityof elements.Proof of Theorem 4.5:
Since m k = k ≤ n −
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