Higher order Massey products and applications
HHIGHER ORDER MASSEY PRODUCTS AND APPLICATIONS
IVAN LIMONCHENKO AND DMITRY MILLIONSHCHIKOV
Abstract.
In this survey, we discuss two research areas related to Massey’s higher operations.The first direction is connected with the cohomology of Lie algebras and the theory of repre-sentations. The second main theme is at the intersection of toric topology, homotopy theory ofpolyhedral products, and the homology theory of local rings, Stanley–Reisner rings of simplicialcomplexes.
Introduction
Higher order Massey operations proved to be a very effective algebraic tool for describingvarious obstructions to the existence or continuation of the diverse topological, geometric oralgebraic structures and their deformations. In this short review, we attempted to systematizea series of results obtained over the past decade concerned with two important topics related toapplications of the Massey higher operations. The starting point for the first topic is the Duady’swork [37] of the 1960, where the relation of Massey products to the theory of deformations wasobserved. This topic has been actively developed in the following decades, we especially notethe works by Palamodov [74], Retakh [77, 78], in which the connection between the higherMassey products and the Kodaira-Spencer theory of deformations was studied.In algebraic topology it is well known that Massey products serve as an obstruction toformality of a topological space (see Section 5). In 1975, Deligne, Griffiths, Morgan and Sullivanproved that simply connected compact K¨ahler manifolds are formal [34]. In particular it meansthat the existence of non-trivial Massey products in the cohomolgy H ∗ ( M, R ) is an obstructionfor a manifold M to be K¨ahler [34]. On the other hand, Halperin and Stasheff [52] constructed anon-formal differential graded algebra such that all Massey products vanish in its cohomology.A blow-up of a symplectic manifold M along its submanifold N inherits non-trivial Masseyproducts [8, 9]. Babenko and Taimanov applied the symplectic blow-up procedure for the con-struction of simply connected non-formal symplectic manifolds in dimensions ≥
10 [8, 9]. How-ever, in the papers of Babenko and Taimanov [9, 10] we were attracted primarily by their
Mathematics Subject Classification.
Primary 13F55, Secondary 52B11, 55S30.
Key words and phrases.
Massey product, Lie algebra cohomology, deformation, polyhedral product, moment-angle manifold, nestohedron, graph-associahedron.The research of the first author was carried out within the University Basic Research Program of the HigherSchool of Economics and was funded by the Russian Academic Excellence Project ‘5-100’. The second author wassupported by the RSF grant 20-11-19998. The work of D.V. Millionshchikov (the second author) is supportedby the Russian Science Foundation under grant 20-11-19998 and performed in Steklov Mathematical Instituteof Russian Academy of Sciences. a r X i v : . [ m a t h . A T ] A p r I.YU. LIMONCHENKO AND D.V. MILLIONSHCHIKOV approach to the definition of Massey products in the language of formal connections and theMaurer-Cartan equation. The idea of such an approach has been encountered in literature be-fore, first May’s [66, 67], then in Palamodov’s article [74], but it was precisely in the articles ofBabenko and Taimanov that this approach was comprehensively developed [9, 10].In the Section 1 we recall the elements of the Babenko-Taimanov approach to the definitionof Massey products [9, 10]. The analogy with the classical Maurer-Cartan equation, whichis especially transparent in the case of Massey products of 1-dimensional cohomology classes (cid:104) ω , . . . , ω n (cid:105) , is discussed in Section 2. The relation of this special case to representation theorywas discovered in [38, 40].The related material, which is presented in the form of a publication for the first time, iscontained in Section 3. We are talking about variations of Massey products, which we called k -step Massey products. It is well known that the classical Massey products are multi-valued andpartially defined operations. We propose to consider successive obstructions (cid:104) ω , . . . , ω n (cid:105) k , k =1 , . . . , n − , arising in constructing a formal connection (defining system) A for the classicalMassey product (cid:104) ω , . . . , ω n (cid:105) as k -step Massey products (see Definition 3.6).The main feature uniting the results that we relate to the first topic is the Massey products inthe cohomology of Lie algebras. Particular attention in this part is given to non-trivial Masseyproducts, it was motivated by applications. Two very important and interesting positivelygraded Lie algebras were chosen as the main examples of this article. Firstly, this is the positivepart W + of the Witt algebra, and secondly, its associated graded algebra with respect tothe filtration by the ideals of the lower central series m . Sometimes m is called the infinitedimensional filiform Lie algebra. We discuss in Section 4 the proof of Buchstaber’s conjecturethat the cohomology H ∗ ( W + ) are generated by means of non-trivial Massey products by theone-cohomology H ( W + ). The corresponding Theorem 4.4 was proved by the second authorin [70]. We consider in Section 4 also the structure results on the Massey products in thecohomology H ∗ ( m ).There is a natural connection between the cohomology of infinite-dimensional positivelygraded (filtered) Lie algebras and the topology of manifolds. The cohomology of finite-dimensional quotients of such algebras with rational structural constants is isomorphic, accord-ing to the Nomizu theorem [72], to the real cohomology of nilmanifolds G/ Γ that correspondto such quotients. It was such a tower of nilmanifolds M n , that correspond to W + (the Wittalgebra) and first introduced by Buchstaber in [29] and later used by Babenko and Taimanovin their construction of symplectic nilmanifolds [8, 9] with simply defined non-trivial tripleMassey products. Namely, the methods of the Lie algebras theory make it possible to efficientlycalculate the first cohomology of nilmanifolds and discover nontrivial Massey products.A well-known conjecture, which dates back to the papers by May [66] and May-Gugenheim [68],asserts that higher differentials in the Eilenberg–Moore spectral sequence of a space are deter-mined by (matric) Massey products of the space. The second central theme of our review isclosely related to that conjecture and originates in the early 1960s, from a pioneering paper ofGolod [43]. He showed that Poincar´e series of a local ring A achieve its (coefficientwise) upperbound, previously identified by Serre, precisely in the case when multiplication and all Masseyproducts, triple and higher, vanish in Koszul homology of A . IGHER ORDER MASSEY PRODUCTS AND APPLICATIONS 3
This result acquired a topological interpretation in toric topology by means of the theoremdue to Buchstaber and Panov [26, 27], who proved that cohomology algebra of a moment-angle-complex Z K over a commutative ring with unit k is isomorphic to Koszul homology ofthe corresponding Stanley-Reisner ring k [ K ]. It gives us a tool to identify a class of simplicialcomplexes with formal moment-angle complexes and it also enables one to construct non-formal(and therefore, non-K¨ahler) moment-angle manifolds, having non-trivial Massey products intheir cohomology.Another classical well-known problem related to this part of our survey is the Steenrodproblem of realization of (rational) cycles in a given space by oriented manifolds. Halperinand Stasheff [52] showed that cohomology ring of a (rationally) formal space is generated byspherical classes. Thanks to toric topology, an example of a non-trivial triple Massey product of3-dimensional spherical classes in cohomology of a 2-connected space (a moment-angle-complex)was contructed by Baskakov [17]. In [62, 63, 64] the first author generalized Baskakov’s con-struction and proved existence of moment-angle manifolds Z P having non-trivial higher Masseyproducts (of any prescribed order) of spherical cohomology classes in H ∗ ( Z P ).In Section 5 we give a survey on the construction of Massey products in the algebraic contextof Koszul homology of local rings and discuss the Golod property for Stanley–Reisner rings ofsimplicial complexes. In Section 6 we deal with the results on non-trivial triple and higherMassey products in cohomology of moment-angle-complexes and moment-angle manifolds, em-phasizing the case of strictly defined (i.e., containing a single element) non-trivial higher Masseyproducts. 1. Massey products in cohomology
Let A = ⊕ l ≥ A l be a differential graded algebra over a field K . It means that the followingoperations are defined: an associative multiplication ∧ : A l × A m → A l + m , l, m ≥ , l, n ∈ Z . such that a ∧ b = ( − lm b ∧ a for a ∈ A l , b ∈ A m , and a differential d, d = 0 d : A l → A l +1 , l ≥ , satisfying the Leibniz rule d ( a ∧ b ) = d a ∧ b + ( − l a ∧ d b for a ∈ A l .Of course, the most natural example of differential graded algebra A is the de Rham complex A = Λ ∗ ( M ) , K = R , of smooth forms of a smooth manifold M . However, in this article we willpay special attention to the following two examples. Example 1.1. A = Λ ∗ ( g ) is the cochain complex of a Lie algebra. Example 1.2.
A Koszul complex K A = Λ A m of a commutative Noetherian local ring ( A, m , k )(see the section 5 for details).For a given differential graded algebra ( A , d ) we denote by T n ( A ) a space of all upper tri-angular ( n + 1) × ( n + 1)-matrices with entries from A , vanishing at the main diagonal. The I.YU. LIMONCHENKO AND D.V. MILLIONSHCHIKOV standard matrix multiplication turns the vector space T n ( A ) into an algebra, we assume thatmatrix entries are multiplying as elements of A . One can define the differential d on T n ( A ) by(1) d A = ( d a ij ) ≤ i,j ≤ n +1 . We extend the involution a → ¯ a = ( − k +1 a, a ∈ A k of A to the involution of T n ( A ) by therule ¯ A = (¯ a ij ) ≤ i,j ≤ n +1 . It satisfies the following properties¯ A = A, AB = − ¯ A ¯ B, d A = − d ¯ A. Also we have the generalized Leibniz rule for the differential (1) d ( AB ) = ( d A ) B − ¯ A ( d B ) . Consider a two-sided ideal I n ( A ) of matrices of the following form . . . τ . . . . . . . . . , τ ∈ A Obviously, the ideal I n ( A ) belongs to the center Z ( T n ( A )) of the algebra T n ( A ). Definition 1.3 ([10]) . A matrix A ∈ T n ( A ) is called the matrix of a formal connection if itsatisfies the Maurer-Cartan equation(2) µ ( A ) = d A − ¯ A · A ∈ I n ( A ) . Proposition 1.4 ([10]) . The generalized Bianchi identity for the Maurer-Cartan operator µ ( A ) = d A − ¯ A · A holds (3) d µ ( A ) = µ ( A ) · A + A · µ ( A ) . Proof.
Indeed it’s easy to verify the following equalities d µ ( A ) = − d ( ¯ A · A ) = − d ¯ A · A + A · d A = d A · A + A · d A == ( µ ( A ) + ¯ A · A ) · A + A ( µ ( A ) + ¯ A · A ) = µ ( A ) · A + A · µ ( A ) . (cid:3) Corollary 1.5 ([10]) . Let A be the matrix of a formal connection, then the entry τ ∈ A of thematrix µ ( A ) ∈ I n ( A ) in the definition (2) is closed. Now let A be the matrix of a formal connection, then the matrix µ ( A ) belongs to the ideal I n ( A ) and hence dµ ( A ) = 0. In a formal sense µ ( A ) plays the role of the curvature matrix of aformal connection A . IGHER ORDER MASSEY PRODUCTS AND APPLICATIONS 5
Let A be an upper triangular matrix from T n ( A ). A = a (1 , a (1 , . . . a (1 , n − a (1 , n )0 0 a (2 , . . . a (2 , n − a (2 , n ) . . . . . . . . . . . . . . . . . . . . . a ( n − , n − a ( n − , n )0 0 0 . . . a ( n, n )0 0 0 . . . . Proposition 1.6.
A matrix A ∈ T n ( A ) is the matrix of a formal connection if and only if thefollowing equalities hold a ( i, i ) = a i ∈ A p i , i = 1 , . . . , n ; a ( i, j ) ∈ A p ( i,j )+1 , p ( i, j ) = j (cid:88) r = i ( p r − d a ( i, j ) = j − (cid:88) r = i ¯ a ( i, r ) ∧ a ( r + 1 , j ) , ( i, j ) (cid:54) = (1 , n ) . (4)The system (4) is just the Maurer-Cartan equation rewritten in terms of the entries of thematrix A and it is a part of the classical definition [60] of the defining system for a Masseyproduct. Definition 1.7 ([60]) . A collection of elements, A = ( a ( i, j )), for 1 ≤ i ≤ j ≤ n and ( i, j ) (cid:54) =(1 , n ) is said to be a defining system for the product (cid:104) a , . . . , a n (cid:105) if it satisfies (4).Under these conditions the ( p (1 , n ) + 2)-dimensional cocycle c ( A ) = n − (cid:88) r =1 ¯ a (1 , r ) ∧ a ( r + 1 , n )is called the related cocycle of the defining system A .One can verify that the notion of the defining system is equivalent to the notion of the formalconnection. We have only to remark that an entry a (1 , n ) of a formal connection A does notbelong to the corresponding defining system. It can be taken as an arbitrary element from A and for the only one nonzero (possibly) entry τ ∈ µ ( A ) we have τ = − c ( A ) + da (1 , n ) . Definition 1.8 ([60]) . The n -fold product (cid:104) a , . . . , a n (cid:105) is defined if there exists at least onedefining system for it (a formal connection A with entries a , . . . , a n at the second diagonal).If it is defined, then the value (cid:104) a , . . . , a n (cid:105) is the set of all cohomology classes α ∈ H p (1 ,n )+2 ( A )for which there exists a defining system A such that the cocycle c ( A ) (or equivalently − τ )represents α . Theorem 1.9 (see [60],[10]) . The product (cid:104) a , . . . , a n (cid:105) depends only on the cohomology classesof the elements a , . . . , a n . I.YU. LIMONCHENKO AND D.V. MILLIONSHCHIKOV
Definition 1.10 ([60]) . A set of closed elements a i , i = 1 , . . . , n from A representing somecohomology classes α i ∈ H p i ( A ) , i = 1 , . . . , n is said to be a defining system for the Massey n -fold product (cid:104) α , . . . , α n (cid:105) if it is one for (cid:104) a , . . . , a n (cid:105) . The Massey n -fold product (cid:104) α , . . . , α n (cid:105) is defined if (cid:104) a , . . . , a n (cid:105) is defined, in which case (cid:104) α , . . . , α n (cid:105) = (cid:104) a , . . . , a n (cid:105) as subsets in H p (1 ,n )+2 ( A ).For n = 2 the matrix A of a formal connection is A = a c b and the generalized Maurer-Cartan equation is equivalent to the system da = 0 , db = 0. Hencea 2-fold Massey product (cid:104) α, β (cid:105) is always defined and (cid:104) α, β (cid:105) = ¯ α ∧ β .Let α , β , and γ be the cohomology classes of closed elements a ∈ A p , b ∈ A q , and c ∈ A r .The Maurer-Cartan equation for A = a f h b g c . is equivalent to the system(5) d f = ( − p +1 a ∧ b, d g = ( − q +1 b ∧ c. Hence the triple Massey product (cid:104) α, β, γ (cid:105) is defined if and only if the following conditions hold α · β = β · γ = 0 in H ∗ ( A ) . The triple Massey product (cid:104) α, β, γ (cid:105) is defined as a subspace H p + q + r − ( A ) of elements (cid:104) α, β, γ (cid:105) = (cid:8) [( − p +1 a ∧ g + ( − p + q f ∧ c ] (cid:9) . Since f and g are defined by (5) up to closed elements from A p + q − and A q + r − respectively, thetriple Massey product (cid:104) α, β, γ (cid:105) is an affine subspace of H p + q + r − ( A ) parallel to α · H q + r − ( A ) + H p + q − ( A ) · γ .Sometimes the triple Massey product (cid:104) α, β, γ (cid:105) is defined as a quotient (cid:104) α, β, γ (cid:105) / ( α · H q + r − ( A ) + H p + q − ( A ) · γ ) [42]. Definition 1.11.
Let an n -fold Massey product (cid:104) α , . . . , α n (cid:105) be defined. It is called trivial if itcontains the trivial cohomology class: 0 ∈ (cid:104) α , . . . , α n (cid:105) . Proposition 1.12.
Let a Massey product (cid:104) α , . . . , α n (cid:105) is defined. Then all Massey products (cid:104) α l , . . . , α q (cid:105) , ≤ l < q ≤ n, q − l < n − are defined and trivial. The triviality of all Massey products (cid:104) α l , . . . , α q (cid:105) , ≤ l < q ≤ n, q − l < n − (cid:104) α , . . . , α n (cid:105) to be defined. It is sufficient only in thecase n = 3. IGHER ORDER MASSEY PRODUCTS AND APPLICATIONS 7
Let us denote by GT n ( K ) a group of non-degenerate upper triangular ( n +1 , n +1)-matricesof the form:(6) C = c , c , . . . c ,n c ,n +1 c , . . . c ,n c ,n +1 . . . . . . . . . . . . c n,n c n,n +1 . . . c n +1 ,n +1 . Proposition 1.13.
Let A ∈ T n ( A ) be the matrix of a formal connection and C an arbitrary ma-trix from GT n ( K ) . Then the matrix C − AC ∈ T n ( A ) and satisfies the Maurer-Cartan equation,i.e. it is again the matrix of a formal connection.Proof. d ( C − AC ) − ¯ C − ¯ A ¯ C ∧ C − AC = C − (cid:0) dA − ¯ A ∧ A (cid:1) C ∈ I n ( A ) . It follows also that the associated classes [ c ( A )] and [ c ( C − AC )] with C from (6) are related[ c ( C − AC )] = c n +1 ,n +1 [ c ( A )] c , . (cid:3) Example 1.14.
Let A ∈ T n ( A ) be the matrix of a formal connection (defining system) for aMassey product (cid:104) α , . . . , α n (cid:105) . Then a matrix C − AC with C = . . . x . . . . . . . . . . . . . . . x . . .x n −
00 0 . . . x . . .x n − x n is a defining system for (cid:104) x α , . . . , x n α n (cid:105) = x . . . x n (cid:104) α , . . . , α n (cid:105) . Definition 1.15.
Two matrices A and A (cid:48) of formal connections are equivalent if there exists anon-degerate scalar matrix C ∈ GT n ( K ) such that A (cid:48) = C − AC.
It is obvious that we can consider only the subgroup of non-degenerate diagonal matricesinstead of the whole GT n ( K ) in the last definition.Following the original Massey’s paper [65], some higher order cohomological operations thatwe call now Massey products were introduced in the 1960s in [60] and [66]. May briefly noticedin [66] that there is a relation between the difinition of the Massey products and Maurer-Cartanequation. However, this analogy was not developed till the Babenko-Taimanov paper [10].In the present article we deal only with Massey products of non-trivial cohomology classes.In general situation it is more natural to consider so-called matric Massey products that werefirst introduced by May in [66] and developed in [10]. I.YU. LIMONCHENKO AND D.V. MILLIONSHCHIKOV Massey products and Lie algebras representations
Consider the cochain complex with trivial coefficients K of an n -dimensional Lie algebra g K d =0 −−−→ g ∗ d −−−→ Λ ( g ∗ ) d −−−→ . . . d n − −−−→ Λ n ( g ∗ ) −−−→ . where d : g ∗ → Λ ( g ∗ ) is a dual mapping to the Lie bracket [ , ] : Λ g → g . The differential d (the whole collection of d p ) is the derivation of the exterior algebra Λ ∗ ( g ∗ ) that continues d d ( ρ ∧ η ) = dρ ∧ η + ( − degρ ρ ∧ dη, ∀ ρ, η ∈ Λ ∗ ( g ∗ ) . It is easy to see that the condition d = 0 is equivalent to the Jacobi identity of g .Continue to use dual language and write with its help the definition of the representation ofa Lie algebra by square ( n + 1 , n + 1)-matrices. Proposition 2.1. A ( n + 1 , n + 1) -matrix A with entries from g ∗ defines a representation ρ : g → g l n ( K ) if and only if A satisfies the strong Maurer-Cartan equation dA − ¯ A ∧ A = 0 . Proof. ( dA − ¯ A ∧ A )( x, y ) = A ([ x, y ]) − [ A ( x ) , A ( y )] , ∀ x, y ∈ g . (cid:3) In the Section 1 the involution of a graded A was defined as ¯ a = ( − k +1 a, a ∈ A k . Thus,for a matrix A with entries in g ∗ we have ¯ A = A . One has to remark that ¯ a differs by the signfrom the definition of the involution ¯ a in [60], however, in [67] there is the same sign rule.From now on, we will consider only representations in upper triangular matrices. So, wedenote by T n ( K ) ⊂ g l n ( K ) the Lie subalgebra of upper triangular ( n + 1 , n + 1)-matrices andconsider representations ρ : g → T n ( K ) of a Lie algebra g for some fixed value n .Consider n = 1 and a linear map ρ : x ∈ g → A ( x ) = (cid:18) a ( x )0 0 (cid:19) . It is evident that ρ is a Lie algebra homomorphism if and only if the linear form a ∈ g ∗ isclosed. Or equivalently, A satisfies the strong Maurer-Cartan equation dA − ¯ A ∧ A = 0.The Lie algebra T n ( K ) has a one-dimensional center I n ( K ) spanned by the matrix . . . . . . . . . . . . . One can consider an one-dimensional central extension0 −−−→ K ∼ = I n ( K ) −−−→ T n ( K ) π −−−→ ˜ T n ( K ) −−−→ . Proposition 2.2 ([40], [38]) . Fixing a Lie algebra homomorphism ˜ ϕ : g → ˜ T n ( K ) is equivalentto fixing a defining system A with elements from g ∗ = Λ ( g ) . The related cocycle c ( A ) iscohomologious to zero if and only if ˜ ϕ can be lifted to a homomorphism ϕ : g → T n ( K ) , ˜ ϕ = πϕ . IGHER ORDER MASSEY PRODUCTS AND APPLICATIONS 9
There is a standard definition.
Definition 2.3.
Two representations ϕ : g → T n ( K ) and ϕ (cid:48) : g → T n ( K ) are called equivalentif there exists a non-degenerate matrix C ∈ GL ( n +1 , K ) such that ϕ (cid:48) ( g ) = C − ϕ ( g ) C, ∀ g ∈ g . It is evident that this definition is equivalent to the Definition 1.15 for Massey products of1-cohomology classes ω , . . . , ω n . Moreover, these linear forms are on the second (top) maindiagonal of the matrix A of the formal connection. Proposition 2.4.
Let a Massey product (cid:104) ω , ω , . . ., ω n (cid:105) be defined and trivial in H ( g ) forsome -cohomology classes ω i ∈ H ( g ) of a Lie algebra g . Then (cid:104) x ω , x ω , . . ., x n ω n (cid:105) is alsodefined and trivial for any choice of non-zero constants x , x , . . ., x n . k -step Massey products in Lie algebra cohomology This paragraph is written largely under the influence of an article by Ido Efrat [35]. His mainobservation was the following theorem.
Theorem 3.1 (Efrat [35]) . Suppose that for every set ω , . . . , ω n of -cocycles from H ( g ) thereis a matrix A of formal connection. Then the Massey product (cid:104)· , . . . , ·(cid:105) : H ( g ) × · · · × H ( g ) (cid:124) (cid:123)(cid:122) (cid:125) n times → H ( g ) is a well-defined single-valued map. However, in this section we want to discuss other ideas of [35] in a revised form, in par-ticular, to define higher order Massey products (cid:104) a , a , . . . , a n (cid:105) as successive obstructions (cid:104) a , a , . . . , a n (cid:105) k , k = 1 , , . . . , n − , to the existence of a formal connection A for a givenset of one-dimensional cocycles a , a , . . . , a k from the differential algebra A .Consider the descending series { T n − kn ( A ) , k = 0 , . . . , n − } of two-sided ideals T n − n ( A ) = T n ( A ) ⊂ T n − n ( A ) ⊂ · · · ⊂ T n − kn ( A ) ⊂ · · · ⊂ T n ( A ) = I n ( A ) ⊂ { } , where T n − kn ( A ) denotes the subspace of upper triangular ( n +1 , n +1)-matrices B with entriesfrom A of the following form B = . . . b (1 , k ) . . . b (1 , n )0 . . . . . . . . . 0 . . . ...... . . . . . . . . . . . . . . . b ( n − k, n )... . . . . . . . . . . . . 0... . . . . . . . . . ...... . . . . . . 00 . . . . . . . . . . . . , b ( i, j ) ∈ A . Obviously for non-negative integers k, m, k + m ≤ n, we have inclusions(7) T n − kn ( A ) · T n − mn ( A ) ⊂ T n − k − mn ( A ) . Definition 3.2.
Consider a non-negative integer k, ≤ k ≤ n −
1. A matrix A k ∈ T n ( A ) iscalled the matrix of a k -step formal connection if it satisfies the k -step Maurer-Cartan equation(8) µ ( A k ) = d A k − ¯ A k · A k ∈ T n − kn ( A ) . Proposition 3.3.
Let A k be the matrix of a k -step formal coonection. Then dµ ( A k ) ∈ T n − k − n ( A ) . Proof.
It follows from (7). Indeed the generalized Bianci identity (3) means d µ ( A k ) = − d ( ¯ A k · A k ) = µ ( A k ) · A k + A k · µ ( A k ) ∈ T n − kn ( A ) · T n − n ( A ) . (cid:3) Corollary 3.4.
Let A k be the matrix of a k -step formal connection, ≤ k ≤ n − . Then allelements on the k -th parallel main diagonal of the matrix µ ( A k ) = ( τ ( i, j )) are closed dτ (1 , k ) = 0 , dτ (2 , k +1) = 0 , . . . , dτ ( n − k, n ) = 0 . From now on, we will consider matrices of formal connections with elements ω from the dualspace g ∗ to the Lie algebra g . It means that they are all 1-forms and ¯ ω = ω for every ω ∈ g . Definition 3.5.
Let A k be the matrix of a k -step formal connection, 0 ≤ k ≤ n −
1. The set of n − k two-dimensional cocycles c ( A k ) = (cid:32) n − (cid:88) r =1 ¯ a (1 , r ) ∧ a ( r +1 , k ) , . . . , n − (cid:88) r =1 ¯ a ( n − k, r ) ∧ a ( r +1 , n ) (cid:33) . is called the related set c ( A k ) of cocycles of the formal k -step connection A k . Definition 3.6.
The k -step n -fold product (cid:104) a , . . . , a n (cid:105) k is defined if there exists at least one k -step formal connection A k for it with entries a , . . . , a n at the second diagonal. If it is defined,then the value (cid:104) a , . . . , a n (cid:105) k is the set of all ( n − k )-tuples of cohomology classes ( α , . . . , α k +1 ) ∈ H ( A ) × . . . H ( A ) (cid:124) (cid:123)(cid:122) (cid:125) n − k for which there exists a k -step formal connection A k such that the sequence c ( A k ) (or equivalently ( − τ , . . . , − τ n − k )) represents ( α , . . . , α n − k ). µ ( A k ) = . . . τ ∗ . . . ∗ τ . . . ...... . . . . . . ∗ ... 0 τ n − k ... 00 . . . . . . . . . ... , τ i ∈ g ∗ , dτ i = 0 , i = 1 , . . . , n − k. IGHER ORDER MASSEY PRODUCTS AND APPLICATIONS 11
Using the standard arguments from [60, 10] one can prove that the product (cid:104) a , . . . , a n (cid:105) k dependsonly on the cohomology classes of the elements a , . . . , a n as it holds in the classic case. Example 3.7.
The 1-step product (cid:104) a , . . . , a n (cid:105) is defined, single-valued and equal to thefollowing ( n − (cid:104) a , . . . , a n (cid:105) = ( a ∧ a , a ∧ a , . . . , a n − ∧ a n − , a n − ∧ a n ) ∈ H ( g ) × . . . H ( g ) (cid:124) (cid:123)(cid:122) (cid:125) n − . Remark.
If we do not pay attention to the subscript k , the n -fold Massey product (cid:104) a , . . . , a n (cid:105) isnow always defined, although we a priori do not know in which space its value will be located.By analogy with the classical case, we give the following definition Definition 3.8.
Let a k -step n -fold Massey product (cid:104) α , . . . , α n (cid:105) k in the Lie algebra cohomologybe defined. It is called trivial if it contains the trivial cohomology class of the vector space H ( g ) × · · · × H ( g ) (cid:124) (cid:123)(cid:122) (cid:125) n − k (0 , . . . , ∈ (cid:104) α , . . . , α n (cid:105) k . Proposition 3.9.
Let a k -step Massey product (cid:104) α , . . . , α n (cid:105) k be defined. Then all Massey prod-ucts (cid:104) α l , . . . , α q (cid:105) s , ≤ l < q ≤ n, q − l < n − , s ≤ k, are defined and trivial. Proposition 3.10.
Let a k -step Massey product (cid:104) α , . . . , α n (cid:105) k be defined and trivial. Then the ( k + 1) -step Massey product (cid:104) α , . . . , α n (cid:105) k +1 is defined. Returning to Example 3.7, we see that a two-step formal connection A for the classes a , . . . , a n , exists if and only if all the cocycle products a i ∧ a i +1 , i = 1 , . . . , n − , are trivial inthe cohomology H ∗ ( A ). Moreover, if we find 1-forms a ( i, i +1) solving the equations a ( i, i ) ∧ a ( i +1 , i +1) = da ( i, i +1) , i = 1 , . . . , n − , we can explicitly write down the formal connection matrix A with their help and (4). Newelements a ( i, i +1) make up the second diagonal of the matrix A . Corollary 3.11.
Let a standard n -fold Massey product (cid:104) α , . . . , α n (cid:105) be defined. Then all k -stepMassey products (cid:104) α , . . . , α n (cid:105) k with k < n are defined and trivial.Proof. It means that the ( n − (cid:104) α , . . . , α n (cid:105) n − is defined. Hence thestatement follows from Proposition 3.9. (cid:3) To conclude this section, we want to note that the technique of k -step Massey productsconstructed in it is not something completely unknown. Anyone who searched for (constructed)the defining systems of Massey products came across this recursive procedure at one levelor another. But it seems however useful to formalize it in this specific way thinking on theapplications to representations theory of nilpotent Lie algebras. Non-trivial Massey products in Lie algebra cohomology
Buchstaber and Shokurov discovered [29] that the tensor product S ⊗ R of the Landweber-Novikov algebra S (the complex cobordism theory) by real numbers S ⊗ R is isomorphic to theuniversal enveloping algebra U ( W + ) of the Lie algebra W + of polynomial vector fields on thereal line R with vanishing non-positive Fourier coefficients. W + is a maximal pro-nilpotentsubalgebra of the Witt algebra.The Witt algebra W is spanned by differential operators on the real line R with a fixedcoordinate x e i = x i +1 ddx , i ∈ Z , [ e i , e j ] = ( j − i ) e i + j , ∀ i, j ∈ Z . Theorem 4.1 (Goncharova [44]) . The Betti numbers dim H q ( W + ) = 2 , for every q ≥ , moreprecisely dim H qk ( W + ) = (cid:26) , if k = q ± q , , otherwise . We will denote by g q ± a basis in the spaces H q q ± q ( W + ). The numbers q ± q = e ± ( q ) are socalled Euler pentagonal numbers. It is easy to verify that the sum e ± ( q ) + e ± ( p ) (cid:54) = e ± ( p + q ) , p, q ∈ N . Hence the cohomology algebra H ∗ ( W + ) has a trivial multiplication. Buchstaber conjecturedthat the algebra H ∗ ( W + ) is generated with respect to some non-trivial Massey products by itsfirst cohomology H ( W + ).Feigin, Fuchs and Retakh [40] represented the basic homogeneous cohomology classes from H ∗ ( W + ) as Massey products. Theorem 4.2 (Feigin, Fuchs, Retakh [40]) . For any q ≥ we have inclusions g q − ∈ (cid:104) g q − , e , . . . , e (cid:124) (cid:123)(cid:122) (cid:125) q − (cid:105) , g q + ∈ (cid:104) g q − , e , . . . , e (cid:124) (cid:123)(cid:122) (cid:125) q − (cid:105) . Feigin, Fuchs and Retakh proposed to consider the defining system that is equivalent to thefollowing matrix of formal connection A = g k − Ω Ω . . . Ω n − ∗ e αe . . .
00 0 0 . . . . . . . . . ...0 0 0 . . . . . . αe e αe e . . . , IGHER ORDER MASSEY PRODUCTS AND APPLICATIONS 13 with homogeneous forms Ω i ∈ Λ k + i − (3( k − +( k − i ( W + ) and parameter α ∈ K . The correspondingcocycle c ( A ) ∈ Λ k +1 (3( k − +( k − n − ( W + ) can be non-trivial only if n = 2 k or n = 3 k thatcorresponds to H k (3 k ± k ) ( W + ). Feigin, Fuchs and Retakh have shown that the triviality of thecocycle c ( A ) is equivalent to the triviality of the differential d n of some spectral sequence E p,qr converging to the cohomology H ∗ ( W + , V ) with coefficients in the graded W + -module V . Theinfinite dimensional module V depends on the parameter α and can be defined by its basis f , f , . . . , f n , . . . and relations e f j = f j +1 , e f j = αf j +2 , j ≥ e k f j = 0 , k, j ∈ N . Feigin, Fuchs and Retakh established that1) the differential d k is trivial if and only if α ∈ { , , . . . , k − } ;2) the differential d k is defined and trivial if and only if:a) α ∈ { , , . . . , k − , k − } in the case of even k ;b) α ∈ { , , . . . , k − , k − } if k is odd. Corollary 4.3.
All Massey products from the Theorem 4.2 are trivial.
The main technical problem in the proof of Theorem 4.2 is to find explicit formulas forthe entries Ω i , i = 1 , . . . , n −
1. One can verify directly that the cocycles g − = e ∧ e and g = e ∧ e − e ∧ span the homogeneous subspaces H ( W + ) and H ( W + ) respectively. Butexplicit formulas for all Goncharova’s cocycles g k ± in terms of exterior forms from Λ ∗ ( W + ) arestill unknown. Fuchs, Feigin and Retakh proposed [40] an elegant way how to establish non-triviality of differentials for some values of the parameter α of the spectral sequence E p,qr thatconverge to the cohomology H ∗ ( W + , V ).Artel’nykh [3] represented a part of basic cocycles in H ∗ ( W + ) by means of non-trivial Masseyproducts, but his very brief article does not contain any sketch of the proof. Theorem 4.4 (Millionshchikov [70]) . The cohomology H ∗ ( W + ) is generated by two elements e , e ∈ H ( W + ) by means of two series of non-trivial Massey products. More precisely therecurrent procedure is organized as follows1) elements e and e span H ( W +) ;2) the triple Massey product (cid:104) e , e , e (cid:105) is single-valued and determines non-trivial cohomologyclass g − = (cid:104) e , e , e (cid:105) ∈ H ( W + ) ;3) the -fold product (cid:104) e , e , e , e , e (cid:105) is non-trivial and it is an affine line { g + tg − , t ∈ K } onthe plane H ( W + ) , where g denotes some generator from H ( W + ) . Denote by ˜ g an arbitraryelement in (cid:104) e , e , e , e , e (cid:105) .Let us suppose that we have already constructed some basis g k − , ˜ g k + of H k ( W + ) , k ≥ , such thatthe cohomology class g k − spans the subspace H k k − k ( W + ) . Then4) the (2 k + 1) -fold Massey product (cid:104) e , . . . , e (cid:124) (cid:123)(cid:122) (cid:125) m , e , e , . . . , e (cid:124) (cid:123)(cid:122) (cid:125) n , ˜ g k + (cid:105) = g k +1 − , m + n = 2 k − , is single-valued and spans the subspace H k +1 (3( k +1) − ( k +1)) ( W + ) .5) the (3 k + 2) -fold product (cid:104) e , . . . , e (cid:124) (cid:123)(cid:122) (cid:125) k , e , e , . . . , e (cid:124) (cid:123)(cid:122) (cid:125) k , ˜ g k + (cid:105) is non-trivial and it is an affine line on the two-dimensional plane H k +1 ( W + ) parallel tothe one-dimensional subspace H k +1 (3( k +1) − ( k +1)) ( W + ) . One can take an arbitrary element in (cid:104) e , . . . , e (cid:124) (cid:123)(cid:122) (cid:125) k , e , e , . . . , e (cid:124) (cid:123)(cid:122) (cid:125) k , ˜ g k + (cid:105) as the second basic element ˜ g k +1+ of the subspace H k +1 ( W + ) . The proof of Theorem 4.4 is based on the technique developed in [40], to which a set offundamental additions was proposed [70]. The first was the construction of the special gradedthread W + -module V gr , which is uniquely determined by its main properites. The non-trivialityof the corresponding Massey products largely follows from such a rigidity. To calculate the dif-ferentials of the spectral sequence associated with the constructed Massey products, explicitformulas for the special vectors of Verma modules over the Virasoro algebra were applied.Finally the nontriviality of the corresponding differentials d k was established by explicit cal-culus of the cohomology H ∗ ( W + , V gr ) using the so-called Feigin-Fuchs-Rocha-Caridi-Wallachresolution [70].As an example, we will reproduce the simplest part of the proof [70] related to the secondcohomology H ∗ ( W + ), but which nevertheless well illustrates the basics of the proof technique.For an arbitrary formal connection A that corresponds to the product (cid:104) e , e , e (cid:105) the cocycle c ( A ) = − e ∧ e + αd ( e ) for some scalar α . Hence the single-valued triple product (cid:104) e , e , e (cid:105) = − [ e ∧ e ] (cid:54) = 0 spans the subspace H ( W + ).Consider (cid:104) e , e , − e , − e , − e (cid:105) instead of (cid:104) e , e , e , e , e (cid:105) and a formal connection AA = e e e e ∗ e e e
00 0 0 − e e − e − te − e e − e . The corresponding cocycle will be c ( A ) = ( e ∧ e − e ∧ e ) + te ∧ e . On the another handfor an arbitrary defining system A (cid:48) the corresponding cocycle will have the form c ( A (cid:48) ) =( e ∧ e − e ∧ e )+ . . . , where dots stand for the summands with the second grading strictlyless than 7. Example 4.5.
The Lie algebra m is defined by its infinite basis e , e , . . . , e n , . . . with com-mutator relations: [ e , e i ] = e i +1 , ∀ i ≥
2; [ e i , e j ] = 0 , i, j (cid:54) = 1 . IGHER ORDER MASSEY PRODUCTS AND APPLICATIONS 15
Let g be a pro-nilpotent or N -graded Lie algebra. The ideals g k of the descending centralsequence define a decreasing filtration of the Lie algebra gg = g ⊃ g ⊃ · · · ⊃ g k = [ g , g k − ] ⊃ . . . ; [ g k , g l ] ⊂ g k + l , k, l ∈ N . Consider the associated graded Lie algebragr g = + ∞ (cid:77) k =1 (gr g ) k , (gr g ) k = g k / g k +1 , k ∈ N . Proposition 4.6.
We have the following isomorphisms: gr W + ∼ = gr m ∼ = m . The corresponding natural grading of m is defined by(gr m ) = Span ( e , e ) , (gr m ) i = Span ( e i +1 ) , i ≥ . Let g = ⊕ α g α be a N -graded (pro-nilpotent) Lie algebra and V is a finite-dimensional nilpo-tent g -module. There is a decreasing filtration of the g -module module VV = V ⊂ V = g V ⊂ . . . V k = g V k − ⊂ . . . One can define the associated graded module gr V over the associated graded Lie algebra gr g gr V = ⊕ + ∞ i =1 ( gr V ) i , gr V i = V i /V i +1 , ( gr g ) i ( gr V ) j ⊂ ( gr V ) i + j , i, j ∈ N . Thus, we came to the problem of description of Massey products in the cohomology H ∗ ( m ).As we saw earlier, it is useful to describe trivial Massey products (cid:104) ω , . . . , ω n (cid:105) of 1-cohomologyclasses ω , . . . , ω n . The purpose of this interest is to consider Massey products of the form (cid:104) ω , . . . , ω n , Ω (cid:105) , where Ω is a cocycle from H p ( g ) for p > p = 2. It was found out in [41] that H ( m ) is spanned by the cohomology classesof the following set of cocycles [41](9) ω ( e k ∧ e k +1 ) = (cid:80) k − l =0 ( − l e k − l ∧ e k +1+ l , k ≥ . All of the cocycles (9) can be represented as Massey products. Namely let us consider thefollowing matrix of a formal connection A = e − e . . . ( − k e k ( − k +1 e k +1
00 0 e . . . e k +1 e . . . e k . . . . . . . . . . . . . . . . . . . . . . . . e e . . . e . . . . For the related cocycle c ( A ) we have c ( A ) = k +1 (cid:88) l =2 ( − l e l ∧ e k +3 − l = 2 ω ( e k ∧ e k +1 ) . So we proved that 2 ω ( e k ∧ e k +1 ) ∈ (cid:104) e , e , . . . , e (cid:124) (cid:123)(cid:122) (cid:125) k − , e (cid:105) , k ≥ . The space H ( m ) is spanned by e and e and therefore an arbitrary n -fold Massey productof elements from H ( m ) has a form (cid:104) α e + β e , α e + β e , . . . , α n e + β n e (cid:105) (cid:124) (cid:123)(cid:122) (cid:125) n . It follows from e ∧ e = de that a triple product (cid:104) ω , ω , ω (cid:105) = (cid:104) α e + β e , α e + β e , α e + β e (cid:105) is defined for all values α i , β i ∈ K , i = 1 , , .A = ω γ e
00 0 ω γ e ω , γ = α β − α β , γ = α β − α β , The related cocycle c ( A ) = γ ω ∧ e − γ ω ∧ e is trivial if and only if(10) β ( α β − α β ) − β ( α β − α β ) = 0 . Let us consider the operator D = ad ∗ e : Λ ∗ ( e , e , . . . ) → Λ ∗ ( e , e , . . . ) acting on the chainsubcomplex Λ ∗ ( e , e , . . . ) ⊂ Λ ∗ ( e , e , e , . . . ) of m .(11) D ( e ) = 0 , D ( e i ) = e i − , ∀ i ≥ ,D ( ξ ∧ η ) = D ( ξ ) ∧ η + ξ ∧ D ( η ) , ∀ ξ, η ∈ Λ ∗ ( e , e , . . . ) . The operator D has the right inverse operator D − : Λ ∗ ( e , e , . . . ) → Λ ∗ ( e , e , . . . ), definedby the formulas D − e i = e i +1 , D − ( ξ ∧ e i ) = (cid:88) l ≥ ( − l D l ( ξ ) ∧ e i +1+ l , (12)where i ≥ ξ stands for an arbitrary form in Λ ∗ ( e , . . . , e i − ). One can verify that D D − = Id on Λ ∗ ( e , e , . . . ).The sum in the definition (12) of D − is always finite because D l strictly decreases the secondgrading by l . For instance, D − ( e i ∧ e k ) = (cid:80) i − l =0 ( − l e i − l ∧ e k + l +1 .There is an explicit formula for D − ( e i ∧ . . . ∧ e i q ∧ e i q ) = D − (0) ω ( e i ∧ . . . ∧ e i q ∧ e i q +1 ) = (cid:88) l ≥ ( − l D l ( e i ∧ · · · ∧ e i q ) ∧ e i q +1+ l , this sum is also always finite and determines a homogeneous closed ( q +1)-form of the secondgrading i + . . . + i q − +2 i q +1. IGHER ORDER MASSEY PRODUCTS AND APPLICATIONS 17
Theorem 4.7 (Fialowski, Millionshchikov [41]) . The bigraded cohomology algebra H ∗ ( m ) = ⊕ k,q H qk ( m ) is spanned by the cohomology classes of the following homogeneous cocycles: e , e , ω ( e i ∧ . . . ∧ e i q ∧ e i q +1 ) = (cid:88) l ≥ ( − l ( ad ∗ e ) l ( e i ∧ e i ∧ . . . ∧ e i q ) ∧ e i q +1+ l , (13) where q ≥ , ≤ i
2, while ring multiplicationin cohomology H ∗ ( m ) is not trivial, although there is a sufficient number of trivial products,in particular, the product mapping H ( m ) ∧ H ( m ) → H ( m ) vanishes.We recall that if an n -fold Massey product (cid:104) ω , ω , . . ., ω n (cid:105) is defined than all ( p + 1)-foldMassey products (cid:104) ω i , ω i +1 , . . . , ω i + p (cid:105) for 1 ≤ i ≤ n − , ≤ p ≤ n − , i + p ≤ n are defined andtrivial.The following theorem shows that cohomology H ∗ ( m ), as well as H ∗ ( W + ) is generated bynon-trivial Massey products, if we include in their number ordinary wedge products, as doubleMassey products. Theorem 4.8 (Millionshchikov [71]) . The cohomology algebra H ∗ ( m ) is generated with respectto the non-trivial Massey products by H ( m ) , namely ω ( e ∧ e i ∧ . . . ∧ e i q ∧ e i q +1 ) = e ∧ ω ( e i ∧ . . . ∧ e i q ∧ e i q +1 ) , ω ( e k ∧ e k +1 ) ∈ (cid:104) e , e , . . . , e (cid:124) (cid:123)(cid:122) (cid:125) k − , e (cid:105) , ( − i ω ( e i ∧ e i ∧ . . . ∧ e i q ∧ e i q +1 ) ∈ (cid:104) e , e , . . . , e (cid:124) (cid:123)(cid:122) (cid:125) i − , ω ( e i ∧ . . . ∧ e i q ∧ e i q +1 ) (cid:105) , (15)First of all we present a graded defining system (a graded formal connection) A for a Masseyproduct (cid:104) e , e , . . . , e , ω ( e i ∧ . . . ∧ e i q ∧ e i q +1 ) (cid:105) . To simplify the formulas we will write ω insteadof ω ( e i ∧ . . . ∧ e i q ∧ e i q +1 ).One can verify that the following matrix A with non-zero entries at the second diagonal, firstline and first row gives us an answer. A = e − e e . . . ( − i e i
00 0 e . . . D i − − ω e . . . D i − − ω. . . . . . e D − ω . . . ω . . . . The proof follows from d ( D k − ω ) = e ∧ D k − − ω, d (( − k e k ) = ( − k − e k − ∧ e . The related cocycle c ( A ) is equal to c ( A ) = i (cid:88) l ≥ ( − l e l ∧ D i − l − ω = ( − i i − (cid:88) k ≥ ( − k D k e i ∧ D k − ω Example 4.9.
We take (cid:104) e , e , ω ( e ∧ e ) (cid:105) . A = e − e
00 0 e D − ω ( e ∧ e )0 0 0 ω ( e ∧ e )0 0 0 0 . Compute the related cocycle c ( A )(16) c ( A ) = e ∧ D − ω ( e ∧ e ) − e ∧ ω ( e ∧ e ) == e ∧ ( e ∧ e − e ∧ e + 3 e ∧ e ) − e ∧ ( e ∧ e − e ∧ e + e ∧ e ) == − e ∧ e ∧ e + e ∧ e ∧ e − e ∧ e ∧ e = − ω ( e ∧ e ∧ e ) . Let us return to Theorem 3.1. Consider the question: for which sets a , . . . , a n of one-dimensional cocycles from H ∗ ( m ), all k -step Massey products (cid:104) a , . . . , a n (cid:105) k are defined andtrivial? The following theorem answers this question. Theorem 4.10 (Millionshchikov [71]) . Up to an equivalence the following trivial n -fold Masseyproducts of non-zero cohomology classes from H ( m ) are defined: IGHER ORDER MASSEY PRODUCTS AND APPLICATIONS 19 name Massey product parameters A n +1 λ (cid:104) αe + βe , αe + βe , . . . , αe + βe (cid:105) (cid:124) (cid:123)(cid:122) (cid:125) n n ≥ , λ = ( α, β ) ∈ K P B n +1 α,β (cid:104) λ e + e , λ e + e , . . ., λ n e + e (cid:105) (cid:124) (cid:123)(cid:122) (cid:125) n n ≥ , λ i = iα + β,α, β ∈ K , α (cid:54) = 0 C n +1 l,α (cid:104) e , . . ., e (cid:124) (cid:123)(cid:122) (cid:125) l , e + αe , e , . . . , e (cid:105) (cid:124) (cid:123)(cid:122) (cid:125) n − l − α ∈ K , n ≥ , ≤ l ≤ n − D k +3 α,β (cid:104) e + αe , e , . . . , e (cid:124) (cid:123)(cid:122) (cid:125) k , e + βe (cid:105) k ≥ , α, β ∈ K It should be noted here that this result is related to the classification of indecomposablegraded thread modules over m from [20].5. Massey products in Koszul homology of local rings
Starting with this section, we turn to a discussion of triple and higher Massey products inKoszul homology of local rings and their applications in toric topology.
Definition 5.1.
Let ( A, m , k ) be a (commutative Noetherian) local ring, its unique maximalideal m having minimal set of generators ( x , . . . , x m ) and its residue field being k = A/ m . A Koszul complex of ( A, m , k ) is defined to be an exterior algebra K A = Λ A m , where A m denotesthe free A -module generated by a set { e , . . . , e m } , which is a differential graded algebra witha differential d acting as follows: d ( e i ∧ . . . ∧ e i k ) = k (cid:88) r =1 ( − r − x i r e i ∧ . . . ∧ (cid:99) e i r ∧ . . . ∧ e i k . Definition 5.2.
Let ( A, m , k ) be a local ring. Then for an A -module M we define its Poincar´eseries to be a formal power series of the type P A ( M ; t ) = ∞ (cid:88) i =0 dim k Tor Ai ( M, k ) t i . By definition, the k -module Tor Ai ( M, k ) is the i th homology of a projective resolution for k (the latter viewed as an A -module via the quotient map A → A/ m = k ) tensored by M .We call P A := P A ( k ; t ) Poincar´e series of the local ring A .A classical problem of homological algebra was to prove a conjecture by Serre and Kaplansky,which asserted that P A is a rational function. Anick [2] (1982) found a counterexample being a quotient ring of a polynomial ring over an ideal generated by a set of monomials alongsidewith one binomial.On the other hand, in 1982 Backelin [11] proved the conjecture is true for monomial rings ,that is, a quotient ring of a polynomial ring by an ideal generated by monomials. More precisely,a Poincar´e series of a monomial ring A = k [ x , . . . , x m ] /I has a form P A = (1 + t ) m Q A, k ( t ) . It remains to be an open problem to determine effectively the coefficients of the denominatorpolynomial Q A, k ( t ) for various important classes of local rings.In fact, homology of local rings ( A, m , k ) and that of loop spaces Ω X are closely related.Namely, when the quotient field k = Q , or R and m = 0, Poincar´e series of A acquirestopological interpretation as Poincar´e series of a loop space for a certain simply connected4-dimensional CW complex X . This enabled to solve the Serre–Kaplansky conjecture in bothalgebraic and topological settings at the same time.The properties of the Poincar´e series of a loop space Ω X on a finite simply connected CWcomplex X were related to the homotopy properties of X , in particular, to the generatingfunction for ranks of the homotopy groups of X , in a series of influential works by Babenko [4,6, 7]. An extensive survey on the problem of rationality and growth for Poincar´e series in thecontext of homological algebra and homotopy theory can be found in [7].Using a spectral sequence associated with a presentation of a local ring as a quotient ringof a regular local ring, Serre showed that for any local ring A the following coefficient-wiseinequality for its Poincar´e series holds:(17) P A ≤ (1 + t ) m − (cid:80) b i t i +1 , where m = dim k m / m and b i = dim k H i ( K A ).In 1962 Golod obtained a key result: he identified an important class of local rings withrational Poincar´e series in terms of vanishing of multiplication and all Massey products in theirKoszul homology. Theorem 5.3 ([43]) . Let A be a local ring. Then Serre’s inequality (17) turns into equality ifand only if multiplication and all Massey products in H + ( K A ) are trivial. In their monograph [51] Gulliksen and Levin proposed to refer to the local rings with theabove property as
Golod rings .In view of the above mentioned correspondence between Poincar´e series of local rings andthat of loop spaces, Golod rings correspond to loop spaces over wedges of spheres of variousdimensions. In this case, spectral sequence of a path–loop fibration degenerates, the Bettinumbers grow as fast as possible, and the Poincar´e series turns out to be equal to a fractionwith denominator as in (17), where b i are the Betti numbers of the base space. IGHER ORDER MASSEY PRODUCTS AND APPLICATIONS 21
Example 5.4 ([43]) . Let A be a free reduced nilpotent algebra, that is, a quotient ring A n,r = k [ x ,...,x n ]( x ,...,x n ) r . Golod [43] observed that multiplication and all Massey products are trivial inKoszul homology of A n,r and, furthermore, its Betti numbers are equal to b i = (cid:0) i + r − r − (cid:1)(cid:0) n + r − i + r − (cid:1) .Therefore, by Theorem 5.3, A n,r is a Golod ring and P A n,r = (1 + t ) n − n (cid:80) i =1 (cid:0) i + r − r − (cid:1)(cid:0) n + r − i + r − (cid:1) t i +1 , which generalizes a computation of Poincar´e series given by Kostrikin and Shafarevich [59].From now on we concentrate mainly on the special class of monomial rings called face rings, orStanley–Reisner rings of simplicial complexes, see Definition 5.7. It is well known that applyingto any monomial ring a procedure called ‘polarization of a monomial ideal’ [53, 54] results in aStanley–Reisner ring (of a certain simplicial complex) carrying the same homological propertiesas the initial ring. To define this special and important class of monomial rings, we need torecall the notions of a simplicial complex and a simple polytope. Definition 5.5.
By an ( abstract ) simplicial complex on a vertex set [ m ] = { , , . . . , m } wemean a subset K of 2 [ m ] such that if σ ∈ K and τ ∈ σ , then τ ∈ K . The singleton elementsof K are called its vertices and the dimension of K is defined to be one less than the maximalnumber of vertices in an element of K .By a ( convex ) n - dimensional simple polytope P with m facets we mean a bounded intersectionof m halfspaces in R n such that the supporting hyperplanes of those halfspaces are in generalposition. The latter condition is equivalent to saying that each vertex of P is an intersection ofexactly n of its facets (i.e. faces of codimension one). Example 5.6.
Let P ∗ be the polytope combinatorially dual to a simple n -dimensional polytope P with m facets. Then P ∗ is a convex hull of m vertices, which are in general position in theambient Euclidean space R n . Therefore, all proper faces of P ∗ are simplices (we say, P ∗ is a simplicial polytope ) and its boundary K P = ∂P ∗ is a simplicial complex of dimension n − m vertices. Definition 5.7.
Let k be a commutative ring with unit and K be a simplicial complex on [ m ].We call a quotient ring k [ K ] = k [ v , . . . , v m ]( v i · · · v i k | { i , . . . , i k } / ∈ K ) , of a polynomial algebra over a monomial ideal a face ring , or a Stanley–Reisner ring of K . Themonomial ideal I = ( v i · · · v i k | { i , . . . , i k } / ∈ K ) is called the Stanley–Reisner ideal of K .By definition, k [ K ] is a finite k -algebra and also a finitely generated k [ m ] = k [ v , . . . , v m ]-module via the natural projection. Now we are going to formulate a graded version of Defini-tion 5.1. Definition 5.8.
Set mdeg( u i ) = ( −
1; 0 , . . . , , . . . , v i ) = (0; 0 , . . . , , . . . ,
0) for1 ≤ i ≤ m , and consider a (0 , a ∈ Z m (this grading will acquire its topological interpretation in Theorem 5.9 and Theorem 5.13 below). Then a multigraded Tor-module of k [ K ] is a direct sum of k -modulesTor − i, ak [ m ] ( k [ K ] , k ) = H − i, a [ k [ K ] ⊗ k Λ[ u , . . . , u m ] , d ] , where in the latter differential graded algebra its differential d acts as follows: d ( u i ) = v i and d ( v i ) = 0 for all 1 ≤ i ≤ m .Denote by K I := K ∩ I the induced subcomplex of K on the vertex set I ⊂ [ m ]. Inthe above notation, the next result due to Hochster [54] describes the k -module structureof Tor − i, ak [ m ] ( k [ K ] , k ) in terms of induced subcomplexes in K . Theorem 5.9 ([54]) . There is a k -module isomorphism Tor − i, Ik [ m ] ( k [ K ] , k ) ∼ = ˜ H | I |− i − ( K I ; k ) , where the j th component of the (0 , -vector I is either 0 or 1, depending on whether or not j ∈ [ m ] is an element of I ⊂ [ m ] , and | I | denotes the cardinality of I . On the right hand sideone has a reduced simplicial cohomology group of an induced subcomplex K I in K . Now let us interprete the Tor-algebra (Koszul homology) K k [ K ] ∼ = ⊕ Tor − i, ak [ m ] ( k [ K ] , k )of a Stanley–Reisner ring k [ K ] of a simplicial complex K as a cohomology ring of a certainfinite CW complex Z K . The space Z K , called a moment-angle-complex of K , is the main objectof study in Toric Topology and is a particular case of the following general construction, whichis of its own interest in modern homotopy theory, see [15].Denote by ( X , A ) = ( X i , A i ) := { ( X i , A i ) } mi =1 an ordered set of topological pairs. The case X i = X, A i = A was firstly introduced in [26]under the name of a K-power and was then intensively studied and generalized in a series ofmore recent works, among which are [12, 49, 55].
Definition 5.10 ([12]) . A polyhedral product over a simplicial complex K on the vertex set [ m ]is a topological space ( X , A ) K = (cid:91) I ∈ K ( X , A ) I ⊆ m (cid:89) i =1 X i , where ( X , A ) I = m (cid:81) i =1 Y i and Y i = X i if i ∈ I , and Y i = A i if i / ∈ I .The term ‘polyhedral product’ was suggested by Browder (cf. [12]). Example 5.11.
Suppose X i = X and A i = A for all 1 ≤ i ≤ m . Then the next spaces areparticular cases of the above construction of a polyhedral product ( X , A ) K = ( X, A ) K .(1) Moment-angle-complex Z K = ( D , S ) K ;(2) Real moment-angle-complex R K = ([ −
1; 1] , {− , } ) K ; IGHER ORDER MASSEY PRODUCTS AND APPLICATIONS 23 (3)
Davis–Januzskiewicz space DJ ( K ) := E T m × T m Z K (cid:39) ( C P ∞ , ∗ ) K ;(4) Cubical complex cc( K ) = ( I , K in the m -dimensional cube I m = [0 , m , which isPL-homeomorphic to a cone over a barycentric subdivision of K .As was shown by Buchstaber and Panov [26], one has a commutative diagram Z K −−−→ ( D ) m (cid:121) r (cid:121) ρ cc( K ) i c −−−→ I m where i c : cc( K ) (cid:44) → I m = ( I , I ) [ m ] is an embedding of a cubical subcomplex, induced by anembedding of pairs: ( I , ⊂ ( I , I ), the maps r and ρ are projection maps onto the orbitspaces of a compact torus T m -action, induced by coordinatewise action of T m on the complexunitary polydisk ( D ) m in C m .When K = K P is a boundary of an n -dimensional simplicial polytope with m vertices,see Definition 5.5, Buchstaber and Panov [26] proved Z K is equivariantly homeomorphic toa smooth compact 2-connected ( m + n )-dimensional manifold Z P with a compact torus T m -action. They called it a moment-angle manifold of the simple n -dimensional polytope P with m facets. This space appeared firstly in the paper by Davis and Januszkiewicz [33].The following definition of Z P given by Buchstaber, Panov, and Ray [28] is equivalent tothe definition of Z P from [33]. Its advantage is that it yields a realization of Z P as a completeintersection of Hermitian quadrics in C m . Definition 5.12. A moment-angle manifold Z P of a polytope P is a pullback defined from thefollowing commutative diagram Z P i Z −−−→ C m (cid:121) (cid:121) µ P i P −−−→ R m ≥ where i P is an affine embedding of P into the nonnegative orthant R m ≥ = { x = ( x , . . . , x m ) | x i ≥ ≤ i ≤ m } , and µ ( z , . . . , z m ) = ( | z | , . . . , | z m | ). The projection map Z P → P in theabove diagram is a quotient map of the canonical T m -action on Z P , induced by the standardcoordinatewise action of T m = { z ∈ C m : | z i | = 1 for i = 1 , . . . , m } on C m . Therefore, T m acts on Z P with an orbit space P and i Z is a T m -equivariant embedding.Moreover, it was proved in [26] that there exists a homotopy fibration of polyhedral products Z K → ( C P ∞ , ∗ ) K → BT m , where the first map is induced by a natural map of pairs ( D , S ) → ( C P , ∗ ) and the secondis induced by inclusion ( C P ∞ , ∗ ) (cid:44) → ( C P ∞ , C P ∞ ). Applying Eilenberg–Moore spectral sequence argument to the above homotopy fibration andanalyzing topology of the polyhedral products involved, Baskakov, Buchstaber and Panov [18]obtained the next fundamental result, which links toric topology with combinatorial commu-tative algebra and, in particular, with homology theory of face rings.
Theorem 5.13 ([27, Theorem 4.5.4]) . Cohomology algebra of Z K over a commutative ring withunit k is given by isomorphisms H ∗ ( Z K ; k ) ∼ = Tor ∗ , ∗ k [ v ,...,v m ] ( k [ K ] , k ) ∼ = H ∗ , ∗ (cid:2) R ( K ) , d (cid:3) ∼ = (cid:77) I ⊂ [ m ] (cid:101) H ∗ ( K I ; k ) , where the differential (multi)graded algebra R ( K ) := k [ K ] ⊗ k Λ[ u , . . . , u m ] / ( v i = u i v i = 0 , ≤ i ≤ m ) and d ( u i ) = v i , d ( v i ) = 0 . Here we denote by (cid:101) H ∗ ( K I ; k ) reduced simplicial cohomologyof a simplicial complex K I and set ˜ H − ( ∅ ; k ) ∼ = k . The last isomorphism above is a sum ofisomorphisms H p ( Z K ; k ) ∼ = (cid:88) I ⊂ [ m ] (cid:101) H p −| I |− ( K I ; k ) . In order to determine a product of two cohomology classes α = [ a ] ∈ ˜ H p ( K I ; k ) and β =[ b ] ∈ ˜ H q ( K I ; k ) , consider an embedding of simplicial complexes i : K I (cid:116) I → K I ∗ K I and acanonical k -module isomorphism of cochains: j : ˜ C p ( K I ; k ) ⊗ ˜ C q ( K I ; k ) → ˜ C p + q +1 ( K I ∗ K I ; k ) . Then a product of the classes α and β is given by: α · β = (cid:40) , if I ∩ I (cid:54) = ∅ ; i ∗ [ j ( a ⊗ b )] ∈ ˜ H p + q +1 ( K I (cid:116) I ; k ) , if I ∩ I = ∅ . It turned out that Golodness of a face ring k [ K ] is closely related to the case when Z K ishomotopy equivalent to a wedge of spheres; by Theorem 5.13, the latter implies the former.However, the opposite statement is not true, see [47, 61, 57].Using methods of stable homotopy theory and toric topology such as the stable homotopydecomposition of polyhedral products due to Bahri, Bendersky, Cohen, and Gitler [12, 13, 14]and the fat wedge filtration method due to Iriye and Kishimoto [56], several authors, amongwhich are Grbi´c and Theriault [48], Iriye and Kishimoto [55], Grbi´c, Panov, Theriault, andWu [47], were able to identify a number of important classes of simplicial complexes K forwhich k [ K ] is a Golod ring over any field k . In the latter case K is called a Golod simplicialcomplex . For a detailed exposition on this problem we refer the reader to a survey article byGrbi´c and Theriault [50].By definition, Golodness of k [ K ] implies triviality of multiplication in its Koszul homology H + ( K k [ K ] ), or equivalently, cup( Z K ) = 1, where cup( X ) denotes cohomology length of a space X , that is, the maximal number of classes of positive dimension in H ∗ ( X ; k ) having a nonzeroproduct. Another related area of research, which also attracts much attention from the scholarsthese days, emerged from a false claim made in [19]: Golodness of K is equivalent to cup( Z K ) = IGHER ORDER MASSEY PRODUCTS AND APPLICATIONS 25
1. The first counterexample was constructed by Katth¨an in 2015. More precisely, he proved thenext result.
Theorem 5.14 ([58]) . The following statements hold. (1) If dim K ≤ , then cup( Z K ) = 1 ⇐⇒ K is a Golod simplicial complex; (2) There exists a 4-dimensional simplicial complex K such that (a) cup( Z K ) = 1 ; (b) There is a non-trivial triple Massey product (cid:104) α , α , α (cid:105) ⊂ H ∗ ( Z K ) ; therefore, K isnot a Golod complex. Since that time there have already appeared several papers devoted to identification of theclass of simplicial complexes K for which Golodness of K is equivalent to triviality of multi-plication in H + ( Z K ), that is, to cup( Z K ) = 1. Such simplicial complexes and face rings arerefered to as quasi-Golod , that is the case when there are no non-trivial triple or higher orderMassey products in their Koszul homology.To state the most general known result characterizing a class of Golod (and quasi-Golod)simplicial complexes and face rings we need the following two notions. A simplicial complex K is called flag if all its minimal non-faces (with respect to inclusion relation) have exactly twovertices. A simple graph Γ is chordal if it does not have any induced cycles of length greaterthan three. Theorem 5.15 ([47, 73]) . Let K be a flag simplicial complex. The following statements areequivalent: (1) The 1-skeleton sk ( K ) of K is a chordal graph; (2) cup( Z K ) = 1 ; (3) Z K is homotopy equivalent to a wedge of spheres; (4) K is a Golod complex; (5) Commutator subgroup π ( R K ) = RC (cid:48) K of the right-angled Coxeter group RC K is a freegroup; (6) Associated graded Lie algebra gr (RC (cid:48) K ) is free. There is an extremely important class of topological spaces arising in homotopy theory, forwhich the rational homotopy type is determined by the singular cohomology ring of a space(over rationals). Namely, a space X is called rationally formal if its Sullivan-de Rham algebra[ A, d ] of PL-forms with coefficients in Q is formal in the category of commutative differentialgraded algebras (CDGA), i.e., there exists a zigzag of quasi-isomorphisms (weak equivalence)between [ A, d ] and its cohomology algebra [ H ∗ ( A ) , K ( π, n ) for n >
1, compactconnected Lie groups G and their classifying spaces BG . Moreover, formality is preserved bywedges, direct products, and connected sums.Polyhedral products of the type ( X, ∗ ) K are formal spaces (over k = Q ), provided the space X is formal (over Q ), see [27, Chapter 8]. In particular, Davis–Januszkiewicz spaces DJ( K ) areformal for any simplicial complex K . It is not hard to see that formality of X implies all triple and higher Massey products vanishin H ∗ ( X ); therefore, Massey products serve as an obstruction to formality of a differentialgraded algebra, or to that of a topological space. The next section is devoted to the case, whenthere exist non-trivial Massey products in (integral) cohomology of moment-angle-complexesand moment-angle manifolds.6. Massey products in Toric Topology
Since for any simplicial complex K its moment-angle-complex Z K is 2-connected, the lowestpossible dimension of cohomology classes in H + ( Z K ), which can form a non-trivial Masseyproduct is equal to three. Note that by a result of Halperin and Stasheff [52], if a space X isformal, then H ∗ ( X ) is generated by spherical classes.Although all 3-dimensional classes in H ∗ ( Z K ) are spherical, which can be easily seen fromTheorem 5.13, there exists a wide class of simplicial complexes K and simple polytopes P suchthat H ∗ ( Z K ) and H ∗ ( Z P ) contain non-trivial triple Massey products of 3-dimensional classes.The case of non-trivial triple Massey products of 3-dimensional classes in H ∗ ( Z K ) was analyzedby Denham and Suciu [36] (strictly defined products) and recently by Grbi´c and Linton [45](products with non-zero indeterminacy). When a simple polytope P is a graph-associahedron,necessary and sufficient conditions for H ∗ ( Z P ) to contain a non-trivial triple Massey productof 3-dimensional classes were obtained by the first author in [63].The first example of a non-trivial triple Massey product in cohomology of a polyhedralproduct was given by Baskakov [17]. His construction was later generalized to higher orderMassey products by the first author [62, 63, 64], by Buchstaber and Limonchenko [25], whodeveloped a theory of direct families of polytopes with non-trivial Massey products (it is beyondthe scope of this survey), and recently by Grbi´c and Linton [46]. The current section is devotedto a discussion of the above mentioned results.At first, let us consider the case of a non-trivial triple Massey product of 3-dimensional classesin H ∗ ( Z K ). Recall that by Theorem 5.13, a 3-dimensional element in H ∗ ( Z K ) corresponds toa pair of vertices of K not connected by an edge, that is, to an induced subcomplex K I in K ,where | I | = 2 and I / ∈ K .In fact, the next criterion shows that non-trivial triple Massey products (cid:104) α , α , α (cid:105) ⊂ ˜ H ∗ ( Z K ) with dim α i = 3 , ≤ i ≤ graph (i.e., the 1-skeleton) sk ( K )of the simplicial complex K . Figure 1.
Non-trivial triple Massey products of 3-dimensional classes in H ∗ ( Z K ) IGHER ORDER MASSEY PRODUCTS AND APPLICATIONS 27
Theorem 6.1 ([36, 45]) . Let (cid:104) α , α , α (cid:105) ⊂ ˜ H ∗ ( Z K ) with dim α i = 3 , ≤ i ≤ be a definedMassey product. Then (a) (cid:104) α , α , α (cid:105) is non-trivial and strictly defined if and only if K contains one of the sixdifferent induced subgraphs described in Figure 1 (a) and α i is a generator of the group ˜ H ( K { i,i +1 } ) for ≤ i ≤ , cf. Theorem 5.13; (b) (cid:104) α , α , α (cid:105) is non-trivial and has non-zero indeterminacy if and only if K contains oneof the two different induced subgraphs described in Figure 1 (b) and α i is a generator ofthe group ˜ H ( K { i,i +1 } ) for ≤ i ≤ , cf. Theorem 5.13.Remark. Statement (a) was proved firstly by Denham and Suciu as [36, Theorem 6.1.1]. How-ever, it was asserted there that any non-trivial triple Massey product of 3-dimensional classesin H ∗ ( Z K ) has such a form; the additional two graphs of Figure 1 (b) were found and thestatement (b) above was proved by Grbi´c and Linton [45]. The result of [45] provided us with afirst example of a non-strictly defined non-trivial Massey product in cohomology of a moment-angle-complex.Now we turn to a discussion of higher order non-trivial Massey products in Koszul homologyof a Stanley–Reisner ring H ∗ ( K k [ K ] ) ∼ = H ∗ ( Z K ; k ). Following [64], we set up the notation asindicated below.Let us fix a set of induced subcomplexes K I j in K on pairwisely disjoint subsets of vertices I j ⊂ [ m ] for 1 ≤ j ≤ k and their cohomology classes α j ∈ ˜ H d ( j ) ( K I j ) of certain dimen-sions d ( j ) ≥ ≤ j ≤ k . If an s -fold Massey product ( s ≤ k ) of consecutive classes (cid:104) α i +1 , . . . , α i + s (cid:105) for 1 ≤ i + 1 < i + s ≤ k is defined, then (cid:104) α i +1 , . . . , α i + s (cid:105) is a subset of˜ H d ( i +1 ,i + s ) ( K I i +1 (cid:116) ... (cid:116) I i + s ), where d ( i + 1 , i + s ) := d ( i + 1) + . . . + d ( i + s ) + 1.Our goal is to determine the conditions sufficient for a Massey product (cid:104) α , . . . , α k (cid:105) of coho-mology classes introduced above to be defined and, furthermore, to be strictly defined.We are going to present a complete proof of the following theorem, in which statements (1)and (2) were originally proved by the first author as [64, Lemma 3.3]. Alongside with statement(3), this result provides an effective tool to determine non-trivial k -fold Massey products for k ≥ H ∗ ( Z K ) for a given simplicial complex K , when one knows multigraded (or, algebraic ) Betti numbers of K (i.e., the dimensions of the (multi)graded components of the Tor-moduleof k [ K ]) and the combinatorial structure of the corresponding induced subcomplex K I (cid:116) ... (cid:116) I k . Theorem 6.2.
Let k ≥ . Then (1) If ˜ H d ( s,r + s ) ( K I s (cid:116) ... (cid:116) I r + s ) = 0 , ≤ s ≤ k − r, ≤ r ≤ k − , then the k -fold Massey product (cid:104) α , . . . , α k (cid:105) is defined; (2) If ˜ H d ( s,r + s ) − ( K I s (cid:116) ... (cid:116) I r + s ) = 0 , ≤ s ≤ k − r, ≤ r ≤ k − and the k -fold Masseyproduct (cid:104) α , . . . , α k (cid:105) is defined, then the k -fold Massey product (cid:104) α , . . . , α k (cid:105) is strictlydefined. (3) In the latter case, there exists a defining system C = ( c i,j ) k +1 i,j =1 for (cid:104) α , . . . , α k (cid:105) such that c s,r + s +1 ∈ C d ( s,r + s ) − ( K I s (cid:116) ... (cid:116) I r + s ) , ≤ s ≤ k − r, ≤ r ≤ k − and (cid:104) α , . . . , α k (cid:105) = { [ a ( C )] } ∈ H d (1 ,k ) ( K I (cid:116) ... (cid:116) I k ) , a ( C ) = − k − (cid:88) j =1 ¯ c , j ∧ c j,k +1 , in the differential graded algebra ⊕ I ⊂ [ m ] C ∗ ( K I ) ∼ = C ∗ ( Z K ) .Proof. The proof goes by induction on k ≥
3. First, we prove statement (1).For k = 3 the condition (1) implies that the 2-fold products (cid:104) α , α (cid:105) and (cid:104) α , α (cid:105) vanishand the triple Massey product (cid:104) α , α , α (cid:105) is defined. By the condition (2), indeterminacy in (cid:104) α , α , α (cid:105) is trivial, and therefore, (cid:104) α , α , α (cid:105) is strictly defined.The inductive hypothesis follows, since all the Massey products of consecutive elements oforders 2 , . . . , k − k ≥ C exists and the corresponding cocycle a ( C ) represents an element in the Massey product (cid:104) α , . . . , α k (cid:105) .Now we prove statement (2).We need to show that [ a ( C )] = [ a ( C (cid:48) )] for any two defining systems C and C (cid:48) . By theinductive assumption, suppose that the statement holds for defined higher Massey products oforders less than k ≥
4. The induction step can be divided into two parts, and in both of themwe also act by induction.I. Let us construct a sequence of defining systems C (1) , . . . , C ( k −
1) for the defined Masseyproduct (cid:104) α , . . . , α k (cid:105) such that the following properties:(1) C (1) = C ;(2) c ij ( r ) = c (cid:48) ij , if j − i ≤ r ;(3) [ a ( C ( r ))] = [ a ( C ( r + 1))], for all 1 ≤ r ≤ k − c i,i +1 ( r ) = a i = a (cid:48) i for all 1 ≤ i ≤ k , and C ( k −
1) = C (cid:48) . We applyinduction on r ≥ C ( r ). Since C (1) = C by (1), we need toprove the induction step assuming that C ( r ) is already defined.For any 1 ≤ s ≤ k − r let us consider a cochain b s = c (cid:48) s,r + s +1 − c s,r + s +1 ( r ) . By definition of a Massey product, d ( b s ) = d ( c (cid:48) s,r + s +1 ) − d ( c s,r + s +1 ( r )) = r + s (cid:88) p = s +1 ¯ c (cid:48) s,p ∧ c (cid:48) p,r + s +1 − r + s (cid:88) p = s +1 ¯ c s,p ( r ) ∧ c p,r + s +1 ( r ) = 0by property (2) of C ( r ) above. It follows that b s is a cocycle, and therefore,[ b s ] ∈ ˜ H d ( s,r + s ) − ( K I s (cid:116) ... (cid:116) I r + s ) = 0 , for all 1 ≤ s ≤ k − r , by condition (2) of our statement, since here we have 1 ≤ r ≤ k − C ( r + 1) will be finished if one is ableto determine a sequence of defining systems C ( r, s ) (0 ≤ s ≤ k − r ) for (cid:104) α , . . . , α k (cid:105) with thefollowing properties: IGHER ORDER MASSEY PRODUCTS AND APPLICATIONS 29 (1’) C ( r,
0) = C ( r );(2’) c ij ( r, s ) = c ij ( r ), if j − i ≤ r , and c ij ( r, s ) = (cid:40) c ij ( r ) , if i > s ; (*) c ij ( r ) + b i , if i ≤ s (**)when j − i = r + 1 ≥ a ( C ( r, s ))] = [ a ( C ( r, s + 1))], for all 0 ≤ s ≤ k − r − j = i + r + 1 implies that c ij ( r, k − r ) = c ij ( r ) +( c (cid:48) i,r +1+ i − c i,r +1+ i ( r )) = c (cid:48) ij , the latter being equal to c ij ( r + 1) by property (2) above, andtherefore, one can take C ( r + 1) = C ( r, k − r ) and the proof will be completed by induction.So, to finish the proof, it suffices to construct a sequence of defining systems C ( r, s ). Now weshall do it acting by induction on s ≥
0. The base of induction s = 0 follows by property (1’).Now assume that we have already constructed C ( r, s ) and let us determine the defining system C ( r, s + 1).If i > s + 1, then one can take c ij ( r, s + 1) = c ij ( r, s ), see (2’)(*). Similarly, one can also take c ij ( r, s + 1) = c ij ( r, s ) if j < s + r + 2, see (2’)(**). Suppose 1 ≤ i ≤ s + 1 < s + 2 + r ≤ j ≤ k + 1.Now, by induction on j − i ≥ r + 1 we shall determine a set of cochains { b ij } such that c ij ( r, s + 1) = c ij ( r, s ) + b ij ( ∗ ∗ ∗ ) . Equality (**) implies that c s +1 ,r +2+ s ( r, s + 1) = c s +1 ,r +2+ s ( r ) + b s +1 and the latter is equalto c s +1 ,r +2+ s ( r, s ) + b s +1 by equality (*). Therefore, one can set b s +1 ,r +2+ s = b s +1 . By inductiveassumption, assume that for all r + 1 ≤ j − i < w the cochains b ij have already been defined.Then equality (***) implies that d ( b ij ) = d ( c ij ( r, s + 1)) − d ( c ij ( r, s )) = j − (cid:88) p = i +1 (¯ c i,p ( r, s ) + ¯ b i,p ) ∧ ( c p,j ( r, s ) + b p,j ) −− j − (cid:88) p = i +1 ¯ c i,p ( r, s ) ∧ c p,j ( r, s ) = s +1 (cid:88) p = i +1 ¯ c i,p ( r, s ) ∧ b p,j + j − (cid:88) p = r + s +2 ¯ b i,p ∧ c p,j ( r, s ) , where the last equality holds, since j − (cid:80) p = i +1 ¯ b i,p ∧ b p,j = 0, because b p,j = 0 when p > s + 1, and b i,p = 0 when p < r + 2 + s , and one gets the following equality for any r + 1 ≤ j − i < w : d ( b ij ) = s +1 (cid:88) p = i +1 ¯ c i,p ( r, s ) ∧ b p,j + j − (cid:88) p = r + s +2 ¯ b i,p ∧ c p,j ( r, s ) (1 . j − i = w the right hand side of the equality (1.1) is a cocycle a representingan element α = [ a ] in −(cid:104) α i , . . . , α s , [ b s +1 ] , α s + r +2 , . . . α j − (cid:105) (1 . j − i ≥ r + 2. Indeed, for j − i = r + 2 and1 ≤ i ≤ s + 1 < s + 2 + r ≤ j ≤ k + 1 we have only two possible cases: (1) i = s + 1 , j = s + r + 3 and the right hand side of (1.1) has the form ¯ b s +1 ,r + s +2 ∧ c s + r +2 ,s + r +3 = ¯ b s +1 a s + r +2 . The lattercocycle represents −(cid:104) [ b s +1 ] , α s + r +2 (cid:105) ; (2) i = s, j = s + r + 2 and the right hand side of (1.1) hasthe form ¯ c s,s +1 b s +1 ,s + r +2 = ¯ a s ∧ b s +1 . The latter cocycle represents −(cid:104) α s , [ b s +1 ] (cid:105) . The inductionstep follows from the equality (1.1), definition of a (higher) Massey product, and the inductiveassumption.Since [ b s +1 ] = 0, one concludes that the Massey product given by the formula (1.2) aboveis trivial. Furthermore, as r ≥ j − − i + 1 = j − i ≤ k . Therefore, we can apply the inductive assumption on k tothis Massey product, since [ b s +1 ] ∈ ˜ H β ( K I s +1 (cid:116) ... (cid:116) I r + s +1 ) for β = d ( s + 1 , r + s + 1) − d ( s + 1) + . . . + d ( r + s + 1). Thus, by the inductive assumption on k , we obtain that the Masseyproduct 0 ∈ (cid:104) α i , . . . , α s , [ b s +1 ] , α s + r +2 , . . . , α j − (cid:105) is strictly defined, that is, it contains only zero. It follows that equality (1.1) above has a solutionfor j − i = w . Therefore, for all 1 ≤ i < j ≤ k + 1 equality (1.1) implies d ( b ij ) = j − (cid:88) p = i +1 ¯ c i,p ( r, s + 1) ∧ c p,j ( r, s + 1) − j − (cid:88) p = i +1 ¯ c i,p ( r, s ) ∧ c p,j ( r, s ) . The above equality means that: (i) C ( r, s + 1) is also a defining system for the Massey product (cid:104) α , . . . , α k (cid:105) and (ii) [ a ( C ( r, s + 1))] = [ a ( C ( r, s ))] (when in the above formula j − i = k ). Thewhole proof of the statement (2) is now completed by induction on the order k of a Masseyproduct.Finally we prove statement (3).We proceed by induction on k again, using the fact that the right hand side in Theorem 5.13admits a multigraded refinement, that is, the differential d respects the multigraded structureon the Tor-module, given by Definition 5.8 and Theorem 5.9. Therefore, induction on j − i = r ≥ C , see condition (3) above, gives us simplicial cochains c s,r + s +1 ∈ C d ( s,r + s ) − ( K I s (cid:116) ... (cid:116) I r + s ) for all 1 ≤ s ≤ k − r, ≤ r ≤ k −
2, satisfying the relations for elementsof a defining system themselves and giving the unique element [ a ( C )] of the k -fold Masseyproduct (cid:104) α , . . . , α k (cid:105) . The latter class is an element of the group H d (1 ,k ) ( K I (cid:116) ... (cid:116) I k ) by definitionof multiplication in H ∗ ( Z K ), see Theorem 5.13. This finishes the proof of the theorem. (cid:3) Remark.
It is easy to see that Theorem 6.2 implies the triple Massey products in [17] and [36]are all strictly defined. On the other hand, in the example of a trivial triple Massey productin H ∗ ( Z P ) when P is a hexagon, see [64, Example 3.4.1] as well as in the case of Theorem 6.1(b) the condition (2) of Theorem 6.2 is not satisfied and those Massey products are not strictlydefined.First examples of non-trivial higher Massey products of any order in H ∗ ( Z K ) were constructedby Limonchenko in a short note [62]. A complete proof of nontriviality and an example ofcomputation of a non-trivial 4-fold Massey product were given by the first author in [63]. Wedescribe this construction below, following [25]. IGHER ORDER MASSEY PRODUCTS AND APPLICATIONS 31
Definition 6.3. [62, 25] Let Q be a point and Q ⊂ R be a segment [0 , I n = [0 , n , n ≥ n -dimensional cube with facets F , . . . , F n labeled in such away that F i , ≤ i ≤ n contains the origin 0, F i and F n + i are parallel for all 1 ≤ i ≤ n . Thenthe face ring of the cube I n has the form: k [ I n ] = k [ v , . . . , v n , v n +1 , . . . , v n ] /I I n , where the Stanley–Reisner ideal is I I n = ( v v n +1 , . . . , v n v n ).Consider a polynomial ring k [ v , . . . , v n , v k (cid:48) ,n + k (cid:48) + i (cid:48) | ≤ i (cid:48) ≤ n − , ≤ k (cid:48) ≤ n − i (cid:48) ]and its monomial ideal, generated by square free monomials: I = ( v k v n + k + i , v k (cid:48) ,n + k (cid:48) + i (cid:48) v n + k (cid:48) + l , v k (cid:48) ,n + k (cid:48) + i (cid:48) v p , v k (cid:48) ,n + k (cid:48) + i (cid:48) v k (cid:48)(cid:48) ,n + k (cid:48)(cid:48) + i (cid:48)(cid:48) ) , where v j corresponds to F j for all 1 ≤ j ≤ n , and0 ≤ i ≤ n − , ≤ k ≤ n − i, ≤ i (cid:48) , i (cid:48)(cid:48) ≤ n − , ≤ k (cid:48) ≤ n − i (cid:48) , ≤ k (cid:48)(cid:48) ≤ n − i (cid:48)(cid:48) , ≤ p (cid:54) = k (cid:48) ≤ k (cid:48) + i (cid:48) , ≤ l (cid:54) = i (cid:48) ≤ n − ,k (cid:48) + i (cid:48) = k (cid:48)(cid:48) or k (cid:48)(cid:48) + i (cid:48)(cid:48) = k (cid:48) . Let us define Q n ⊂ R n to be a simple polytope such that for its Stanley–Reisner ideal: I Q n = I .Note that Q n has a natural realization as a , that is, a simple n -dimensionalpolytope obtained from an n -dimensional cube as a result of performing truncations of faces ofcodimension two by generic hyperplanes in R n , see [30]. Moreover, its combinatorial type doesnot depend on the order in which the faces of the cube I n are truncated (the generators v i,j correspond to the truncated faces F i ∩ F j of I n ).Then by [64, Theorem 3.6], for any n ≥ n -foldMassey product (cid:104) α , . . . , α n (cid:105) ⊂ H ∗ ( Z Q n ), where α i is a generator of ˜ H ( F i (cid:116) F n + i ).Finally, using the above construction, Theorem 6.2, and the simplicial multiwedge operation(or, J - construction ), introduced in the framework of toric topology by Bahri, Bendersky, Cohen,and Gitler [12], the first author proved the following result. Theorem 6.4 ([64]) . For any n ≥ there exists a strictly defined non-trivial Massey productof order k in H ∗ ( Z Q n ) for all k , ≤ k ≤ n . Furthermore, there exists a family of moment-anglemanifolds F such that for any given l, r ≥ there is an l -connected manifold M ∈ F with astrictly defined non-trivial r -fold Massey product in H ∗ ( M ) . Applying the theory of nestohedra [39, 76] and the theory of the differential ring of poly-topes, introduced by Buchstaber [22], the previous result was generalized by Buchstaber andLimonchenko as follows.
Theorem 6.5 ([25]) . For any given (cid:96) ≥ and n , . . . , n r ≥ , r ≥ there exists a polyhedralproduct of the type ( D j i , S j i − ) K = Z P B ( J ) , where K := K P B and J = J ( (cid:96), n , . . . , n r ) :=( j , . . . , j m ) for a certain flag nestohedron P B with m vertices on a connected building set B ,such that • The moment-angle manifold Z P B ( J ) is (cid:96) -connected; • There exist strictly defined non-trivial Massey products of orders n , . . . , n r in H ∗ ( Z P B ( J ) ) . Another way to generalize Theorem 6.4 has been recently suggested by Grbi´c and Linton.Their approach is based on a careful investigation, on the level of cochains, of cup and Masseyproducts of the cohomology classes occured in Theorem 6.2. Note that below the non-trivialMassey products, which are stated to exist, are no longer strictly defined, in general.
Theorem 6.6 ([46]) . The following statements hold. (1)
For any given simplicial complexes K , . . . , K n there exists a sequence of stellar subdi-visions of their join K ∗ . . . ∗ K n resulting in a simplicial complex K such that thereexists a non-trivial n -fold Massey product in H ∗ ( Z K ) ; (2) If a simplicial complex K (cid:48) is obtained from a simplicial complex K by a sequence ofa special type edge truncations and there exists a non-trivial n -fold Massey product in H ∗ ( Z K (cid:48) ) , then the same property holds in H ∗ ( Z K ) .Remark. The case of n = 3 in Theorem 6.6 (1) coincides with the construction due toBaskakov [17].Using Theorem 6.6 (2) and Theorem 6.1 (1), Grbi´c and Linton obtain a result due to Zhu-ravleva [79], who proved that for any Pogorelov polytope P (see [75, 1, 23, 24]) there exists anon-trivial triple Massey product in H ∗ ( Z P ).Moreover, using Theorem 6.6 (2) and Theorem 6.4, Grbi´c and Linton showed that for any n ≥ H ∗ ( Z P e n ) of any order k , 2 ≤ k ≤ n , where P e n is an n -dimensional permutohedron.It was earlier proved by the first author [64, Lemma 4.9, Theorem 4.10] that if P is a graph-associahedron , in particular, an n -dimensional permutohedron, see [31], then the followingconditions are equivalent: 1) Z P is rationally formal; 2) there exist no non-trivial strictly definedtriple Massey products in H ∗ ( Z P ); 3) P is a product of segments, pentagons, and hexagons.Finally, it should be mentioned that there exists a simple polytope P such that there are nonon-trivial triple Massey products of 3-dimensional classes in H ∗ ( Z P ), but there exists a non-trivial strictly defined 4-fold Massey product in H ∗ ( Z P ). This was proved by Barali´c, Grbi´c,Limonchenko, and Vuˇci´c [16] for P being the dodecahedron using Theorem 6.2.7. Acknowledgements
The authors wish to thank Victor Buchstaber for many fruitful discussions, encouragement,and advice. The first author is also grateful to the Fields Institute for Research in MathematicalSciences, University of Toronto (Canada) for providing excellent research conditions and sup-port while working on this paper at the Thematic Program on Toric Topology and PolyhedralProducts.
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Faculty of Computer Science, National Research University Higher School of Economics,Moscow, Russia
E-mail address : [email protected] The Fields Institute for Research in Mathematical Sciences, Department of Mathematics,University of Toronto, Toronto, Canada
E-mail address : [email protected] Department of Mechanics and Mathematics, Moscow State University, 119992 Moscow, Rus-sia
E-mail address : mitia [email protected] [email protected]