Ivan Limonchenko, Dmitry Millionshchikov
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 representations. The second main theme is at the intersection of toric topology, homotopy theory of polyhedral products, and the homology theory of local rings, Stanley-Reisner rings of simplicial complexes.
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Massey products for homotopy inner products
本文在$A_\infty$代数上定义同伦内积的Massey积，给出普通Massey积的显式描述，并通过透镜空间、链环和低维流形展示其包含超越普通Massey积的信息。
Cyclic ABC Massey Products
本文将循环 Massey 积推广至双分次情形，提出循环 ABC Massey 积，其通常非平凡且携带不同于四重 ABC Massey 积的信息。
We define a new invariant in the homology of a differential graded algebra. This invariant is the obstruction to defining a fourfold Massey product.
We study the relationship between the higher Massey products on the cohomology $H$ of a differential graded algebra, and the $A_\infty$ structures induced on $H$ via homotopy transfer techniques.
We show that, in general, there exist non-vanishing triple Massey products in the cohomology with finite field coefficients of a complex hypersurface complement. In contrast, the Massey products, triple and higher, in the rational cohomology of such a space are all known to vanish.
We give a combinatorial classification of non-trivial triple Massey products of three dimensional classes in the cohomology of a moment-angle complex. This work improves on a result by Denham and Suciu (2007) by considering triple Massey products with non-trivial indeterminacy.
On the behavior of Massey products under field extension
本文研究了交换微分分次代数上 Massey 乘积在域扩张下的行为，证明其整体消没性不保持不变，并揭示了三重与高阶乘积在代数闭域扩张下的不同表现。
The paper is devoted to study of Massey products in symplectic manifolds. Theory of generalized and classical Massey products and a general construction of symplectic manifolds with nontrivial Massey products of arbitrary large order are exposed. The construction uses the symplectic blow-up and is based on the author results, which describe conditions under which the blow-up of a symplectic manifold X along its submanifold Y inherits nontrivial Massey products from X ot Y. This gives a general construction of nonformal symplectic manifolds.
In this work we construct nontrivial Massey products in the cohomology of moment-angle manifolds corresponding to polytopes from the Pogorelov class. This class includes the dodecahedron and all fullerenes, i. e. simple 3-polytopes with only 5-gonal and 6-gonal facets. The existence of a nontrivial Massey product implies the nonformality of the space in the sense of rational homotopy theory.
We show how and when it is possible to detect and recover higher Massey products on the cohomology $H$ of a differential graded algebra $A$ with higher multiplications on quasi-isomorphic $A_\infty$ structures on $H$.
The purpose of this paper is to compare two spectral sequences converging to the cohomology of a configuration space. The collapsing of these spectral sequences is established, in some cases, using Massey products.
In this note I use cup-products and higher Massey products to find topological lower bounds on the number of geometrically distinct critical points of any closed 1-form in a given cohomology class.
We define an obstruction to the formality of a differential graded algebra over a graded operad defined over a commutative ground ring. This obstruction lives in the derived operadic cohomology of the algebra. Moreover, it determines all operadic Massey products induced on the homology algebra, hence the name of derived universal Massey product.
We provide a uniform definition of higher order Toda brackets in a general setting, covering the known cases of long Toda brackets for topological spaces and chain complexes and Massey products for differential graded algebras, among others.
We discuss Massey products in a N-graded Lie algebra cohomology. One of the main examples is the positive part L_1 of the Witt algebra $W$. We consider an associated graded algebra m_0 of L_1 with respect to the descending central series and prove that H*(m_0) is generated with respect to non-trivial Massey products by one cohomology H^1(m_0).
We apply algebraic Morse theory to the Taylor resolution of a monomial ring $R = S/I$ to obtain an $A_{\infty}$-structure on the minimal free resolution of $R$. Using this structure we describe the vanishing of higher Massey products in case the minimal free resolution is simplicial. Under this assumption, we show that R is Golod if and only if the product on $\text{Tor}^S(R, k)$ vanishes. Lastly, we give two combinatorial characterizations of the Golod property in this case.
As part of various obstruction theories, non-trivial Massey products have been studied in symplectic and complex geometry, commutative algebra and topology for a long time. We introduce a general approach to constructing non-trivial Massey products in the cohomology of moment-angle complexes, using homotopy theoretical and combinatorial methods. Our approach sets a unifying way of constructing higher Massey products of arbitrary cohomological classes and generalises all existing examples of non-trivial Massey products in moment-angle complexes. As a result, we obtain explicit constructions of …
Using the combinatorics of the underlying simplicial complex $K$, we give various upper and lower bounds for the Lusternik-Schnirelmann (LS) category of moment-angle complexes $\zk$. We describe families of simplicial complexes and combinatorial operations which allow for a systematic description of the LS category. In particular, we characterise the LS category of moment-angle complexes $\zk$ over triangulated $d$-manifolds $K$ for $d\leq 2$, as well as higher dimension spheres built up via connected sum, join, and vertex doubling operations. %This characterisation is given in terms of the co…
Working in the context of symmetric spectra, we prove higher homotopy excision and higher Blakers-Massey theorems, and their duals, for algebras and left modules over operads in the category of modules over a commutative ring spectrum (e.g., structured ring spectra).
Poincaré dualization and Massey products
本文研究了庞加莱对偶空间中马塞产品的性质，揭示了非零度映射下四重马塞产品可能为平凡的现象，并探讨了形式性与庞加莱对偶空间的关系。
We give a correct statement and a complete proof of the criterion obtained by Grbić, Panov, Theriault and Wu for the face ring $\Bbbk[K]$ of a simplicial complex $K$ to be Golod over a field $\Bbbk$. (The original argument depended on the main result of a paper by Berglund and Jöllenbeck, which was shown to be false by Katthän.) We also construct an example of a minimally non-Golod complex $K$ such that the cohomology of the corresponding moment-angle complex $\mathcal Z_K$ has trivial cup product and a non-trivial triple Massey product.
Massey iterated products and closed geodesics
该研究揭示有限型1-连通CW复形上同调中含零的两组Massey迭代乘积序列，与其自由环路空间Betti数无界增长存在直接关联，推进了代数拓扑与闭测地线相关问题的研究。
Holomorphic linking numbers, ABC Massey products, and Calabi-Yau 3-folds
本文通过将复解析循环的 ABC 马斯奇产品与全纯链接数关联，构造出具有非消失 ABC 马斯奇产品的简单连通投影3折面族。
We study the twisted cohomology groups of $A_\infty$-algebras defined by twisting elements and their behavior under morphisms and homotopies using the bar construction. We define higher Massey products on the cohomology groups of general $A_\infty$-algebras and establish the naturality under morphisms and their dependency on defining systems. The above constructions are also considered for $C_\infty$-algebras. We construct a spectral sequence converging to the twisted cohomology groups an show that the higher differentials are given by the $A_\infty$-algebraic Massey products.
We define a generalization of Massey products for algebras over a Koszul operad in characteristic zero, extending Massey's and Allday's and Retah's in the associative and Lie cases, respectively. We establish connections with minimal models and with Dimitrova's universal operadic cohomology class. We compute a Gerstenhaber algebra example and a hypercommutative algebra example related to the Chevalley-Eilenberg complex of the Heisenberg Lie algebra.
We construct splitting varieties for triple Massey products. For a,b,c in F^* the triple Massey product < a,b,c> of the corresponding elements of H^1(F, mu_2) contains 0 if and only if there is x in F^* and y in F[\sqrt{a}, \sqrt{c}]^* such that b x^2 = N_{F[\sqrt{a}, \sqrt{c}]/F}(y), where N_{F[\sqrt{a}, \sqrt{c}]/F} denotes the norm, and F is a field of characteristic different from 2. These varieties satisfy the Hasse principle by a result of D.B. Lee and A.R. Wadsworth. This shows that triple Massey products for global fields of characteristic different from 2 always contain 0.
This paper shows that the operad encoding modular operads is Koszul. Using this result we construct higher composition operations on (hairy) graph homology which characterize its rational homotopy type.
Non-realizability of a triple Massey product for the algebra $\mathbb{F}_2[a,b,c]/(ab,bc)$
该研究证明了特定代数的三重Massey乘积无法被实现，通过计算Hochschild上同调群揭示了其本质形式特性。
The problem of existence of nontrivial Massey products in cohomology of a space is well-known in algebraic topology and homological algebra. A number of problems in complex geometry, symplectic geometry, and algebraic topology can be stated in terms of Massey products. One of such problems is to establish formality of smooth manifolds in rational homotopy theory. There have already been constructed a few classes of spaces with nontrivial triple Massey products in cohomology. Until now, very few examples of manifolds $M$ with nontrivial higher Massey products in $H^*(M)$ were known. In this wor…
We extend Massey products from cohomology to differential cohomology via stacks, organizing and generalizing existing constructions in Deligne cohomology. We study the properties and show how they are related to more classical Massey products in de Rham, singular, and Deligne cohomology. The setting and the algebraic machinery via stacks allow for computations and make the construction well-suited for applications. We illustrate with several examples from differential geometry and mathematical physics.
We employ Massey products to find sharper lower bounds for the Schwarz genus of a fibration than those previously known. In particular we give examples of non-formal spaces $X$ for which the topological complexity $\TC(X)$ (defined to be the genus of the free path fibration on $X$) is greater than the zero-divisors cup-length plus one.
Strong minimal model theorem and Massey products
本研究证明了比Kadeishvili定理更强的、同痕意义下唯一的极小模型存在性，揭示了该结果与三重及高阶Massey积的对应关系，阐明了二者间的关联。
Let $G$ be a finite 2-group of maximal nilpotency class, and let $BG$ be its classifying space. We prove that iterated Massey products in $H^*(BG;\F_2)$ do characterize the homotopy type of $BG$ among 2-complete spaces with the same cohomological structure. As a consequence we get an alternative proof of the modular isomorphism problem for 2-groups of maximal nilpotency class.
Bigraded notions of formality and Aeppli-Bott-Chern-Massey products
本文定义复流形微分形式代数的双分次形式性，并研究其与高阶 Aeppli–Bott–Chern–Massey 乘积的关系，证明低次乘积可非平凡拉回到沿复子流形的爆破。
Associated to every connected, topological space $X$ there is a Hopf algebra - the Pontrjagin ring of the based loop space of the configuration space of two points in X. We prove that this Hopf algebra is not a homotopy invariant of the space. We also exhibit interesting examples of H-spaces, which are homotopy equivalent as spaces, which either lead to isomorphic rational Hopf algebras or not, depending crucially on the existence of Whitehead products. Moreover, we investigate a (naturally motivated) twisted version of these Pontrjagin rings in the various aforementioned contexts. In all of t…
In this paper we introduce a direct family of simple polytopes $P^{0}\subset P^{1}\subset\ldots$ such that for any $k$, $2\leq k\leq n$ there are non-trivial strictly defined Massey products of order $k$ in the cohomology rings of their moment-angle manifolds $\mathcal Z_{P^n}$. We prove that the direct sequence of manifolds $\ast\subset S^{3}\hookrightarrow\ldots\hookrightarrow\mathcal Z_{P^n}\hookrightarrow\mathcal Z_{P^{n+1}}\hookrightarrow\ldots$ has the following properties: every manifold $\mathcal Z_{P^n}$ is a retract of $\mathcal Z_{P^{n+1}}$, and one has inverse sequences in cohomolo…
Massey products, sphere bundles and T-duality
研究迭代球丛的球面T-对偶，通过Massey积重打包Gysin序列的上同调数据，并证明在特定条件下存在反向Massey积对应的T-对偶迭代球丛。
Spectral sequences, Massey products and homology of covering spaces
本研究重新探讨Papadima-Suciu的等变谱序列，证明其微分由高阶Massey乘积计算，扩展Pajitnov结果，给出覆盖空间同调相关上界并改进现有结果，还将结论应用于超平面排列补空间。
Nontrivial Massey products on compact Kähler manifolds
该研究揭示紧Kähler流形的双分次双微分代数拟同构型信息更丰富，证明低维及高维流形上存在非平凡ABC-Massey乘积，指出其在多复变同伦论意义下通常非形式。
We show that the existence of a nontrivial Massey product in the cohomology ring H^*(X) imposes global constraints upon the Riemannian geometry of a manifold X. Namely, we exhibit a suitable systolic inequality, associated to such a product. This generalizes an inequality proved in collaboration with Y. Rudyak, in the case when X has unit Betti numbers, and realizes the next step in M. Gromov's program for obtaining geometric inequalities associated with nontrivial Massey products. The inequality is a volume lower bound, and depends on the metric via a suitable isoperimetric quotient. The proo…
Motivated by the construction of Steenrod cup-$i$ products in the singular cochain algebra of a space and in the algebra of non-commutative differential forms, we define a category of binomial cup-one differential graded algebras over the integers and over prime fields of positive characteristic. The Steenrod and Hirsch identities bind the cup-product, the cup-one product, and the differential in a package that we further enhance with a binomial ring structure arising from a ring of integer-valued rational polynomials. This structure allows us to define the free binomial cup-one differential g…
Associated to every finite simplicial complex K there is a "moment-angle" finite CW-complex, Z_K; if K is a triangulation of a sphere, Z_K is a smooth, compact manifold. Building on work of Buchstaber, Panov, and Baskakov, we study the cohomology ring, the homotopy groups, and the triple Massey products of a moment-angle complex, relating these topological invariants to the algebraic combinatorics of the underlying simplicial complex. Applications to the study of non-formal manifolds and subspace arrangements are given.
Higher order Massey products for algebras over algebraic operads
本文为代数operad上的代数引入高阶Massey积，推广了Muro的次级积，研究其性质、形式性联系，并证明其表示operadic Eilenberg--Moore谱序列的微分，同时探讨与P∞结构的关系，涵盖Koszul及非Koszul情形。
Universal first-order Massey product of a prefactorization algebra
本文研究预因子化代数的泛一阶Massey积，在局部常值情形下显式计算，应用于因子化包络和线性Chern-Simons理论紧化。
Let pi be a pro-l completion of a free group, and let G be a profinite group acting continuously on pi. First suppose the action is given by a character. Then the boundary maps delta_n: H^1(G, pi/[pi]_n) -> H^2(G, [pi]_n/[pi]_{n+1}) are Massey products. When the action is more general, we partially compute these boundary maps. Via obstructions of Jordan Ellenberg, this implies that pi_1 sections of P^1_k-{0,1,infty} satisfy the condition that associated nth order Massey products in Galois cohomology vanish. For the pi_1 sections coming from rational points, these conditions imply that < (1-x)^…
For $x$ an element of a field other than $0$ or $1$, we compute the order $n$ Massey products $$\langle (1-x)^{-1}, x^{-1}, \ldots, x^{-1}, (1-x)^{-1} \rangle$$ of $n-2$ factors of $x^{-1}$ and two factors of $(1-x)^{-1}$ by embedding $\mathbb{P}^1 - \{0,1,\infty\}$ into its Picard variety and constructing $\operatorname{Gal}(k^s/k)$ equivariant maps from $π_1$ applied to this embedding to unipotent matrix groups. This method produces obstructions to $π_1$-sections of $\mathbb{P}^1 - \{0,1,\infty\}$, partial computations of obstructions of Jordan Ellenberg, and also computes the Massey product…
Closed (and simply-connected) manifolds whose dimensions are larger than 4 are central geometric objects in classical algebraic topology and differential topology. They have been classified via algebraic and abstract objects. On the other hand, It is difficult to understand them in geometric and constructive ways. In the present paper, we show such studies via explicit fold maps, higher dimensional versions of Morse functions. The author captured information of the topologies and the differentiable structures of closed (and simply-connected) manifolds which are not so complicated with respect …
We show that for any face $F$ of a simple polytope $P$ the canonical equivariant homeomorphisms $h_P:\,\mathcal Z_P\to\mathcal Z_{K_P}$ and $h_F:\,\mathcal Z_F\to\mathcal Z_{K_F}$ are linked in a pentagonal commutative diagram with the maps of moment-angle manifolds and moment-angle-complexes, induced by a face embedding $i_{F,P}:\,F\to P$ and a simplicial embedding $Φ_{F,P}:\,K_F\to K_{F,P}\to K_P$, where $K_{F,P}$ is the full subcomplex of $K_P$ on the same vertex set as $Φ_{F,P}(K_F)$. We introduce the explicit constructions of the maps $i_{F,P}$, $Φ_{F,P}$ and show that a polytope $P$ is f…
Let us say that a map of arcwise connected topological spaces (having the homotopy type of CW-complexes) is a pseudo-homeomorphism if it induces an isomorphism of the first integer homology groups and an epimorphism of the second integer homology groups. We prove that any invariant of a topological space w.r.t. pseudo-homeomorphisms is an invariant of the fundamental group of this space. We also describe a necessary condition for the fundamental groups to be distinguished by such invariants. As an example we show that the invariant used in math.AG/9805056 to distinguish the fundamental groups …
Let L_1 denotes the Lie algebra of formal vector fields on the line which vanish at the origin together with their first derivatives. Buchstaber and Shokurov have shown that the universal enveloping algebra U(L_1) is isomorphic to the tensor product of the Landweber-Novikov algebra S in complex cobordism theory by reals. The cohomology H*(L_1) has trivial multiplication. Buchstaber conjectured that H*(L_1) is generated with respect to non-trivial Massey products by H^1(L_1). Feigin, Fuchs and Retakh found representation of H*(L_1) by trivial Massey products. In the present article we prove tha…
Let $ϕ: M\to M$ be a diffeomorphism of a $C^\infty$ compact connected manifold, and $X$ its mapping torus. There is a natural fibration $p:X\to S^1$, denote by $ξ\in H^1(X, \mathbb{Z})$ the corresponding cohomology class. Let $λ\in \mathbb{Z}^*$. Consider the endomorphism $ϕ_k^*$ induced by $ϕ$ in the cohomology of $M$ of degree $k$, and denote by $J_k(λ)$ the maximal size of its Jordan block of eigenvalue $λ$. Define a representation $ρ_λ: π_1(X)\to\mathbb{C}^*$ by $ρ_λ(g) = λ^{p_*(g)}.$ Let $H^*(X,ρ_λ)$ be the corresponding twisted cohomology of $X$. We prove that $J_k(λ)$ is equal to the ma…
Let $ϕ: M\to M$ be a diffeomorphism of a $C^\infty$ compact connected manifold, and $X$ its mapping torus. There is a natural fibration $p:X\to S^1$, denote by $ξ\in H^1(X, \mathbb{Z})$ the corresponding cohomology class. Let $ρ:π_1(X)\to GL(n,\mathbb{C})$ be a representation, denote by $H^*(X,ρ)$ the corresponding twisted cohomology of $X$. Denote by $ρ_0$ the restriction of $ρ$ to $π_1(M)$, and by $ρ^*_0$ the antirepresentation conjugate to $ρ_0$. We construct from these data an automorphism of the group $H_*(M,ρ^*_0)$, that we call the twisted monodromy homomorphism $ϕ_*$. The aim of the pr…
We construct a family of manifolds, one for each $n\geq 2$, having a nontrivial Massey $n$-product in their cohomology. These manifolds turn out to be smooth closed 2-connected manifolds with a compact torus action called moment-angle manifolds $\mathcal Z_P$, whose orbit spaces are simple $n$-dimensional polytopes $P$ obtained from a $n$-cube by a sequence of truncations of faces of codimension 2 only (2-truncated cubes). Moreover, the polytopes $P$ are flag nestohedra but not graph-associahedra. We compute some bigraded Betti numbers $β^{-i,2(i+1)}(Q)$ for an associahedron $Q$ in terms of it…
Let X be a finite CW-complex, denote its fundamental group by G. Let R be an n-dimensional complex repesentation of G. Any element A of the first cohomology group of X with complex coefficients gives rise to the exponential deformation of the representation R, which can be considered as a curve in the space of representations. We show that the cohomology of X with local coefficients corresponding to the generic point of this curve is computable from a spectral sequence starting from the cohomology of X with R-twisted coefficients. We compute the differentials of the spectral sequence in terms …
We construct closed $(k-1)$-connected manifolds of dimensions $\ge 4k-1$ that possess non-trivial rational Massey triple products. We also construct examples of manifolds $M$ such that all the cup-products of elements of $H^k(M)$ vanish, while the group $H^{3k-1}(M;\Q)$ is generated by Massey products: such examples are useful for theory of systols.
We prove that certain conditions on multigraded Betti numbers of a simplicial complex $K$ imply existence of a higher Massey product in cohomology of a moment-angle-complex $\mathcal Z_K$, which contains a unique element (a strictly defined product). Using the simplicial multiwedge construction, we find a family $\mathcal{F}$ of polyhedral products being smooth closed manifolds such that for any $l,r\geq 2$ there exists an $l$-connected manifold $M\in\mathcal F$ with a nontrivial strictly defined $r$-fold Massey product in $H^{*}(M)$. As an application to homological algebra, we determine a wi…
We investigate some topological properties, in particular formality, of compact Sasakian manifolds. Answering some questions raised by Boyer and Galicki, we prove that all higher (than three) Massey products on any compact Sasakian manifold vanish. Hence, higher Massey products do obstruct Sasakian structures. Using this we produce a method of constructing simply connected K-contact non-Sasakian manifolds. On the other hand, for every $n \geq 3$, we exhibit the first examples of simply connected compact Sasakian manifolds of dimension $2n + 1$ which are non-formal. They are non-formal because …
Highly connected non-formal Milnor fibers via polyhedral products
通过结合Fernández de Bobadilla的实现定理和Grbić-Linton的系统Massey积构造，产生高度连接且非形式的Milnor纤维。
We give a construction of the universal enveloping $A_\infty$ algebra of a given $L_\infty$ algebra, alternative to the already existing versions. As applications, we derive a higher homotopy algebras version of the classical Milnor-Moore theorem, proposing a new $A_\infty$ model for simply connected rational homotopy types, and uncovering a relationship between the higher order rational Whitehead products in homotopy groups and the Pontryagin-Massey products in the rational loop space homology algebra.
We provide two criteria for discarding the formality of a differential graded Lie algebra in terms of higher Whitehead brackets, which are the Lie analogue of the Massey products of a differential graded associative algebra. We also show that formality of a differential graded Lie algebra is not equivalent to the collapse of the Quillen spectral sequence. Finally, we use $L_\infty$ algebras and Quillen's formulation of rational homotopy theory to recover and improve a classical theorem for detecting higher Whitehead products in Sullivan minimal models and give some applications.
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