Triple Massey Products Research Articles

On the behavior of Massey products under field extension

We show that global vanishing of Massey products on a commutative differential graded algebra is not invariant under field extension. Non-vanishing triple Massey products remain non-vanishing upon field extension, while higher Massey products can generally vanish. If the field being extended is algebraically closed, all non-vanishing Massey products remain non-vanishing on a finite type commutative differential graded algebra.

Massey products and elliptic curves

AbstractWe study the vanishing of Massey products of order at least 3 for absolutely irreducible smooth projective curves over a field with coefficients in . We mainly focus on elliptic curves, for which we obtain a complete characterization of when triple Massey products do not vanish.

Минимально неголодовы кольца граней и произведения Масси

We give a correct statement and a complete proof of the criterion obtained by Grbi\'c, 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 J\"ollenbeck, which was shown to be false by Katth\"an.) 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.

Differential graded algebras, Steenrod cup-one products, binomial operations, and Massey products

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 graded algebra generated by a set and derive its basic properties. It also provides the context for defining restricted triple Massey products, which have a smaller indeterminacy than the classical ones, and hence, give stronger homotopy type invariants.

Lowest-degree triple Massey products in moment-angle complexes

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.

Lowest-degree triple Massey products in moment-angle complexes

Lowest-degree triple Massey products in moment-angle complexes

Higher Triple Massey products and symbols

Fix an arbitrary prime p. Let F be a field containing a primitive p-th root of unity, with absolute Galois group GF, and let Hn denote its mod p cohomology group Hn(GF,Z/pZ). The triple Massey product of weight (n,k,m)∈N3 is a partially defined, multi-valued function 〈⋅,⋅,⋅〉:Hn×Hk×Hm→Hn+k+m−1. In this work we prove that for an arbitrary prime p, any defined 3MP of weight (n,1,m), where the first and third entries are symbols, contains zero; and that any defined 3MP of weight (1,k,1), where the middle entry is a symbol, contains zero.

Associative Yang–Baxter equation and Fukaya categories of square-tiled surfaces

We show that all strongly non-degenerate trigonometric solutions of the associative Yang–Baxter equation (AYBE) can be obtained from triple Massey products in the Fukaya categories of square-tiled surfaces. Along the way, we give a classification result for cyclic A∞-algebra structures on a certain Frobenius algebra associated with a pair of 1-spherical objects in terms of the equivalence classes of the corresponding solutions of the AYBE. As an application, combining our results with homological mirror symmetry for punctured tori (cf. [17]), we prove that any two simple vector bundles on a cycle of projective lines are related by a sequence of 1-spherical twists and their inverses.

On mod 3 triple Milnor invariants and triple cubic residue symbols in the Eisenstein number field

We introduce mod 3 triple Milnor invariants and triple cubic residue symbols for certain primes of the Eisenstein number field $$\mathbb {Q}(\sqrt{-3})$$ , following the analogies between knots and primes. Our triple symbol generalizes both the cubic residue symbol and Redei’s triple symbol, and describes the decomposition law of a prime in a mod 3 Heisenberg extension of degree 27 over $$\mathbb {Q}(\sqrt{-3})$$ with restricted ramification, which we construct concretely in the form similar to Redei’s dihedral extension over $$\mathbb {Q}$$ . We also give a cohomological interpretation of our symbols by triple Massey products in Galois cohomology.

Triple Massey products of weight (1,n,1) in Galois cohomology

Fix an arbitrary prime p. Let F be a field containing a primitive p-th root of unity, with absolute Galois group GF, and let Hn denote its mod p cohomology group Hn(GF,Z/pZ). The triple Massey product (abbreviated 3MP) of weight (n,k,m)∈N3 is a partially defined, multi-valued function 〈⋅,⋅,⋅〉:Hn×Hk×Hm→Hn+k+m−1. In this work we prove that for an odd prime p, any defined 3MP of weight (1,k,1) contains zero.

Triple Massey products and absolute Galois groups

Let p be a prime number, F a field containing a root of unity of order p , and G_F the absolute Galois group. Extending results of Hopkins, Wickelgren, Mináč and Tân, we prove that the triple Massey product H^1(G_F)^3\to H^2(G_F) contains 0 whenever it is non-empty. This gives a new restriction on the possible profinite group structure of G_F .

Topology of moment-angle manifolds arising from flag nestohedra

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 $\beta^{-i,2(i+1)}(Q)$ for an associahedron $Q$ in terms of its graph structure and relate it to the structure of the loop homology (Pontryagin algebra) $H_{*}(\Omega\mathcal Z_Q)$. We also study triple Massey products in $H^{*}(\mathcal Z_Q)$ for a graph-associahedron $Q$.

Triple Massey products and Galois theory

We show that any triple Massey product with respect to prime 2 contains 0 whenever it is defined over any field. This extends the theorem of M. J. Hopkins and K. G. Wickelgren, from global fields to any fields. This is the first time when the vanishing of any n -Massey product for some prime p has been established for all fields. This leads to a strong restriction on the shape of relations in the maximal pro-2-quotients of absolute Galois groups, which was out of reach until now. We also develop an extension of Serre's transgression method to detect triple commutators in relations of pro- p -groups, where we do not require that all cup products vanish. We prove that all n -Massey products, n \geq 3 , vanish for general Demushkin groups. We formulate and provide evidence for two conjectures related to the structure of absolute Galois groups of fields. In each case when these conjectures can be verified, they have some interesting concrete Galois theoretic consequences. They are also related to the Bloch–Kato conjecture.

Triple Massey products vanish over all fields

We show that the absolute Galois group of any field has the vanishing triple Massey product property. Several corollaries for the structure of maximal pro-$p$-quotient of absolute Galois groups are deduced. Furthermore, the vanishing of some higher Massey products is proved.

On formality of Sasakian manifolds

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 they have a non-zero triple Massey product. We also prove that arithmetic lattices in some simple Lie groups cannot be the fundamental group of a compact Sasakian manifold.

Vanishing of Massey Products and Brauer Groups

AbstractLet p be a prime number and F a field containing a root of unity of order p. We relate recent results on vanishing of triple Massey products in the mod-p Galois cohomology of F, due to Hopkins, Wickelgren, Mináč, and Tân, to classical results in the theory of central simple algebras. We prove a stronger form of the vanishing property for global fields.

Graph 4-braid groups and Massey products

We first show that the braid group over a graph topologically containing no Θ-shape subgraph has a presentation related only by commutators. Then using discrete Morse theory and triple Massey products, we prove that a graph topologically contains none of four prescribed graphs if and only if its 4-braid groups is a right-angled Artin group.

Simply connected K-contact and Sasakian manifolds of dimension 7

We construct a compact simply-connected 7-dimensional manifold admitting a K-contact structure but not a Sasakian structure. We also study rational homotopy properties of such manifolds, proving in particular that a simply-connected 7-dimensional Sasakian manifold has vanishing cup-product on the second cohomology and that it is formal if and only if all its triple Massey products vanish.

Triple Massey products over global fields

Let K be a global field which contains a primitive p -th root of unity, where p is a prime number. M. J. Hopkins and K. G. Wickelgren showed that for p=2 , any triple Massey product over K with respect to \mathbb F_p , contains 0 whenever it is defined. We show that this is true for all primes p .

Splitting varieties for triple Massey products

We construct splitting varieties for triple Massey products. For a,b,c∈F⁎ the triple Massey product 〈a,b,c〉 of the corresponding elements of H1(F,μ2) contains 0 if and only if there are x∈F⁎ and y∈F[a,c]⁎ such that bx2=NF[a,c]/F(y), where NF[a,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. Leep and A.R. Wadsworth. This shows that triple Massey products for global fields of characteristic different from 2 always contain 0.