Massey Products Research Articles

Two families of pro-𝑝 groups that are not absolute Galois groups

AbstractLet 𝑝 be a prime. We produce two new families of pro-𝑝 groups which are not realizable as absolute Galois groups of fields. To prove this, we use the 1-smoothness property of absolute Galois pro-𝑝 groups. Moreover, we show in these families, one has several pro-𝑝 groups which may not be ruled out as absolute Galois groups employing the quadraticity of Galois cohomology (a consequence of the norm residue theorem), or the vanishing of Massey products in Galois cohomology.

Non-trivial higher Massey products in moment-angle complexes

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 infinitely many non-formal manifolds that appear in topology, complex geometry and algebraic geometry.

Non-formality in PIN(2)-monopole Floer homology

In previous work, we introduced a natural \mathcal{A}_{\infty} -structure on the Pin(2)-monopole Floer chain complex of a closed, oriented three-manifold Y , and showed that it is non-formal in the simplest case in which Y is the three-sphere S^3 . In this paper, we explore further this non-formality phenomenon. Specifically, we provide explicit descriptions of several Massey products induced on homology, and discuss applications to the computation of the Pin(2)-monopole Floer homology of connected sums.

Some Secondary Differentials on the Fourth Line of the Adams Spectral Sequence

In this paper, we determine some nontrivial secondary Adams differentials on the fourth line $${\rm{Ext}}_{\cal A}^{4,*}(/p,/p)$$ of the classical Adams spectral sequence. Specially, among these differentials, two of them are obtained via the matrix Massey products.

Massey Products for Graph Homology

Abstract 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.

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.

Iterated Extensions and Uniserial Length Categories

In this paper, we study length categories using iterated extensions. We fix a field k, and for any family S of orthogonal k-rational points in an Abelian k-category mathcal {A}, we consider the category Ext(S) of iterated extensions of S in mathcal {A}, equipped with the natural forgetful functor mathbf {Ext}(mathsf {S}) to mathbf {mathcal {A}}(mathsf {S}) into the length category mathbf {mathcal {A}}(mathsf {S}). There is a necessary and sufficient condition for a length category to be uniserial, due to Gabriel, expressed in terms of the Gabriel quiver (or Ext-quiver) of the length category. Using Gabriel’s criterion, we give a complete classification of the indecomposable objects in mathbf {mathcal {A}}(mathsf {S}) when it is a uniserial length category. In particular, we prove that there is an obstruction for a path in the Gabriel quiver to give rise to an indecomposable object. The obstruction vanishes in the hereditary case, and can in general be expressed using matric Massey products. We discuss the close connection between this obstruction, and the noncommutative deformations of the family S in mathcal {A}. As an application, we classify all graded holonomic D-modules on a monomial curve over the complex numbers, obtaining the most explicit results over the affine line, when D is the first Weyl algebra. We also give a non-hereditary example, where we compute the obstructions and show that they do not vanish.

The rational homotopy type of (n−1)‐connected manifolds of dimension up to 5n−3

We define the Bianchi-Massey tensor of a topological space X to be a linear map from a subquotient of the fourth tensor power of H*(X). We then prove that if M is a closed (n-1)-connected manifold of dimension at most 5n-3 (and n > 1) then its rational homotopy type is determined by its cohomology algebra and Bianchi-Massey tensor, and that M is formal if and only if the Bianchi-Massey tensor vanishes. We use the Bianchi-Massey tensor to show that there are many (n-1)-connected (4n-1)-manifolds that are not formal but have no non-zero Massey products, and to present a classification of simply-connected 7-manifolds up to finite ambiguity.

The rank of Mazur’s Eisenstein ideal

We use pseudodeformation theory to study Mazur’s Eisenstein ideal. Given prime numbers N and p>3, we study the Eisenstein part of the p-adic Hecke algebra for Γ0(N). We compute the rank of this Hecke algebra (and, more generally, its Newton polygon) in terms of Massey products in Galois cohomology, thereby answering a question of Mazur and generalizing a result of Calegari and Emerton. We also give new proofs of Merel’s result on this rank and of Mazur’s results on the structure of the Hecke algebra.

Relations in the maximal pro-𝑝 quotients of absolute Galois groups

We observe that some fundamental constructions in Galois theory can be used to obtain interesting restrictions on the structure of Galois groups of maximal p p -extensions of fields containing a primitive p p -th root of unity. This is an extension of some significant ideas of Demushkin, Labute, and Serre from local fields to all fields containing a primitive p p -th root of unity. Our techniques use certain natural simple Galois extensions together with some considerations in Galois cohomology and Massey products.

Lowest-degree triple Massey products in moment-angle complexes

Lowest-degree triple Massey products in moment-angle complexes

Toric objects associated with the dodecahedron

In this paper we illustrate a tight interplay between homotopy theory and combinatorics within toric topology by explicitly calculating homotopy and combinatorial invariants of toric objects associated with the dodecahedron. In particular, we calculate the cohomology ring of the (complex and real) moment-angle manifolds over the dodecahedron, and of a certain quasitoric manifold and of a related small cover. We finish by studying Massey products in the cohomology ring of moment-angle manifolds over the dodecahedron and how the existence of nontrivial Massey products influences the behaviour of the Poincar? series of the corresponding Pontryagin algebra.

Cocycles of nilpotent quotients of free groups

We focus on the cohomology of the $k$-th nilpotent quotient of a free group. We describe all the group 2-, 3-cocycles in terms of the Massey product and give expressions for some of the 3-cocycles. We also give simple proofs of some of the results on the Milnor invariant and Johnson–Morita homomorphisms.

$$A_\infty $$ Structures and Massey Products

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.

Massey products, toric topology and combinatorics of polytopes

In this paper we introduce a direct family of simple polytopes such that for any there are non-trivial strictly defined Massey products of order in the cohomology rings of their moment-angle manifolds . We prove that the direct sequence of manifolds has the following properties: every manifold is a retract of , and one has inverse sequences in cohomology (over and , where as ) of the Massey products constructed. As an application we get that there are non-trivial differentials , for arbitrarily large as , in the Eilenberg–Moore spectral sequence connecting the rings and with coefficients in a field, where .

Formality criteria in terms of higher Whitehead brackets

We provide two criteria for establishing the non-formality of a differential graded Lie algebra in terms of higher Whitehead brackets, which are the Lie analogues 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 its associated Quillen spectral sequence. Finally, we use L∞ 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.

Extensions of unipotent groups, Massey products and Galois theory

We study the vanishing of four-fold Massey products in mod p Galois cohomology. First, we describe a sufficient condition, which is simply expressed by the vanishing of some cup-products, and is directly analogous to the work of Guillot, Mináč and Topaz for p=2. For local fields with enough roots of unity, we prove that this sufficient condition is also necessary, and we ask whether this is a general fact.We provide a simple splitting variety, that is, a variety which has a rational point if and only if our sufficient condition is satisfied. It has rational points over local fields, and so, if it satisfies a local-global principle, then the Massey Vanishing Conjecture holds for number fields with enough roots of unity.At the heart of the paper is the construction of a finite group U˜5(Fp), which has U5(Fp) as a quotient. Here Un(Fp) is the group of unipotent n×n-matrices with entries in the field Fp with p elements; it is classical that Un+1(Fp) is intimately related to n-fold Massey products. Although U˜5(Fp) is much larger than U5(Fp), its definition is very natural, and for our purposes, it is easier to study.

Weight decompositions of Thom spaces of vector bundles in rational homotopy

Motivated by the theory of representability classes by submanifolds, we study the rational homotopy theory of Thom spaces of vector bundles. We first give a Thom isomorphism at the level of rational homotopy, extending work of Félix-Oprea-Tanré by removing hypothesis of nilpotency of the base and orientability of the bundle. Then, we use the theory of weight decompositions in rational homotopy to give a criterion of representability of classes by submanifolds, generalising results of Papadima. Along the way, we study issues of formality and give formulas for Massey products of Thom spaces. Lastly, we link the theory of weight decompositions with mixed Hodge theory and apply our results to motivic Thom spaces.

On Higher Massey Products and Rational Formality for Moment—Angle Manifolds over Multiwedges

We prove that certain conditions on multigraded Betti numbers of a simplicial complex K imply the existence of a higher Massey product in the cohomology of a moment-angle complex $${{\cal Z}_K}$$ , and this product contains a unique element (a strictly defined product). Using the simplicial multiwedge construction, we find a family $${\cal F}$$ of polyhedral products being smooth closed manifolds such that for any l, r ≥ 2 there exists an l-connected manifold $$M \in {\cal F}$$ with a nontrivial strictly defined r-fold Massey product in H*(M). As an application to homological algebra, we determine a wide class of triangulated spheres K such that a nontrivial higher Massey product of any order may exist in the Koszul homology of their Stanley–Reisner rings. As an application to rational homotopy theory, we establish a combinatorial criterion for a simple graph Γ to provide a (rationally) formal generalized moment-angle manifold $$\mathcal{Z}_{P}^{J}=\left(\underline{D}^{2 j_{i}}, \underline{S}^{2 j_{i}-1}\right)^{\partial P^{*}}$$ J = (j1,…,jm), over a graph-associahedron P = PΓ, and compute all the diffeomorphism types of formal moment-angle manifolds over graph-associahedra.

Massey Products and Fujita decompositions on fibrations of curves

Let $$f:S\rightarrow B$$ be a fibration of curves and let $$f_*\omega _{S/B}={{\mathcal {U}}}\oplus {{\mathcal {A}}}$$ be the second Fujita decomposition of f. In this paper we study a kind of Massey products, which are defined as infinitesimal invariants by the cohomology of a curve, in relation to the monodromy of certain subbundles of $${{\mathcal {U}}}$$. The main result states that their vanishing on a general fibre of f implies that the monodromy group acts faithfully on a finite set of morphisms and is therefore finite. In the last part we apply our result in terms of the normal function induced by the Ceresa cycle. On the one hand, we prove that the monodromy group of the whole $${{\mathcal {U}}}$$ of hyperelliptic fibrations is finite (giving another proof of a result due to Luo and Zuo). On the other hand, we show that the normal function is non torsion if the monodromy is infinite (this happens e.g. in the examples shown by Catanese and Dettweiler).