Massey Triple Product Research Articles

Generalized Bockstein maps and Massey products

Abstract Given a profinite group G of finite p-cohomological dimension and a pro-p quotient H of G by a closed normal subgroup N, we study the filtration on the Iwasawa cohomology of N by powers of the augmentation ideal in the group algebra of H. We show that the graded pieces are related to the cohomology of G via analogues of Bockstein maps for the powers of the augmentation ideal. For certain groups H, we relate the values of these generalized Bockstein maps to Massey products relative to a restricted class of defining systems depending on H. We apply our study to prove lower bounds on the p-ranks of class groups of certain nonabelian extensions of $\mathbb {Q}$ and to give a new proof of the vanishing of Massey triple products in Galois cohomology.

An algorithm for a lifted Massey triple product of a smooth projective plane curve

We provide an explicit algorithm to compute a lifted Massey triple product relative to a defining system for a smooth projective plane curve [Formula: see text] defined by a homogeneous polynomial [Formula: see text] over a field. The main idea is to use the description (due to Carlson and Griffiths) of the cup product for [Formula: see text] in terms of the multiplications inside the Jacobian ring of [Formula: see text] and the Cech–deRham complex of [Formula: see text]. Our algorithm gives a criterion whether a lifted Massey triple product vanishes or not in [Formula: see text] under a particular nontrivial defining system of the Massey triple product and thus can be viewed as a generalization of the vanishing criterion of the cup product in [Formula: see text] of Carlson and Griffiths. Based on our algorithm, we provide explicit numerical examples by running the computer program.

The $${\mathcal {L}}_B$$ L B -cohomology on compact torsion-free $$\mathrm {G}_2$$ G 2 manifolds and an application to ‘almost’ formality

We study a cohomology theory $H^{\bullet}_{\varphi}$, called the $\mathcal L_B$-cohomology, on compact torsion-free $\mathrm{G}_2$-manifolds. We show that $H^k_{\varphi} \cong H^k_{\mathrm{dR}}$ for $k \neq 3, 4$, but that $H^k_{\varphi}$ is infinite-dimensional for $k = 3,4$. Nevertheless there is a canonical injection $H^k_{\mathrm{dR}} \to H^k_{\varphi}$. The $\mathcal L_B$-cohomology also satisfies a Poincar\'e duality induced by the Hodge star. The establishment of these results requires a delicate analysis of the interplay between the exterior derivative $\mathrm{d}$ and the derivation $\mathcal L_B$, and uses both Hodge theory and the special properties of $\mathrm{G}_2$-structures in an essential way. As an application of our results, we prove that compact torsion-free $\mathrm{G}_2$-manifolds are 'almost formal' in the sense that most of the Massey triple products necessarily must vanish.

On geometric Bott–Chern formality and deformations

A notion of geometric formality in the context of Bott–Chern and Aeppli cohomologies on a complex manifold is discussed. In particular, by using Aeppli–Bott–Chern–Massey triple products, it is proved that geometric Aeppli–Bott–Chern formality is not stable under small deformations of the complex structure.

Dolbeault–Massey triple products of low degree

Let A = ( A • , • , ∂ ¯ A ) be a differential bigraded algebra. We characterize non-vanishing Dolbeault–Massey triple products of low degree (see Theorem 3.1 , Theorem 3.2 ). We give some applications for the Dolbeault complex on a compact complex manifold.

Controlling indeterminacy in Massey triple products

A technique is discussed to control the indeterminacy in cohomology Massey triple products. It involves finding cohomology classes with certain properties. Poincare duality spaces always have such classes if the coefficients are in a field.

An adelic resolution for homology sheaves

We propose a generalization of the ordinary idele group by constructing certain adelic complexes for sheaves of -groups on schemes. Such complexes are defined for any abelian sheaf on a scheme. We focus on the case when the sheaf is associated with the presheaf of a homology theory with certain natural axioms satisfied, in particular, by -theory. In this case it is proved that the adelic complex provides a flabby resolution for this sheaf on smooth varieties over an infinite perfect field and that the natural morphism to the Gersten complex is a quasi-isomorphism. The main advantage of the new adelic resolution is that it is contravariant and multiplicative. In particular, this enables us to reprove that the intersection in Chow groups coincides (up to a sign) with the natural product in the corresponding -cohomology groups. Also, we show that the Weil pairing can be expressed as a Massey triple product in -cohomology groups with certain indices.

Examples of non-formal closed (𝑘-1)-connected manifolds of dimensions ≥4𝑘-1

We construct closed ( k − 1 ) (k-1) -connected manifolds of dimensions ≥ 4 k − 1 \ge 4k-1 that possess non-trivial rational Massey triple products. We also construct examples of manifolds M M such that all the cup-products of elements of H k ( M ) H^k(M) vanish, while the group H 3 k − 1 ( M ; Q ) H^{3k-1}(M;\mathbb {Q}) is generated by Massey products: such examples are useful for the theory of systols.

Towards a third-order topological invariant for magnetic fields

An expression for a third-order link integral of three magnetic fields is presented. It is a topological invariant and therefore an invariant of ideal magnetohydrodynamics. The integral generalizes existing expressions for third-order invariants which are obtained from the Massey triple product, where the three fields are restricted to isolated flux tubes. The derivation and interpretation of the invariant shows a close relationship with the well-known magnetic helicity, which is a second-order topological invariant. Using gauge fields with an SU(2) symmetry, helicity and the new third-order invariant originate from the same identity, an identity which relates the second Chern class and the Chern-Simons three-form. We present an explicit example of three magnetic fields with non-disjunct support. These fields, derived from a vacuum Yang-Mills field with a non-vanishing winding number, possess a third-order linkage detected by our invariant.

A differential in the Lyndon—Hochschild—Serre spectral sequence

We consider the Lyndon-Hochschild-Serre spectral sequence with F n coefficients for a central extension with kernel cyclic of order a power of p and arbitrary discrete quotient group. For this spectral sequence d 2 and d 3 are known, and we give a description for d 4. Using this result we deduce a similar formula for the Serre spectral sequence for a fibration with fibre BC p n . The differential from odd rows to even rows involves a Massey triple product, so we describe the calculation of such products in the cohomology of a finite abelian group. As an example we determine the Poincaré series for the mod-3 cohomology of various 3-groups.