Massey Products Research Articles

Symmetric Massey Products and a Hirsch Formula In Homology

A Hirsch formula is proved for the singular chains of a second loop space and is applied to show that the symmetric Massey produce ${\langle x\rangle ^p}$ is defined for $x$ an odd dimensional $\bmod p$ homology class of a second loop space with $p$ an odd prime. ${\langle x\rangle ^p}$ is then interpreted in terms of the Dyer-Lashof and Browder operations.

A duality between transpotence elements and Massey products

A duality between transpotence elements and Massey products

Functional Massey products and homological algebra

0. Introduction. In (8, 9) certain higher order operations, called K-theory Massey Products, were introduced and developed. These operations were designed to investigate the Kunneth formula spectral sequence in equivariant K-theory, constructed in (4). In that application the important feature of Massey products was that they gave operations on certain Tor-algebras which were well-behaved with respect to the algebraic coboundary, δa.

Massey products in K-theory. II

0. Introduction: In (14), higher order operations in K-theory, called Massey products, were introduced. These were motivated by the construction, in (7), of a spectral sequence in equivariant K-theory

Massey products in K-theory

The aim of this paper is to reformulate the work of Massey(6), and Spanier(9, 10), on higher order cohomology operations, in terms of vector bundles in such a way as to produce geometrically some higher order operations in K-theory, which we will call Massey products.

Secondary compositions and the Adams spectral sequence

In each term of the Adams spectral sequence [1, 2] Massey products [10] can be formed and in the E2-term it has been found convenient to describe specific elements by means of these products. In Theorem (1.1) of this paper a description is given of the action of the differential d r on the Massey product in E r. In homotopy an analogous construction to the Massey product is the Toda bracket [15] and Theorem (1.2) discusses the convergence of Massey products in the Adams spectral sequence to Toda brackets in homotopy. It states that, under favorable conditions, a Massey product that is formed from permanent cycles will itself contain a permanent cycle that is realized in homotopy by an element of the appropriate Toda bracket. The statement of both Theorems (1.1) and (1.2) will be found in w 1 and the proofs are given in w167 5, 6. The intervening sections contain various geometric and algebraic preliminaries. A previous account of Theorem (1.2) was given in [12] and I am grateful to Professor J.F.Adams for suggesting the problem to which this theorem provides a partial solution. In 1966, Ivanovskil announced, at the Moscow congress, various results concerning the Adams spectral sequence and there appears to be some overlap between these results and Theorem (1.2). More recently, in Chicago, Lawrence has developed an account of the higher Massey product behaviour in the Adams spectral sequence and his results generalize the main theorems of this paper.

Matric Massey products

It has long been recognized that the homologies of differential algebras and of differential modules over differential algebras have not only products but also higher order operations, namely Massey products. These operations have largely been ignored because of the difficulty in computing them and because of their seeming lack of conceptual interest. We shall here introduce and study generalizations of these operations; the new operations will be defined on n-tuples of matrices rather than on n-tuples of elements. Of course, the generalization does nothing to simplify the computations, although the results on spectral sequences in this paper will have this effect. The larger class of operations does, however, have essential conceptual interest. There are a variety of situations

Rational Cohomology Operations and Massey Products

Rational Cohomology Operations and Massey Products

Rational cohomology operations and Massey products

Let Q be the group of rational numbers. Then H*(Q, n; Q) is either an exterior algebra or polynomial algebra on a class u of dimension n. By the Kunneth formula, if PX>=K(Q, nj), that is if P is a rational generalized Eilenberg-MacLane space (GEM), then every class in Hk(P; Q) is a polynomial on the fundamental classes {ui}. Thus every rational primary cohomology operation on (xi, ** *, x) can be written d) {Xj I} = ZXjxj+ Eyrzr when XjE Q and yr, zrEI Q). The decomposable term is the twofold matrix Massey product

The Decomposition of Stable Homotopy

The purpose of this paper is to show that, under the operation of higher Toda bracket (which will be defined in ? 2), certain classes in the stable homotopy ring of spheres w*(S) generate all of w*(S). These classes, called the Hopf classes, are those detected by primary cohomology operations. We show further that if X is any spectrum, then w*(X) is generated by w*(S) acting by higher Toda brackets on those elements of w*(X) not in the kernel of the Hurewicz homomorphism. The proofs provide on algorithm for finding such brackets. These results answer conjectures raised recently by J. P. May [11] in connection with his decomposition of the E2-term of the Adams spectral sequence into Massey higher products on certain terms. Our definition of 3-fold Toda bracket is equivalent to Toda's toral construction [15]. The definition of the higher brackets is believed to be equivalent (at least in the stable range) to those of Spanier [13]. (Our notation is the reverse of these authors, but is more in keeping with that of Massey products.) The author wishes to thank Professor A. Liulevicius and Professor B. Gray for many fruitful disscussions. Furthermore, the author expresses his appreciation of the clarifying suggestions of both the reference and the editor. The results of this paper have been announced in [6].

THE DECOMPOSITION OF STABLE HOMOTOPY

The purpose of this paper is to show that, under the operation of higher Toda bracket (which will be defined in ? 2), certain classes in the stable homotopy ring of spheres w*(S) generate all of w*(S). These classes, called the Hopf classes, are those detected by primary cohomology operations. We show further that if X is any spectrum, then w*(X) is generated by w*(S) acting by higher Toda brackets on those elements of w*(X) not in the kernel of the Hurewicz homomorphism. The proofs provide on algorithm for finding such brackets. These results answer conjectures raised recently by J. P. May [11] in connection with his decomposition of the E2-term of the Adams spectral sequence into Massey higher products on certain terms. Our definition of 3-fold Toda bracket is equivalent to Toda's toral construction [15]. The definition of the higher brackets is believed to be equivalent (at least in the stable range) to those of Spanier [13]. (Our notation is the reverse of these authors, but is more in keeping with that of Massey products.) The author wishes to thank Professor A. Liulevicius and Professor B. Gray for many fruitful disscussions. Furthermore, the author expresses his appreciation of the clarifying suggestions of both the reference and the editor. The results of this paper have been announced in [6].

The Cohomology of Augmented Algebras and Generalized Massey Products for DGA-Algebras

The Cohomology of Augmented Algebras and Generalized Massey Products for DGA-Algebras

The cohomology of augmented algebras and generalized Massey products for 𝐷𝐺𝐴-Algebras

paper, involves the application of a spectral sequence defined for any DGAalgebra U to the special case U = C(A), the cobar construction of A. In the second half of the paper we define Massey products in U and relate them to the differentials in the cited spectral sequence.

Moment-angle Complexes, Monomial Ideals and Massey Products

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.