Steenrod squares in spectral sequences. I

Mathematical proceedings of the Cambridge Philosophical Society, 1981

Transactions of the American Mathematical Society, 1998

Using methods developed by W. Singer and J. P. May, we describe a systematic approach to showing that many spectral sequences, determined by a filtration on a complex whose homology has an action of operations, possess a compatible action of the same operations. As a consequence, we obtain W. Singer's result for Steenrod operations on Serre spectral sequence and extend A. Bahri's action of Dyer-Lashof operations on the second quadrant Eilenberg-Moore spectral sequence.

Mathematical Notes, 1992

International Journal of Mathematics and Mathematical Sciences, 2003

For a commutative Hopf algebra A over Z/p, where p is a prime integer, we define the Steenrod operations P i in cyclic cohomology of A using a tensor product of a free resolution of the symmetric group S n and the standard resolution of the algebra A over the cyclic category according to . We also compute some of these operations.

Journal of Pure and Applied Algebra, 2009

Homology, Homotopy and Applications, 2011

It is proved that Kaygun's Hopf-Hochschild cochain complex for a module-algebra is a brace algebra with multiplication. As a result, an analogue of Deligne's Conjecture holds for modulealgebras, and the Hopf-Hochschild cohomology of a modulealgebra has a Gerstenhaber algebra structure.

Journal of K-theory: K-theory and its Applications to Algebra, Geometry, and Topology, 2014

In the world of chain complexes En-algebras are the analogues of based n-fold loop spaces in the category of topological spaces. Fresse showed that operadic En-homology of an En-algebra computes the homology of an n-fold algebraic delooping. The aim of this paper is to construct two spectral sequences for calculating these homology groups and to treat some concrete classes of examples such as Hochschild cochains, graded polynomial algebras and chains on iterated loop spaces. In characteristic zero we gain an identification of the summands in Pirashvili's Hodge decomposition of higher order Hochschild homology in terms of derived functors of indecomposables of Gerstenhaber algebras and as the homology of exterior and symmetric powers of derived Kähler differentials.

Journal of Pure and Applied Algebra, 1977

So we are well motivated to attack Problem (*). Let S" ("n-sphere") be the unique JICA object for which (S"), = 22; (S"), = 0 if p# n. In this paper we determine completely the unstable A-modules a f S", for those cases in which k G n + s-1. The results form an easily grasped whole, and offer clues to the general solution. We can state them in terms of the A-algebras Vt, that are introduced in [9]. VS = Z2(u0,.. ., us+) is a polynomial algebra on generators uk of dimension 2k, and its structure as an A-module is determined by formula (0.1) of [9]. For each i, 1 G i s s, we will define in this paper an element o1 in V, of dimension 2"-2'-' ((r i.1) below). We will prove: Theorem 0.1. The monomial .j {a? l. l ai 1 iI a 0,. . ., iS a 0) are &nearly independent in V, and the polydomiaJ' sub-algebra L, E V, thut they span is closed under A-action. We will show further that the ideal in V, generated by the elements { fl+s-1 n+s-2 uo ,ut ?'..9 uL} is closed under A-action for any n 3 0. We denote by Vi the quotient of V, by this ide ; by L," the image of L, in Vt, and prove: Theorem 0.2. The monomials {a> 8 l l CT)/ i, + ... + is s n-I} are a Z2-b&s for the unstable A-module L:,

Proceedings of the American Mathematical Society, 1991

Let Ps be the mod-2 cohomology of the elementary abelian group (Z/2Z) x • ■ • x (Z/2Z) (s factors). The mod-2 Steenrod algebra A acts on Ps according to well-known rules. If A C A denotes the augmentation ideal, then we are interested in determining the image of the action A ® Ps-* Ps: the space of elements in Ps that are hit by positive dimensional Steenrod squares. The problem is motivated by applications to cobordism theory [PI] and the homology of the Steenrod algebra [S]. Our main result, which generalizes work of Wood [W], identifies a new class of hit monomials. Theorem 1.1 (R. Wood, [W]). Suppose x e Ps is a monomial of degree ô, and suppose a[ô + s] > s. Then x is hit.

arXiv (Cornell University), 2024

We describe the action of the mod 2 Steenrod algebra on the cohomology of various polyhedral products and related spaces. We carry this out for Davis-Januszkiewicz spaces and their generalizations, for moment-angle complexes as well as for certain polyhedral joins. By studying the combinatorics of underlying simplicial complexes, we deduce some consequences for the lowest cohomological dimension in which non-trivial Steenrod operations can appear. We present a version of cochain-level formulas for Steenrod operations on simplicial complexes. We explain the idea of "propagating" such formulas from a simplicial complex K to polyhedral joins over K and we give examples of this process. We tie the propagation of the Steenrod algebra actions on polyhedral joins to those on moment-angle complexes. Although these are cases where one can understand the Steenrod action via a stable homotopy decomposition, we anticipate applying this method to cases where there is no such decomposition.