Related papers: Steenrod operations on polyhedral products
The Steenrod squares are cohomology operations with important applications in algebraic topology. While these operations are well-understood classically, little is known about them in the setting of homotopy type theory. Although a…
In this paper we construct an analog of Steenrod operations in motivic cohomology and prove their basic properties including the Cartan formula, the Adem relations and the realtions to characteristic classes.
In this paper we completely classify which graded polynomial R-algebras in finitely many even degree variables can occur as the singular cohomology of a space with coefficients in R, a 1960 question of N. E. Steenrod, for a commutative ring…
This note explores the interaction between cohomology operations in a generalized cohomology theory and a string topology loop coproduct dual to the Chas--Sullivan loop product. More precisely, we ask for a description for the failure of a…
We study a naturally occurring $E_{\infty}$-subalgebra of the full $E_2$-Hochschild cochain complex arising from coherent cochains. For group rings and certain category algebras, these cochains detect $H^*(B {\cal{C}})$, the simplicial…
A polyhedral product is a natural subspace of a Cartesian product, which is specified by a simplicial complex K. The automorphism group Aut(K) of K induces a group action on the polyhedral product. In this paper we study this group action…
Steenrod homotopy theory is a framework for doing algebraic topology on general spaces in terms of algebraic topology of polyhedra; from another viewpoint, it studies the topology of the lim^1 functor (for inverse sequences of groups). This…
Given a unital action $\theta $ of an inverse monoid $S$ on an algebra $A$ over a filed $K$ we produce (co)homology spectral sequences which converge to the Hochschild (co)homology of the crossed product $A\rtimes_\theta S$ with values in a…
In this note, working in the context of simplicial sets, we give a detailed study of the complexity for computing chain level Steenrod squares, in terms of the number of face operators required. This analysis is based on the combinatorial…
Explicit extensions representing cocycles $x \in Ext_{A}^{s,t}(F_2,F_2)$ are useful in calculating Steenrod operations $Sq^i : Ext_{A}^{s,t}(F_2,F_2) \longrightarrow Ext_{A}^{s+i,2t}(F_2,F_2)$ by a method devised by the second author. This…
The polyhedral product is a space constructed from a simplicial complex and a collection of pairs of spaces, which is connected with the Stanley Reisner ring of the simplicial complex via cohomology. Generalizing the previous work Grbic and…
We describe the variety of `symmetric' left actions of the mod 2 Steenrod algebra $\mathcal{A}$ on its subalgebra $\mathcal{A}(2)$. These arise as the cohomology of $\text{v}_2$ self maps $\Sigma^7 Z \longrightarrow Z$, as in…
Let $\mathbb S^{\infty}/\mathbb Z_2$ be the infinite lens space and $\mathscr A$ be the Steenrod algebra over the binary field $\mathbb F_2.$ The cohomology $H^{*}((\mathbb S^{\infty}/\mathbb Z_2)^{\oplus s}; \mathbb F_2)$ is known to be…
Power operations in the homology of infinite loop spaces, and $H_\infty$ or $E_\infty$ ring spectra have a long history in Algebraic Topology. In the case of ordinary mod p homology for a prime p, the power operations of Kudo, Araki, Dyer…
We construct and analyze the "syntomic Steenrod algebra", which acts on the mod $p$ syntomic cohomology (also known as etale-motivic cohomology) of algebraic varieties in characteristic $p$. We then apply the resulting theory to resolve the…
First, by inspiration of the results of Wood \cite{differential,problems}, but with the methods of non-commutative geometry and different approach, we extend the coefficients of the Steenrod squaring operations from the filed $\mathbb{F}_2$…
We present here a combinatorial method for computing cup-$i$ products and Steenrod squares of a simplicial set $X$. This method is essentially based on the determination of explicit formulae for the component morphisms of a higher diagonal…
We give a new construction of a weak form of Steenrod operations for Chow groups modulo a prime number p for a certain class of varieties. This class contains projective homogeneous varieties which are either split or over a field admitting…
This two-page note gives a non-computational derivation of the dual Steenrod algebra as the automorphisms of the formal additive group. Instead of relying on computational tools like spectral sequences and Steenrod operations, the argument…
In this paper, we compute the homology group and cohomology algebra of various polyhedral product objects uniformly from the point of view of diagonal tensor product. As applications, we introduce the polyhedral product method into…