Related papers: Extended powers and Steenrod operations in algebra…

The Steenrod operations (mod p) in Chow theory are defined for any prime p for a quasi-projective scheme, without appealing to the results of any domain but Milnor's K-theory. The new definition also gives a direct formula that depends only…

The study of the action of the Steenrod algebra on the mod $p$ cohomology of spaces has many applications to the topological structure of those spaces. In this paper we present combinatorial formulas for the action of Steenrod operations on…

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…

Building up on work of Epstein, May and Drury, we define and investigate the mod $p$ Steenrod operations on the de Rham cohomology of smooth algebraic stacks over a field of characteristic $p>0$. We then compute the action of the operations…

Based on a relative Wu theorem in \'etale cohomology, we study the compatibility of Steenrod operations on Chow groups and on \'etale cohomology. Using the resulting obstructions to algebraicity, we construct new examples of non-algebraic…

Using the recent work of Frankland and Spitzweck, we define Steenrod operations $P^{n}$ on the mod $p$ motivic cohomology of smooth varieties defined over a base field of characteristic $p$. We show that $P^{n}$ is the $p$th power on…

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.

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…

Let S be an essentially smooth scheme over a field and l a prime number invertible on S. We show that the algebra of bistable operations in the mod l motivic cohomology of smooth S-schemes is generated by the motivic Steenrod operations.…

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…

Steenrod defined in 1947 the Steenrod squares on the mod 2 cohomology of spaces using explicit cochain formulae for the cup-$i$ products; a family of coherent homotopies derived from the broken symmetry of Alexander--Whitney's chain…

We define a chain map of the form $\E(k)\otimes BA^{\otimes k}\longrightarrow BA$, where $\E$ is a combinatorial $E_\infty$-operad called the sequence operad, and $BA$ is the bar complex of an $\E$-algebra $A$. We see that Steenrod-type…

We construct motivic power operations on the mod-$p$ motivic cohomology of $\Fb_p$-schemes using a motivic refinement of Nizio{\l}'s theorem. The key input is a purity theorem for motivic cohomology established by Levine. Our operations…

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 characterize primary operations in differential cohomology via stacks, and illustrate by differentially refining Steenrod squares and Steenrod powers explicitly. This requires a delicate interplay between integral, rational, and mod p…

In this article we construct Symmetric operations for all primes (previously known only for p=2). These unstable operations are more subtle than the Landweber-Novikov operations, and encode all p-primary divisibilities of characteristic…

We describe the dualization of the algebra of secondary cohomology operations in terms of generators extending the Milnor dual of the Steenrod algebra. In this way we obtain explicit formulae for the computation of the E_3-term of the Adams…

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…

We exploit a uniform recursive procedure using preferred contractions of targets $C_*$ to construct morphisms $B_* \to C_*$ between chain complexes in a wide variety of situations. Examples include classical Alexander-Whitney and…

The purpose of this paper is to establish a correspondence between the higher Bruhat orders of Yu. I. Manin and V. Schechtman, and the cup-$i$ coproducts defining Steenrod squares in cohomology. To any element of the higher Bruhat orders we…