A Combinatorial Method for Computing Steenrod Squares

New formulas for cup-$i$ products and fast computation of Steenrod squares

Operations on the cohomology of spaces are important tools enhancing the descriptive power of this computable invariant. For cohomology with mod 2 coefficients, Steenrod squares are the most significant of these operations. Their effective computation relies on formulas defining a cup-$i$ construction, a structure on (co)chains which is important in its own right, having connections to lattice field theory, convex geometry and higher category theory among others. In this article we present new formulas defining a cup-$i$ construction, and use them to introduce a fast algorithm for the computat…

Steenrod 长度与 Vakil 的一个问题

Steenrod Lengths and a Problem of Vakil

本文通过组合方法研究实射影空间的 Steenrod 长度，推导出 $f(n)$ 的显式公式，并解决 Vakil 提出的问题。

奇Khovanov同调的二阶Steenrod平方

Two second Steenrod squares for odd Khovanov homology

本研究受Sarkar-Scaduto-Stoffregen构造的奇Khovanov同调稳定同伦型启发，提出二阶Steenrod平方的计算方法，证明其为链环不变量且可细化Rasmussen $s$-不变量，预期与前者构造的二阶Steenrod平方相关。

Steenrod squares on Intersection cohomology and a conjecture of M. Goresky and W. Pardon

We prove a conjecture raised by M. Goresky and W. Pardon, concerning the range of validity of the perverse degree of Steenrod squares in intersection cohomology. This answer turns out of importance for the definition of characteristic classes in the framework of intersection cohomology. For this purpose, we present a construction of ${\rm cup}_{i}$-products on the cochain complex, built on the blow-up of some singular simplices and introduced in a previous work. We extend to this setting the classical properties of the associated Steenrod squares, including Adem and Cartan relations, for any g…

链Floer同调的Steenrod平方

A Steenrod Square for Link Floer Homology

本文基于Manolescu-Sarkar的稳定同调类型，通过显式给出模空间的框架，构建了包含足够信息计算第二Steenrod平方的1-流范畴，并提出计算所有网格同调版本第二Steenrod平方的算法。

关于偶和奇Khovanov同调的Steenrod平方

On Steenrod squares for even and odd Khovanov homology

本文计算了Sarkar-Scaduto-Stoffregen构造的偶和奇Khovanov同调的空间细化上的Sq²，确定了所有不超过11个交叉点的结的稳定同伦型，并证明了Schütz定义的Steenrod平方Sq₀²和Sq₁²出现在这些空间上。

Higher Steenrod squares for Khovanov homology

We describe stable cup-i products on the cochain complex with $F^2$ coefficients of any augmented semi-simplicial object in the Burnside category. An example of such an object is the Khovanov functor of Lawson, Lipshitz and Sarkar. Thus we obtain explicit formulas for cohomology operations on the Khovanov homology of any link.

Computing Cocycles on Simplicial Complexes

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 formulation given in [R. Gonzalez-Diaz, P. Real. A Combinatorial Method for Computing Steenrod Squares. J. of Pure and Applied Algebra, 139 (1999) 89-108]. As an application, we give here an algorithm for computing cup-$i$ products over integers on a simplicial complex at chain level.

Khovanov同调上的Steenrod平方与cup-i乘积

A Steenrod square on Khovanov homology and a cup-i product

本文研究了Khovanov同调上的Steenrod平方与cup-i乘积，证明了Lipshitz-Sarkar的Sq²与Morán的sq²一致。

通过无序连接构造斯廷罗德平方运算

The Steenrod squares via unordered joins

本文在同伦类型论中基于无序连接高阶归纳类型，为布鲁内里定义的斯廷罗德平方运算证明了稳定性、嘉当公式和阿德姆关系，并提供了立方阿格达形式化。

A Steenrod Square on Khovanov Homology

In a previous paper, we defined a space-level version X(L) of Khovanov homology. This induces an action of the Steenrod algebra on Khovanov homology. In this paper, we describe the first interesting operation, Sq^2:Kh^{i,j}(L) -> Kh^{i+2,j}(L). We compute this operation for all links up to 11 crossings; this, in turn, determines the stable homotopy type of X(L) for all such links.

Steenrod coalgebras

This paper shows that a functorial version of the "higher diagonal" of a space used to compute Steenrod squares actually contains far more topological information --- including (in some cases) the space's integral homotopy type.

Unstable Modules with the Top $k$ Squares

Unstable modules over the Steenrod algebra with only the top $k$ operations are introduced in the language of ringoids. We prove the category of such modules has homological dimension at most $k$. A pratical method, which generalizes the $Λ$ complex, to compute the $\mathrm{Ext}$ group from such modules to spheres is proposed. We are also able to establish several functors to relate such modules and unstable modules over the Steenrod algebra, and to describe the connections between the $\mathrm{Ext}$ groups in them.

Khovanov homotopy calculations using flow category calculus

The Lipshitz-Sarkar stable homotopy link invariant defines Steenrod squares on the Khovanov cohomology of a link. Lipshitz-Sarkar constructed an algorithm for computing the first two Steenrod squares. We develop a new algorithm which implements the flow category simplification techniques previously defined by the authors and Dan Jones. We give a purely combinatorial approach to calculating the second Steenrod square and Bockstein homomorphisms in Khovanov cohomology, and flow categories in general. The new method has been implemented in a computer program by the third author and applied to lar…

关于motivic Peterson猜想的探讨

On the motivic Peterson conjecture

本文研究了motivic上同调中Steenrod平方作用的Peterson猜想的类比是否成立，发现其不成立。

Geometrically Interpreting Higher Cup Products, and Application to Combinatorial Pin Structures

We provide a geometric interpretation of the formulas for Steenrod's $\cup_i$ products, giving an explicit construction for a conjecture of Thorngren. We construct from a simplex and a branching structure a special frame of vector fields inside each simplex that allow us to interpret cochain-level formulas for the $\cup_i$ as a generalized intersection product on the dual cellular decomposition. It can be thought of as measuring the intersection between a collection of dual cells and thickened, shifted version of another collection, where the vector field frame determines the thickening and sh…