Kai Xu
Let D_n be the Dickson invariant ring of F_2[X_1,...,X_n] acted by the general linear group GL(n,\F_2). In this paper, we provide an elementary proof of the conjecture by [Hung]: each element in D_n is in the image of the Steenrod square in F_2[X_1,...,X_n], where n>3.
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Let $p$ be an odd prime number. We study the problem of determining the module structure over the mod $p$ Steenrod algebra $\mathcal A(p)$ of the Dickson algebra $D_n$ consisting of all modular invariants of general linear group $GL(n,\mathbb F_p)$. Here $\mathbb F_p$ denotes the prime field of $p$ elements. In this paper, we give an explicit answer for $n=2$. More precisely, we explicitly compute the action of the Steenrod-Milnor operations $St^{S,R}$ on the generators of $D_n$ for $n=2$ and for either $S=\emptyset, R=(i)$ or $S=(s), R=(i)$ with $s,i$ arbitrary nonnegative integers.
Normalized Derivations for Steenrod--Milnor Operations on the Dickson Algebra and Applications
本文研究了斯蒂恩罗德-米尔诺操作对迪克森代数的作用，通过归一化得到导出，推导出高阶迭代的闭式表达，并应用归一化系数构造了结构类比于马格利斯同调的普通柯兹尔型构造。
Let $p$ be an odd prime number. Denote by $GL_n = GL(n,\mathbb F_p)$ the general linear group over the prime field $\mathbb F_p$. Each subgroup of $GL_n$ acts on the algebra $P_n=E(x_1,\ldots,x_n)\otimes \mathbb F_p(y_1,\ldots,y_n)$ in the usual manner. We grade $P_n$ by assigning $\dim x_i=1$ and $\dim y_i=2.$ This algebra is a module over the mod $p$ Steenrod algebra $\mathcal A_p$. The purpose of the paper is to compute the action of the Steenrod-Milnor operations on the generators of $P_2^{GL_2}$. More precisely, we explicitly determine the action of $St^{(i,j)}$ on the Dickson invariants …
The spaces BG_2 and BDI(4) have the property that their mod 2 cohomology is given by the rank 3 and 4 Dickson invariants respectively. Associated with these spaces one has for q odd the classifying spaces of the finite groups BG_2(q)and the exotic family of classifying spaces of 2-local finite groups BSol(q). In this article compute the mod 2 loop space homology of the 2-completed classifying space of G_2(q) and of BSol(q) for all odd primes q, as algebras over the Steenrod algebra, and the associated Bockstein spectral sequences.
Steenrod closed parameter ideals in the mod-$2$ cohomology of $A_4$ and $SO(3)$
本文分类了A₄和SO(3)模2上同调中Steenrod闭参数理想，进而得到这两个群可自由作用于Sⁿ×Sᵐ时维度n,m的限制。
We prove that the coefficients of the so-called conjugation equation for conjugation spaces in the sense of Hausmann-Holm-Puppe are completely determined by Steenrod squares. This generalises a result of V.A. Krasnov for certain complex algebraic varieties. It also leads to a generalisation of a formula given by Borel and Haefliger, thereby largely answering an old question of theirs in the affirmative.
In 1947, N.E. Steenrod defined the Steenrod Squares, which are mod 2 cohomology operations, using explicit cochain formulae for cup-i products of cocycles. He later recast the construction in more general homological terms, using group homology and acyclic model methods, rather than explicit cochain formulae, to define mod p operations for all primes p. Steenrod's student J. Adem applied the homological point of view to prove fundamental relations, known as the Adem relations, in the algebra of cohomology operations generated by the Steenrod operations. In this paper we give a proof of the mod…
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…
Lipshitz and Sarkar recently introduced a space-level refinement of Khovanov homology. This refinement induces a Steenrod square operation $\Sq^2$ on Khovanov homology which they describe explicitly. This paper presents some computations of $\Sq^2$. In particular, we give examples of links with identical integral Khovanov homology but with distinct Khovanov homotopy types.
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.
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 approximation (i.e., a family of morphisms measuring the lack of commutativity of the cup product on the cochain level) in terms of face operators of $X$. A generalization of this method to Steenrod reduced powers is sketched. This description can be considered as a translation of the most ancient definition of Steenrod squares to the general setting of the …
We show that there are prime knots so that the Steenrod operations of Lipshitz and Sarkar arXiv:1204.5776 are non trivial on their Khovanov homology. This answers a question posed by Lipshitz and Sarkar in their paper arXiv:1709.03602. We then go on to show that there are hyperbolic and satellite knots so that the Steenrod operations are non trivial on their Khovanov homology.
It has long been envisioned that the strength of the barcode invariant of filtered cellular complexes could be increased using cohomology operations. Leveraging recent advances in the computation of Steenrod squares, we introduce a new family of computable invariants on mod 2 persistent cohomology termed $Sq^k$-barcodes. We present a complete algorithmic pipeline for their computation and illustrate their real-world applicability using the space of conformations of the cyclo-octane molecule.
The groups of differential characters of Cheeger and Simons admit a natural multiplicative structure. The map given by the squares of degree 2k differential characters reduces to a homomorphism of ordinary cohomology groups. We prove that the homomorphism factors through the Steenrod squaring operation of degree 2k. A simple application shows that five-dimensional Chern-Simons theory for pairs of B-fields is SL(2, Z)-invariant on spin manifolds.
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。
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…
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。
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。
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.
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。
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.
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。
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.
Graph colouring and Steenrod's problem for Stanley-Reisner rings
针对代数拓扑中Steenrod提出的分次环作为空间上同调环的经典问题，定义图着色的代数版本跨度着色，探究其与分次Stanley-Reisner环的Steenrod问题的关联。
Steenrod problem and some graded Stanley-Reisner rings
本文针对Steenrod提出的环的空间奇异上同调表示问题，研究分次Stanley-Reisner环的可实现性，在一定假设下给出了其可实现的充要条件。
On Modular Invariants of Truncated Polynomial Rings in low ranks
本文通过构造显式生成元验证了Lewis-Reiner-Stanton关于低秩截断多项式环不变环Hilbert级数的猜想，并提出关于Steenrod代数与Dickson代数在自然滤过上作用的新猜想。
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…
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.
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。
In this note, we examine the right action of the Steenrod algebra $\mathcal{A}$ on the homology groups $H_*(BV_s, \F_2)$, where $V_s = \F_2^s$. We find a relationship between the intersection of kernels of $Sq^{2^i}$ and the intersection of images of $Sq^{2^{i+1}-1}$, which can be generalized to arbitrary right $\mathcal{A}$-modules. While it is easy to show that $\bigcap_{i=0}^{k} \mathrm{im}\,Sq^{2^{i+1}-1} \subseteq \bigcap_{i = 0}^k \mathrm{ker}\,Sq^{2^i}$ for any given $k \geq 0$, the reverse inclusion need not be true. We develop the machinery of homotopy systems and null subspaces in or…
The $2$-primary Hopf invariant $1$ elements in the stable homotopy groups of spheres form the most accessible family of elements. In this paper we explore some properties of the $\mathcal{E}_\infty$ ring spectra obtained from certain iterated mapping cones by applying the free algebra functor. In fact, these are equivalent to Thom spectra over infinite loop spaces related to the classifying spaces $B\mathrm{SO},\,B\mathrm{Spin},\,B\mathrm{String}$. We show that the homology of these Thom spectra are all extended comodule algebras of the form $\mathcal{A}_*\square_{\mathcal{A}(r)_*}P_*$ over th…
Binary Minor Certificates for the $\mathcal A(1)$-hit problem of $MO(2)$
本文通过构造二元 hit 矩阵的行列式为一的子式及对偶泛函，严格证明了 $MO(2)$ 的 $\mathcal A(1)$-hit 问题中各次数的精确秩，确立了 Shih 的猜想，并给出 Steenrod 平方的闭式公式、Hilbert 级数及模的直和分解。
Geometric realization via unoriented bordism and a counterexample to Singer's conjecture for the sixth algebraic transfer
本文通过开发计算Kameko同态核的GL(q)-不变量的新算法，证明了在双次数(6,6+36)上Singer猜想不成立。
The rank-five Peterson hit problem, the Singer transfer, and canonical Milnor-operation layers
本文研究了$H^{*}(V^{\oplus 5})$的hit问题，验证了Singer猜想在特定度数下的成立，并提供了新算法的实现。
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