Daniel C. Isaksen
We study the Mahowald operator $M = \langle g_2,h_0^3, - \rangle$ in the cohomology of the Steenrod algebra. We show that the operator interacts well with the cohomology of $A(2)$, in both the classical and $\mathbb{C}$-motivic contexts. This generalizes previous work of Margolis, Priddy, and Tangora.
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Mahowald's conjecture arose as part of a program attempting to view chromatic phenomena in stable homotopy theory through the lens of the classical Adams spectral sequence. The conjecture predicts the existence of nonzero classes in the cohomology of the finite sub-Hopf algebras $A(n)$ of the mod 2 Steenrod algebra that correspond to generators in the homotopy rings of certain periodic spectra. The purpose of this note is to present a proof of the conjecture.
In this paper we use Peter May's algebraic approach to Steenrod operations to construct Steenrod's reduced power operations in simplicial Bredon-Illman cohomology with local coefficients of a one vertex G-Kan complex, G being a discrete group.
Let NG0 denote the category of all pointed numerically generated spaces and continuous maps preserving base-points. In [SYH], we described a passage from bivariant functors to generalized homology and cohomology theories. In this paper, we construct a bivariant functor such that the associated cohomology is the Cech cohomology and the homology is the Steenrod homology (at least for compact metric spaces).
The purpose of this article is to compute the cohomology of the motivic Steenrod algebra over Spec C through the geometric 70-stem. The main computational tool is the motivic May spectral sequence. Everywhere in this article, we are working only over Spec C, and we are always computing at the prime p = 2.
We compute the cotorsion product of the mod 2 cohomology of spinor group spin(n), which is the E_2-term of the Rothenberg-Steenrod spectral sequence for the mod 2 cohomology of the classifying space of the spinor group spin(n). As a consequence of this computation, we show the non-collapsing of the Rothenberg-Steenrod spectral sequence for n > 16.
A minimal resolution of the mod 2 Steenrod algebra in the range $0 \leq s \leq 128$, $0 \leq t \leq 200$, together with chain maps for each cocycle in that range and for the squaring operation $Sq^0$ in the cohomology of the Steenrod algebra.
We show that the Rothenberg--Steenrod spectral sequence converging to the mod 3 cohomology of the classifying space of the exceptional Lie group E_8 does not collapse at the E_2-level.
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 the cohomology of Grassmannians, both in the Borel and the Schubert picture. We consider integral lifts of Steenrod operations, which lie in a certain Hopf algebra of differential operators. The latter has been considered recently as a realization of the Landweber-Novikov algebra in complex cobordism theory; it also has connections with the action of the Vi…
A sparse periodic family in the cohomology of the $\mathbb{C}$-motivic Steenrod algebra
本文研究$\mathbb{C}$-动机Steenrod代数上同调中一个具有异常周期性的元素族，并揭示其与$h_1$-局部化及动机模形式谱的代数Hurewicz像的联系。
Weighted polyhedral products and Steenrod's problem
本文构造了加权多面体乘积并计算其特殊情形的上同调，进而解决了与球面乘积相关的Steenrod上同调实现问题。
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.
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…
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…
We describe an algorithm that allows to compute a minimal resolution of the Steenrod algebra. The algorithm has built-in knowledge about vanishing lines for the cohomology of sub Hopf algebras of the Steenrod algebra which makes it both faster and more economical than the generic approach.
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 approximation to the diagonal. He later defined his homonymous operations for all primes using the homology of symmetric groups. This approach enhanced the conceptual understanding of the operations and allowed for many advances, but lacked the concreteness of their definition at the even prime. In recent years, thanks to the development of new applications of coho…
Steenrod closed $C_3$-invariant parameter ideals in the mod 2 cohomology of $\mathbb{Z}/2\times\mathbb{Z}/2$
本文分类了 F₂[a,b] 上 C₃ 非平凡作用下 Steenrod 闭的 C₃-不变参数理想，并将其用于研究可扩张为 A₄ 作用的 Klein 四元群自由作用。
We describe algebraic obstruction theories for realizing an abstract coalgebra K_* over the mod p Steenrod algebra as the homology of a topological space, and for distinguishing between the p-homotopy types of different realizations. The theories are expressed in terms of the Quillen cohomology of K_*.
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 uses a few simple universal properties of certain cohomology theories.
The Singer transfer provides an interesting connection between modular representation theory and the cohomology of the Steenrod algebra. We discuss a version of `Quillen stratification' theorem for the Singer transfer and its consequences.
Let $V$ be an elementary abelian $2$-group and $X$ be a finite $V$-CW-complex. In this memoir we study two cochain complexes of modules over the mod2 Steenrod algebra $\mathrm{A}$, equipped with an action of $\mathrm{H}^{*}V$, the mod2 cohomology of $V$, both associated with $X$. The first, which we call the "topological complex", is defined using the orbit filtration of $X$. The second, which we call the "algebraic complex", is defined just in terms of the unstable $\mathrm{A}$-module $\mathrm{H}^*_V X$, the mod2 equivariant cohomology of $X$. Our study makes intensive use of the theory of un…
Equivariant Steenrod Operations
本文提出了一种针对有限群的等变斯坦诺普运算构造方法，基于 $\mathcal{N}_\infty$-环谱的 $\mathrm{R}$-欧拉序列理论。
Geometric Constructions of Mod $p$ Cohomology Operations
本文基于Brown可表示性和Almgren同构定理，构造了从球面循环空间到圆盘相对循环空间的显式映射，用以表示模$p$上同调中的所有Steenrod平方、Steenrod幂和Bockstein同态。
The aim of this paper is to study sub-algebras of the $\mathbb{Z}/2$-equivariant Steenrod algebra (for cohomology with coefficients in the constant Mackey functor $\mathbb{F}_2$) which come from quotient Hopf algebroids of the $\mathbb{Z}/2$-equivariant dual Steenrod algebra. In particular, we study the equivariant counterpart of profile functions, exhibit the equivariant analogues of the classical $\mathcal{A}(n)$ and $\mathcal{E}(n)$ and show that the Steenrod algebra is free as a module over these.
In this paper, we compute ${\rm Ext}_{A}^{s}(\widetilde{H}^*(B\mathbb{Z}/p),\mathbb{F}_p)$ for $s\leq 1$. Using this result, we investigate the behavior of $φ_3^{\mathbb{F}_p}$ and $φ_s^{\widetilde{H}^*(B\mathbb{Z}/p)}\ (s\leq1)$ for an odd prime $p$.
Steenrod operations have been defined by Voedvodsky in motivic cohomology in order to show the Milnor and Bloch-Kato conjectures. These operations have also been constructed by Brosnan for Chow rings. The purpose of this paper is to provide a setting for the construction of the Steenrod operations in algebraic geometry, for generalized cohomology theories whose formal group law has order two. We adapt the methods used by Bisson-Joyal in studying Steenrod and Dyer-Lashof operations in unoriented cobordism and mod 2 cohomology.
This article fills some gaps in Voevodsky's construction of the Steenrod operations acting on the motivic cohomology with coefficients in Z/lZ of motivic spaces in the sense of Morel and Voevodsky over a perfect field of characteristic different from l. Moreover, as a consequence of the method of proof of a theorem by Voevodsky on stable cohomology operations, we show that the spectrum that represents motivic cohomology with coefficients Z/lZ has no nonzero "superphantom" endomorphism.
The algebra S of symmetric invariants over the field with two elements is an unstable algebra over the Steenrod algebra A, and is isomorphic to the mod two cohomology of BO, the classifying space for vector bundles. We provide a minimal presentation for S in the category of unstable A-algebras, i.e., minimal generators and minimal relations. From this we produce minimal presentations for various unstable A-algebras associated with the cohomology of related spaces, such as the BO(2^m-1) that classify finite dimensional vector bundles, and the connected covers of BO. The presentations then show …
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…
The purpose of these notes is to provide an introduction to the Steenrod algebra in an algebraic manner avoiding any use of cohomology operations. The Steenrod algebra is presented as a subalgebra of the algebra of endomorphisms of a functor. The functor in question assigns to a vector space over a Galois field the algebra of polynomial functions on that vector space: the subalgebra of the endomorphisms of this functor that turns out to be the Steenrod algebra if the ground field is the prime field, is generated by the homogeneous components of a variant of the Frobenius map.
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。
On the Steenrod module structure of $\mathbb{R}$-motivic Spanier-Whitehead duals
本文研究了 $\mathbb{R}$-motivic 谱的 Spanier-Whitehead 对偶的上同调作为 Steenrod 代数模的结构恢复方法，并证明了 $\mathcal{A}^{\mathbb{R}}(1)$ 的 128 种模结构中有 16 种是自对偶的。
Noncommutative Gröbner Bases and Ext groups; Application to the Steenrod Algebra
本文提出递减过滤代数上的非交换Gröbner基理论并迁移相关算法，高效计算了Steenrod代数在素数2下的Ext群，将其上同调计算至总次数261并包含环结构。
In this paper we begin the study of the (dual) Steenrod algebra of the motivic Witt cohomology spectrum $H_W\mathbb{Z}$ by determining the algebra structure of ${H_W\mathbb{Z}}_{**}H_W\mathbb{Z}$ over fields $k$ of characteristic not $2$ which are extensions of fields $F$ with $K^M_2(F)/2=0$. For example, this includes all fields of odd characteristic, as well as fields that are extensions of quadratically closed fields of characteristic $0$. After inverting $η$, this computes the $HW:=H_W\mathbb{Z}[η^{-1}]$-algebra ${HW}_{**}HW$. In particular, for the given base fields, this implies the $HW$…
We study the structure of the $E_2$-term of the Rothenberg-Steenrod spectral sequence converging to the mod 3 cohomology of the classifying space of the compact, connected, simply connected, exceptional Lie group of rank 6.
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 R satisfying mild conditions. In the fundamental case R = Z, our result states that the only polynomial cohomology rings over Z which can occur, are tensor products of copies of H^*(CP^\infty;Z) = Z[x_2], H^*(BSU(n);Z) = Z[x_4,x_6,...,x_{2n}], and H^*(BSp(n):Z) = Z[x_4,x_8,...,x_{4n}] confirming an old conjecture. Our classification extends Notbohm's soluti…
Prismatic Steenrod operations and arithmetic duality on Brauer groups
本文构建并分析了
Steenrod operations on polyhedral products
该研究刻画了模2 Steenrod代数在多面体乘积等空间上同调中的作用，提出上链层公式及从单纯复形到多面体join的传播方法，建立其与矩角复形作用的关联，为无稳定同伦分解的情形提供研究思路。
This paper extends the relation established for group cohomology by Green, Hunton and Schuster between chromatic phenomena in stable homotopy theory and certain natural subrings of singular cohomology. This exploits the theory due to Henn, Lannes and Schwartz of unstable algebras over the Steenrod algebra localized away from nilpotents.
We determine the mod $2$ cohomology over the Steenrod algebra of the classifying spaces of the free loop groups $LG$ for compact groups $G=Spin(7)$, $Spin(8)$, $Spin(9)$, and $F_4$. Then, we show that they are isomorphic as algebras over the Steenrod algebra to the mod $2$ cohomology of the corresponding Chevalley groups of type $G(q)$, where $q$ is an odd prime power. In a similar manner, we compute the cohomology of the free loop space over $BDI(4)$ and show that it is isomorphic to that of $BSol(q)$ as algebras over the Steenrod algebra.
We examine the dual of the so-called "hit problem", the latter being the problem of determining a minimal generating set for the cohomology of products of infinite projective spaces as module over the Steenrod Algebra $\mathcal{A}$ at the prime 2. The dual problem is to determine the set of $\mathcal {A}$-annihilated elements in homology. The set of $\mathcal{A}$-annihilateds has been shown by David Anick to be a free associative algebra. In this note we prove that, for each $k \geq 0$, the set of {\it $k$ partially $\mathcal{A}$-annihilateds}, the set of elements that are annihilated by $Sq^i…
Primary cohomology operations, i.e., elements of the Steenrod algebra, are given by homotopy classes of maps between Eilenberg--MacLane spectra. Such maps (before taking homotopy classes) form the topological version of the Steenrod algebra. Composition of such maps is strictly linear in one variable and linear up to coherent homotopy in the other variable. To describe this structure, we introduce a hierarchy of higher distributivity laws, and prove that the topological Steenrod algebra satisfies all of them. We show that the higher distributivity laws are homotopy invariant in a suitable sens…
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. This was previously proved by Voevodsky for S a field of characteristic zero. We follow Voevodsky's proof but remove its dependence on characteristic zero by using étale cohomology instead of topological realization and by replacing resolution of singularities with a theorem of Gabber on alterations.
The Hochschild homology and cohomology of A(1)
本文利用若干 May 型谱序列，计算了由 $Sq^1$、$Sq^2$ 生成的 2-素 Steenrod 代数子代数 $A(1)$ 的 Hochschild 同调与上同调。
Let $F_2$ be the prime field of two elements and let $GL_s:= GL(s, F_2)$ be the general linear group of rank $s.$ Denote by $\mathscr A$ the Steenrod algebra over $F_2.$ The (mod-2) Lambda algebra, $Λ,$ is one of the tools to describe those mysterious "Ext-groups". In addition, the $s$-th algebraic transfer of William Singer \cite{Singer} is also expected to be a useful tool in the study of them. This transfer is a homomorphism $Tr_s: F_2 \otimes_{GL_s}P_{\mathscr A}(H_{*}(B\mathbb V_s, F_2))\to {\rm Ext}_{\mathscr {A}}^{s,s+*}(F_2, F_2),$ where $\mathbb V_s$ denotes the elementary abelian $2$…
Let G be an exceptional Lie group with a maximal torus T. Based on Schubert calculus on the flag manifold G/T we have described the integral cohomology ring H*(G) by explicitely constructed generators in [DZ2], and determined the structure of H*(G;F_{p}) as a Hopf algebra over the Steenrod algebra in[DZ3]. In this sequel to [DZ2], [DZ3] we obtain the near--Hopf ring structure on H*(G).
We present a formula describing the action of a generalised Steenrod operation of $\Z_2$-type on the cohomology class represented by a proper self-transverse immersion $f\co M\imm X$, in terms of the equivariant double points of $f$ and the characteristic classes of its normal bundle. This generalises a classical result of R.\ Thom: If $α\in H^k(X;\Z_2)$ is the ordinary cohomology class represented by $f\co M\imm X$, then $\mathrm{Sq}^i(α)=f_* w_i(ν_f)$.
Given a simply connected space $X$ with the cohomology $H^*(X;{\mathbb Z}_2)$ to be polynomial, we calculate the loop cohomology algebra $H^*(ΩX;{\mathbb Z}_2)$ by means of the action of the Steenrod cohomology operation $Sq_1$ on $H^*(X;{\mathbb Z}_2).$ As a consequence we obtain that $H^*(ΩX;{\mathbb Z}_2)$ is the exterior algebra if and only if $Sq_1$ is multiplicatively decomposable on $H^{\ast}(X;{\mathbb Z}_2).$ The last statement in fact contains a converse of a theorem of A. Borel.
Let $P_{k}=H^{*}((\mathbb{R}P^{\infty})^{k})$ be the modulo-$2$ cohomology algebra of the direct product of $k$ copies of infinite dimensional real projective spaces $\mathbb{R}P^{\infty}$. Then, $P_{k}$ is isomorphic to the graded polynomial algebra $\mathbb{F}_{2}[x_{1},\ldots,x_{k}]$ of $k$ variables, in which each $x_{j}$ is of degree 1, and let $GL_k$ be the general linear group over the prime field $\mathbb{F}_2$ which acts naturally on $P_k$. Here the cohomology is taken with coefficients in the prime field $\mathbb F_2$ of two elements. We study the {\it hit problem}, set up by Frank P…
The mod p cohomology of a space comes with an action of the Steenrod Algebra. L. Schwartz [A propos de la conjecture de non realisation due a N. Kuhn, Invent. Math. 134, No 1, (1998) 211--227] proved a conjecture due to N. Kuhn [On topologicaly realizing modules over the Steenrod algebra, Annals of Mathematics, 141 (1995) 321--347] stating that if the mod $p$ cohomology of a space is in a finite stage of the Krull filtration of the category of unstable modules over the Steenrod algebra then it is locally finite. Nevertheless his proof involves some finiteness hypotheses. We show how one can re…
We transport Steenrod's cup-i products from the singular cochains on the free loop space Maps(S^1, BG) to Hochschild's original cochain complex Hom (k[G]^*, k[G]) defining Hochschild cohomology. Here G is a discrete group, k an arbitrary coefficient ring, and BG the classifying space of G. For cochains supported on BG, we prove that Gerstenhaber's cup product agrees with the simplicial cup product and Gerstenhaber's pre-Lie product agrees with Steenrod's cup-one product. As a consequence, for cocycles f and g supported on BG, the Gerstenhaber bracket [f, g] = 0 in Hochschild cohomology. This i…
We generalize some of the fundamental results of algebraic topology from topological spaces to Čech's closure spaces, also known as pretopological spaces. Using simplicial sets and cubical sets with connections, we define three distinct singular (relative) simplicial and six distinct singular (relative) cubical (co)homology groups of closure spaces. Using acyclic models we show that the three simplicial groups have isomorphic cubical analogues among the six cubical groups. Thus, we obtain a total of six distinct singular (co)homology groups of closure spaces. Each of these is shown to have a c…
Let us consider the prime field of two elements, $\mathbb F_2\equiv \mathbb Z_2.$ It is well-known that the classical "hit problem" for a module over the mod 2 Steenrod algebra $\mathscr A$ is an interesting and important open problem of Algebraic topology, which asks a minimal set of generators for the polynomial algebra $\mathcal P_m:=\mathbb F_2[x_1, x_2, \ldots, x_m]$, regarded as a connected unstable $\mathscr A$-module on $m$ variables $x_1, \ldots, x_m,$ each of degree 1. The algebra $\mathcal P_m$ is the $\mathbb F_2$-cohomology of the product of $m$ copies of the Eilenberg-MacLan comp…
In this paper, we show that the finite subalgebra $\mathcal{A}^{\mathbb{R}}(1)$, generated by $\mathrm{Sq}^1$ and $\mathrm{Sq}^2$, of the $\mathbb{R}$-motivic Steenrod algebra $\mathcal{A}^{\mathbb{R}}$ can be given $128$ different $\mathcal{A}^{\mathbb{R}}$-module structures. We also show that all of these $\mathcal{A}^{\mathbb{R}}$-modules can be realized as the cohomology of a $2$-local finite $\mathbb{R}$-motivic spectrum. The realization results are obtained using an $\mathbb{R}$ -motivic analogue of the Toda realization theorem. We notice that each realization of $\mathcal{A}^{\mathbb{R}…
We compute the cohomology of the subalgebra $A^{C_2}(1)$ of the $C_2$-equivariant Steenrod algebra $A^{C_2}$. This serves as the input to the $C_2$-equivariant Adams spectral sequence converging to the $RO(C_2)$-graded homotopy groups of an equivariant spectrum $ko_{C_2}$. Our approach is to use simpler $\mathbb{C}$-motivic and $\mathbb{R}$-motivic calculations as stepping stones.
We characterize Hopf spaces with finitely generated cohomology as an algebra over the Steenrod algebra. We "deconstruct" the original space into an H-space Y with finite mod p cohomology and a finite number of p-torsion Eilenberg-Mac Lane spaces. We give a precise description of homotopy commutative H-spaces in this setting.
Bosonic SPT and invertible phases and its relation to Steenrod's problem
研究玻色子可逆和SPT相，发现高维中首个主要非平凡性是模3现象，指出这与施泰因罗德问题相关，通过托姆的配边理论分析得出迪杰格拉夫 - 维滕相在一般单纯复形和流形上的不同平凡性表现。
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…
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 cohomology of the classifying space of the underlying group or category, $\cal{C}$. In this setting the simplicial cup product of cochains on $B{\cal{C}}$ agrees with the Gerstenhaber product and Steenrod's cup-one product of cochains agrees with the pre-Lie product. We extend the idea of coherent cochains to algebras more general than category algebras and dub …
The cohomology of the connective spectra for {K}-theory revisited
针对连通实与复K-理论谱的稳定模2上同调现有基表示复杂的问题，本文基于Adem关系，用Steenrod运算的可容许序列构造了更简单的基，并明确了复情形基的次数限制条件。
We show that H-spaces with finitely generated cohomology, as an algebra or as an algebra over the Steenrod algebra, have homotopy exponents at all primes. This provides a positive answer to a question of Stanley.
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