Dual Steenrod Algebra Research Articles

Connectedness of graphs arising from the dual Steenrod algebra

We establish connectedness criteria for graphs associated to monomials in certain quotients of the mod 2 dual Steenrod algebra $$\mathscr {A}^*$$ . We also investigate questions about trees and Hamilton cycles in the context of these graphs. Finally, we improve upon a known connection between the graph theoretic interpretation of $$\mathscr {A}^*$$ and its structure as a Hopf algebra.

Quotient rings of 𝐻𝔽₂∧ℍ𝔽₂

We study modules over the commutative ring spectrum H F 2 ∧ H F 2 H\mathbb F_2\wedge H\mathbb F_2 , whose coefficient groups are quotients of the dual Steenrod algebra by collections of the Milnor generators. We show that very few of these quotients admit algebra structures, but those that do can be constructed simply: killing a generator ξ k \xi _k in the category of associative algebras freely kills the higher generators ξ k + n \xi _{k+n} . Using new information about the conjugation operation in the dual Steenrod algebra, we also consider quotients by families of Milnor generators and their conjugates. This allows us to produce a family of associative H F 2 ∧ H F 2 H\mathbb F_2\wedge H\mathbb F_2 -algebras whose coefficient rings are finite-dimensional and exhibit unexpected duality features. We then use these algebras to give detailed computations of the homotopy groups of several modules over this ring spectrum.

The Steenrod algebra from the group theoretical viewpoint

In the paper “The Steenrod algebra and its dual” [2], J. Milnor determined the structure of the dual Steenrod algebra which is a graded commutative Hopf algebra of finite type. We consider the affine group scheme Gp represented by the dual Hopf algebra of the mod p Steenrod algebra. Then, Gp assigns a graded commutative algebra A⁎ over a prime field of finite characteristic p to a set of isomorphisms of the additive formal group law over A⁎, whose group structure is given by the composition of formal power series ([4] Appendix). The aim of this paper is to show some group theoretic properties of Gp by making use of this presentation of Gp(A⁎). We give a decreasing filtration of subgroup schemes of Gp which we use for estimating the length of the lower central series of finite subgroup schemes of Gp. We also give a successive quotient maps Gp→ρ0Gp〈1〉→ρ1Gp〈2〉→ρ2⋯→ρk−1Gp〈k〉→ρkGp〈k+1〉→ρk+1⋯ of affine group schemes over a prime field Fp such that the kernel of ρk is a maximal abelian subgroup.

Secondary power operations and the Brown--Peterson spectrum at the prime 2

The dual Steenrod algebra has a canonical subalgebra isomorphic to the homology of the Brown-Peterson spectrum. We will construct a secondary operation in mod-2 homology and show that this canonical subalgebra is not closed under it. This allows us to conclude that the 2-primary Brown-Peterson spectrum does not admit the structure of an E_n-algebra for any n greater than or equal to 12, answering a question of May in the negative.

The mod 2 dual Steenrod algebra as a subalgebra of the mod 2 dual Leibniz-Hopf algebra

The mod 2 Steenrod algebra $\mathcal{A}_2$ can be defined as the quotient of the mod 2 Leibniz--Hopf algebra $\mathcal{F}_2$ by the Adem relations. Dually, the mod 2 dual Steenrod algebra $\mathcal{A}_2^*$ can be thought of as a sub-Hopf algebra of the mod 2 dual Leibniz--Hopf algebra $\mathcal{F}_2^*$. We study $\mathcal{A}_2^*$ and $\mathcal{F}_2^*$ from this viewpoint and give generalisations of some classical results in the literature.

Mathcal {E}_infty ring spectra and elements of Hopf invariant\xa01

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 Bmathrm {SO},,Bmathrm {Spin},,Bmathrm {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 the dual Steenrod algebra mathcal {A}_* with mathcal {A}_*square _{mathcal {A}(r)_*}mathbb {F}_2 as an algebra retract. This suggests that these spectra might be wedges of module spectra over the ring spectra Hmathbb {Z}, kmathrm {O} or mathrm {tmf}; however, apart from the first case, we have no concrete results on this.

Power Operations and Coactions in Highly Commutative Homology Theories

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 and Lashof interact with Steenrod operations via the Nishida relations, but for many purposes this leads to complicated calculations once iterated applications of these functions are required. On the other hand, the homology coaction turns out to provide tractable formulae better suited to exploiting multiplicative structure. We show how to derive suitable formulae for the interaction between power operations and homology coactions in a wide class of examples; our approach makes crucial use of modern frameworks for spectra with well behaved smash products. In the case of mod p homology, our formulae extend those of Bisson and Joyal to odd primes. We also show how to exploit our results in sample calculations, and produce some apparently new formulae for the Dyer-Lashof action on the dual Steenrod algebra.

Subalgebras of the $\mathbb{Z}/2$-equivariant Steenrod algebra

The aim of this paper is to study sub-algebras of the Z/2-equivariant Steenrod algebra (for cohomology with coefficients in the constant Mackey functor F 2) which come from quotient Hopf algebroids of the Z/2-equivariant dual Steenrod algebra. In particular, we study the equivariant counterpart of profile functions, exhibit the equivariant analogues of the classical A(n) and E(n) and show that the Steenrod algebra is free as a module over these.

PROPERTIES OF DUAL MODULES OVER STEENROD ALGEBRA

Properties of annulators and modules generated by annulators, including dual modules over Steenrod algebra are studied. Properties of Kroneker pairing are proved using general properties of Steenrod algebra and dual algebra as graded connected Hopf algebras. Isomorphisms between modules generated by annulators and dual modules over dual Stennrod algebra are proved. It is shown that these modules are Hopf comodules induced by coproduct in dual Steenrod algebra. All generators of these modules are found. The method of finding basis of module of indecomposable elements, viewed as vector space over cyclic field for some of the studied modules

Some quotient Hopf algebras of the dual Steenrod algebra

Fix a prime p p , and let A A be the polynomial part of the dual Steenrod algebra. The Frobenius map on A A induces the Steenrod operation P ~ 0 \widetilde {\mathscr {P}}^{0} on cohomology, and in this paper, we investigate this operation. We point out that if p = 2 p=2 , then for any element in the cohomology of A A , if one applies P ~ 0 \widetilde {\mathscr {P}}^{0} enough times, the resulting element is nilpotent. We conjecture that the same is true at odd primes, and that “enough times” should be “once.” The bulk of the paper is a study of some quotients of A A in which the Frobenius is an isomorphism of order n n . We show that these quotients are dual to group algebras, the resulting groups are torsion-free, and hence every element in Ext over these quotients is nilpotent. We also try to relate these results to the questions about P ~ 0 \widetilde {\mathscr {P}}^{0} . The dual complete Steenrod algebra makes an appearance.

Corrigendum to ‘Hit polynomials and conjugation in the dual Steenrod algebra’

Corrigendum to ‘Hit polynomials and conjugation in the dual Steenrod algebra’

On conjugation invariants in the dual Steenrod algebra

We investigate the canonical conjugation, , of the mod 2 dual Steenrod algebra, A, with a view to determining the subspace, A ,o f ele- ments invariant under . We give bounds on the dimension of this subspace for each degree and show that, after inverting 1, it becomes polynomial on a natural set of generators. Finally we note that, without inverting 1,A is far from being polynomial.

Problems in the Steenrod Algebra

This article contains a collection of results and problems about the Steenrod algebra and related algebras acting on polynomials which non-specialists in topology may find of some interest. Although there are topological allusions throughout the article, the emphasis is on the algebraic development of the Steenrod algebra and its connections to the various topics indicated below. Contents 1 Historical background 452 2 Differential operators and integral Steenrod squares 460 3 Symmetric functions and differential operators 471 4 Bases, excess and conjugates 476 5 The stripping technique 483 6 Iteration theory and nilpotence 490 7 The hit problem and invariant theory 495 8 The dual of A(n) and graph theory 504 9 The Steenrod group 507 10 Computing in the Steenrod algebra 508 References 511 In Section 1 the scene is set with a few remarks on the early history of the Steenrod algebra A at the prime 2 from a topologist's point of view, which puts into context some of the problems posed later. In Section 2 the subject is recast in an algebraic framework, by citing recent work on integral versions of the Steenrod algebra defined in terms of differential operators. In Section 3 there is an explanation of how the divided differential operator algebra D relates to the classical theory of symmetric functions. In Section 4 some comments are made on a few of the recently discovered bases for the Steenrod algebra. The stripping technique in Section 5 refers to a standard action of a Hopf algebra on its dual, which is particularly useful in the case of the Steenrod algebra for deriving relations from relations when implemented on suitable bases. In Section 6 a parallel is drawn between certain elementary aspects of the iteration theory of quadratic polynomials and problems about the nilpotence height of families of elements in the Steenrod algebra. The hit problem in Section 7 refers to the general question in algebra of finding necessary and sufficient conditions for an element in a graded module over a graded ring to be decomposable into elements of lower grading. Equivariant versions of this problem with respect to general linear groups over finite fields have attracted attention in the case of the Steenrod algebra acting on polynomials. Similar problems arise with respect to the symmetric groups and the algebra D. This subject relates to topics in classical invariant theory and modular representation theory. In Section 8 a number of statements about the dual Steenrod algebra are transcribed into the language of graph theory. In Section 9 a standard method is employed for passing from a nilpotent algebra over a finite field of characteristic p to a p-group, and questions are raised about the locally finite 2-groups that arise in this way from the Steenrod algebra. Finally, in Section 10 there are a few comments on the use of a computer in evaluating expressions and testing relations in the Steenrod algebra. 1991 Mathematics Subject Classification 55S10.

Hit polynomials and conjugation in the dual Steenrod algebra

Let [Ascr ]* be the mod-2 Steenrod algebra of cohomology operations and χ its canonical antiautomorphism. For all positive integers f and k, we show that the excess of the element χ[Sq (2k−1f)· Sq (2k−2f)… Sq (2f)·Sq (f)] is (2k−1)μ(f), where μ(f) denotes the minimal number of summands in any representation of f as a sum of numbers of the form 2i−1. We also interpret this result in purely combinatorial terms. In so doing, we express the Milnor basis representation of the products Sq (a1)…Sq (an) and χ[Sq (a1)…Sq (an)] in terms of the cardinalities of certain sets of matrices.For s[ges ]1, let ℙs=[ ]2= [x1, …, xs] be the mod-2 cohomology of the s-fold product of ℝP∞ with itself, with its usual structure as an [Ascr ]*-module. A polynomial P∈ℙs is hit if it is in the image of the action [Ascr ]*×ℙs→ℙs, where [Ascr ]* is the augmentation ideal of [Ascr ]*. We prove that if the integers e, f, and k satisfy e<(2k−1)μ(f), then for any polynomials E and F of degrees e and f respectively, the product E·F2k is hit. This generalizes a result of Wood conjectured by Peterson, and proves a conjecture of Singer and Silverman.

Stripping and conjugation in the Steenrod algebra

Let S( k; f) = Sq(2 k−1 f) · Sq(2 k−2 f)… Sq(2 f) · Sq( f) in the mod-2 Steenrod algebra A ∗, and let χ denote the canonical antiautomorphism of A ∗. Given positive integers k, Λ and j with 1 ≤ j ≤ Λ, we prove that χS( k;2 Λ − j) = S( Λ − ( j − 1); 2 j − 1 (2 k − 1)) · χS( k; 2 j−1 − j), generalizing formulae of Davis and the author. Our proof relies on the “stripping” action of the dual Steenrod algebra A ∗ or A ∗ itself, which we identify as a special case of a general Hopf algebra phenomenon. Given a positive integer f, denote by μ(f) the minimal number of summands in any representation of f in the form ∑(2 i k − 1). The antiautomorphism formula above implies that for f = 2 Λ − j, 1 ≤ j ≤ Λ + 2, the excess of χS( k; f) satisfies ex( χS( k; f)) = (2 k − 1) μ( f) for all k, confirming the conjecture of the author (Silverman, 1993) for such f. We also prove that ex( χS( k; f)) ≤ (2 k − 1) μ( f) for all f and k.

The mod 2 homology of the space of loops on the exceptional Lie group

SynopsisThe Hopf algebra structure of H*(ΩG, F2) and the action of the dual Steenrod algebra are completely and explicitly determined when G isone of the connected, simply connected, exceptional, simple Lie groups. The approach is homological, using connected coverings and spectral sequences.

The Homotopy Type of MSU

0. Introduction. This paper examines the homotopy type of the Thom spectrum MSU associated with special unitary cobordism. For odd primes p, standard methods show that the p-localization MSU(p) is equivalent to a wedge of suspensions of the Brown-Peterson spectrum BP. Forp = 2, however, this is not the case, and our work is devoted to determining the 2-primary homotopy type of MSU. This involves a new indecomposable spectrum, and our main results are the following. There is an indecomposable 2-local spectrum, which we call BoP, such that MSU(2) is equivalent to a wedge of suspensions of BoP and BP. Under the equivalence, the Thom class lies in a BoP summand. As a comodule over the dual Steenrod algebra A [11], H*(BoP; Z/2) is a sum of 4 '2 2 suspensions of B = Z/2[?1, t2, ..., j, . ... C A, where tj is the conjugate of Milnor's generator (j. There is one suspension of B beginning in each nonnegative dimension divisible by 8. BoP bears strong similarities to BP and the (1)-connected K-theory spectra bo and bu. In particular, in Section 6 we show there is a map BoP bo(2) inducing an epimorphism v* of homotopy groups. In fact, v* induces an isomorphism of torsion subgroups, and its torsion free kernel is concentrated in even dimensions. A brief summary of our methods is as follows. In Sections 1 and 2 we describe the Adams spectral sequence for 7r*MSU(2), including a computation of the differentials, with particular attention paid to the product structure. Anderson, Brown, and Peterson [4] gave a computation for these differentials, but their proof requires some correction, and in any case we will need the more extensive knowledge of the product structure. In Sections 3 to 5, we construct BoP and show it is indecomposable. To produce BoP, first the Sullivan-Baas construction is applied to MSU, yielding a spectrum representing a bordism theory of SU-manifolds with