On the Action of Steenrod Squares on Polynomial Algebras (original) (raw)

A note on the hit problem for the Steenrod algebra and its applications

2021

Let Pk = H((RP )) be the modulo-2 cohomology algebra of the direct product of k copies of infinite dimensional real projective spaces RP . Then, Pk is isomorphic to the graded polynomial algebra F2[x1, . . . , xk] of k variables, in which each xj is of degree 1, and let GLk be the general linear group over the prime field F2 which acts naturally on Pk. Here the cohomology is taken with coefficients in the prime field F2 of two elements. We study the hit problem, set up by Frank Peterson, of finding a minimal set of generators for the polynomial algebra Pk as a module over the mod-2 Steenrod algebra, A. In this Note, we explicitly compute the hit problem for k = 5 and the degree 5(2 − 1) + 24.2 with s an arbitrary non-negative integer. These results are used to study the Singer algebraic transfer which is a homomorphism from the homology of the mod-2 Steenrod algebra, TorAk,k+n(F2, F2), to the subspace of F2 ⊗A Pk consisting of all the GLk-invariant classes of degree n. We show that ...

On the hit problem for the Steenrod algebra in some generic degrees and applications

2021

Let Pn := H((RP )) ∼= F2[x1, x2, . . . , xn] be the graded polynomial algebra over the prime field of two elements, F2. We investigate the Peterson hit problem for the polynomial algebra Pn, viewed as a graded left module over the mod-2 Steenrod algebra, A. For n > 4, this problem is still unsolved, even in the case of n = 5 with the help of computers. The purpose of this paper is to continue our study of the hit problem by developing a result in [17] for Pn in the generic degree ks = r(2 − 1) + m.2 with r = n = 5, m = 13, and s an arbitrary non-negative integer. Note that for s = 0, k0 = 5(20 −1)+13.20 = 13, and s = 1, k1 = 5(21 −1)+13.21 = 31, these problems have been studied by Phuc [16], and [17], respectively. Moreover, as an application of these results, we get the dimension result for the graded polynomial algebra in the generic degree d = (n − 1).(2n+u−1 − 1) + l.2n+u−1 with u an arbitrary non-negative integer, l ∈ {23, 67}, and in the case n = 6. One of the major applica...

The Cohomology of the Universal Steenrod Algebra

manuscripta mathematica, 2005

The mod 2 universal Steenrod algebra Q is a homogeneous quadratic algebra closely related to the ordinary mod 2 Steenrod algebra and the Lambda algebra introduced in [1]. In this paper we show that Q is Koszul. It follows by [7] that its cohomology, being purely diagonal, is isomorphic to a completion of Q itself with respect to a suitable chain of two-sided ideals.

A note on the new basis in the mod 2 Steenrod algebra

Journal of Linear and Topological Algebra, 2018

‎The Mod 222 Steenrod algebra is a Hopf algebra that consists of the primary cohomology operations‎, ‎denoted by SqnSq^nSqn‎, ‎between the cohomology groups with mathbbZ_2mathbb{Z}_2mathbbZ2 coefficients of any topological space‎. ‎Regarding to its vector space structure over mathbbZ2mathbb{Z}_2mathbbZ_2‎, ‎it has many base systems and some of the base systems can also be restricted to its sub algebras‎. ‎On the contrary‎, ‎in addition to the work of Wood‎, ‎in this paper we define a new base system for the Hopf subalgebras mathcalA(n)mathcal{A}(n)mathcalA(n) of the mod 222 Steenrod algebra which can be extended to the entire algebra‎. ‎The new base system is obtained by defining a new linear ordering on the pairs (s+t,s)(s+t,s)(s+t,s) of exponents of the atomic squares Sq2s(2t−1)Sq^{2^s(2^t-1)}Sq2s(2t−1) for the integers sgeq0sgeq 0sgeq0 and tgeq1tgeq 1tgeq1‎.

The cohomology of the Steenrod algebra and representations of the general linear groups

Transactions of the American Mathematical Society, 2005

Let T r k be the algebraic transfer that maps from the coinvariants of certain GL k-representations to the cohomology of the Steenrod algebra. This transfer was defined by W. Singer as an algebraic version of the geometrical transfer tr k : π S * ((BV k) +) → π S * (S 0). It has been shown that the algebraic transfer is highly nontrivial, more precisely, that T r k is an isomorphism for k = 1, 2, 3 and that T r = k T r k is a homomorphism of algebras. In this paper, we first recognize the phenomenon that if we start from any degree d and apply Sq 0 repeatedly at most (k − 2) times, then we get into the region in which all the iterated squaring operations are isomorphisms on the coinvariants of the GL k-representations. As a consequence, every finite Sq 0-family in the coinvariants has at most (k − 2) nonzero elements. Two applications are exploited. The first main theorem is that T r k is not an isomorphism for k ≥ 5. Furthermore, for every k > 5, there are infinitely many degrees in which T r k is not an isomorphism. We also show that if T r detects a nonzero element in certain degrees of Ker(Sq 0), then it is not a monomorphism and further, for each k > , T r k is not a monomorphism in infinitely many degrees. The second main theorem is that the elements of any Sq 0-family in the cohomology of the Steenrod algebra, except at most its first (k − 2) elements, are either all detected or all not detected by T r k , for every k. Applications of this study to the cases k = 4 and 5 show that T r 4 does not detect the three families g, D 3 and p , and that T r 5 does not detect the family {h n+1 g n | n ≥ 1}.

On triviality of Dickson invariants in the homology of the Steenrod algebra

Mathematical Proceedings of the Cambridge Philosophical Society, 2003

Let A be the mod 2 Steenrod algebra and D k the Dickson algebra of k variables. We study the Lannes-Zarati homomorphisms ϕ k : Ext k,k+i A (F 2 , F 2) −→ (F 2 ⊗ A D k) * i , which correspond to an associated graded of the Hurewicz map H: π s * (S 0) % π * (Q 0 S 0) → H * (Q 0 S 0). An algebraic version of the long-standing conjecture on spherical classes predicts that ϕ k = 0 in positive stems, for k > 2. That the conjecture is no longer valid for k = 1 and 2 is respectively an exposition of the existence of Hopf invariant one classes and Kervaire invariant one classes. This conjecture has been proved for k = 3 by H , ung [9]. It has been shown that ϕ k vanishes on decomposable elements for k > 2 [14] and on the image of Singer's algebraic transfer for k > 2 [9, 12]. In this paper, we establish the conjecture for k = 4. To this end, our main tools include (1) an explicit chain-level representation of ϕ k and (2) a squaring operation Sq 0 on (F 2 ⊗ A D k) * , which commutes with the classical Sq 0 on Ext k A (F 2 , F 2) through the Lannes-Zarati homomorphism.

On the Steenrod operations in cyclic cohomology

International Journal of Mathematics and Mathematical Sciences, 2003

For a commutative Hopf algebra A over Z/p, where p is a prime integer, we define the Steenrod operations P i in cyclic cohomology of A using a tensor product of a free resolution of the symmetric group S n and the standard resolution of the algebra A over the cyclic category according to . We also compute some of these operations.