Dyadic 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 ...

Journal of Homotopy and Related Structures

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 2 Adem relations at the cochain level. Specifically, given a mod 2 cocycle, we produce explicit cochain formulae whose coboundaries are the Adem relations among compositions of Steenrod Squares applied to the cocycle, using Steenrod's original cochain definition of the Square operations.

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.

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...

Mathematical Notes, 1992

Applied Mathematical Sciences, 2014

As recently observed by the second author, the mod2 universal Steenrod algebra Q has a fractal structure given by a system of nested subalgebras Q s , for s ∈ N, each isomorphic to Q. In the present paper we provide an alternative presentation of the subalgebras Q s through suitable derivations δ s , and give an invariant-theoretic description of them.

We investigate the Hopf algebra conjugation, χ, of the mod 2 Steenrod algebra, A 2 , in terms of the Hopf algebra conjugation, χ ′ , of the mod 2 Leibniz-Hopf algebra. We also investigate the fixed points of A 2 under χ and their relationship to the invariants under χ ′ .

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}.

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.

Journal of Linear and Topological Algebra, 2019

‎Let $mathcal{A}_p$ be the mod $p$ Steenrod algebra‎, ‎where $p$ is an odd prime‎, ‎and let $mathcal{A}$ be the‎ subalgebra $mathcal{A}$ of $mathcal{A}_p$ generated by the Steenrod $p$th powers‎. ‎We generalize the $X$-basis in $mathcal{A}$ to $mathcal{A}_p$‎.