The squaring operartion and the Singer algebraic transfer

Abstract

Let $P_k$ be the graded polynomial algebra ${\mathbb{F}}_2[x_1,x_2,\dots ,x_k]$, with the degree of each $x_i$ being 1, regarded as a module over the mod-2 Steenrod algebra $\mathcal{A}$, and let $G{L}_k$ be the general linear group over the prime field ${\mathbb{F}}_2$ which acts regularly on $P_k$. We study the algebraic transfer constructed by Singer using the technique of the hit problem. This transfer is a homomorphism from the homology of the mod-2 Steenrod algebra, ${\text{Tor}}_{k,k+n}^{\mathcal{A}}({\mathbb{F}}_2,{\mathbb{F}}_2)$, to the subspace of ${\mathbb{F}}_2{\otimes }_{\mathcal{A}}{P}_k$ consisting of all the $G{L}_k$-invariant classes of degree $n$. In this paper, we extend a result of Hung on the relation between the Singer algebraic transfer and the squaring operation on the cohomology of the Steenrod algebra. Using this result, we show that Singer's conjecture for the algebraic transfer is true in the case $k=5$ and the degree $5(2^s-1)$ with $s$ an arbitrary positive integer.

38 pages. Theorems 1.3 and 1.8 of this paper have already been announced in Comptes Rendus Mathematique, Volume 354, Issue 9, 2016. The readers can find some papers related to this paper, available online at arXiv:1609.02250, arXiv:1607.01095, arXiv:1502.05569, arXiv:1412.3309, arXiv:1412.1709. arXiv admin note: substantial text overlap with arXiv:1609.02250

References in corpus (5)

• On the generators of the polynomial algebra as a module over the Steenrod algebra
• Sub-Hopf algebras of the Steenrod algebra and the Singer transfer
• On the Peterson hit problem
• On a minimal set of generators for the polynomial algebra of five variables as a module over the Steenrod algebra
• On behavior of the fifth algebraic transfer

Cited by in corpus (1)

• On a construction for the generators of the polynomial algebra as a module over the Steenrod algebra