The squaring operartion and the Singer algebraic transfer

TL;DR

This paper investigates the Singer algebraic transfer in the context of the Steenrod algebra, extending known results and confirming a conjecture for specific cases involving the squaring operation.

Contribution

It extends Hung's results on the relation between the algebraic transfer and the squaring operation, proving Singer's conjecture for k=5 and degrees of the form 5(2^s -1).

Findings

Confirmed Singer's conjecture for k=5 and degrees 5(2^s -1).

Extended the relation between algebraic transfer and squaring operation.

Provided new insights into the structure of the Steenrod algebra and its invariants.

Abstract

Let $P_k$ be the graded polynomial algebra $\mathbb F_2[x_1,x_2,\ldots ,x_k]$, with the degree of each $x_i$ being 1, regarded as a module over the mod-2 Steenrod algebra $\mathcal A$, and let $GL_k$ 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, $\text{Tor}^{\mathcal A}_{k,k+n} (\mathbb F_2,\mathbb F_2)$, to the subspace of $\mathbb F_2{\otimes}_{\mathcal A}P_k$ consisting of all the $GL_k$-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.