The Squaring Operation and the Singer Algebraic Transfer

Abstract

Let Pk be the graded polynomial algebra $\mathbb {F}_{2}[x_{1},x_{2},{\ldots } ,x_{k}]$ , with the degree of each xi being 1, regarded as a module over the mod-2 Steenrod algebra $\mathcal A$ , and let GLk be the general linear group over the prime field $\mathbb F_{2}$ which acts regularly on Pk. 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, $\text {Tor}^{\mathcal A}_{k,k+d} (\mathbb F_{2},\mathbb F_{2})$ , to the subspace of $\mathbb F_{2}{\otimes }_{\mathcal A}P_{k}$ consisting of all the GLk-invariant classes of degree d. In this paper, we extend a result of Hưng on the relation between the Singer algebraic transfer and the classical 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(2s − 1) with s an arbitrary positive integer.