On the determination of the Singer transfer

TL;DR

This paper proves the Singer algebraic transfer is an isomorphism for up to three variables, explicitly determines the fourth transfer in some degrees, and discusses computational methods related to the Peterson hit problem.

Contribution

It establishes the isomorphism of the Singer transfer for k ≤ 3 and explicitly computes the transfer for k=4 in certain degrees, advancing understanding of the algebraic transfer.

Findings

Proves the Singer algebraic transfer is an isomorphism for k ≤ 3.

Explicitly determines the fourth transfer in some degrees.

Provides computational evidence related to the image of the transfer.

Abstract

Let $P_k$ be the graded polynomial algebra $\mathbb F_2[x_1,x_2,\ldots ,x_k]$ with the degree of each generator $x_i$ being 1, where $\mathbb F_2$ denote the prime field of two elements, and let $GL_k$ be the general linear group over $\mathbb F_2$ which acts regularly on $P_k$. We study the algebraic transfer $Tr_k^*$ constructed by Singer using the technique of the Peterson hit problem. This transfer is a homomorphism from the homology of the mod-2 Steenrod algebra $\mathcal A$, $\text{Tor}^{\mathcal A}_{k,k+d} (\mathbb F_2,\mathbb F_2)$, to the subspace of $\mathbb F_2{\otimes}_{\mathcal A}P_k$ consisting of all the $GL_k$-invariant classes of degree $d$. In this paper, by using the results on the Peterson hit problem we present the proof of the fact that the Singer algebraic transfer is an isomorphism for $k \leqslant 3$. We also explicitly determine the fourth Singer algebraic transfer in some degrees. The new results in the paper are different from the ones of Bruner, Ha and Hung [5], Chon and Ha [6,7,8], Ha [9], Hung and Quynh [12], Nam [16]. To illustrate the fact that $d_0 \in \mbox{Im}(Tr_4)$, we present the computations of Ha [9, Page 102] for this result. We can easily verify that these computations are correct. So, it is possible the algorithm in Phuc [29] is flawed.