Determination of the fifth Singer algebraic transfer in some degrees

TL;DR

This paper proves that the fifth Singer algebraic transfer is an isomorphism in degrees 20 and 30, using Peterson hit problem results, correcting a previous proof error for degree 20.

Contribution

It establishes the isomorphism of the fifth Singer algebraic transfer in specific degrees, providing new proofs and correcting prior inaccuracies.

Findings

Proves the fifth Singer algebraic transfer is an isomorphism at degree 20.

Proves the fifth Singer algebraic transfer is an isomorphism at degree 30.

Refutes previous proof for degree 20 in Phúc [17].

Abstract

Let $P_k$ be the graded polynomial algebra $\mathbb F_2[x_1,x_2,\ldots ,x_k]$ over the prime field $\mathbb F_2$ with two elements and the degree of each variable $x_i$ being 1, and let $GL_k$ be the general linear group over $\mathbb F_2$ which acts on $P_k$ as the usual manner. The algebra $P_k$ is considered as a module over the mod-2 Steenrod algebra $\mathcal A$. In 1989, Singer [22] defined the $k$-th homological algebraic transfer, which is a homomorphism $$\varphi_k :{\rm Tor}^{\mathcal A}_{k,k+d} (\mathbb F_2,\mathbb F_2) \to (\mathbb F_2\otimes_{\mathcal A}P_k)_d^{GL_k}$$ from the homological group of the mod-2 Steenrod algebra $\mbox{Tor}^{\mathcal A}_{k,k+d} (\mathbb F_2,\mathbb F_2)$ to the subspace $(\mathbb F_2\otimes_{\mathcal A}P_k)_d^{GL_k}$ 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 of the Peterson hit problem we present the proof of the fact that the Singer algebraic transfer of rank five is an isomorphism in the internal degrees $d= 20$ and $d = 30$. Our result refutes the proof for the case of $d=20$ in Ph\'uc [17].