On the determination of the Singer 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 generator $x_i$ being 1, where ${\mathbb{F}}_2$ denote the prime field of two elements, and let $G{L}_k$ be the general linear group over ${\mathbb{F}}_2$ which acts regularly on $P_k$. We study the algebraic transfer $T{r}_k^{\ast }$ 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 $\mathcal{A}$, ${\text{Tor}}_{k,k+d}^{\mathcal{A}}({\mathbb{F}}_2,{\mathbb{F}}_2)$, to the subspace of ${\mathbb{F}}_2{\otimes }_{\mathcal{A}}{P}_k$ consisting of all the $G{L}_k$-invariant classes of degree $d$. 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 $k⩽3$. 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.

19 pages. In this version, we correct some errors in the statement of Theorem 4.1 of the original paper [VJSTE, 60 (1)(2018), 3-16] and clarify some details in its proof. arXiv admin note: text overlap with arXiv:1609.02250 by other authors

References in corpus (3)

• Sub-Hopf algebras of the Steenrod algebra and the Singer transfer
• On the Peterson hit problem
• The hit problem for the polynomial algebra of four variables

Cited by in corpus (6)

• The "hit" problem of five variables in the generic degree and its application
• On a construction for the generators of the polynomial algebra as a module over the Steenrod algebra
• The affirmative answer to Singer's conjecture on the algebraic transfer of rank four
• On Singer's conjecture for the fourth algebraic transfer in certain generic degrees
• A note on the modular representation on the $\mathbb{Z}/2$-homology groups of the fourth power of real projective space and its application
• Structure of the space of $G{L}_4({\mathbb{Z}}_2)$-coinvariants ${\mathbb{Z}}_2{\otimes }_{G{L}_4({\mathbb{Z}}_2)}P{H}_{\ast }({\mathbb{Z}}_2^4,{\mathbb{Z}}_2)$ in some generic degrees and its application