On the hit problem for the Steenrod algebra in some generic degrees and applications

Abstract

Let ${\mathcal{P}}_n:=H^{\ast }((\mathbb{R}{P}^{\infty }{)}^n)\cong {\mathbb{F}}_2[x_1,x_2,\dots ,x_n]$ be the polynomial algebra over the prime field of two elements, ${\mathbb{F}}_2.$ We investigate the Peterson hit problem for the polynomial algebra ${\mathcal{P}}_n,$ viewed as a graded left module over the mod-$2$ Steenrod algebra, $\mathcal{A}.$ For $n>4,$ this problem is still unsolved, even in the case of $n=5$ with the help of computers. The purpose of this paper is to continue our study of the hit problem by developing a result in \cite{ph31} for ${\mathcal{P}}_n$ in the generic degree $r(2^s-1)+m.2^s$ where $r=n=5,m=13,$ and $s$ is an arbitrary non-negative integer. Note that for $s=0,$ and $s=1,$ this problem has been studied by Phuc \cite{ph20ta}, and \cite{ph31}, respectively. As an application of these results, we get the dimension result for the polynomial algebra in the generic degree $d=(n-1).(2^{n+u-1}-1)+\ell .2^{n+u-1}$ where $u$ is an arbitrary non-negative integer, $\ell \in \{23,67\},$ and $n=6.$ One of the major applications of hit problem is in surveying a homomorphism introduced by Singer, which is a homomorphism $$T{r}_n:{\text{Tor}}_{n,n+d}^{\mathcal{A}}({\mathbb{F}}_2,{\mathbb{F}}_2)⟶({\mathbb{F}}_2{\otimes }_{\mathcal{A}}{\mathcal{P}}_n{)}_d^{GL(n;{\mathbb{F}}_2)}$$ from the homology of the Steenrod algebra to the subspace of $({\mathbb{F}}_2{\otimes }_{\mathcal{A}}{\mathcal{P}}_n{)}_d$ consisting of all the $GL(n;{\mathbb{F}}_2)$-invariant classes. It is a useful tool in describing the homology groups of the Steenrod algebra, ${\text{Tor}}_{n,n+d}^{\mathcal{A}}({\mathbb{F}}_2,{\mathbb{F}}_2).$ The behavior of the fifth Singer algebraic transfer in degree $5(2^s-1)+13.2^s$ was also discussed at the end of this paper.

12 pages. arXiv admin note: text overlap with arXiv:2103.04393

References in corpus (1)

• The "hit" problem of five variables in the generic degree and its application