A note on the $\mathcal{A}$-generators of the polynomial algebra of six variables and applications

Abstract

Let $ \mathcal P_{n}:=H^{*}((\mathbb{R}P^{\infty})^{n}) \cong \mathbb Z_2[x_{1},x_{2},\ldots,x_{n}]$ be the polynomial algebra of $n$ generators $x_1, x_2, \ldots, x_n$ with the degree of each $x_i$ being 1. We investigate the Peterson hit problem for the polynomial algebra $ \mathcal P_{n},$ regarded as a module over the mod-$2$ Steenrod algebra, $ \mathcal{A}.$ For $n>4,$ this problem remains unsolvable, even with the aid of computers in the case of $n=5.$ In this article, we study the hit problem for the case $n=6$ in degree $d_s=6(2^s -1)+3.2^s,$ with $s$ an arbitrary nonnegative integer. By considering $ \mathbb Z_2$ as a trivial $ \mathcal A$-module, then the hit problem is equivalent to the problem of finding a basis of $ \mathbb Z_2$-vector space $ \mathbb Z_2 {\otimes}_{\mathcal{A}}\mathcal P_{n}.$ The main goal of the current article is to explicitly determine an admissible monomial basis of the $ \mathbb Z_2$vector space $ \mathbb Z_2{\otimes}_{\mathcal{A}}\mathcal P_n$ for $n=6$ in some degrees. One of the most important applications of the hit problem is to investigate homomorphism introduced by Singer, which is a homomorphism $$ \varphi_n :\text{Tor}^{\mathcal A}_{n, n+d} (\mathbb Z_2,\mathbb Z_2) \longrightarrow (\mathbb{Z}_{2}{\otimes}_{\mathcal{A}}\mathcal P_{n})_d^{GL(n; \mathbb Z_2)}$$ from the homology of the Steenrod algebra to the subspace of $(\mathbb{Z}_{2}{\otimes}_{\mathcal{A}}\mathcal P_{n})_d$ consisting of all the $GL(n; \mathbb Z_2)$-invariant classes. It is a useful tool in describing the homology groups of the Steenrod algebra, $\text{Tor}^{\mathcal A}_{n, n+d}(\mathbb Z_2,\mathbb Z_2).$ The behavior of the sixth Singer algebraic transfer in degree $d_s=6(2^s -1)+3.2^s$ was also discussed at the end of this paper.