On a minimal set of generators for the algebra $H^*(BE_6; \mathbb F_2)$ as a module over the Steenrod algebra and applications

Abstract

Let $\mathcal P_n \cong H^{*}\big(BE_n; \mathbb F_2 \big)$ be the graded polynomial algebra over the prime field of two elements $\mathbb F_2$, where $E_n$ is an elementary abelian 2-group of rank $n,$ and $BE_n$ is the classifying space of $E_n.$ We study the {\it hit problem}, set up by Frank Peterson, of finding a minimal set of generators for the polynomial algebra $\mathcal P_{n},$ viewed as a module over the mod-2 Steenrod algebra $\mathcal{A}$. This problem remains unsolvable for $n>4,$ even with the aid of computers in the case of $n=5.$ By considering $\mathbb F_2$ as a trivial $\mathcal A$-module, then the hit problem is equivalent to the problem of finding a basis of $\mathbb F_2$-graded vector space $\mathbb F_2 {\otimes}_{\mathcal{A}}\mathcal P_{n}.$ This paper aims to explicitly determine an admissible monomial basis of the $ \mathbb{F}_{2}$-vector space $\mathbb{F}_{2}{\otimes}_{\mathcal{A}}\mathcal P_{n}$ in the generic degree $n(2^{r}-1)+2\cdot 2^{r},$ where $r$ is an arbitrary non-negative integer, and in the case of $n=6.$ As an application of these results, we obtain the dimension results for the polynomial algebra $\mathcal P_n$ in degrees $(n-1)\cdot(2^{n+u-1}-1)+\ell\cdot2^{n+u},$ where $u$ is an arbitrary non-negative integer, $\ell =13,$ and $n=7.$ Moreover, for any integer $r>1,$ the behavior of the sixth Singer algebraic transfer in degree $6(2^{r}-1)+2\cdot2^r$ is also discussed at the end of this paper. Here, the Singer algebraic transfer is a homomorphism from the homology of the Steenrod algebra to the subspace of $\mathbb{F}_{2}{\otimes}_{\mathcal{A}}\mathcal P_{n}$ 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, $\text{Tor}^{\mathcal A}_{n, n+*}(\mathbb F_2,\mathbb F_2).$