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

Authors : Nguyen Khac Tin

Pages : 1135-1150

Doi:10.15672/hujms.1127140

View : 415 | Download : 333

Publication Date : 2023-10-31

Article Type : Research Paper

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, $\\text{Tor}^{\\mathcal A}_{n, n+*}(\\mathbb F_2,\\mathbb F_2).$
Keywords : Steenrod algebra, polynomial algebra, hit problem, graded rings, Steenrod algebra, polynomial algebra, hit problem, graded rings

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