On modules over the mod 2 Steenrod algebra and hit problems

Abstract

<p>Let us consider the prime field of two elements, $\mathbb F_2\equiv \mathbb Z_2.$ The classical "hit problem" in Algebraic Topology, which is widely recognized as an important and intriguing open problem, asks for a minimal set of generators for the polynomial algebra, $\mathcal P_m:=\mathbb F_2[x_1, x_2, \ldots, x_m]$, on $m$ variables $x_1, \ldots, x_m$, each of which has degree one, regarded as a connected unstable $\mathscr A$-module. The algebra $\mathcal P_m$ is the cohomology with $\mathbb F_2$-coefficients of the product of $m$ copies of the Eilenberg-MacLan complex $K(\mathbb F_2, 1)$. Despite extensive study over the past three decades, the hit problem remains unresolved for $m\geq 5$. </p> <p>In this article, we develop our previous work [Commun. Korean Math. Soc. 35 (2020), 371-399] on the hit problem for $\mathscr A$-module $\mathcal P_5$ in generic degree $n_s = 5(2^{s}-1) + 18.2^{s}$ with $s$ an arbitrary non-negative integer. As a result, a local version of Kameko's conjecture, which concerns the dimension of the cohit space $\mathbb F_2\otimes_{\mathscr A}\mathcal P_m$ in relation to weight vectors, has been confirmed for the case where $m = 5$ and the degree is $n_s.$ Also, we show that this conjecture also holds for any $m\geq 1$ and degrees $\leq 12.$ This study has two important applications: (1) it establishes the dimension result for the cohit space $\mathbb F_2\otimes_{\mathscr A}\mathcal P_m$ for $m = 6$ in generic degree $5(2^{s+4}-1) + n_1.2^{s+4}$ with $s > 0;$ and (2) it describes the representations of the general linear group of rank $5$ over $\mathbb F_2.$ As a result, we prove that the algebraic transfer, defined by W. Singer [Math. Z. 202 (1989), 493-523], is an isomorphism in bidegrees $(5, 5+n_s)$ with $s\geq 0.$ Besides, we obtain new results on the behavior of this algebra transfer for all homological degrees $m$. Specifically, we demonstrate that Singer's transfer is a trivial isomorphism in bidegree $(m, m+12)$ for any $m > 0$.</p>