On modules over the mod 2 Steenrod algebra and hit problems

Abstract

Let us consider the prime field of two elements, $\mathbb F_2\equiv \mathbb Z_2.$ It is well-known that the classical "hit problem" for a module over the mod 2 Steenrod algebra $\mathscr A$ is an interesting and important open problem of Algebraic topology, which asks a minimal set of generators for the polynomial algebra $\mathcal P_m:=\mathbb F_2[x_1, x_2, \ldots, x_m]$, regarded as a connected unstable $\mathscr A$-module on $m$ variables $x_1, \ldots, x_m,$ each of degree 1. The algebra $\mathcal P_m$ is the $\mathbb F_2$-cohomology of the product of $m$ copies of the Eilenberg-MacLan complex $K(\mathbb F_2, 1).$ Although the hit problem has been thoroughly studied for more than 3 decades, solving it remains a mystery for $m\geq 5.$ Our intent in this work is of studying the hit problem of five variables. More precisely, 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 a degree of the generic form $n_t:=5(2^t-1) + 18.2^t,$ for any non-negative integer $t.$ An efficient approach to solve this problem had been presented. Two applications of this study are to determine the dimension of $\mathcal P_6$ in the generic degree $5(2^{t+4}-1) + n_1.2^{t+4}$ for all $t > 0$ and to describe the modular representations of the general linear group of rank 5 over $\mathbb F_2.$ As a corollary, the cohomological "transfer", defined by William Singer [Math. Z. 202 (1989), 493-523], is an isomorphism in bidegree $(5, 5+n_0).$ Singer's transfer is one of the relatively efficient tools to approach the structure of mod-2 cohomology of the Steenrod algebra.