On modules over the mod 2 Steenrod algebra and hit problems

TL;DR

This paper advances understanding of the hit problem for modules over the mod 2 Steenrod algebra, specifically focusing on five variables, providing new methods and applications in algebraic topology and modular representation theory.

Contribution

It develops an efficient approach to solve the hit problem for five variables in specific degrees and explores applications to dimensions and representations related to the Steenrod algebra.

Findings

Solved the hit problem for $ extbf{P}_5$ in certain degrees

Determined the dimension of $ extbf{P}_6$ in related degrees

Showed the cohomological transfer is an isomorphism in specific bidegrees

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.