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On modules over the mod 2 Steenrod algebra and hit problems
  • Đặng Võ Phúc
Đặng Võ Phúc
University of Khanh Hoa, University of Khanh Hoa, University of Khanh Hoa, University of Khanh Hoa, University of Khanh Hoa, University of Khanh Hoa, University of Khanh Hoa, University of Khanh Hoa, University of Khanh Hoa, University of Khanh Hoa, University of Khanh Hoa, University of Khanh Hoa, University of Khanh Hoa, University of Khanh Hoa, University of Khanh Hoa, University of Khanh Hoa

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Abstract

We will look at the binary field $\mathbb F_2$.  The classical “hit problem” in algebraic topology, which is widely considered to be an important and fascinating open problem that has yet to be resolved, 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 module over the 2-primary Steenrod algebra $\mathscr A.$ 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 $m\geq 5$.
In this article, we develop our previous work [Commun. Korean Math. Soc. \textbf{35} (2020), 371-399] on the hit problem for the $\mathscr A$-module $\mathcal P_5$ in the generic degree $n_s = 5(2^{s}-1) + 18.2^{s}$ with $s$ an arbitrary non-negative integer. As a consequence, a localized variation  of the Kameko conjecture, which concerns the dimension of the cohit space $\mathbb F_2\otimes_{\mathscr A}\mathcal P_m$ in relation to parameter vectors, has been claimed to be veracious in the instance where $m = 5$ and the degree is $n_s.$ Also, we demonstrate that this conjecture remains valid for all $m\geq 1$ 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 the 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 William Singer [Math. Z. \textbf{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 algebraic transfer for all homological degrees $m$. Specifically, we show that Singer’s transfer is a trivial isomorphism in bidegree $(m, m+12)$ for any $m > 0$. At the end of this work, we discuss the hit problem for the symmetric polynomial algebra $\mathcal P_m^{\Sigma_m}.$ This topic has been previously studied by Ali Janfada and Reginald Wood for $m\leq 3.$