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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

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 A is an interesting and important open problem of Algebraic topology, which asks a minimal set of generators for the polynomial algebra $\mathcal P_m:=\mathbb F_2[x_1, x_2, \ldots, x_m]$ on $m$ variables $x_1, \ldots, x_m,$ each of 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 $K(\mathbb F_2, 1).$ Although the hit problem has been thoroughly studied for more than 3 decades, solving it remains a mystery for any $m\geq 5.$ 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 $n:=n_s = 5(2^{s}-1) + 18.2^{s}$ with $s\geq 0.$ As an immediate consequence, a local version of Kameko’s conjecture for the dimension of the cohit space $\mathbb F_2\otimes_{\mathscr A}\mathcal P_m$ associated with weight vectors is true for $m = 5$ and degree $n_s.$ Also, we show that this conjecture also holds for any $m\geq 1$ and degrees $\leq 12.$ Two applications of this study are to establish 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 to describe the modular representations of the general linear group of rank $5$ over $\mathbb F_2.$ Our results then show that the algebraic transfer, defined by W. Singer [Math. Z. 202 (1989), 493-523], is an isomorphism in bidegrees $(5, 5+n_s)$ with $s\geq 0.$ Also, we prove that Singer’s transfer is a trivial isomorphism in bidegrees $(m, m+12)$ with $m\geq 1.$