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

Corresponding Author:[email protected]

Author Profile## 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 $m\leq 3.$