On the hit problem for the polynomial algebra and the algebraic transfer

Let $\mathcal A$ be the classical, singly-graded
Steenrod algebra over the prime order field $\mathbb
F_2$ and let $P^{\otimes h}: =
\mathbb F_2[t_1, \ldots, t_h]$
denote the polynomial algebra on $h$ generators, each of degree $1.$
Write $GL_h$ for the usual general linear group of rank $h$ over
$\mathbb F_2.$ Then, $P^{\otimes
h}$ is an $\mathcal A[GL_h]$-module. As is well
known, for all homological degrees $h \geq 6$, the
cohomology groups ${\rm
Ext}_{\mathcal A}^{h,
h+\bullet}(\mathbb F_2,
\mathbb F_2)$ of the algebra $\mathcal
A$ are still shrouded in mystery. The algebraic transfer
$Tr_h^{\mathcal A}: (\mathbb
F_2\otimes_{GL_h}{\rm
Ann}_{\overline{\mathcal
A}}[P^{\otimes
h}]^{*})_{\bullet}\longrightarrow
{\rm Ext}_{\mathcal A}^{h,
h+\bullet}(\mathbb F_2,
\mathbb F_2)$ of rank $h,$ constructed by W. Singer
[Math. Z. \textbf{202} (1989), 493-523], is a
beneficial technique for describing the Ext groups. Singer’s conjecture
about this transfer states that \textit{it is always a
one-to-one map}. Despite significant effort, neither a complete proof
nor a counterexample has been found to date. The unresolved nature of
the conjecture makes it an interesting topic of research in Algebraic
topology in general and in homotopy theory in particular.

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The objective of this paper is to investigate Singer’s conjecture, with
a focus on all $h\geq 1$ in degrees
$n\leq 10 = 6(2^{0}-1) + 10\cdot
2^{0}$ and for $h=6$ in the general degree
$n:=n_s=6(2^{s}-1) + 10\cdot
2^{s},\, s\geq 0.$ Our methodology
relies on the hit problem techniques for the polynomial algebra
$P^{\otimes h}$, which allows us to investigate
the Singer conjecture in the specified degrees. Our work is a
continuation of the work presented by Mothebe et al. [J. Math. Res.
\textbf{8} (2016), 92-100] with regard to the hit
problem for $P^{\otimes 6}$ in degree $n_s$,
expanding upon their results and providing novel contributions to this
subject. More generally, for $h\geq 6,$ we show that
the dimension of the cohit module $\mathbb
F_2\otimes_{\mathcal
A}P^{\otimes h}$ in degrees $2^{s+4}-h$ is
equal to the order of the factor group of $GL_{h-1}$ by the Borel
subgroup $B_{h-1}$ for every $s\geq h-5.$
Especially, for the Galois field $\mathbb F_{q}$
($q$ denoting the power of a prime number), based on Hai’s recent work
[C. R. Math. Acad. Sci. Paris \textbf{360} (2022),
1009-1026], we claim that the dimension of the space of the
indecomposable elements of $\mathbb F_q[t_1,
\ldots t_h]$ in general degree $q^{h-1}-h$ is
equal to the order of the factor group of
$GL_{h-1}(\mathbb F_q)$ by a subgroup of the Borel
group $B_{h-1}(\mathbb F_q).$ As applications, we
establish the dimension result for the cohit module
$\mathbb
F_2\otimes_{\mathcal
A}P^{\otimes 7}$ in degrees
$n_{s+5},\, s > 0.$ Simultaneously, we
demonstrate that the non-zero elements $h_2^{2}g_1 =
h_4Ph_2\in {\rm
Ext}_{\mathcal A}^{6,
6+n_1}(\mathbb F_2, \mathbb F_2)$ and
$D_2\in {\rm
Ext}_{\mathcal A}^{6,
6+n_2}(\mathbb F_2, \mathbb F_2)$ do
not belong to the image of the sixth Singer algebraic transfer,
$Tr_6^{\mathcal A}.$ This discovery holds
significant implications for Singer’s conjecture concerning algebraic
transfers. We further deliberate on the correlation between these
conjectures and antecedent studies, thus furnishing a comprehensive
analysis of their implications.

Jan 2023