On the hit problem for the polynomial algebra and the algebraic transfer
Authors: Phúc Đặng Võ 
Published: September 2022

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.\medskipThe 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 $n_{s+5},\, s &gt; 0.$ 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 imp