On the hit problem for the polynomial algebra and the algebraic transfer
Authors: Võ Phúc Đặng 
Published: August 2023

<p>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.</p>
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<p>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.</p>