Singer's algebraic transfer for all ranks $s\geq 5$ and the negation of the dimension results for the graded spaces $\mathbb F_2\otimes_{\mathscr A} \mathbb F_2[x_1, \ldots, x_s]$ in degrees $s+5$

Abstract

Let $P_s:= \mathbb F_2[x_1,x_2,\ldots ,x_s]$ be the graded polynomial algebra over the prime field of two elements, $\mathbb F_2$, in $s$ variables $x_1, x_2, \ldots , x_s$, each of degree one. This algebra is considered as a graded module over the mod-2 Steenrod algebra, $\mathscr {A}$. The classical "hit problem", initiated by Frank Peterson [Abstracts Amer. Math. Soc., 833 (1987), 55-89], concerned with seeking a minimal set of $\mathscr A$-module $P_s.$ Equivalently, when $\mathbb F_2$ is an $\mathscr A$-module concentrated in degree 0, one can write down explicitly a monomial basis for the $\mathbb Z$-graded vector space over $\mathbb F_2$:$$ QP_s:= \mathbb F_2 \otimes_{\mathscr A} P_s.$$The problem is unresolved in general. The paper is devoted to refute the results of Moetele and Mothebe [East-West J. of Mathematics 18 (2016), 151-170] on the dimension of $QP_s$ in degree $13$ and the dimension of $QP_9$ in degree $14.$ Further, we have computed explicitly the dimension of $QP_s$ in degree $14$ for all $s.$ Then, by the facts of the $s$-th cohomology groups of the Steenrod algebra ${\rm Ext}_{\mathscr A}^{s, s+13}(\mathbb F_2, \mathbb F_2)$ and ${\rm Ext}_{\mathscr A}^{s, s+14}(\mathbb F_2, \mathbb F_2),$ we show that that the algebraic transfer of rank $s> 0$, defined by W. Singer [Math. Z., 202 (1989), 493-523], is an isomorphism in degrees $13$ and $14.$