A note on the hit problem for the Steenrod algebra and its applications

Let $P_{k}=H^{*}((\mathbb{R}P^{\infty})^{k})$ be the modulo-$2$ cohomology algebra of the direct product of $k$ copies of infinite dimensional real projective spaces $\mathbb{R}P^{\infty}$. Then, $P_{k}$ is isomorphic to the graded polynomial algebra $\mathbb{F}_{2}[x_{1},\ldots,x_{k}]$ of $k$ variables, in which each $x_{j}$ is of degree 1, and let $GL_k$ be the general linear group over the prime field $\mathbb{F}_2$ which acts naturally on $P_k$. Here the cohomology is taken with coefficients in the prime field $\mathbb F_2$ of two elements. We study the {\it hit problem}, set up by Frank Peterson, of finding a minimal set of generators for the polynomial algebra $P_k$ as a module over the mod-2 Steenrod algebra, $\mathcal{A}$. In this Note, we explicitly compute the hit problem for $k = 5$ and the degree $5(2^s-1)+24.2^s$ with $s$ an arbitrary non-negative integer. These results are used to study the Singer algebraic transfer which is a homomorphism from the homology of the mod-$2$ Steenrod algebra, $\mbox{Tor}^{\mathcal{A}}_{k, k+n}(\mathbb{F}_2, \mathbb{F}_2),$ to the subspace of $\mathbb{F}_2\otimes_{\mathcal{A}}P_k$ consisting of all the $GL_k$-invariant classes of degree $n.$ We show that Singer's conjecture for the algebraic transfer is true in the case $k=5$ and the above degrees. This method is different from that of Singer in studying the image of the algebraic transfer. Moreover, as a consequence, we get the dimension results for polynomial algebra in some generic degrees in the case $k=6.$