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 \(k=6.\)