A note on the Peterson hit problem for the Steenrod algebra

Abstract

Let $P_{k}=\oplus_{n\geqslant 0} (P_{k})_{n} \cong \mathbf{F}_{2}[x_{1},x_{2},\ldots ,x_{k}]$ be the graded polynomial algebra over the prime field of two elements $\mathbf{F}_{2}$, in $k$ generators $x_{1}, x_{2}, \ldots , x_{k}$, each of degree 1. Being the mod-2 cohomology of the classifying space $B(\mathbf{Z}/2)^{k}$, the algebra $P_{k}$ is a module over the mod-2 Steenrod algebra $\mathcal{A}$. In this Note, we explicitly compute the hit problem of some generic degrees $r(2^{s}-1)+2^{s}m$ in $P_{k}$, where $r=k-1=4, m \in \{8; 10; 11 \}$ and $s$ an arbitrary non-negative integer. Moreover, as a consequence, we get the dimension results for polynomial algebra in some generic degrees and in the cases $k=5$ and 6.