On $\mathbb{A}$-generators of the cohomology $H^{*}(V^{\oplus 5})=\mathbb{Z}/2[u_1,\ldots,u_5]$ and the cohomological transfer of rank 5

Abstract: Computing the cohomology of the 2-primary Steenrod algebra $\mathbb{A}$ is a central problem in algebraic topology, as it forms the $E_2$-term of the Adams spectral sequence converging to the stable homotopy groups of spheres. The Singer cohomological transfer, $\varphi_n$, is a key homomorphism for characterizing this cohomology. Singer conjectured that $\varphi_n$ is always a monomorphism. The Singer transfer is closely linked to the Peterson hit problem, which seeks a minimal generating set for the $\mathbb{A}$-module $H^{*}(V^{\oplus n}) = \mathbb{Z}/2[u_1, \ldots, u_n]$, also unsolved for $n \geq 5$. In this paper, we study the hit problem for $H^{*}(V^{\oplus 5})$ and verify Singer's conjecture for the case $n=5$ in the general degree $d = 2^{t+5} + 2^{t+2} + 2^{t+1}-5$ for any non-negative integer $t$. We demonstrate that the Singer cohomological transfer is an isomorphism for $n=5$ in degree $d$. This provides a positive answer to Singer's conjecture in these specific cases.