The fifth algebraic transfer in generic degrees and validation of a localized Kameko's conjecture

This paper develops our previous works concerning the classical Peterson hit problem for the polynomial algebra on five variables over the mod–2 Steenrod algebra $\mathcal{A}$ in a generic family of degrees, together with applications to the fifth Singer algebraic transfer and a localized variation of Kameko’s conjecture. As a topological illustration of the usefulness of the Steenrod algebra, we prove that $\mathbb{C}{P}^4\mathrm{/}\mathbb{C}{P}^2$ and ${\mathbb{S}}^6\vee {\mathbb{S}}^8$ are not homotopy equivalent by showing that their mod–2 cohomologies are not isomorphic as $\mathcal{A}$-modules, and we further determine the homotopy type of the quotient $\mathbb{C}{P}^n\mathrm{/}\mathbb{C}{P}^{n-2}$ for all $n\ge 3$. For the generic degrees under consideration, we determine the relevant cohit spaces and describe the associated $GL(5,{\mathbb{F}}_2)$-module structure. As a consequence, the fifth algebraic transfer is an isomorphism in an explicit infinite family of internal degrees. These results were independently verified by implementations in $\mathtt{\text{SageMath}}$ and $\mathtt{\text{OSCAR}}$. We also study a localized form of Kameko’s conjecture concerning the dimensions of the indecomposables ${\mathbb{F}}_2{\otimes }_{\mathcal{A}}{\mathbb{F}}_2[x_1,\dots ,x_m]$ relative to parameter vectors, and prove that this conjecture holds for all $m\ge 1$ in certain degrees.