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

Abstract: This paper develops our previous works concerning the classical Peterson hit problem of five variables for the 2-primary Steenrod algebra $\mathscr A$ in generic degrees and related applications. To illustrate the utility of the Steenrod algebra, we give a proof that $\mathbb{C}P^4/\mathbb{C}P^2$ and $\mathbb{S}^6\vee \mathbb{S}^8$ are not homotopy equivalent by demonstrating that they are not isomorphic as $\mathscr A$-modules. The results are used to describe the representations of the general linear group of rank $5$ over the binary field $\mathbb F_2.$ As a consequence, the fifth algebraic transfer is an isomorphism at the internal degrees under consideration. These results have also been completely verified based on our novel algorithmic programs implemented in the computer algebra systems SageMath and OSCAR, announced in [arXiv:2507.10108] and [arXiv:2509.09455]. We also study a localized variation of Kameko's conjecture concerning the dimension of the indecomposables $\mathbb F_2\otimes_{\mathscr A}\mathbb F_2[x_1, x_2, \ldots, x_m]$ relative to parameter vectors and prove that this conjecture is valid for all $m\geq 1$ in certain degrees.