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

TL;DR

This paper advances the understanding of the algebraic transfer in generic degrees, confirms a localized version of Kameko's conjecture, and applies Steenrod algebra techniques to topological problems.

Contribution

It determines the fifth algebraic transfer as an isomorphism in specific degrees and verifies a localized Kameko's conjecture for all m in certain degrees.

Findings

Fifth algebraic transfer is an isomorphism in an explicit infinite family of degrees.

Topological distinctions between certain spaces are established via Steenrod algebra modules.

Localized Kameko's conjecture holds for all m in specific degrees.

Abstract

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 $\mathscr A$ 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 $\mathbb{C}P^4/\mathbb{C}P^2$ and $\mathbb{S}^6\vee \mathbb{S}^8$ are not homotopy equivalent by showing that their mod--2 cohomologies are not isomorphic as $\mathscr A$-modules, and we further determine the homotopy type of the quotient $\mathbb{C}P^n/\mathbb{C}P^{\,n-2}$ 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 \texttt{SageMath} and \texttt{OSCAR}. We also study a localized form of Kameko's conjecture concerning the dimensions of the indecomposables $\mathbb F_2\otimes_{\mathscr A}\mathbb F_2[x_1,\ldots,x_m]$ relative to parameter vectors, and prove that this conjecture holds for all $m\ge 1$ in certain degrees.