On the mod-2 cohomology of the product of the infinite lens space and the space of invariants in a generic degree

TL;DR

This paper investigates the kernel of the Kameko squaring operation for specific degrees in the mod-2 cohomology of infinite lens spaces, correcting previous inaccuracies and introducing algorithms for explicit computation of indecomposables.

Contribution

It provides new results on the kernel of the Kameko operation for s=5 and generic degrees, rectifies earlier errors, and develops algorithms in SageMath for computing indecomposables.

Findings

Determined the kernel of the Kameko operation for s=5 and degrees N_d.

Corrected main results from previous publication by Nguyen Khac Tin.

Developed algorithms in SageMath for explicit basis computation of indecomposables.

Abstract

Let $\mathbb S^{\infty}/\mathbb Z_2$ be the infinite lens space and $\mathscr A$ be the Steenrod algebra over the binary field $\mathbb F_2.$ The cohomology $H^{*}((\mathbb S^{\infty}/\mathbb Z_2)^{\oplus s}; \mathbb F_2)$ is known to be isomorphic to the graded polynomial ring $\mathcal {P}_s:= \mathbb F_2[x_1, \ldots, x_s]$ on $s$ generators of degree 1, viewed as an unstable $\mathscr A$-module. The Kameko squaring operation $(\widetilde {Sq^0_*})_{(s; N)}: (\mathbb F_2\otimes_{\mathscr A} \mathcal {P}_s)_{2N+s} \longrightarrow (\mathbb F_2\otimes_{\mathscr A} \mathcal {P}_s)_{N}$ is rather useful in studying an open problem of determining the dimension of the indecomposables $(\mathbb F_2\otimes_{\mathscr A} \mathcal {P}_s)_N.$ As a continuation of our recent works, this paper deals with the kernel of the Kameko $(\widetilde {Sq^0_*})_{(s; N_d)}$ for the case where $s = 5$ and the "generic" degree $N_d$ is of the form $N_d = 5(2^{d} - 1) + 11.2^{d+1}$ for arbitrary $d > 0.$ We then rectify almost all of the main results that were inaccurate in an earlier publication [Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 116:81 (2022)] by Nguyen Khac Tin. We have also constructed several advanced algorithms in SAGEMATH to validate our results. These new algorithms make an important contribution to tackling the intricate task of explicitly determining both the dimension and the basis for the indecomposables $\mathbb F_2 \otimes_{\mathscr A} \mathcal {P}_s$ at positive degrees, a problem concerning algorithmic approaches that had not previously been addressed by any author. Also, the present study encompasses an investigation of the behavior of the cohomological transfer in bidegrees $(5, 5+N_d)$, with the internal degree $N_d$ mentioned above.