On the computation of $\mathrm{Ext}_{\mathcal A}^{k,k+*}(\mathbb{Z}/2,\mathbb{Z}/2)$ for $k \leq 5$

TL;DR

This paper introduces a computational algorithm to determine the basis of the mod 2 Steenrod algebra cohomology for degrees up to 5, verifying previous hand calculations and aiding in algebraic transfer analysis.

Contribution

It presents a new algorithm for computing Ext groups over the Steenrod algebra for low degrees, based on known generators and relations, to verify and extend prior results.

Findings

Verified hand-computed Ext groups for k ≤ 5

Developed an algorithm for basis computation

Facilitated algebraic transfer verification

Abstract

This Note presents a computational algorithm for determining a basis of the cohomology of the mod 2 Steenrod algebra, $\mathrm{Ext}_{\mathcal A}^{k, k+*}(\mathbb{Z}/2, \mathbb{Z}/2)$ for $k \leq 5$, based on the well-known generators and the Adams relations given in Chen's work [2]. The purpose of this algorithm is to verify the hand-computed results for $\mathrm{Ext}_{\mathcal A}^{4, 4+*}(\mathbb{Z}/2, \mathbb{Z}/2)$ presented in our corrected papers [8, 9, 10]. Combining our most recent works [11, 12], the verification of the domain and the codomain of the fourth algebraic transfer in specific degrees can now be completed.