On the computation of Ext_{\mathcal A}^k,k+*(ℤ/2,ℤ/2) for k ≤ 5

2025/07/21 by Phuc, Dang Vo
#55S05 #55S10 #55T15 #Algebraic Topology (math.AT) #FOS: Mathematics #Geometric Topology (math.GT) #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.2507.15611

Abstract

This Note presents a computational algorithm for determining a basis of the cohomology of the mod 2 Steenrod algebra, Ext_{\mathcal A}^{k, k+*}(ℤ/2, ℤ/2) for k ≤ 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 Ext_{\mathcal A}^{4, 4+*}(ℤ/2, 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.

Citations

• A matrix criterion and algorithmic approach for the Peterson hit problem: Part I
• On the algebraic transfers of ranks 4 and 6 at generic degrees
• On Singer's conjecture for the fourth algebraic transfer in certain generic degrees
• The affirmative answer to Singer's conjecture on the algebraic transfer of rank four
• The A-decomposability of the Singer construction

Related

• On modules over the mod 2 Steenrod algebra and hit problems
• The affirmative answer to Singer's conjecture on the algebraic transfer of rank four
• On the dimension of the "cohits" space \mathbb Z₂⊗_{\mathcal A2} H^*((\mathbb RP(∞))^{× t}, \mathbb Z₂) and some applications
• The cohomology of the Steenrod algebra and the mod p Lannes-Zarati homomorphism
• Extended powers and Steenrod operations in algebraic geometry