The squaring operartion and the Singer algebraic transfer

2016/09/10 by Nguyễn Sum, Sum, Nguyen
Computer Science · Mathematics · #55S05 #55S10 (Primary) #55T15 (Secondary) #Algebraic Topology (math.AT) #FOS: Mathematics #Logic, programming, and type systems #Mathematics and Applications #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.1609.03006

openalex publication_date 2016/09/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let P_k be the graded polynomial algebra \mathbb F₂[x₁,x₂,… ,x_k], with the degree of each x_i being 1, regarded as a module over the mod-2 Steenrod algebra \mathcal A, and let GL_k be the general linear group over the prime field \mathbb F₂ which acts regularly on P_k. We study the algebraic transfer constructed by Singer using the technique of the hit problem. This transfer is a homomorphism from the homology of the mod-2 Steenrod algebra, Tor^{\mathcal A}_k,k+n (\mathbb F₂,\mathbb F₂), to the subspace of \mathbb F₂⊗_{\mathcal A}P_k consisting of all the GL_k-invariant classes of degree n. In this paper, we extend a result of Hung on the relation between the Singer algebraic transfer and the squaring operation on the cohomology of the Steenrod algebra. Using this result, we show that Singer's conjecture for the algebraic transfer is true in the case k=5 and the degree 5(2^s -1) with s an arbitrary positive integer.

Related

• On Singer's conjecture for the fifth algebraic transfer
• On the determination of the Singer transfer
• Determination of the fifth Singer algebraic transfer in some degrees
• An application of the hit problem to the algebraic transfer
• A note on the hit problem for the Steenrod algebra and its applications