On Singer's conjecture for the fifth algebraic transfer

2016/09/08 by Nguyễn Khắc Tín, Tin, Nguyen Khac · 1 citation
Biochemistry, Genetics and Molecular Biology · Mathematics · #55S05 #55S10 (Primary) #55T15 (Secondary) #Algebraic Topology (math.AT) #FOS: Mathematics #Glycosylation and Glycoproteins Research #Homotopy and Cohomology in Algebraic Topology #Sphingolipid Metabolism and Signaling

paper · pdf · doi:10.48550/arxiv.1609.02250

openalex publication_date 2016/09/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let P_k:= \mathbbF₂[x₁,x₂,… ,x_k] be the polynomial algebra in k variables with the degree of each x_i being 1, regarded as a module over the mod-2 Steenrod algebra A, and let GL_k be the general linear group over the prime field \mathbbF₂ which acts naturally on P_k. We study the hit problem, set up by Frank Peterson, of finding a minimal set of generators for the polynomial algebra P_k as a module over the mod-2 Steenrod algebra, A. These results are used to study the Singer algebraic transfer which is a homomorphism from the homology of the mod-2 Steenrod algebra, Tor^A_{k, k+n}(\mathbbF₂, \mathbbF₂), to the subspace of \mathbbF₂⊗_AP_k consisting of all the GL_k-invariant classes of degree n. In this paper, we explicitly compute the hit problem for k = 5 and the degree 7.2^s-5 with s an arbitrary positive integer. Using this result, we show that Singer's conjecture for the algebraic transfer is true in the case k=5 and the above degree.

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• On Singer's conjecture for the fourth algebraic transfer in certain generic degrees

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• Determination of the fifth Singer algebraic transfer in some degrees
• On the determination of the Singer transfer
• The squaring operartion and the Singer algebraic transfer
• A counter-example to Singer's conjecture for the algebraic transfer
• An application of the hit problem to the algebraic transfer