2025/01/01 by Đặng Võ Phúc, Phuc, Dang Vo
Mathematics · Computer Science · #Homotopy and Cohomology in Algebraic Topology #Advanced Combinatorial Mathematics #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.2511.11745
Computing the cohomology of the 2-primary Steenrod algebra \mathbbA is a central problem in algebraic topology, as it forms the E2-term of the Adams spectral sequence converging to the stable homotopy groups of spheres. The Singer cohomological transfer, φn, is a key homomorphism for characterizing this cohomology. Singer conjectured that φn is always a monomorphism. The Singer transfer is closely linked to the Peterson hit problem, which seeks a minimal generating set for the \mathbbA-module H*(V⊕ n) = ℤ/2[u1, …, un], also unsolved for n ≥ 5. In this paper, we study the hit problem for H*(V⊕ 5) and verify Singer's conjecture for the case n=5 in the general degree d = 2t+5 + 2t+2 + 2t+1-5 for any non-negative integer t. We demonstrate that the Singer cohomological transfer is an isomorphism for n=5 in degree d. This provides a positive answer to Singer's conjecture in these specific cases. The appendix provides our new algorithm implemented on the computer algebra system OSCAR, through which all principal results of this paper have been completely verified.