2025/12/13 by Đặng Võ Phúc, Phuc, Dang Vo
Mathematics · #Homotopy and Cohomology in Algebraic Topology #Geometric and Algebraic Topology #Advanced Operator Algebra Research
paper · pdf · doi:10.48550/arxiv.2512.12195
Let Gk denote the gauge group of the principal G2--bundle over S4 classified by k∈ π4(BG2)≅ \mathbb Z. Motivated by the p--local homotopy classification of these gauge groups, due to Kishimoto--Theriault--Tsutaya and Kameko, we study the low-degree mod~2 cohomology of the classifying spaces BGk as unstable modules over the Steenrod algebra. Using the evaluation fibration Ω30G2\longrightarrow BGk \xrightarrow ev BG2 and its Serre spectral sequence, we analyze Hs(BG2;Ht(Ω30G2;\mathbb F2)) \Longrightarrow Hs+t(BGk;\mathbb F2) in total degree at most 10. We show that Hj(Ω30G2;\mathbb F2)=0 (1≤ j≤ 4), H5(Ω30G2;\mathbb F2)≅\mathbb F2, so the first positive-degree fibre class is a generator u5∈ H5(Ω30G2;\mathbb F2). In this range, the only possible Serre differential with source u5 is d6(u5)=ε(k)x6, where x6∈ H6(BG2;\mathbb F2) and ε(k)∈\mathbb F2. We also prove that, 2--locally, ε(k) depends only on k\bmod 8, and that ε(k)=0 whenever 8| k.