Low-degree mod 2 cohomology of classifying spaces of $G_2$-gauge groups

TL;DR

This paper investigates the mod 2 cohomology of classifying spaces of $G_2$-gauge groups, revealing a specific differential in the spectral sequence and periodicity properties related to the bundle class.

Contribution

It provides a detailed analysis of the low-degree mod 2 cohomology of $B ext{G}_k(G_2)$, identifying the key differential and its dependence on the bundle class mod 4.

Findings

Identification of the first positive-degree generator in degree 5.

Determination of the differential $d_6(u_5) = ext{epsilon}(k) x_6$ in the spectral sequence.

Proof of 4-periodicity of epsilon(k) and its vanishing for $k mod 4 = 0$.

Abstract

Let $G$ be a simply connected compact simple Lie group and let $\mathcal{G}_k$ denote the gauge group of a principal $G$--bundle over $S^4$ with second Chern class $k\in \pi_4(BG)\cong \mathbb Z$. For $G=G_2$, the $p$--local homotopy types of the gauge groups have been completely classified by Kishimoto--Theriault--Tsutaya and Kameko in terms of the order of the fundamental Samelson product $\langle i_3,1\rangle\in [\Sigma^3G_2,G_2]$. In this paper, we begin a complementary study of the mod $2$ cohomology of the classifying spaces $B\mathcal{G}_k(G_2)$. Our goal is to understand the structure of $H^*(B\mathcal{G}_k;\mathbb{F}_2)$ as an unstable module over the mod~$2$ Steenrod algebra in a low range of degrees. Using the evaluation fibration \[ \Omega_0^3 G_2 \longrightarrow B\mathcal{G}_k \xrightarrow{\;\mathrm{ev}\;} BG_2 \] together with Serre and Eilenberg--Moore spectral sequences, we study the Serre spectral sequence \[ H^s(BG_2;H^t(\Omega^3_0G_2)) \Longrightarrow H^{s+t}(B\mathcal{G}_k) \] in total degree $\le 10$. A careful analysis of the homotopy groups of $G_2$ shows that \[ H^j(\Omega^3_0G_2;\mathbb{F}_2)=0\quad\text{for }1\le j\le 4, \qquad H^5(\Omega^3_0G_2;\mathbb{F}_2)\neq 0, \] so the first positive-degree generator of the fibre cohomology occurs in degree $5$. As a consequence, there is a distinguished class \[ u_5\in H^5(\Omega^3_0G_2;\mathbb{F}_2) \] whose only possible Serre differential in total degree $\le 10$ is a $d_6$--differential \[ d_6(u_5) = \epsilon(k)\,x_6 \] from $u_5$ to the degree-$6$ generator $x_6\in H^6(BG_2;\mathbb{F}_2)$, for a scalar $\epsilon(k)\in\mathbb{F}_2$ encoding the low-degree effect of the bundle class. In addition, $2$--locally we prove that $\epsilon(k)$ is $4$--periodic in $k$ (i.e. it depends only on $k\bmod 4$) and that $\epsilon(k)=0$ for all $k\equiv 0\pmod 4$.