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

Abstract

Let ${\mathcal{G}}_k$ be the gauge group of the principal $G_2$-bundle over $S^4$ classified by $k\in {\pi }_4(B{G}_2)\cong \mathbb{Z}$. In this paper, we determine the first positive fibre row in the mod $2$ Serre spectral sequence of ${\Omega }_0^3{G}_2\to B{\mathcal{G}}_k\stackrel{p_k}{\to }B{G}_2$. The fibre cohomology vanishes in degrees $1$ through $4$ and has a unique nonzero class $u_5$ in degree $5$. We prove that its transgression is $d_6(u_5)=(k\mathrm{mod}2)x_6$. The proof uses the Stiefel--Whitney classes of the seven-dimensional representation and maps of evaluation fibrations induced by the pinch map of $S^4$. The same comparison maps show that the possible incoming differential $d_4:H^8({\Omega }_0^3{G}_2;{\mathbb{F}}_2)\to {\mathbb{F}}_2\{x_4{u}_5\}$ vanishes. Consequently, the classes $u_5$ and $x_4{u}_5$ survive precisely when $k$ is even. In degree six, a nonzero linear functional $\lambda :H^6({\Omega }_0^3{G}_2;{\mathbb{F}}_2)\to {\mathbb{F}}_2$ determines the next transgression by $d_7(z)=(k\mathrm{mod}2)\lambda (z)x_7$. Both $d_7$ maps to the degree-seven and degree-eleven base classes vanish for even $k$ and are surjective for odd $k$; in particular, $p_k^{\ast }{x}_7\ne 0$ exactly when $k$ is even. We derive further forced vanishings, cohomological consequences, and Steenrod-square relations, and delimit the remaining differentials with source of total degree at most $10$. The calculation is independent of the exact order of the fundamental Samelson product.

arXiv comment13 pages. The Acknowledgment has been updated. To appear in Vietnam Journal of Mathematics (2026)