A Note on the 2-Local Homotopy Types of $G_2$-Gauge Groups

Abstract

In a recent preprint \cite{Kameko2026} , Kameko presented a substantial completion of the 2-local classification of $G_2$-gauge groups over $S^4$, extending earlier work by Kishimoto, Theriault, and Tsutaya. The central strategy relies on reducing the 2-local classification to the order of the Samelson product $\langle i_3,1\rangle$ and separating specific gauge group homotopy types. The purpose of this note is to provide necessary mathematical refinements and localization clarifications to several key steps in the proof of the classification theorem. Specifically, we refine the integral isomorphism claim for gauge group homotopy to its correct 2-local form, resolve an EHP sequence extension regarding the injectivity of the Hopf invariant, and make explicit the Postnikov layer conventions required for the mod 2 Leray--Serre spectral sequence calculations. We confirm that with these adjustments, the main 2-local classification theorem in \cite{Kameko2026} also holds as claimed.