Modular invariants detecting the cohomology of BF_4 at the prime 3

TL;DR

This paper verifies Adams' conjecture for the exceptional Lie group F_4 at the prime 3 by analyzing its mod-3 cohomology using invariant theory and elementary abelian subgroups.

Contribution

It demonstrates that the mod-3 cohomology of BF_4 is detected by elementary abelian 3-subgroups, confirming the conjecture for this specific case.

Findings

Confirmed Adams' conjecture for F_4 at p=3

Identified invariant subrings in cohomology

Mapped elementary abelian subgroups via Steenrod algebra

Abstract

Attributed to J F Adams is the conjecture that, at odd primes, the mod-p cohomology ring of the classifying space of a connected compact Lie group is detected by its elementary abelian p-subgroups. In this note we rely on Toda's calculation of H^*(BF_4;F_3) in order to show that the conjecture holds in case of the exceptional Lie group F_4. To this aim we use invariant theory in order to identify parts of H^*(BF_4;F_3) with invariant subrings in the cohomology of elementary abelian 3-subgroups of F_4. These subgroups themselves are identified via the Steenrod algebra action on H^*(BF_4;F_3).