A counterexample to a conjecture of Adams

Feifei Fan

algebraic topology

A counterexample to a conjecture of Adams

arXiv:2501.07797

summary

The paper disproves J. F. Adams’s conjecture that the mod‑p cohomology of classifying spaces of compact Lie groups is always detected by elementary abelian p‑subgroups, by exhibiting a counterexample at p = 3 for the group PU(9), and derives several related algebraic results.

Abstract

A conjecture due to J. F. Adams says that, for any odd prime $p$ , 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 paper, we show that the mod $3$ cohomology ring of the classifying space of the projective unitary group $P U (9)$ is not detected by its elementary abelian $3$ -subgroups, providing a counterexample to this conjecture. We also obtain many algebraic results as byproducts.

31 pages, typo corrections and more details added

Topics & keywords

#cohomology #classifying spaces #compact Lie groups #elementary abelian subgroups #counterexamplemod 3 cohomologyPU(9)Adams conjecturedetection theoremelementary abelian 3-subgroups