Counterexamples to a conjecture of Adams

Abstract

A conjecture due to J. F. Adams states 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 for any odd prime $p$ the mod $p$ cohomology ring of the classifying space of the projective unitary group $PU(p^2)$ is not detected by its elementary abelian $p$-subgroups, providing counterexamples to this conjecture. We also obtain several algebraic results as byproducts.

arXiv comment30 pages. Main theorem generalized