Counterexamples to a conjecture of Adams

TL;DR

This paper provides counterexamples to a conjecture by J. F. Adams, showing that the mod p cohomology of classifying spaces of certain projective unitary groups is not fully detected by elementary abelian p-subgroups, with applications to Milnor operations and Brown-Peterson cohomology.

Contribution

It constructs explicit counterexamples to Adams' conjecture for specific cases of PU(n), expanding understanding of cohomology detection.

Findings

Counterexamples for p^2 dividing n with p odd prime

Mod p cohomology not fully detected by elementary abelian p-subgroups

Applications to Milnor operations and Brown-Peterson cohomology

Abstract

For any odd prime $p$ and any integer $n>0$ with $p^2|n$, we show that the mod $p$ cohomology ring of the classifying space of the projective unitary group $PU(n)$ is not completely detected by elementary abelian $p$-subgroups, providing counterexamples to a conjecture due to J. F. Adams. We also give an application involving Milnor operations and Brown-Peterson cohomology.