In the paper On the mod p cohomology of BPU(p), the authors say that there is a conjecture of J. F. Adams as follows:

**Conjecture** (J. F. Adams) Let $G$ be a compact connected Lie group, and let $p$ be an odd prime. Then the mod $p$ cohomology of $BG$ is detected by elementary abelian $p$-subgroups.

In other words, the homomorphism $$H^*(BG;\mathbb{F}_p)\to \operatorname{invlim}H^*(BE;\mathbb{F}_p) $$ where the inverse limit is taken over the category of elementary abelian $p$-subgroups is injective. In particular, $E$ runs over elementary abelian $p$ subgroups of $G$.

However, I was not able to find the original document containing this conjecture. Is there one?

What is the status of this conjecture?

10more comments