# A conjecture of Adams on the mod p cohomology of classifying spaces of Lie groups

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?

• Is the statement supposed to mean that that homomorphism is injective? If so, the analogous question allowing $E$ to range over all finite subgroups came up not too long ago here on MO (that question was for integral cohomology, but I think the second answer is general enough to cover mod $p$). The answer was yes, and I don't think it used anything unavailable to Adams. So I guess the hard part is going from finite subgroups to elementary ones? – Tim Campion Feb 7 at 4:02
• The analogous question for $K$-theory instead of mod-$p$ cohomology, and for finite cyclic subgroups rather than elementary ones, is a theorem of McClure: see Theorem C of Restriction maps in equivariant K-theory. This result was apparently the inspiration for the above MO question. I should mention that that question had previously been asked on MSE, where most of the work was actually done. – Tim Campion Feb 7 at 4:19
• The point of using elementary abelian subgroups comes from the fact that the map is known as Quillen's map and had been studied. Furthermore the cohomology of finite group is in general not computable, meaning that the target is not a useful approximation of the source. And yes, the finite to elementary abelian part is far from obvious, the Quillen's map for finite groups is not in general injective. – user43326 Feb 7 at 8:49
• What the Quillen map is? Is it in his paper on the Adams’ conjecture? – user51223 Feb 7 at 12:49
• @TimCampion Oh, I also should have mentioned that not only the cohomology of finite group is in general messy, but also the category of all finite subgroups of a Lie group is not easy... – user43326 Feb 7 at 17:57