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Quillen proved in 1971 ("The Spectrum of an Equivariant Cohomology Ring: I,II") for a large class of groups $G$ including

  • compact Lie groups
  • groups of finite virtual cohomological dimension
  • compact topological groups with a finite number of conjugacy classes of elementary abelian $p$-subgroups

that the map $$\text{res}: H^\ast(BG,\mathbb{F}_p) \to \varprojlim_E H^\ast(BE,\mathbb{F}_p)$$ where $E$ runs through the elementary abelian $p$-subgroups of $G$ (ordered by inclusion and conjugacy) is an F-isomorphism (i.e. it has finite kernel and for each $x$ in the RHS, a $p$-power of $x$ is in the image).

Question 1: Has this theorem been generalized to other classes of groups in the meantime ?

Conversely, I'm also interested in counterexamples to the theorem, i.e.

Question 2: What is an example of a topological group $G$ s.t. the map above is no F-isomorphism ?

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A version of this result for finite group schemes is proven here: – Dylan Wilson Mar 23 '13 at 15:12

This might be a rather different direction than you're interested in, but analogues of Quillen's theorem have been studied for graded cocommutative Hopf algebras. These generalizations were originally motivated by computations in stable homotopy theory that used finite subalgebras of the Steenrod algebra. Wilkerson ("The cohomology algebras of finite dimensional Hopf algebras") proved that an analogue of Quillen's theorem holds for all finite subalgebras of the mod 2 Steenrod algebra, but gave counterexamples for more general connected graded cocommutative Hopf algebras, including subalgebras of the mod $p$ Steenrod algebra for odd $p$. Palmieri ("A note on the cohomology of finite dimensional Hopf algebras") gave a generalization of the theorem that holds for any finite-dimensional connected graded cocommutative Hopf algebra, but Palmieri uses a very general class of algebras in the place of elementary abelian subgroups, to the point that the theorem essentially becomes purely formal.

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Yes, I'm more interested in groups, but it's nevertheless interesting to learn that there are analogues for graded cocommutative Hopf algebras. Why are the Hopf algebras required to be graded connected ? ( the natural generalization of the finite group case seems to be finite dimensional cocommutative Hopf algebras. But these aren't connected when considered as graded Hopf algebras conncentrated in degree 0). – Demin Hu Mar 23 '13 at 7:47
Generally speaking, connectedness makes it possible to prove things using some form of induction. The most basic form of this is that an element of minimal positive degree is automatically primitive, and in particular it is easy to construct primitive elements. Variations on this argument can be used to show that any such Hopf algebras can be filtered into "monogenic elementary algebras", similar to how a $p$-group can be filtered into cyclic groups of order $p$. – Eric Wofsey Mar 23 '13 at 16:04
I should admit that I haven't thought much about what you can say if you don't assume connectedness, but the papers I've looked at tend to always assume connectedness. This may simply be because connectedness always holds in the examples from topology that motivate this. – Eric Wofsey Mar 23 '13 at 16:08
Thanks for your answer and the comments. Palmieri's paper has the no-graded case as a conjecture (Conjecture 1.5). So the graded connected and the ungraded case are probably of rather different nature. – Demin Hu Mar 25 '13 at 14:57

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