Consider the restriction of the group cohomology $H^\*(BG,\mathbb{Z})$, where $G$ is a compact Lie group and $BG$ is its classifying space, to finite subgroups $F \le G$. If we consider the product of all such restrictions
$$H^\*(BG,\mathbb{Z}) \to \prod\_F H^\*(BF,\mathbb{Z}),$$
is this map injective?


Note, according to [McClure - Restriction maps in equivariant K-theory](https://doi.org/10.1016/0040-9383(86)90019-4) a similar result holds in equivariant K-theory. Perhaps there is a way to derive the above from McClure's theorem?


I asked this question [on stackexchange](https://math.stackexchange.com/questions/3821407/is-lie-group-cohomology-determined-by-restriction-to-finite-subgroups) and didn't manage to get a (complete) answer nor solve it myself. However, Qiaochu Yuan offered a proof for the non-torsion elements.



Is Lie group cohomology determined by restriction to finite subgroups?
After the heavy lifting done by people on MSE and in the comments, I think it's not too bad to finish off the proof that the answer is *yes*.


As argued by Ben Wieland in the [comments](https://mathoverflow.net/questions/372106/is-lie-group-cohomology-determined-by-restriction-to-finite-subgroups#comment941496_372106), we reduce to showing that for any short exact sequence of topological groups
$$U(1)^n \to G \to W $$
where $W$ is finite, we have that $H^\ast(BG;\mathbb Z)$ injects into the product of $H^\ast(BF;\mathbb Z)$ over all finite subgroups $F \subseteq G$. The jist of the argument is going to be replacing $U(1)^n$ with $(Q/\mathbb Z)^n$, and then arguing that every finitely-generated subgroup of the resulting extension $G\_\ast$ is finite.


The first thing to note is that for some $m \in \mathbb Z$, there exist maps of exact sequences
$$\require{AMScd} \begin{CD} (\mathbb Z/m)^n @>>> G\_m @>>> W \\ @VVV @VVV @VV{=}V\\(\mathbb Q /\mathbb Z)^n @>>> G\_\ast @>>> W \\ @VVV @VVV @VV{=}V\\ U(1)^n @>>> G @>>> W \end{CD}$$


The middle line exists and maps to the bottom line because (1) $\mathbb Q/\mathbb Z$ is the torsion subgroup of $U(1)$, and so must be preserved by the action of $W$, and (2) the discrete group quotient $U(1)^\delta / (\mathbb Q/\mathbb Z)$ is a rational vector space, so no matter the action of $W$, the cohomology of $W$ with values in this quotient vanishes by Maschke's theorem. Thus $H^\ast(BW; \underline{(\mathbb Q/\mathbb Z)^n}) \to H^\ast(BW; \underline{U(1)^{\delta,n}})$ is an isomorphism and in particular the class classifying the extension is hit.


The top line exists and maps to the middle line because when we choose a cocycle $W \times W \to (\mathbb Q/\mathbb Z)^n$, we see that because $W$ is finite, the cocyle has finite image, and every finitely-generated subgroup of $(\mathbb Q/\mathbb Z)^n$ is finite — thus the cocycle lives in some finite, $W$-invariant subgroup $(\mathbb Z/m)^n \subseteq (\mathbb Q / \mathbb Z)^n$ (using that $(\mathbb Z/m)^n \subseteq (\mathbb Q/\mathbb Z)^n$ is the $m$-torsion subgroup and so must be $W$-invariant).


By similar reasoning, we see that every finitely-generated subgroup of $G\_\ast$ is finite. Therefore, because homology commutes with filtered colimits, we have $H\_\ast(BG\_\ast) = \varinjlim\_{G' \subseteq G\_\ast} H\_\ast(BG')$, where the colimit is over finite subgroups (or even just those finite subgroups $G' = G\_m$ of the form given above), and we use any constant coefficients.


Now, because $B(\mathbb Q/\mathbb Z)^n \to BU(1)^n$ is a homology isomorphism with any finite coefficients (this can be checked in several ways), we see that $BG\_\ast \to BG$ is likewise a homology isomorphism with finite coefficients by the Serre spectral sequence. So the composite map $\varinjlim H\_\ast(BG\_m) \to H\_\ast(BG\_\ast) \to H\_\ast(BG)$ is an isomorphism with finite coefficients. By the universal coefficient theorem, the map $H^\ast(BG) \to \prod H^\ast(BG\_m)$ is injective with finite coefficients. Since Qiaochu has already [shown](https://math.stackexchange.com/a/3821423/45082) that it is an injection on nontorsion elements, it follows that this map is an injection on integral cohomology (noting that these things are sufficiently finite to be safe).




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Note that most of the above boiled down to facts about the extension $U(1)^n \to G \to W$, not really facts about (co)homology — the only thing we really needed was the isomorphism $H\_\ast(B(\mathbb Q/\mathbb Z)) \cong H\_\ast(BU(1))$ with finite coefficients.


In fact, Theorem 5.7 of Becker and Gottlieb's original paper [The transfer map and fiber bundles](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.452.4032&rep=rep1&type=pdf) ([DOI](https://doi.org/10.1016/0040-9383(75)90029-4)) is actually stated for general cohomology theory, and implies by Ben Wieland's [argument](https://mathoverflow.net/questions/372106/is-lie-group-cohomology-determined-by-restriction-to-finite-subgroups#comment941496_372106) that $\Sigma^\infty BG$ splits off of $\Sigma^\infty BN(T)$, so the reduction to extensions $T \to N \to W$ with $T$ a torus and $W$ finite holds for an arbitrary homology or cohomology theory $E$.


It's not hard to show that if $E$ is a spectrum with trivial rationalization, then $B(\mathbb Q/\mathbb Z)^n \to BU(1)^n$ is and $E$-homology or cohomology equivalence. So the above argument shows that in this case, we have that $N\_\ast \to N$ is an $E$-homology and $E$-cohomology equivalence, where $N\_\ast$ fits in the extension $(\mathbb Q/\mathbb Z)^n \to N\_\ast \to W$ as above. Moreover, since $BN\_\ast = \varinjlim BN'$ where the colimit is over finite subgroups, and since this is a homotopy colimit, we have $E\_\ast(BN) = \varinjlim E\_\ast(BN')$, so that $\bigoplus E\_\ast(BN') \to E\_\ast(BN)$ is surjective. For cohomology, there are potential $\varprojlim^1$ issues.


Thus the statement we get is:


**Theorem:** Let $E$ be a spectrum with trivial rationalization, and let $G$ be a compact Lie group. Then $\bigoplus\_{F \subseteq G} E\_\ast(BF) \to E\_\ast(BG)$ is surjective, where the sum is over finite subgroups $F \subseteq G$.


It would be nice if this could be upgraded to a statement about all spectra by considering also the rationalization, but that seems unpromising because of Maschke's theorem — Qiaochu's argument for nontorsion classes is more subtle, it seems.


Probably also some statement about cohomology is possible ….




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In fact, it's not hard to extend the statement to arbitrary $G$-spaces. That is:


**Theorem:** Let $E$ be a spectrum with trivial rationalization, let $G$ be a compact Lie group, and let $X$ be a $G$-space. Then $\bigoplus\_{F \subseteq G} E\_\ast(X\_{hF}) \to E\_\ast(X\_{hG})$ is surjective, where the sum is over finite subgroups $F \subseteq G$.


