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Let $G$ be a discrete group acting freely and cellularily on a CW-complex $X$.

I am interested in the Cartan-Leray spectral sequence from Eilenberg and Cartan's Homological Algebra, Theorem XVI.8.4, which goes $$E^2_{p,q}=H^p(G,H^q(X;M))\Rightarrow H^{p+q}(X/G;M)\,,$$ where $M$ is a left $G$-module intepreted as a local coefficient system on $X$ rsp. $X/G$. The proof presented there is quite explicit, and I was wondering if there is a more abstract way to prove this using the Leray spectral sequence: For a continous map $f:A\to B$ and a sheaf $\mathcal F$ on $A$, there is a spectral sequence $$E^2_{p,q}=H^p(B;R^q(f_\ast)(\mathcal F))\Rightarrow H^{p+q}(A;\mathcal F)\,.$$

My idea was to construct a map $f:X/G\to K(G,1)$, where $K(G,1)$ is an Eilenberg-MacLane complex for $G$, and set $\mathcal F:=\underline{M}_{X/G}$. Then $H^{p+q}(A;\mathcal F)=H^{p+q}(X/G;M)$, which is good. But $$H^p(B;R^q(f_\ast)(\mathcal F))=H^p(K(G,1);\operatorname{Sheafification of}H^q(f^{-1}(-);M))\,,$$ of which I am not sure why it should equal $H^p(K(G,1);\underline{H^q(X;M)}_{K(G,1)})=H^p(G,H^q(X;M))$.

Is there a way to make this work? Did I go wrong somewhere along?

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    $\begingroup$ Nice question, but I think it differs from the title question. Do you want to see the Cartan--Leray-spectral sequence as a special case of the Leray-, or the much more general Grothendieck-spectral sequence? $\endgroup$
    – Ben
    Mar 21, 2023 at 11:01
  • $\begingroup$ Ah, I think I see what you mean. I meant to ask how the Cartan-Leray spectral sequence can be derived from the Leray spectral sequence, because I (perhaps mistakenly) assumed there was a pre-established connection between the two. How the Leray sequence can be derived from the Groethendieck sequence is clear to me. But if there is some other way to arrive at the Cartan-Leray sequence from the Groethendieck sequence, skipping on the Leray sequence, I'd very much appreciate that too. $\endgroup$
    – dryope
    Mar 21, 2023 at 12:48
  • $\begingroup$ I would start a bounty to this question to raise its visibility, if you would restrict to the Leray sequence-bit, which I find intriguing. (By changing the title accordingly.) I think the derivation from Grothendieck's spectral sequence is possible rather easily and wouldn't justify a bounty. (I would write this up in case no one can answer the Leray-part of the question once the bounty period is over, if that is of any help.) $\endgroup$
    – Ben
    Mar 29, 2023 at 11:03
  • $\begingroup$ I have changed the question according to your suggestion, although I would still also be interested in the easy part, if you find the time. $\endgroup$
    – dryope
    Mar 29, 2023 at 13:09
  • $\begingroup$ This can be delineated into two steps. 1.) The Leray spectral sequence implies a restricted version of the Serre spectral sequence. 2.) The Cartan-Leray spectral sequence is a special case of the Serre spectral sequence. The hard part of this is in 1.), where you have to somehow translate sheaf cohomology in a locally constant sheaf to singular cohomology with local coefficients. This should probably be doable in the same way sheaf cohomology in a constant sheaf is compared to singular cohomology, though I've never seen it done. $\endgroup$
    – Thorgott
    Mar 29, 2023 at 14:17

1 Answer 1

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Your idea of using the Leray spectral sequence to recover the Cartan-Leray spectral sequence for group cohomology is on the right track. However, to make the connection work, you need to choose an appropriate map $f$ and the correct sheaf $\mathcal{F}$. Let's go through the steps to see if we can make this work.

  1. Define the map $f: X \to K(G, 1)$: To make the connection between the two spectral sequences, you can use the fact that there is a natural map $f: X/G \to K(G, 1)$, which is a classifying map for the principal $G$-bundle $X \to X/G$. This map is continuous, and $K(G, 1)$ is an Eilenberg-MacLane space for the group $G$.

  2. Define the sheaf $\mathcal{F}$: Since $X$ is a CW-complex with a free and cellular action of $G$, we have that the action of $G$ on the cochain complex $C^*(X; M)$ is free. Therefore, the sheaf $\mathcal{F}$ should be the constant sheaf on $X$ associated to the $G$-module $M$.

  3. The Leray spectral sequence: Now we have a continuous map $f: X/G \to K(G, 1)$ and a sheaf $\mathcal{F}$ on $X$. We can compute the Leray spectral sequence associated to this data: $$E^{2}_{p,q} = H^{p}(K(G,1); R^{q}(f_{\ast})(\mathcal{F})) \Rightarrow H^{p+q}(X/G;\mathcal{F}).$$

  4. Connecting the two spectral sequences: To make the connection between the Leray spectral sequence and the Cartan-Leray spectral sequence, we need to show that the $E^2$-term in the Leray spectral sequence is isomorphic to the $E^2$-term in the Cartan-Leray spectral sequence. For this, we need to show that $H^p(K(G, 1); R^q(f_\ast)(\mathcal{F})) = H^p(G, H^q(X; M))$. To do this, observe that $R^q(f_\ast)(\mathcal{F})$ is a sheaf on $K(G, 1)$, which can be identified with the group cohomology $H^q(X; M)$, viewed as a local system on $K(G, 1)$. This identification comes from the fact that the fibers of $f$ are isomorphic to the quotient $X/G$, and the action of $G$ on $H^q(X; M)$ is free.

Now, we have: $$H^p(K(G,1); R^q(f_{\ast})(\mathcal{F})) = H^p(K(G,1); \underline{H^q(X; M)}_{K(G, 1)}) = H^p(G, H^q(X; M)). $$ Thus, the $E^2$-term of the Leray spectral sequence is isomorphic to the $E^2$-term of the Cartan-Leray spectral sequence, and the Leray spectral sequence converges to the cohomology of the quotient $X/G$ with coefficients in $M$. Therefore, you can recover the Cartan-Leray spectral sequence for group cohomology from the Leray spectral sequence for sheaf cohomology by making the appropriate choices of the map $f$.

Some clarifications as for 4:

Connecting the two spectral sequences: To make the connection between the Leray spectral sequence and the Cartan-Leray spectral sequence, we need to show that the $E^2$-term in the Leray spectral sequence is isomorphic to the $E^2$-term in the Cartan-Leray spectral sequence. For this, we need to show that $H^p(K(G, 1); R^q(f_\ast)(\mathcal{F})) = H^p(G, H^q(X; M))$. To do this, observe that $R^q(f_\ast)(\mathcal{F})$ is a sheaf on $K(G, 1)$, which can be identified with the group cohomology $H^q(X; M)$, viewed as a local system on $K(G, 1)$. This identification comes from the fact that the fibers of $f$ are isomorphic to the quotient $X/G$, and the action of $G$ on $H^q(X; M)$ is free.

The identification of $R^q(f_\ast)(\mathcal{F})$ with $H^q(X; M)$ as a local system on $K(G, 1)$ can be made more explicit as follows:

Consider the map $f^{-1}(U) \to U$, where $U$ is an open set in $K(G, 1)$. Since $f$ is the classifying map for the principal $G$-bundle $X \to X/G$, the inverse image $f^{-1}(U)$ is a disjoint union of copies of the quotient $X/G$, one for each element of the fiber of $f$. The action of $G$ on $f^{-1}(U)$ is free and transitive on these copies, which means that the induced action of $G$ on the cohomology of $f^{-1}(U)$ with coefficients in $M$ is also free and transitive. Therefore, $R^q(f_\ast)(\mathcal{F})(U) \cong H^q(f^{-1}(U); M)$ can be identified with the group cohomology $H^q(X; M)$ as a local system on $K(G, 1)$.

Now, we have:

Thus, the $E^2$-term of the Leray spectral sequence is isomorphic to the $E^2$-term of the Cartan-Leray spectral sequence. Note that it is not enough to conclude that the spectral sequences are isomorphic based only on this comparison; however, the isomorphism of the $E^2$-terms indicates that the two spectral sequences contain equivalent information.

To ensure that the Leray spectral sequence converges to the cohomology of the quotient $X/G$ with coefficients in $M$, we can verify that the conditions for convergence of the Leray spectral sequence are satisfied. In this case, since the action of $G$ on the cochain complex $C^*(X; M)$ is free and $X$ is a CW-complex, the conditions for convergence are met, and the Leray spectral sequence converges to the desired cohomology group.

To summarize, by making the appropriate choices of the map $f: X/G \to K(G, 1)$ and the sheaf $\mathcal{F}$, as well as providing a more explicit identification of $R^q(f_\ast)(\mathcal{F})$ with $H^q(X; M)$ as a local system on $K(G, 1)$, we have shown that the $E^2$-term of the Leray spectral sequence is isomorphic to the $E^2$-term of the Cartan-Leray spectral sequence. Furthermore, we have verified that the conditions for convergence of the Leray spectral sequence are satisfied, ensuring that it converges to the cohomology of the quotient $X/G$ with coefficients in $M$. This demonstrates that you can recover the Cartan-Leray spectral sequence for group cohomology from the Leray spectral sequence for sheaf cohomology by making the appropriate choices of the map $f$.

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  • $\begingroup$ Besides the fact that, strictly speaking, it's not enough to compare the second page to conclude that the spectral sequences are isomorphic, I think you are hiding the interesting bit the OP wasn't aware of in the seemingly innocuous sentence "The identification comes from the fact that ..." Could you elaborate on that point? $\endgroup$
    – Ben
    Apr 5, 2023 at 19:45
  • $\begingroup$ Hey Alon! Thank you, this is exactly what I had in mind. Still, I have to reiterate Ben: could you give a few more hints on how the identification in step 4 is done? $\endgroup$
    – dryope
    Apr 5, 2023 at 20:26
  • $\begingroup$ @dryope I have expanded my answer, hope that is clearer :) $\endgroup$
    – Alon Yariv
    Apr 25, 2023 at 16:36

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