Is it possible to recover the Cartan-Leray Spectral Sequence for Group Cohomology from the Leray Spectral Sequence for Sheaf Cohomology?

Question

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?

Answer 1

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 $f$.