"Rotated" version of the Atiyah-Hirzebruch spectral sequence

Question

Let $G$ be a group, $X$ a topological space with $G$-action. For an Abelian group $A$, let $\mathcal{C}^n(X,A)$ be the group of $n$-cochains on $X$ with $A$ coefficients. We can treat this as a $G$-module (Abelian group with compatible $G$-action) with the $G$-action inherited from the $G$-action on $X$.

Now for any $G$-module $B$, we can introduce the Abelian group of "group cochains" $\mathcal{C}^m(G, B)$ that are used to define group cohomology. For example, we can define $\mathcal{C}^m(G, B) = \mathrm{Hom}_G(F_n, B)$, where $$\cdots F_n \to F_{n-1} \to \cdots \to F_0 \to \mathbb{Z} \to 0$$ is a projective $\mathbb{Z}[G]$-resolution of the integers.

In particular, $\mathcal{C}^m(G, \mathcal{C}^n(X,A))$ defines a double complex. Now for any double complex there are two spectral sequences we can use to compute the total cohomology. In this case, the total cohomology is equivalent to the equivariant cohomology $H^{\bullet}_G(X,A) = H^{\bullet}(X//G,A)$. One of the two spectral sequences can be identified as the Serre spectral sequence associated with the fibration $X \to X//G \to BG$. The other one does not appear to have a name.

My question now is, suppose that we now replace ordinary cohomology with generalized cohomology, that is, we want to compute $\hat{H}^{\bullet}(X//G)$ for some generalized cohomology theory $\hat{H}$. The Serre spectral sequence generalizes to the Atiyah-Hirzebruch spectral sequence. But does the other, unnamed, spectral sequence have any analogous generalization?

Answer 1

Good question. I think the answer is yes.

The unnamed spectral sequence is usually referred to as the isotropy spectral sequence. For a group $G$ acting on $X$ and an abelian group $A$ of coefficients as in your question, it has $E_2$-page given by the sheaf cohomology $H^p(X/G; \mathcal{H}^q)$ of the orbit space, and converges under favourable circumstances to the cohomology $H^*(EG\times_G X;A)$ of the homotopy orbit space. If $X$ is a regular $G$-complex, then the sheaf $\mathcal{H}^q$ of coefficients takes the value $H^q(G_\sigma;A)$ over a simplex $[\sigma]$ of the orbit space $X/G$. Summarising, the spectral sequence goes from cohomology of the orbit space with coefficients in the cohomology of the isotropy subgroups, to the equivariant cohomology.

This can be identified with the Leray spectral sequence of the map $EG\times_G X\to X/G$ from the homotopy orbit space to the (genuine) orbit space, given by projecting $EG$ to a point. This is not a fibration in general, but one still gets a spectral sequence, at the expense of replacing cohomology with local coefficients by sheaf cohomology. A decent reference for the Leray spectral sequence of map is the book of Bott and Tu.

Now, the Leray spectral sequence of a map works just as well for generalized cohomology theories. The details appear in the thesis of Richard Cain,

Cain, R.N., The Leray spectral sequence of a mapping for generalized cohomology, Commun. Pure Appl. Math. 24, 53-70 (1971). ZBL0205.53002.

So for a generalized cohomology theory $F^*$ and a regular $G$-complex $X$, you should get a spectral sequence with $E_2$-page $H^p(X/G; \mathcal{F}^q)$ converging to $F^*(EG\times_G X)$, where $\mathcal{F}^q$ takes the value $F^q(G_\sigma)$ over a simplex $[\sigma]$ of $X/G$.