Spectral sequence for computing the homotopy fixed points in unstable equivariant homotopy theory

Question

I'm reading Carlsson's "A survey of equivariant homotopy theory" and I have a question.

Let $G$ be a topological group and $X$ be a $G$-space (for a nice notion of "space"). He defines

$$X_{hG}=EG\times_G X$$ the homotopy orbit space, and

$$X^{hG}=F(EG,X)^G$$ the homotopy fixed point space.

He claims there are spectral sequences

$$E^2_{p,q}=H_p(G;H_q(X))\Rightarrow H_{p+q}(X_{hG})$$

and

$$E_2^{p,q}=H^{-p}(G;\pi_q(X))\Rightarrow \pi_{p+q}(X^{hG}).$$

Now, if I'm not mistaken the first one follows from the Serre spectral sequence applied to the Borel fibration $X\to X_{hG}\to BG$ (take the fiber bundle with fiber $X$ associated to the action of $G$ on $X$ and to the $G$-principal bundle $EG\to BG$).

But where does the second one come from?

Answer 1

It should be a special case of the Bousfield-Kan spectral sequence for homotopy limits. You can think of it as a "Grothendieck spectral sequence" associated to the "derived functors" of taking fixed points and taking $\pi_0$ (which are, respectively, taking homotopy fixed points / group cohomology and taking homotopy groups).

Answer 2

This is equivalent to Qiaochu's answer, but maybe it provides another point of view. The answer is: the spectral sequence arises from the general context of Federer spectral sequences for section spaces. See this paper of Kupers and Randal-Williams for precise constructions.

For a fibration $\pi:E\to B$ of (sufficiently nice) spaces with $1$-connected fiber $F$ (probably also holds replacing this assumption by simple) with a section $s$, denote its section space by $\Gamma(\pi)$. There exists a spectral sequence of the form

\begin{equation} E^2_{p,q}=H^p(B,\underline{\pi_q}(\pi,s))\Rightarrow \pi_{q-p}(\Gamma(\pi),s) \end{equation}where $\underline{\pi_q}(\pi,s)$ is the local system on $B$ assigning to a point $b\in B$ the group $\pi_q(\pi^{-1}(b),s(b)).$

In our context, if $X,Y$ are $G$-spaces, we can view the equivariant mapping space $F(X,Y)^G$ as the section space of the fibration $X\times_GY\to X\times_G EG=X_{hG}.$ This fibration has fiber $Y$, so, under the assumptions on Y, we have

\begin{equation} E^2_{p,q}=H^p(X_{hG},\underline{\pi_q}(Y,s))\Rightarrow \pi_{q-p}(F(X,Y)^G,s) \end{equation}and your spectral sequence follows when you take $X=EG.$

Let me just remark that this construction recovers an equivariant version of the Federer spectral sequence defined by Moller in this paper.