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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?

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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).

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  • $\begingroup$ How is it a homotopy limit? $\endgroup$ – user11417 Jun 30 '16 at 12:22

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