 # 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 XhG=EG*GX the homotopy orbit space, and XhG=F(EG,X)G the homotopy fixed point space. He claims there are spectral sequences E2p,q=Hp(G;Hq(X))=>Hp+q(XhG) and Ep,q2=H−p(G;πq(X))=>πp+q(XhG). Now, if I'm not mistaken the first one follows from the Serre spectral sequence applied to the Borel fibration X->XhG->BG (take the fiber bundle with fiber X associated to the action of G on X and to the G-principal bundle EG->BG). But where does the second one come from? reinzogoq 2022-09-14 Answered
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
${X}_{hG}=EG{×}_{G}X$
the homotopy orbit space, and
${X}^{hG}=F\left(EG,X{\right)}^{G}$
the homotopy fixed point space.
He claims there are spectral sequences
${E}_{p,q}^{2}={H}_{p}\left(G;{H}_{q}\left(X\right)\right)⇒{H}_{p+q}\left({X}_{hG}\right)$
and
${E}_{2}^{p,q}={H}^{-p}\left(G;{\pi }_{q}\left(X\right)\right)⇒{\pi }_{p+q}\left({X}^{hG}\right).$
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?
You can still ask an expert for help

• Live experts 24/7
• Questions are typically answered in as fast as 30 minutes
• Personalized clear answers

Solve your problem for the price of one coffee

• Math expert for every subject
• Pay only if we can solve it yamalwg
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 π0 (which are, respectively, taking homotopy fixed points / group cohomology and taking homotopy groups).

We have step-by-step solutions for your answer!