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This is a general question for the homotopy theory crowd: How does one go about computing the homology and homotopy groups of homotopy fixed point spaces $X^{hG}:= Map^G(EG, X)$ for the action of a group $G$ on a space $X$? There seem to be some tools:

  1. Lannes' theory: which allows you to compute (or at least say something about) $H_*(X^{hG}, \mathbb{F}_p)$ when $G$ is a $p$-group.
  2. Homotopy fixed point spectral sequences, which allow you to compute the stable homotopy groups of homotopy fixed point spectra.

Are there other tools out there? I feel like (1.) should be the harder version of a fact that I'm missing about computing $H_*(X^{hG}, \mathbb{F}_p)$ when $|G|$ is coprime to $p$. Regarding (2.), is there any hope of an unstable homotopy fixed point spectral sequence?

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    $\begingroup$ Can you be a bit more explicit what your $G$ is (discrete, Lie, finite, p, p-prime, etc.) and what your $X$ is (finite CW, p-complete, ...)? There is an unstable homotopy fixed point spectral sequence, a version of the Bousfield spectral sequence of a cosimplicial space, coming from looking at $map(EG,X)$ as a cosimplicial $G$-space by the canonical simplicial structure of $EG$. Whether or not that helps depends on your particular case -- in general, it's hard to describe $E^2$, it'll be a fringed spectral sequence, and convergence will be an issue. $\endgroup$
    – Tilman
    Commented Mar 14, 2011 at 7:31
  • $\begingroup$ I guess I'm happy starting with $G$ being finite and of order coprime to $p$, whereas $X$ has, say, finite $\mathbb{F}_p$ homology. But in the end, I's like to have a general picture of all of the tools available. $\endgroup$
    – Craig Westerland
    Commented Mar 14, 2011 at 9:47
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    $\begingroup$ "2." has an analog for spaces, namely, the space of sections of the fibration $EG x_G X \to BG$ has a Federer spectra sequence which converges to the homotopy of the function space. Also if $BG$ is finitely dominated and $X$ is a spectrum then one has the norm equivalence $D_G \wedge_G X \simeq X^{hG}$ which expresses the homotopy of $X^{hG}$ has the homotopy of $X$ with coefficients twisted by the dualizing spectrum $D_G$. This can be computed in some cases... $\endgroup$
    – John Klein
    Commented Mar 14, 2011 at 16:05
  • $\begingroup$ John, I'd love a version of the latter statement when $X$ is a space, and not a spectrum. I suppose I can get it from the latter when $X$ is an infinite loop space, but is there any hope of that happening when it's not? $\endgroup$
    – Craig Westerland
    Commented Mar 15, 2011 at 4:50
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    $\begingroup$ Addendum to my penultimate comment: if $X$ is a $G$-finitely dominated spectrum then the norm equivalence is valid for all $G$. $G$ finitely dominated means that $X$ is an equivariant retract up to homotopy of a $G$-finite spectrum $Y$, i.e., $Y$ is built up from the trivial spectrum by attaching a finite number of free cells. $\endgroup$
    – John Klein
    Commented Mar 15, 2011 at 16:25

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Hej Craig,

Re (2) as Tilman says in his comment, there is an unstable homotopy fixed point spectral sequence, a special case of the spectral sequence of a homotopy limit as described by Bousfield and others.

Re (1) when X is finite (and more generally), Lannes theory should be seen as generalization of ordinary Smith theory. Smith theory only works for p-groups, so I don't think it is a harder version of a prime-to-p statement.

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