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:

- 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. - 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?

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 Mar 15 '11 at 16:25