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?
Best Answer
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.