7
$\begingroup$

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?

$\endgroup$
6
  • 5
    $\begingroup$ 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
    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. $\endgroup$ Mar 14, 2011 at 9:47
  • 2
    $\begingroup$ "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
    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? $\endgroup$ Mar 15, 2011 at 4:50
  • 1
    $\begingroup$ 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
    Mar 15, 2011 at 16:25

1 Answer 1

4
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.