Apologies for the title.


Miller's theorem (formerly [Sullivan's conjecture](https://en.wikipedia.org/wiki/Sullivan_conjecture)) gives that for a finite group $G$ and a finite dimensional connected CW complex $X$, the based mapping space $\operatorname{Map}\_\*(BG,X)$ is weakly contractible. In particular, by consideration of $\pi\_0$ we see that every map $BG\to X$ is null-homotopic.


A generalization of Sullivan's conjecture to $G$-CW complexes $X$ with non-trivial action was proved Dwyer-Miller-Neisendorfer, Lannes and Carlsson. It states that if $G$ is a $p$-group and $X$ is a finite dimensional $G$-CW complex, then the natural map 
$$ 
X^G \to \operatorname{Map}\_G(EG,X)
$$
from the fixed points to the homotopy fixed points is a weak equivalence after $p$-completion. In particular, this map induces an isomorphism on mod $p$ homology and consequently induces a bijection
$$
\pi\_0(X^G) \cong \pi\_0(\operatorname{Map}\_G(EG,X))
$$ on path components. This could be viewed as a weakened generalized Sullivan's conjecture.



> 
> Does the weakened Sullivan conjecture hold for all finite groups? That is, let $G$ be a finite group and $X$ a finite dimensional $G$-CW complex. Is it true that 
>  $$
> \pi\_0(X^G)\cong\pi\_0(\operatorname{Map}\_G(EG,X))?
> $$
>  If not, what extra hypothesis could be put on $X$ so that this holds?
> 
> 
> 



Extending a weak version of Sullivan's generalized conjecture
The answer to the first question is negative by a result of Dror Farjoun and Zabrodsky. In [Fixed points and homotopy fixed points](https://eudml.org/doc/140123 "Fixed points and homotopy fixed points"), they prove that if a finite group $G$ is not a $p$-group, then there exists a finite $G$-CW complex $X$ with empty fixed points $X^G$, but non-empty homotopy fixed points $\mathrm{Map}\_G(EG,X)$.


