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Apologies for the title.

Miller's theorem (formerly Sullivan's 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?

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The answer to the first question is negative by a result of Dror Farjoun and Zabrodsky. In 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 $\mathrm{Map}_G(EG,X)$.

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  • $\begingroup$ This is great, thanks! It shows that at least the additional hypothesis that $X$ has fixed points is needed. In fact, I am most interested in the case when $X$ is a based $G$-complex, and the fixed point sets $X^H$ for $H\leq G$ are all non-empty, connected and aspherical. $\endgroup$ – Mark Grant May 25 '18 at 18:42

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